Multi-Body Dynamics: Monitoring and Simulation Techniques III

Multi-body Dynamics: Monitoring and Simulation Techniques - 111

Edited by

Professor Homer Rahnejat and Dr Steve Rothberg

Professional Engineering Publishing Professional Engineering Publishing Limited London and Bury St Edmunds, UK

First Published 2004

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0 2004 The Institute of Measurement and Control, unless otherwise stated.

ISBN 1 86058 463 2

A CIP catalogue record for this book is available from the British Library. Printed and bound in Great Britain by Antony Rowe Limited, Chippenham, Wiltshire, UK.

Front cover: Transient elasto-multi-body model of single cylinder racing engine (courtesy of Perfect Bore Ltd and Loughborough University) Back cover: Laser torsional vibrometry off a spinning tappet (courtesy of SKF and Loughborough University)

The Publishers are not responsible for any statement made in this publication. Data, discussion, and conclusions developed by Authors are for information only and are not intended for use without independent substantiating investigation on the part of potential users. Opinions expressed are those of the Authors and are not necessarily those of the Institution of Mechanical Engineers or its Publishers.

Preface This volume contains the papers presented at the 3rd International Symposium on Multi-body Dynamics: Monitoring and Simulation Techniques. The papers were chosen from a number of submissions made, reviewed by a panel of learned referees and members of the Organizing Committee in a rigorous and fair manner. The quality of the papers are very high, contributing to advancement of knowledge, as well as maintaining the high quality of the Symposium. We would like to thank all the contributors, as well as the members of the Organizing Committee and the learned reviewers. In particular, sincere thanks are extended to the Keynote Speakers, Professors Werner Schiehlen and Ahmed Shabana, and Doctors Gothard Rainer and Heinz Foellinger, for their outstanding contributions to the Symposium. We were all honoured by the presence of the Symposium's Guests of Honour. Firstly, Professor Duncan Dowson (CBE, FRS, FREng) who gave the opening address, and presented our esteemed colleague, Professor Schiehlen, with commemoration for his long standing and significant contributions to the field. Secondly, Professor Richard Parry-Jones for his enlightening speech at the Gala Dinner, held at the Prestwold Hall, Leicestershire. We were also delighted to have many participants from among the members of the Editorial Board of the Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, particularly Professor Nicolae Orlandea, the Associate Editor and a leading light in the field. The organization of such successful events often requires dedicated and hard working administrative staff, who on this occasion have also dealt with a sizeable exhibition from many leading industrial concerns, logos of many of whom appear on the back-cover of this volume. Sincere thanks are, therefore, due to Jackie Baseley and Christine Biggs. From the outset, in 1997, the Symposium has enjoyed the sponsorship of the Institute of Measurement and Control, for which the Organizing Committee are most grateful. The same is also extended to other co-sponsoring Professional Institutions. The presentation, publication, and dissemination of the Proceedings are a major professional achievement. This task has been camed out from the outset (on three occasions) by the team at the Professional Engineering Publishing. We are very grateful for their painstaking thoroughness, particularly on this occasion by Lynsey Gathercole, S h e d Leich, and Jo Oxford. Homer Rahnejat and Steve Rothberg MBBMST 2004

xi

Contents Preface Foreword by Homer Rahnejat - A tribute to Jean D'Alembert and Albert Einstein Geometrical interpretationof Motion: an ironic legacy of apparently irreconcilable atomistic and continuum philosophies

xi

xiii

Multi-body Methodology Multi-body dynamics -fundamentals and applications W Schiehlen

3

Computer methodsfor the analysis of large deformations in multi-body system dynamics A A Shabana

15

An index zero formulation of the general dynamic differential equations using the transmissionfunctions N V Orlandea

31

Marionetteposture analysis by particle swarm optimization algorithm M Okuma and G M Germain

51

Visualizationof dynamic multi-body simulation data A Siemers and D Fritzson

57

Fatigue analysis on a virtual test rig based on multi-body simulation S Dietz and A Eichberger

73

Representation and visualization of surface related multi-body simulation data A Siemers and D Fritzson

83

Sub-surface visualization and parallel simulation A Siemers and D Fritzson

91

Influence of modelling and numerical parameters on the performance of aflexible MBS formulation J Cuadrado and R Gutierrez

99

The multi-particle system (MPS) model as a toolfor simulation of mechanisms with rigid and elastic bodies D Talaba and Cs Antonya

111

Structural Dynamics Stability and Chaotic Response of Elastic plate with large defection L Dai, Q Han, and A Liu

123

Dynamic simulation of civil engineering structures in virtual reality environment Cs Antonya and D Talaba

137

Windforce time-history generation by discrete Fourier transform (DFT) P J Murtagh, B Basu, and B M Broderick

147

FRFs for wind turbine lattice towers subjected to rotor mass inbalance P J Murtagh, B Basu, and B M Broderick

155

Periodic motions in a periodicallyforced, piecewise linear system ACJLuo

163

Dynamic modelsfor components considered in the design of a solar concentrator H R Hamidzadeh and L C Moxey

175

A simple beam elementfor large rotation problems K KerkkSinen, J Sopanen, and A Mikkola

191

Engine Dynamics Advanced CAE simulation and prediction of drivetrain attributes H Foellinger

207

Virtual testing supports reliability engineering of engine prototypes G Rainer

22 1

Periodic response and stability of reciprocating engines I Goudas, P Metallidis, I Stavrakis, and S Natsiavas

23 1

Coupled torsional and transverse vibration of engines A L Guzzomi, S J Drew, and B J Stone

243

Quality and validation of cranktrain vibration predictions - effect of hydrodynamicjournal bearing models G Offner, H H Priebsch, M T Ma, U Karlsson, A Wikstrom, and B Loibnegger

255

Multi-body analysis and measurement of valve train motions M Teodorescu, H Rahnejat, and S J Rothberg

273

Drivetrain Dynamics The torsional vibration of gear boxes with backlash M L Coltrona and B J Stone

287

Coupled torsional and transverse vibration of gearboxes M Sargeant and B J Stone

299

Development of a simulation toolfor the prediction of dynamic transmission error, the source of transmission whine D Parkin-Moore, G Davis, D Bell, C H Lu, P Brooks, and A Leavitt

311

Low-noise automotive transmissions - investigations of rattling and clattering S N Dogan, J Ryborz, and B Bertsche

323

NVH-behaviour of side shaft-systems J W Biermann

339

Friction models of automotive transmissions equipped with tripodjoints J-P Mariot, J-Y K’nevez, and B Barbedette

351

Using Taguchi methods to aid understanding of a multi-body clutch pedal noise and vibration phenomenon P Kelly and J W Bierrnann

361

Combined multi-body dynamics, structural modal analysis, and boundary element mefhod to predict multi-physics interactions of driveline clonk S Theodossiades, M Gnanakumarr, H Rahnejat, and M Menday

373

Vehicle Dynamics, Ride and Handling Dynamic analysis of semi-active suspension systems using a co-simulation approach R Ramli, M Pownall, M Levesley, and D A Crolla

391

Smart driver: a research projectfor closed loop vehicle simulation in MSCADAMS R Frezza, A Saccon, D Minen, and C Ortmann

40 1

Non-linear response of an all-terrain vehicle on a rough terrain L Dai, J Wu, and M Dong

415

Multi-objective optimization of quarter car models with passive and semi-active suspensions G Verros, M Kazantzis, S Natsiavas, and C Papadimitriou

429

Torque steer influences on McPhersonfront axles J Dornhege

439

Computer-based development of control strategiesfor ground vehicles M A Naya and J Cuadrado

447

Machines and Mechanisms Modelling of a smart spindle unit P Hynek, M Jackson, R Parkin, and N Brown

46 1

A dynamic modelling and simulation of cutting process in turning W S E Moughith, A A Abdul-Ameer, and A Khanipour

475

Modelling and simulation of a precision pneumatic actuation system R C Richardson, M Brown, B Bhakta, and M Levesley

485

Transient dynamic behaviour of deep-groove ball bearings W W Sum,E J Williams, S McWilliam, and D R A s h o r e

497

Determination of the effect of contact kinematics of squeeze caving phenomenon through general covariance M Kushwaha and H Rahnejat

507

Authors’ Index

520

Multi-body Methodology

Multi-body dynamics - fundamentals and applications W SCHIEHLEN Institute B of Mechanics, University of Stuttgart, Germany

SYNOPSIS The close connection between the fundamentals of multibody dynamics and their applications in engineering sciences is shown. Based on the history and recent activities in multibody dynamics, engineering systems are modelled and classified according to their vibration phenomena. Multiple and/or nonlinear vibration systems are analysed by matrix methods, nonlinear dynamics approaches and simulation techniques. Applications are shown from low frequency vehicles dynamics including comfort and safety requirements to high frequency structural vibrations generating noise and sound, and from controlled limit cycles of mechanisms to periodic nonlinear oscillations of biped walkers. 1 HISTORY AND RECENT ACTIVITIES

The roots of multibody dynamics date back to the origins of analytical mechanics starting with Newton’s Principia [l], Euler’s Corporum Rigidanun [2] and Lagrange’s Mdcanique Analytique [3]. Even more important for the computational aspects of multibody dynamics are the contributions of D’Alembert [4] in his Trait6 de Dynamique, Jourdain [5] with his Analogue at Gauss’ Principle and the work of Kane and Levinson [ 6 ] .Multibody dynamics was also promoted at the beginning of the 201h century by the theory of gyroscopes, see e.g. Grammel [7], and mechanism theory by the early work of Wittenbauer [SI. During the middle of the last century spacecrafts and biomechanics pushed the development of multibody dynamics as documented by Roberson and Wittenburg [9] and Huston and Passerello [IO]. Multibody dynamics as a new branch of mechanics was set up in 1977 by a IUTAM Symposium chaired by Magnus [I 11. Twenty years later MULTIBODY SYSTEM DYNAMICS was established as the first scientific journal fully devoted to multibody

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dynamics. And in 2003 an ASME Technical Committee on Multibody Systems and Nonlinear Dynamics was formed with the task to organize biannual conferences starting in 2005. Recent research topics may be listed as follows. 1. Datamodels from CAD 2. Parameter identification 3. Optimal design 4. Dynamic strength analysis (Flexibility) 5. Contact and impact problems (Impact) 6. Extension to control and mechatronics (Control) 7. Nonholonomic systems 8. Integration codes 9. Real time simulation 10. Challenging applications In particular, elastic or flexible rnultibody systems, respectively, contact and impact problems and actively controlled mechatronic systems represent key issues for researchers worldwide. The focus of multibody systems is shown in Figure 1.

Figure 1. Focus of multibody systems

2 FUNDAMENTAL DYNAMICS In the section the essential steps for generation the equations of motion in multibody dynamics will be summarized.

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2.1 Mechanical modelling First of all the engineering or natural system has to be replaced by the elements of the multibody system approach: rigid and/or flexible bodies, joints, gravity, springs, dampers and position and/or force actuators The system constrained by bearings and joints is disassembled as free body system using an appropriate number of inertial, moving reference and body fixed frames for the mathematical description.

2.2 Kinematics A system of p rigid bodies holds f vectors and rotation tensors as

=

6p degrees of freedom characterized by translation

Thus, the position vector x of the free system can be written as

The system’s position remains as

ri = r i ( x ) ,si = S,(x).

(3)

Assembling the system by q holonomic, rheonomic constraints reduces the number of degrees of freedom to f = 6p -q . The corresponding constraint equations may be written in explicit or implicit form, respectively, as x = x(y, t ) or ~ ( xt ),= o

(4)

where the position vector y summarizes the f generalized coordinates of the holonomic system = [YI

Y2 Y3

”‘

Y,

I

(5)

Then, for the system’s position it remains

By differentiation the translational and rotational velocity vectors are found

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where s means a vector of infinitesimal rotations following from the corresponding rotation tensor, see e.g. Ref. [12]. Further, the Jacobian matrices J , and J,, for translation and rotation are defined by Eqs. (7) and (8). The system may be subject to additional r nonholonomic constraints which do not affect the f = 6p - q positional degrees of freedom. But they reduce the velocity dependent degrees of freedom to g = f - r = 6p - q - r . The corresponding constraint equations can be written explicitly or implicitly, too, j , = j , ( y , z , t ) or ly(y,jJ,t)=O,

(9)

where the g generalized velocities are summarized by the vector

z(t)= [z, 22

ZJ

... ZJ

For the system’s translational and rotational velocities it follows from Eqs.(7) to (9) V I = Vi(Y>Z,t) and

w, = w,(y,z,r)

(1 1)

By differentiation the acceleration vectors are obtained, e.g., the translational acceleration as

av.

av ayT

a . =‘i+i+---L

azT

at

(y , z , t ) Z + ~ ( y , z , t ) .

A similar equation yields for the rotational acceleration. The Jacobian matrices L are related to the generalized velocities, for translations as well as for rotations. 2.3 Newton-Euler Equations Newton’s equations and Euler’s equations are based on the velocities and accelerations from Section 2.2 as well as on the applied forces and torques, and the constraint forces and torques acting on all the bodies. The reactions or constraint forces and torques, respectively, can be reduced to a minimal number of generalized constraint forces also known as Lagrange’s multipliers. In matrix notation the following equations are obtained, see also Ref. [ 121. Free body system kinematics and holonomic constraint forces:

Holonomic system kinematics and constraints:

Nonholonomic system kinematics and constraints:

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On the left hand side of Eqs. (13) to (15) the inertia forces appear characterized by the inertia -

matrix

, the global Jacobian matrices

the right hand side the vector global distribution matrix

7

,xand the vector 4‘ of the Coriolis forces. On

of the applied forces and the constraint forces composed by a and the vector of the generalized constraint forces 1 are found.

Each of the Eqs. (13) to (15) represents 6p scalar equations. However, the number of unknowns is different. In Eq. (13) there are 6p + q unknowns resulting from the vectors x and 1 . In Eq. (14) the number of unknowns is exactly 6p = f + q by the vectors y and 1, while in Eq. (1 5) the number of unknowns is 12p - q due to the additional velocity vector z and an extended constraint vector 1. Obviously, the Newton-Euler equations have to be supplemented for the simulation of motion. 2.4 Equations of Motion The equations of motion are complete sets of equations to be solved by vibration analysis andor numerical integration. There are two approaches used resulting in differential-algebraic equations (DAE) or ordinary differential equations (ODE), respectively.

For the DAE approach the implicit constraint equations (4) are differentiated twice and added to the Newton-Euler equations (1 3) resulting in

Eqs. (16) are numerically unstable due to a double zero eigenvalue originating from the differentiation of the constraints. During the last decade great progress was achieved in the stabilization of the solutions of Eqs. (16) well documented by Eich-Soellner and Ftihrer [13]. The ODE approach is based on the elimination of the constraint forces using the orthogonality -T -

of generalized motions and constraints, J Q = 0 , also known as D’Alembert’s principle for holonomic systems. Then, it remains a minimal number of equations

-T -

The orthogonality may also be used for nonholonomic systems, L Q = 0 , corresponding to Jourdain’s principle and Kane’s equations. However, the explicit form of the nonholonomic constsraints (9) has to be added, Y = Y(Y>ZJ)? M(y,z,t)i +k(y,z,t)= q(y,z,t)

.

(18)

Eqs. (17) and (1 8) can now be solved by any standard time integration code. The equations presented can also be extended to flexible bodies. For the analysis of small structural vibration a floating frame of reference is used while for large deformations the

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absolute nodal coordinate formulation turned out to be very efficient. For more details see, e.g., Shabana [14,15] .

3 VIBRATION ANALYSIS Most important for engineering applications are the mechanical vibrations of holonomic, rheonomic systems. The vibration phenomena are classified according to the equations of nonlinear and linear motion. Starting with Eq. (1 7) , nonlinear time-variant mechanical systems, even withf = 1 degree of freedom, may show chaotic vibrations. For small motions Eq. (1 7) can be linearized resulting in

This system may feature parametrically excited vibrations due to the time-varying often periodic matrices. In the case of time-invariant matrices with symmetric and skew-symmetric characteristic one gets M j ; + ( D + C ) j + (K + N ) y = h(t),

(20)

a system which performs forced vibrations due to the external excitation on the right hand side. In the case of h(t) = 0 only free vibrations remain. Furthermore, if the damping matrix D ,the gyroscopic matrix G , and the circulatory matrix N are missing, a conservative system

Mj;+Ky=O

(21)

with free undamped vibrationsis found. 3.1 Linear Vibration Analysis The special structure of Eqs. (20) and (21) simplifies the analysis. Marginal stability of Eqs. (21) is guaranteed if the stiffness matrix K is positive definitive. The free damped vibrations of Eqs. (20) with G = N = 0 are asymptotically stable if both, the stiffness matrix K is positive definite and the damping matrix D is positive definite or pervasively positive semidefinite, respectively, see Ref. [ 161. Moreover, Eqs. (20) is asymptotically stable is all eigenvalues have a negative real part. The general solution of Eqs. (21) reads as

At) = y , t ) Y , + y2 ( 4 j O

(22)

where the transition matrices Y,(t),Yv,(t)are found from a real eigenvalue analysis of dimension f .The general solution of Eq. (20) is more easily written in the state space with the state vector x(t) summarizing the system’s state given by the generalized coordinates and their first time derivatives as

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Then, it yields simply

where the state transition matrix @(t) follows from a complex eigenvalue problem of dimension 2 f . Matrix methods for linear system with harmonic excitation h(r) lead to the concept of frequency response matrices while random excitation processes require spectral density matrices or covariance matrices, respectively. In the case of Eqs. (19) with periodically timevarying coefficients Floquet’s theory allows closed form solutions.

3.2 Nonlinear vibration analysis Chaotic vibrations can be analyzed by time integration only resulting in the solution

which is very sensitive to the initial conditions. Powerful characteristics of chaotic vibrations are the phase portrait, the power spectral density, the Ljapunov exponents and the dimensions. In addition to the chaotic vibrations also periodic motions may be found depending on the parameters of the system.

As an example some results of Bestle [I71 are presented here for the Duffig oscillator. Parameter Set a allows a periodic motion, often called a limit cycle, while Set d represents chaotic behaviour resulting in a strange attractor, Figure 2. The Ljapunov exponents for Set a are computed as cr, = 0 , cr2 = -0.10 , crj = -0. IO what means a periodic motion, for Set d on gets cr, = 0.1 7 , cr2 = 0 ,cr3 = -0.37. The positive Ljapunov exponents identifies a chaotic motion. The same behaviour is found from the dimension, Set a results in D, = I , and for Set done gets D, = 2.46 .

A chaotic multibody system is represented by the chaos pendulum consisting of p=3 bodies with f = 3 degrees of freedom, see Ref. [ 181. 4 VEHICLE VIBRATIONS AND CONTROL

Vehicle dynamics is a major application field of multibody dynamics. The corresponding software tools have been highlighted in Ref. [19]. These tools are most successful used for detailed models representing the vehicle motion by simulation. For the control design such models are too complex, additional more simple models are helpful.

As an example the lateral dynamics of vehicle convoy with the second vehicle following autonomously the leading vehicle is considered, Figure 3. The simulation model consists of

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p=19 bodies with f = I 9 degrees of freedom, McPherson front wheel strut, semi-trailing rear wheel suspension, Pacejka’s magic formula tire model and driver models by Legouis and Power spectral density

Phase portrait

f

3.1

I

I -4.

-2.

0.

2.

X

4.

Figure 2. Characteristics of a Duffing oscillator (from Bestle [17])

Figure 3. Vehicle convoy as simulation and control design model

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Donges. The control design model is restricted to a plane motion of lateral and yaw dynamics, the two tires of each axis are replaced by one tire in the middle of the axis (bicycle model), a linear tire model is used and the longitudinal velocity is constant. More details of the models, the corresponding equations and simulation results are available from Ref. [20].

5 STRUCTURAL VIBRATIONS AND CONTACT Structural vibrations occur often after collisions representing dynamical contact modelled as impacts between rigid and/or elastic bodies,respectively. Contact can be considered as a multiscale problem as shown in Ref. [21]. On the fast time scale the energy loss can be computed by an elastodynamic or finite element model, respectively. Then, from the momentum balance the Coefficient of restitution is found and fed back to the multibody dynamics analysis. Using a linear motion of the two colliding bodies with masses mI , m2 it yields in the compression and the restitution phase

Poisson's law of momentum reads as A p = d ~ +c d P r = d ~ c O + e ) . From Eqs. (26) and (27) it follows the coefficient of restitution as

h+ m2 P P

(28)

The coefficient of restitution depends on the shape of the bodies, their material and their relative velocity. Computational and experimental results are shown in Figure 4 for rods, plates, balls and beams made from aluminium. Ball ( * measured)

1

"0

0.2

0.4

0.6

Velocity

[Ns]

0.8

Figure 4. Coefficient of restitution for bodies of different shape

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The structural vibrations superimposed to the rigid body motion are shown in Figure 5 . For more details see Ref. [22].

Ball impact 0.1

'

-0.2 0

2

4

Time [SI

6

a

Figure 5. Slow time scale simulation 6 MECHANISMS AND BIPED WALKERS Robots and manufacturing systems as well as walking devices are characterized by mechanisms with some or all mechanical degrees of freedom controlled resulting in prescribed motions or rheonomic constraints, respectively. These motions are usually periodic vibrations and due to the control effort for accelerating and decelerating of the bodies a considerable amount of energy may be consumed. By using storage springs, the motion may be adjusted to the limit cycle of periodic nonlinear vibrations. The first example is a robot arm with f = 2 degrees of freedom and the task of a horizontal motion featuring a limit cycle, Figure 6 . The storage springs with stiffness c, , c2 support the motion in a natural way reducing the energy consumption as shown in Ref. [23]. Reduction of the energy consumption may reach more than 90 %. This principle can also be applied to walking machines. Passive walking devices are very efficient just powered by a small slope of the ground. In this case the potential energy is stored in the gravitational field by the vertical vibrations of the machine's centre of mass. The passive motion is then used as prescribed motion of a fully active walking machine, see Figure 7. The equation of motion of the active machine with f = 9 degrees of freedom reads as

where W(y,t)Z.represent the reaction forces due to the feet contact points and the locking knee, B is the control input matrix and u = [uo u , u2 ug ug u,] means the control input vector. As shown in Ref. [24] the actively controlled biped model is as efficient as human walking what is superior to walking machine operating for comfort reasons without vertical vibrations of the centre of mass.

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Rotating crank x(f) Oscillating crank x(r)

= L

+ R coswr ,

= a

+ b COSO~

y(r) = 0

between a=30° and a=150°

Figure 6. Robot arm with two prescribed horizontal motions

Figure 7. Acitvely controlled biped walking machine

CONCLUSIONS Multibody dynamics is an excellent foundation for multivariable vibration analysis and sophisticated control design. Multibody systems show all kinds of motion: harmonic oscillations, periodic limit cycles and chaotic attractors as well as instabilities. Simulations with software tools for multibody dynamics are more trustworthy knowing the potential vibration phenomena. Applications of multibody systems include machine dynamics, vehicle dynamics, aerospace engineering, manufacturing, robotics and biomechanics of locomotion and sports. Recent research activities are devoted to large deformations in flexible multibody systems, to contact and impact problems requiring multi-time-scale modeling and all kinds of actively control mechanical systems often denoted as mechatronic systems.

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REFERENCES 1. Newton, I., Philosophiae Naturalis Principia Mathematica, Royal Society, London, 1687. 2. Euler, L. ‘Nova methodi motum corporum rigidarum determinandi’, Novi Commentarii Academiae Scientiarum Petropolitanae 20, 1776,208-238. See also Euler, L., Opera Omnia, Series 11, Vo1.9,99-125. 3. Lagrange, J.-L., Mbcanique Analytique, L’AcadBmie Royal des Sciences, Paris, 1788. 4. D’Alembert, J., Traitd de Dynamique, Paris, 1743. 5. Jourdain, P.E.B., ‘Note on an analogue at Gauss’ principle of least constraint’, Quarterly Journal on Pure AppliedMathematics 40, 1909, 153-197. 6. Kane, T.R. and Levinson, D.A., Dynamics: Theory and Applications, McGraw-Hill, New York, 1985. 7. Grammel, R., Der Kreisel- Seine Theorie und seine Anwendungen, First ed., Vieweg, Braunschweig 1920. 8. Wittenbauer, F. Graphische Dynamik Springer-Verlag. Berlin 1923. 9. Roberson, R.E. and Wittenburg, J., ‘A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control’, Proceedings 3”’ Congr. Int. Fed. Autom. Control, Butterworth, Vol. 1, Book 3, Paper 46 D, London, 1967. IO. Huston, R.L. and Passerello, C.E.,’On the dynamics of a human body model’, Journal on Biomechanics 4, 1971,369-378. 11. Magnus, K.(ed.), Dynamics of Multibody Systems, Springer-Verlag, Berlin 1978. 12. Schiehlen, W., Multibody System Dynamics: Roots and Perspectives, Multibody System Dynamics 1, 1997, 149- 188. 13. Eich-Soellner, E. and Fuhrer, C., Numerical Methods in Multibody Dynamics, Teubner-Verlag, Stuttgart, 1998. 14. Shabana, A.A., Dynamics of Multibody Systems, Second ed., Cambridge University Press, Cambridge, 1998. 15. Shabana, A.A.(ed.), ‘Special Issue on Flexible Multibody Dynamics’, Nonlinear Dynamics 34,2003, 1-232. 16. Muller, P.C. and Schiehlen, W., Linear Vibrations, Martinus Nijhoff Publ., Dordrecht, 1985. 17. Bestle, D., Beurteilungskriterien f i r chaotische Bewegungen nichtlinearer Schwingungssysteme, VDIVerlag, DUsseldorf 1988. 18. Schiehlen, W.,’Control of chaos for pendulum systems’, New Applications of Nonlinear and Chaotic Dynamics in Mechanics, Moon, F.C. (ed.), Kluwer, Dordrecht, 1999,363-370. 19. Kortilm, W., Schiehlen, W. and Arnold, M. ‘Software tools: From multibody system analysis to vehicle system dynamics’,Mechanicsfor a New Millenium, Aref, H. and Phillips, J.W. (eds.), Kluwer, Dordrecht, 2001,225-238. 20. Schiehlen, W. and Petersen, U.,’Control concepts for lateral motion of road vehicles in convoy’, Interaction between Dynamics and Control in Advanced Mechanical Systems, Van Campen, D.-H. (ed.), Kluwer, Dordrecht, 1997,345-354. 21. Schiehlen, W. and Bin Hu, ‘Contact problems in multibody dynamics’, Multibody Dynamics: Monitoring and Simulation Techniques 11, Rahnejat, H., Ebrahimi, M. and Whalley, R. (eds.), Prof. Eng. Publ., London, 2000,3-14. 22. Schiehlen,W. and Seifried, R.,’Multiscale Impact Models: Multibody Dynamics and Wave Propagation’, Nonlinear Stochastic Dynamics, Namachchivaya, N. Sri and Lin, Y.K. (eds.), Kluwer, Dordrecht, 2003, 353-362. 23. Guse, N. and Schiehlen, W.,’Effcient inverse dynamics control of multibody systems’, Proc. dhInt. Conj Motion Vibration Control, Mizumo, T. and Suda, Y.(eds.), Japan Mech. Society, Tokyo, 2002, Vol.1, 502507. 24. Gruber, S . and Schiehlen, W.,’Biped Walking machines -a challenge to dynamics and mechatronics’, Proc. 5“ World Congr. Comp. Mech.. Mang, H.A., Rammerstorfer, F.G.and Eberhardsteiner, J. (eds.), Vienna Univ. Techn., Vienna, 2002, Paper ID 81426, 1 1 pages (http://wccm.tuwien.ac.at).

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Computer methods for the analysis of large deformations in multi-body system dynamics A A SHABANA Department of Mechanical Engineering, University of Illinois at Chicago, USA

ABSTRACT The objective of this paper is to discuss generalization of existing multi-body system algorithms to solve large deformation problems. Existing multi-body system computer programs can systematically solve small deformation problems. The techniques and formulations implemented in these codes, however, assume that the shape of the body deformation remains simple, allowing the use of simple functions or linear modes to define the displacement field. In the case of large deformations, other finite element techniques must be employed. In this paper, the finite element absolute nodal coordinate formulation is discussed. This formulation does not require the interpolation of finite rotations, and as a consequence, it does not suffer from the problem of coordinate redundancy that characterizes large rotation vector formulations. Furthermore, in the case of the absolute nodal coordinate formulation, no special measures are needed in the numerical integration in order to satisfy the principle of work and energy. NOMENCLATURE Transformation matrix of body i. Vector of kinematic constraint equations. Constraint Jacobian matrix. Vector of absolute nodal coordinates. Stiffness matrix. Stiffness matrix associated with the elastic coordinates. System mass matrix. Vector of applied forces.

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Q” 9 4r

e S

t V A

Vector of Coriolis and centrifugal forces. Vector of system generalized coordinates. Vector of reference coordinates. Vector of elastic coordinates. Vector of system non-generalized coordinates. Time. Volume. Mass density.

1. INTRODUCTION Most of the research in the area of flexible multi-body system dynamics has been focused on small deformation problems. In this case, the deformation of the system components can be approximated using simple functions or linear mode shapes. The floating frame of reference formulation has been widely used to solve small deformation problems that characterize many of the multi-body system applications including automotive systems, robotics, bio-mechanics, and railroad vehicle systems [I-51. In the floating frame of reference formulation [6-IO], the motion of the body reference is defined using absolute coordinates, while the small deformation of the body is defined with respect to the body coordinate system using polynomial functions or linear modes that can be identified using the finite element method [11-15]. In the early eighties, a procedure was proposed [ 16, 171 to systematically couple general purpose finite element computer programs with the newly developed at this time flexible multi-body computer programs. Finite element codes are used as preprocessor to obtain a specific set of inertia shape integrals that are required to formulate the nonlinear mass matrix and the centrifugal and Coriolis forces of the flexible body. Since in some finite element codes, information about the element shape functions used may not be readily available, a lumped mass formulation was first proposed in [I71 in order to allow the development of an interface between finite element computer programs and flexible multi-body codes without knowledge of the finite element shape function used. This procedure which was implemented in several finite element codes has become common and standardized to the point that complex flexible multi-body systems subject to small deformations are systematically analyzed and efficiently solved without the need to impose the assumptions of the linear theory of elasto-dynamics [ 18-22]. Solving small deformation problems in multi-body system applications has become common practice that requires systematic steps and well developed algorithms accepted by the research community and practicing engineers. The analysis of large deformation problems in multi-body system applications, on the other hand, remains a challenging problem that will require the development of new generations of flexible multi-body system computer programs. In large deformation applications, the shape of the body deformation can not, in general, be described using simple polynomial functions or linear modes. The formulation of large deformation problem may also require the use of finite element nodal coordinates representation as compared to the reduced set modal coordinates used for the solution of small deformation problems [23]. For this reason, large deformation problems require the use of formulations that are conceptually and fundamentally different from the formulations used for small deformation problems. The use of the nodal coordinates and the complexity of the shape of deformation lead to significant increase in the dimensionality of the mathematical model required to describe the dynamics of systems that experience large deformations.

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Multi-body system applications that are characterized by the finite reference rotations and existence of the kinematic constraints that describe mechanical joints are inherently different from structural system applications. The solution of multi-body system applications requires developing finite element formulations that are fundamentally different from those formulations used to solve structural problems. For instance, the finite element floating frame of reference formulation that is widely used to solve small deformation problems is based on the concept of the intermediate element coordinate system [16,23] which has not been used by the finite element community. The use of the intermediate element coordinate system ensures exact representation of the rigid body dynamics and accurate formulation of the nonlinear inertia forces. Attempts have been made by many researchers to import finite element formulations to solve flexible multi-body system applications. Most of these attempts, however, have failed to identify a reliable procedure that can be an alternative to the floating frame of reference formulation which remains the most widely and accepted method for solving small deformation problems. It is also expected that existing large deformation finite element formulations will have serious limitations when considered for solving multi-body system applications in which finite rotations are experienced. For this reason, there is a need to use a new finite element procedure that addresses the fundamental problems that characterize flexible multi-body system applications. Some of the problems that hinder importing existing finite element formulations to solve multi-body system applications are discussed in this paper. In recent years, a new non-incremental finite element procedure, called the absolute nodal coordinate formulation, was introduced to solve large deformation problems in flexible multibody system applications. Absolute position coordinates and gradients are used as nodal coordinates. No finite or infinitesimal rotations are used as nodal coordinates. The absolute nodal coordinate formulation ensures exact representation of the rigid body dynamics. Unlike large rotation vector formulations, the absolute nodal coordinate formulation leads to a constant mass matrix and does not suffer from the coordinate redundancy problem that can be encountered when other large rotation finite element formulations are used with absolute coordinates. For this reason, the absolute nodal coordinate formulation does not require the use of special measures in the numerical integration algorithm in order to satisfy the principle of work and energy. The absolute nodal coordinate formulation has been successfully used to solve large deformation problems in multi-body system applications, and has been implemented in a general purpose flexible multi-body computer program (SAMS2000). In this computer program, the system may consist of three different types of bodies; rigid, flexible, and very flexible. The equations of motion of the rigid bodies are formulated using Newton-Euler equations and a set of absolute reference coordinates. The equations of motion of flexible bodies are developed using the floating frame of reference formulation and a mixed set of absolute reference and local elastic coordinates. The equations of motion of very flexible bodies are formulated using the absolute nodal coordinate formulation that employs absolute position and gradient coordinates. These three formulations; Newton-Euler, floating frame of reference, and the absolute coordinate; represent the basis for developing a new generation of flexible multi-body computer programs that can be efficiently and systematically used to solve a large class of mechanical and aerospace system applications. One may argue that one large deformation finite element based formulation is sufficient since this formulation can also be used to solve small deformation problems. In response to this argument, one must realize that large deformation formulations can be inefficient in solving applications that include rigid bodies or flexible bodies that experience small

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deformations. For this reason, the use of the above mentioned three formulations becomes necessary if an efficient and general simulation tool is to be developed. In this article, the coordinate redundancy problem that characterizes large deformation vector formulations is discussed in Section 2 in order to shed light on some of the serious limitations that arise from the use of these formulations in the non-incremental solution algorithms used in the simulation of flexible multi-body systems. In Section 3, the floating frame of reference formulation that is widely used in the analysis of small deformation problems is briefly reviewed and it is explained why such a formulation does not suffer from the coordinate redundancy problem. The large deformation finite element absolute nodal coordinate formulation is discussed in Section 4. In Section 5, the implementation of the large deformation formulations in flexible multi-body computer algorithms is explained. Numerical examples are presented in Section 6 in order to examine the convergence of the new method and demonstrate that it automatically satisfies the principle of work and energy. Summary and conclusions drawn from this investigation are presented in Section 7. 2. COORDINATE REDUNDANCY The matrix of the position vector gradients used in the continuum mechanics literature [24-251 defines the state of rotation and strain at any point on the flexible body. This matrix has nine independent components. According to the Polar Decomposition Theorem,the matrix of position vector gradients can be written as the product of an orthogonal matrix that defines the rotation and a stretch matrix that defines the deformation. The orthogonal rotation matrix can always be defined in terms of three independent parameters, while the symmetric stretch matrix can be defined in terms of six independent parameters. Given a field vector that defines the location of arbitrary infinitesimal volumes on the deformable body in terms of the spatial coordinates, this vector can be differentiated to systematically define the matrix of position vector gradients that define the rotations and deformation of an arbitrary infinitesimal volume on the flexible body. That is, the rotation of an infinitesimal volume on the deformable body can be uniquely defined by differentiating the filed vector that defines the displacements of the arbitrary points. The rotation parameters are, in general, nonlinear functions of the position vector gradients. For this reason, the continuum mechanics theory does not require introducing any field rotation parameters in addition to the field vector that defines the position and position vector gradients of the material points. This brief introduction can be used to shed light on some of the problems associated with the use of some finite element formulations in flexible multi-body system applications. In particular, we focus in this section on two finite element approaches which are widely used in the finite element commercial codes to solve large rotation problems. The first is the approach that employs non-isoparametric beam, plate and shell elements with the incremental solution procedure to solve large rotation problems. This approach will be referred to in this paper as injnitesimal nodal rotation jnite element formulations. The second approach, on the other hand, employs elements that are developed using what is called large rotation vecforformulations. This approach differs from the infinitesimal nodal rotation approach in several aspects, some of which are summarized below. 1. Finite elements developed using large rotation vector formulations can provide exact representation of the rigid body motion, while infinitesimal nodal rotation formulations are, in general, not capable of providing exact representation of rigid body motion since a rigid body motion does not always lead to zero strains.

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2. In large rotation vector formulations, finite rotation parameters are used as nodal coordinates. The corresponding rotation parameters within the finite element are treated as field variables which are, in the most formulations, interpolated independently from the position vector that defines the gradients. In most infinitesimal nodal rotation formulations, the infinitesimal rotations are not interpolated independently from the position vector, and in many cases are related to the gradients. 3. In the case of finite rotation applications, infinitesimal nodal rotation formulations must be used in the case of large rotations with an incremental solution procedure that employs a co-rotational formulation. Clearly, the use of such a procedure has its serious drawbacks in multi-body system applications because of the linearization of the rigid body rotation. While, on the other hand, large rotation vector formulations can describe rigid body motion, the use of finite rotation parameters as nodal coordinates leads to a problem of coordinate redundancy that makes impossible the use of these formulations in multi-body system applications with non-incremental solution procedures. For this reason, many of the large rotation vector formulations employ the classical co-rotational finite element approach to solve the resulting dynamic equations. This problem will be briefly discussed later in this section 4. Both infinitesimal nodal rotations and large rotation vector approaches do not lead to a constant mass matrix in two- and three-dimensional applications. The problem of coordinate redundancy associated with the use of the large rotation vector formulation should be clear after the brief introduction previously made in this section. As previously pointed out, in the case of the finite element analysis, the matrix of the position vector gradients can be obtained using the displacement field that describes the displacements of the material points. The rotations of the material points can be uniquely defined using the position vector gradients determined using the position or displacement field. Introducing finite rotation parameters without properly relating them to the position vector gradients using kinematic constraints can lead to serious problems in multi-body system applications, as evident by the fact that large rotation vector formulations lead to a violation of the principle of work and energy if no special measures are taken in the numerical integration routines. It is also important to point out that a first generation of rigid multi-body computer programs miserably failed and did not stay in the market because of the problem of coordinate redundancy. In multibody system applications, kinematic relationships must be properly imposed at the position, velocity and acceleration level. FINITE ELEMENT FLOATING FRAME OF REFERENCE 3. In this section, the floating frame of reference formulation which is widely in multi-body computer programs is briefly discussed in order to explain the fundamental differences between this nonlinear formulation and the infinitesimal nodal rotation formulations and large rotation vector formulations. This brief discussion will help explaining why the floating frame of reference formulation does not lead to violation of the principle of work and energy or to nonzero strains under an arbitrary rigid body motion. Furthermore, the floating frame of reference formulation does not suffer from the problem of coordinate redundancy. It is important, however, to point out that the floating frame of reference formulation has been used in the most part to solve small deformation problems.

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Figure 1. Floating Frame of Reference Formulation In the floating frame of reference formulation, the global position vector of an arbitrary point on the flexible body is written as the sum of two vectors; the first vector defines the global position of the reference point, as shown in Fig. 1, while the second vector defines the position of the point with respect to the body coordinate system. The global position vector of an arbitrary point on the flexible body i can then be written as follows: r' = R'+ A'$ (1) where R' is the vector that defines the global position of the origin of the body coordinate system, A' is the transformation matrix that defines the orientation of the body coordinate system, and ii'is the local position of the arbitrary point with respect to the body coordinate system. The vector i i l can be further decomposed as the sum of two vectors; one vector defines the local position of the point in the undeformed state, while the other defines the deformation vector. The deformation vector can be written as explained in the literature in terms of the finite element nodal coordinates using the concept of the finite element intermediate element coordinate system. Using the kinematic description of the finite element floating frame of reference formulation, the configuration of a flexible body in the multi-body system can be expressed in terms of the body generalized coordinates q' which can be written in the following partitioned form:

where q: is the vector of reference coordinates that define the location and orientation of the body coordinate system, and q; is the vector of elastic nodal coordinates that defines the deformation of the bodies with respect to the body coordinate system. Comments on the Kinemutic Description The kinematic description used in the floating frame of reference formulation is conceptually different from the one used in the classical co-rotational finite element formulation. This description which does not lead to a separation between the rigid body motion and the elastic deformations leads to zero strains under an arbitrary rigid body displacement when non-isoparametric finite elements are used. This is clear since the reference

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motion is described using absolute Cartesian and orientation reference coordinates instead of the finite element nodal coordinates. It is also important to note that the rigid body motion is not, in general, the reference motion since different reference conditions can be used to define the coordinate system of the flexible body in the floating frame of reference formulation. It is also clear that the kinematic description used in the floating frame of reference formulation does not lead to the problem of coordinate redundancy since the deformations are defined with respect to the body coordinate system; that is, the parameters used to describe the reference rotations are different from those used to describe the rotations due to deformations. For this reason, the floating frame of reference formulation does not lead to a violation in the principle of work and energy, and such a problem has not been a concern in the multi-body system research community in which this formulation is widely used. On the contrary, importing finite element techniques that were developed for structural systems can lead to violation of the principle of work and energy since, in some of these formulations, the finite rotation parameters are not consistently defined. Nonlinear Equations ofMotion Using the kinematic description presented in this section and the techniques of classical mechanics, one can show that the nonlinear equations of motion of the flexible body i can be written as follows: M'q' + K'q' = QL + QL (3) where Mi and K' are the symmetric mass and stiffness matrices of the body, Qe is the vector of externally applied forces, and Q: is the vector of Coriolis and centrifugal forces. The floating frame of reference formulation in the case of small deformation problems leads to a highly nonlinear mass matrix and a simple expression for the stiffness matrix. Because of the nonlinearity of the stiffness matrix, the preceding equation of motion includes the Coriolis and centrifugal forces which are quadratic in the velocities. The floating frame of reference formulation as described by the preceding equation also leads to accurate representation of the nonlinear dynamic coupling between the rigid body motion and the elastic deformations. In order to demonstrate this dynamic coupling, the preceding equation can be written in the following known partitioned form [23]:

where subscripts r and f refer respectively to reference and elastic coordinates. The highly nonlinear nature of the inertia forces is attributed to the use of two sets of coordinates; the reference and the elastic coordinates. Using these two sets of coordinates, the kinematic equations are explicitly written in terms of the rotation matrix that defines the orientation of the floating frame of reference. As will be discussed later in this paper, the use of absolute coordinates in the case of large deformation formulations can lead to a constant mass matrix, and as a result, the Coriolis and Centrifugal forces are identically equal to zero. Implementation and Coupling With Finite Element Codes The finite element floating from of reference formulation is implemented in several commercial and research computer codes which are widely used in industry and research and educational institutions. The success of this formulation can be attributed to its generality and to the fact that it can be systematically implemented in general purpose flexible multi-body computer programs. Since its introduction in the early eighties [16, 171, several problems related to establishing the interface between flexible

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multi-body codes and existing commercial finite element codes have been addressed and successfully solved. The procedure for coupling flexible multi-body codes and finite element computer programs using the nonlinear finite element floating frame of reference formulation was first established in the early eighties [17]. Such a procedure does not require knowledge of the shape functions of the finite element used in commercial finite element computer programs. The inertia shape integrals that enter into the formulation of the nonlinear dynamic coupling between the reference motion and the elastic deformation can be evaluated using a lumped mass approach and summation instead of integration [17, 231. The use of this approach allows to systematically develop a computer procedure in which finite element codes are used as preprocessors for general purpose multi-body computer codes. This procedure also allows for using component mode synthesis techniques to reduce the number of elastic degrees of freedom in the case of small deformation problems. The inertia shape integrals that represent the dynamic coupling between the reference motion and the elastic deformation can be expressed in a modal form, thereby eliminating high frequency modes that do not significantly contribute to the solutions. It is important to point out that the finite element floating frame of reference formulation will remain a powerful technique to solve small deformation problems. Large deformation formulations which do not allow for coordinate reduction are not a substitute for the floating frame of reference formulation since these formulations are not, in general, efficient in solving small deformation problems. The floating frame of reference formulation has been widely used in solving many challenging engineering problems as the tracked vehicle shown in Fig. 2.

Figure 2. Tracked Vehicles

4. ABSOLUTE NODAL COORDINATE FORMULATION There are many important multi-body system applications in which the system components experience large deformations. In these cases, the shape of deformation can be complex (Fig. 3) and can not be described using simple linear modes. The kinematic description used in the floating frame of reference formulation, therefore, can not be successfully or efficiently used in many of these cases. A larger number of degrees of freedom is often required in order to be able to accurately represent the deformations. For this reason, a finite element nodal formulation capable of capturing the details of the large deformations must be used. In this formulation, the large rigid body displacements must be accurately represented and consistent sets of kinematic

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relationships must be used in order to avoid many of the problems encountered when existing finite element formulations are used.

Figure 3. Large Deformation In this section, a large deformation formulation, the absolute nodal coordinate formulation, that can be systematically used in the analysis of flexible multi-body systems is briefly discussed [23, 26-29]. In the absolute nodal coordinate formulation, one set of absolute coordinates that consist of displacements and slopes are used to describe the configuration of the finite element. The global position of an arbitrary point, as shown in Fig. 4, on an element e of the flexible body i can be written as follows: = vefe

, . I

(5)

where S" is the element shape function, and ere is the vector of element nodal coordinates that consists of absolute position coordinates and slopes. In the case of the three-dimensional analysis, the nine slope coordinates represent the position vector gradients which uniquely define the state of rotation and deformation of the material points within the element. Using the preceding equation, the finite element equations of motion can be written as follows: Mieefe = Q (6) where M" is the element mass matrix, and Q I e is the vector of nodal forces. The mass matrix MIe is constant and is given by M" =

J-p"S"TS"dY"

(7)

V1'

In this equation, V."'and dieare, respectively, the volume and density of the finite element. The mass matrix in Eq. 7 is constant in the both cases of two- and three-dimensional analysis. Unlike the floating frame of reference formulation, because the mass matrix is constant in the absolute nodal coordinate formulation, the Coriolis and centrifugal forces are identically equal to zero. On the other hand, the form of the elastic forces takes a highly nonlinear form.

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Node 2

Figure 4. Absolute Nodal Coordinate Formulation Perhaps, it is important at this point to revisit the problem of coordinate redundancy from which some existing large deformation finite element formulations suffer. Using the kinematic description of the absolute nodal coordinate formulation, it is clear that such a problem is not encountered in the case of the absolute nodal coordinate formulation because infinitesimal or finite rotations are not used as nodal coordinates. Furthermore, in the absolute nodal coordinate formulation, no finite rotations are interpolated, and therefore, there is only one assumed displacement field that defines the position vector of the material points. Information on the nodal rotations or the rotations of the material points within the elements are readily available from the matrix of the position vector gradients that can always be decomposed to an orthogonal matrix and a stretch matrix. The orthogonal matrix defines the rotation of the material points and the stretch matrix describes the state of deformation. Since the absolute nodal coordinate formulation does not suffer from the problem of coordinate redundancy and leads to exact modeling of the rigid body dynamics, this formulation automatically satisfies the principle of work and energy. 5.

IMPLEMENTATION OF LARGE DEFORMATION FORMULATIONS

The new generation of general purpose flexible multi-body computer programs will be able to model systems that consist of rigid, flexible and very flexible bodies. Rigid bodies can be modeled using Newton-Euler equations, flexible bodies can be modeled using the floating frame of reference formulation, and very flexible bodies can be modeled using the absolute nodal coordinate formulation. Therefore, these new codes will employ different formulations and different sets of generalized coordinates. The dynamic coupling between these different generalized coordinates as the result of the joint connectivity will introduce new challenging implementation problems that must be addressed. Some of multi-body system applications may require the use of non-generalized coordinates in the dynamic formulation. Examples of these applications are contact problems that require introducing the surface parameters that describe the geometry of the contact surfaces. One may choose to eliminate these surface parameters or keep them in the dynamic formulation as a

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set of non-generalized coordinates that have no inertia or forces associated with them. In the later case, the equations of motion and the constraint equations are explicitly expressed in terms of the system generalized and non-generalized coordinates. If the dynamic equations of the multi-body system are formulated in terms of the non-generalized coordinates, the augmented Lagrangian form of the equations of motion must be modified as explained in this section. As previously pointed out, a new general flexible multi-body algorithm must allow joint connectivity between rigid bodies, flexible bodies, and very flexible bodies as well as the general treatment of the contact problem. In this case, the joint constraints must be expressed in terms of the reference, elastic and absolute nodal coordinates as well as the non-generalized coordinates s. Considering also driving constraints that can depend explicitly on time, the vector of nonlinear constraint functions can be expressed in terms of the system reference, elastic and absolute nodal coordinates as well as the non-generalized surface parameters as follows: C(q,, q,,q,,s,t)=O (8) where q, is the vector of system absolute coordinates, and t is time. As previously discussed, the floating frame of reference formulation leads to nonlinear expressions for the joint constraints as the result of using the body coordinate system that introduces geometric nonlinearities. The constraint equations are formulated in terms of the body transformation matrices and their derivatives. The formulation of some of these joints, however, becomes much simpler when the absolute nodal coordinate formulation is used. Nonetheless, since slopes are used as coordinates in the absolute nodal coordinate formulation, the formulations of some joints require the development of new special techniques for defining the kinematics of selected reference frames at the joint definition points in terms of the absolute nodal coordinates. The kinematic constraints that describe mechanical joints, contact conditions and specified motion trajectories can be adjoined to the system differential equations of motion using the technique of Lagrange multipliers. This leads to the following augmented form of the system equations of motion:

Q, Q, =

Q.

(9)

0

_Q,

where M refers to a mass sub-matrix, subscripts r, f; a, and s refer, respectively, to reference, elastic, absolute nodal coordinates, and non-generalized surface parameters, C, is the constraint Jacobian matrix associated with the generalized coordinates, C, is the constraint Jacobian matrix associated with the non-generalized surface parameters s, 8 is the vector of Lagrange multipliers, Qr, Q and Q, are the generalized forces associated with reference, elastic, and absolute nodal coordinates, and Q, is a quadratic velocity vector that results from the differentiation of the kinematic constraint equations twice with respect to time [23]. The augmented form of the equations of motion can be solved in order to obtain the second time derivative of the vectors of reference, elastic, absolute nodal coordinates and surface parameters as well as the vector of Lagrange multipliers. Lagrange multipliers can be used to determine the generalized constraint forces associated with the reference, elastic, and absolute nodal coordinates. The reference,

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elastic, and absolute nodal accelerations and the time derivatives of the surface parameters can be integrated forward in time in order to determine the coordinates and velocities. The numerical algorithm proposed in this investigation ensures that the algebraic constraint equations are not violated. The vector of Lagrange multipliers can also be used to determine the normal contact forces. These normal contact forces can be used to determine the creep forces required for accurate modeling of railroad vehicle system applications. Cholesky Coordinates Since the mass matrix M, associated with the absolute nodal coordinates is constant, a Cholesky transformation can be used to obtain a generalized identity mass matrix. This will lead to an optimum sparse matrix structure for the augmented form of the equations of motion of the system. The resulting augmented form of the equations of motion can be written as follows:

where I is an identity matrix, and subscript ch refers to Cholesky coordinates. NUMERICAL RESULTS In this section, results obtained using the absolute nodal coordinate formulation are presented in order to examine the convergence of the method and to show that the method satisfies the principle of work and energy. Convergence ofthe Method In order to test the convergence of the absolute nodal coordinate formulation in solving large deformation and rotation problem, the simple pendulum example shown in Fig. 5 is considered. The data for this example are presented in the literature [26]. Figure 6 shows the convergence of the method when different numbers of finite elements are used, while Fig. 7 shows the animation of the motion of the pendulum. 6.

Figure 5 . Pendulum Example

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Figure 6. Convergence of the Method

Figure 7. Motion Animation

Principle of Work and Energy Unlike many of the finite element large rotation vector formulation, the absolute nodal coordinate formulation automatically satisfies the principle of work and energy without the need to take special measures in the numerical integration. In order to demonstrate this fact, the four bar mechanism shown in Fig. 8 is used. The crankshaft of the mechanism is subjected to the moment shown in Fig. 9. The data of this mechanism are presented in the literature [27].The results presented in Fig. 10 show that the absolute nodal coordinate formulation automatically satisfies the principle of work and energy.

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Figure 8. Four Bar Mechanism

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Figure 10. Principle of Work and Energy SUMMARY AND CONCLUSIONS The new generation of flexible multi-body computer programs will allow modeling rigid, flexible and very flexible bodies. The development of these codes will require the computer implementation of large deformation finite element formulations. Existing large rotation vector formulations suffer from the problem of coordinate redundancy. This problem can result in a 7.

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solution that violates the principle of work and energy. The multi-body system methodologies, on the other hand, require the use of consistent kinematic relationships in order to have reliable and robust algorithms. The problem of coordinate redundancy associated with the large rotation vector formulations was discussed in this paper. It is explained that the displacement field associated with the position vector can be used to determine the matrix of position vector gradients. This matrix can be used to define the state of rotation and deformation of the material points on the flexible body. It is also explained why the problem of coordinate redundancy is not encountered when the floating frame of reference formulation is used. This formulation which is widely used to solve multi-body system applications employs two different sets of generalized coordinates; reference and elastic coordinates. As the result of this choice, the finite rotation of the body is not described using the element nodal coordinates. This description leads to a unique displacement field, and the solutions obtained using this nonlinear formulation automatically satisfy the principle of work and energy. The computer implementation of the floating frame of reference formulation and the coupling between finite element computer programs and general purpose flexible multi-body computer codes are among the issues discussed in this study [ 171. The methodology proposed in this investigation to solve large deformation problems in multi-body system applications is based on the absolute nodal coordinate formulation. The proposed methodology does not require the use of special measures to satisfy the principle of work and energy, as demonstrated by the results presented in previous publications. Accurate results for highly nonlinear systems can be obtained using explicit integrators commonly used in the multi-body simulation codes. As pointed out in this paper and in the literature, the absolute nodal coordinate formulation does not require interpolation of rotations or slopes, automatically captures the effect of geometric centrifugal stiffening, and can be systematically applied to beam, plate and shell elements. Continuity of the displacement gradients is ensured, and as a consequence, the stress calculations are accurate. Another important feature of the proposed absolute nodal coordinate formulation is that it leads to a constant mass matrix in two- and threedimensional applications. Furthermore, many of the assumptions of Euler-Bernoulli, Timoshenko, and Mindlin beam and plate theories are relaxed. This important property of the constant mass matrix allows for introducing Cholesky coordinates that lead to an identity generalized inertia matrix associated with the Cholesky coordinates. The result is an optimum sparse matrix structure for the augmented form of the multibody equations of motion. REFERENCES 1. Greenwood, D.T. (1988) Principles of Dynamics, Second Edition, Prentice Hall. 2. Fowles, G.R. (1 986) Analytical Mechanics, Fourth Edition, Saunders College Publishing. Goldstein, H. (1 950) Classical Mechanics, Addison-Wesley. 3. 4. Nikravesh, P.E. (1 988) Computer Aided Analysis ofMechanical Systems, Prentice Hall. Shabana, A.A.. (2001) Computational Dynamics, Second Edition. 5. 6. Agrawal, O.P.,and Shabana, A.A. (1985) Dynamic analysis of multibody systems using component modes, Computers and Structures, 21(6), 1301-1312. 7. Ashley, H. (1967) Observations on the dynamic behavior of large flexible bodies in orbit, AIM Journal, 5(3), 460-469. 8. Cavin, R.K., and Dusto, A.R. (1977) Hamilton's principle: finite element methods and flexible body dynamics, AM Journal, 15(2), 1684-1690.

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9. 10.

11.

12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22. 23. 24. 25. 26. 27. 28. 29.

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De Veubeke, B.F. (1976) The dynamics of flexible bodies, Znt. J. Eng. Sci., 14, 895-913. Hughes, P.C. (1979) Dynamics of chain of flexible bodies, J. Astronaut. Sci., 27(4), 359380. Cook, R.D. (1 98 1) Concepts and Applications of Finite Element Analysis, Second Edition. Huebner, K.H., Thomton, E.A., and Byrom, T.G. (1995) The Finite Element Method for Engineers, Third Edition, Wiley & Sons. Tong, P., and Rossettos, J.N. (1977) Finite Element Method, The MIT Press. Bathe, K.J. (1981) Finite Element Procedures in Engineering Analysis, Prentice Hall. Zienkiewicz, O.C. (1979) The Finite Element Method, McGraw-Hill. Shabana, A.A. (1982) Dynamics of large scale flexible mechanical systems", Ph.D. Thesis, University of Iowa, Iowa City. Shabana, A.A., (1985) Automated Analysis of Constrained Inertia-Variant Flexible Systems, ASME Journal of Vibration, Acoustics, Stress and Reliability in Design, 107(4), 43 1-440. Winfrey, R.C. (1971) Elastic link mechanism dynamics, ASME J. Eng. Industry, 93,268272. Winfrey, R.C. (1972) Dynamic analysis of elastic link mechanisms by reduction of coordinates, ASME J. Eng. Industry, 94,557-582. Erdman, A.G., and Sandor, G.N. (1 972) Kineto-Elastodynamics: A review of the state of the art and trends, Mechanism and Machine Theory, 7, 19-33. Bahgat, B., and Willmert, K.D. (1973) Finite element vibration analysis of planar mechanisms", Mechanism and Machine Theory, 8,497-5 16. Lowen, G.G.,and Chassapis, C. (1986) The elastic behavior of links: An update, Mechanism and Machine Theory, 21(1), 33-42. Shabana, A. (1998) Dynamics of Multibody Systems, Second Edition, Cambridge University Press. Spencer, A.J.M. (1 980) ContinuumMechanics, Longman. Fung, Y.C. (1977) A First Course in Continuum Mechanics, Second Edition, Prentice Hall. Omar, M.A., and Shabana, A.A. (2001) A two-dimensional shear deformable beam for large rotation and deformation problems, Sound and Vibration, 243(3), 565-576. Campanelli, M., Berzeri, M., and Shabana, A.A. (2000) Performance of the incremental and non-incremental finite element formulations in flexible multibody problems, ASME Journal of Mechanical Design, 122( 4), 498-507. Shabana, A.A., and Yakoub, R.Y. (2001) Three-dimensional absolute nodal coordinate formulation for beam elements: Theory, ASME Journal of Mechanical Design, 123 (4), 606-61 3. Yakoub, R.Y., and Shabana, A.A. (2001) Three-dimensional absolute nodal coordinate formulation for beam elements: implementation and applications, ASME Journal of Mechanical Design, 123 (4), 614-621.

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An index zero formulation of the general dynamic differential equations using the transmission functions N V ORLANDEA

University of Michigan and MSc Software, USA

This paper harmonizes the relation between the numerical integration methods and dynamic formulation of the equation of motion, keeping in mind the robustness and accuracy of the numerical solutions. It is based on the results obtained in papers [1],[2] and [3]. The formulation specifically refers to articulated mechanisms andplanetaiy .systems. However, it can be applied to any multi degrees of freedom system for which the transmissionfunctions can be defined as in the references [4] and [SI. AJer the definitions of index and transmissionfinctions are introduced and explained the formulation is implemented and applied to mechanisms. The results indicate that for some systems such as planetary systems and some articulated mechanisms the method can be applied to real time digital simulation. Key words: Equation of motion, formulation index, transmission function, energy, generalized coordinates, numerical integration 1. Introduction.

The Index 3(I3) formulation based on the “FreedomAxiom” is well known by now and it stands at the basis of the ADAMS(Automatic Dynamic Analysis of Mechanical Systems) computer program. A natural extension of this formulation was the way how the DAE (Differential and Algebraic Equations) system is solved by using sparse matrix techniques and variable time step and order backward numerical integration methods as described in [l]. The formulation has advantages: The equations are simple, understandable and there is a one to one mapping of the 1. physical mechanical system to be simulated and the input data. This makes it easy to prepare the input data and to understand the system’s dynamic behavior.

Y 004/028/2004

31

2. 3.

When the state of forces and mechanical stresses are required this is the formulation to be desired because it also solves for internal forces in the system. If lower index is considered then the system is automatically stabilized. This will ensure that the drift of the constraints will not take place.

However, this formulation also has some disadvantages: Sometimes it solves for variables of no interest such as LaGrange Multipliers. 1. 2. The results can be guaranteed only if the numerical integration time step is constant or the integration order is equal or higher than four. To solve this last disadvantage Orlandea and Coddington [2] and Orlandea [3] introduced lower Index formulations for ADAMS by solving together with the geometric constraints their first derivative with respect to time for index 2 formulation. For the index 1 formulation the second order derivative of the constraints with respect to time is also employed. Under these conditions a first order method with variable integration time step is able to guarantee good results. By continuing with a third order derivative of the constraints with respect to time the LaGrange Multipliers can be eliminated and the system of equation will have only differential equations. This will be an index zero formulation. Hence, the index number is given by the highest order of the derivative of the constraints with respect to time necessary to completely eliminate the algebraic variables and equations. Another way to define the index of a formulation is “the measure of how far a formulation is from a differential equation system”. Some of the main properties of an index zero formulation concerning this paper are: An index zero formulation contain the minimum number of differential equations. It is assumed that any numerical integration method, explicit or implicit, can be applied to index zero formulation There are other methods to formulate an index zero system of differential equations. For systems having holonomic, schleronomic and holonomic rehonomic constraints such as articulated mechanisms one can use the transmission functions. 1. 2.

The transmission functions were defined and discussed in [5]. However, a small attempt will be made here to refresh their definition. Let us consider a system having one degree of freedom with the inputs p,(f),yi, and @,, and outputs p2(pl),yi, and 8,. Taking the derivative of p2(pI)with respect to time the result is

second order transmission functions. The main properties of the transmission functions are:

I.

32

They are functions only of the position of a mechanism. This may facilitate the precomputation and development of look up tables for the dynamic parameters such as reduced mass and moment of inertia.

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2.

For gear trains the first order transmission functions are the transmission ratios between gears that are constant, hence the second order transmission functions are equal to zero .

In the following using the transmission functions an index zero formulation will be worked out with applications to a two degrees of freedom mechanisms. Also, using this formulation for planetary mechanisms will be discussed.

Formulation of the equation of motion.

2.

Before defining the equation of motion some definitions are in order. In Figure 1 is represented a body i that is part of a kinematic scheme as shown in the Figure 2. The coordinates that define the kinematic and dynamic state of body i are x,,y,,z,,v/,,Bf,p, and their time derivatives. The Euler angles v/,,p, are oriented angles as shown on Figure 1. The state of the velocities and accelerations are defined by the time derivatives of these coordinates.

Figure 1. Dejinilion of a body frred system of reference

Figure 2. Kinematic chain definition

The LaGrange’s dynamic equation is given by

where: E is the kinetic energy of the system q, and q, are generalized velocities and position

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33

Q, are the conservative and neoconservative generalized forces. For a system that has n bodies and k degrees of freedom or k generalized coordinates q 1 , q 2.......qk , as represented in the Figure 2, the angular velocity of body i is given by the three components qi,myi,coz,that are the components of the body’s angular velocity on the mobile

system of reference i. The relation between time derivatives of the Euler angles I,+, ,b,,4, and w,,,my,,q,are given by

where, A,

= sin

e, sin 4,, B, = COS^^, C, = sin 0, COS^,, 0,= sin 4,, H, = cose,

(3)

Much simpler than the angular velocities are the linear velocities of the body’s center of mass,

hence, the body’s kinetic energy has the expression: 1

E , = -m,(x,? + y: + i:) + I,,@:, + Iw,w:, + Ia,mii 2 where, I,, , I , , I,, are the principal moments of inertia of body i. The kinetic energy of the entire system is given by :

(5)

or by considering (2),(3),(4) and (6) the system’s kinetic energy as function of the generalized coordinates has the expression:

34

Applying the transformation (1 A) from the Appendix A, the general form of the kinetic energy expressed as a function of the transmission function is:

or by reorganizing the expression of the kinetic energy E the result is,

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35

Introducing the notations

the relation (9) becomes:

/=I

Further, introducing another notation, m,P,i + Irn,RF + Iw,R; + In,R; = F;, the kinetic energy of the entire system is given by E

1 "

=-CCF:,9,9, 8

(12)

/=I

I=I

where Fj, is the generalized kinetic energy function. The fact that must be underlined is that the functions '' F;l depends only on the position of the system This form of the kinetic energy function will be used to determine the index zero formulation of the general equation of motion of the mechanical systems. 'I.

For this the LaGrange's equation ( I ) will be considered. The first term that will be found is:

36

YOO4/028/2004

further, ; s = (1,2,3.......k)

(1 5 )

and the second term of the LaGrange’s equation is given by,

Considering the fact that

(17) After a series of calculations the first term of the LaGrange’s equation or equation (15) becomes: r

1

Substituting “ j ”for “s” in the second summation of equation (18) and substituting “I” for “s” in the third summation the result is r

Due to equations (16) and (19), equation (1) changes to

or by simplifying the notations,

the relation (20) takes the form

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37

Setting

1'

[I

where

is the Cristoffel symbol of the first kind, equation (22) can also be

rewritten in a much simpler symbolical form:

The physical meaning of the functions F, is that they are reduced mass or moment of inertia. Equation (24) is linear with respect with q, ,hence it can be written in the matrix form: Fq=Q-Cr -

-

where,

(26) Expressing the vector q by multiplying equation (25) by F-I, the result is:

cr

q = F-' Q- F-' - -

,

(27)

where F-' is the inverse inertia matrix and has the following expression:

38

Y004/028/2004

4 41 ...

Dl2

"'

D22

.'.

DM

I

Dzk

... ... ...

, where A is the determinant of F and D,,, D12,...Du are the

4 2 .'* Dkk 1 nors of the elements of matrix F

4

Special attention will be given to the two vectors from the right hand side of equation (27). The first vector that will be discussed now is:

p PSID/=I ,=I @ r

Defining now,

LA ip=l D'['

where

'I={

']q,q,J

'}

{' '}

represents the Cristoffel's second symbol. Here again, the important

characteristic of the first and second Cristoffel's symbols is that they represent functions dependent only on the positions of the mechanism or mechanical system. Different methods can be implemented to pre-compute these functions and create look up tables for increasing the efficiency of the simulation. Another vector that is ,art of the right hand side of the equation (27) is:

ip=lQ p D l p ip=1Q P Q P

P=l

Hence, the general expression of the equation of motion is given below :

Y 004102812004

39

l k

9, =,ZQpDv p 4

-c' f /=I

']q,q,;

r=1,2,...........,k

j=l

As it can be seen the accelerations can always be expressed explicitly in the most general case. This must be taken into account when choosing a numerical integration method to solve such a problem. Also this formulation is an index zero formulation. To put this in a simple way the total number of equations of the system (31) is equal with the number of the degrees of freedom of a mechanical system. Thus, there are not algebraic variables or algebraic equations to solve. All the equations and variables are differential equations and differential variables.

To define the generalized forces Q the equivalence of the mechanical power of the generalized forces and the mechanical power of the applied forces and torques is employed. The following assumptions are made: 1 - The set of applied forces includes also the forces derived from potential (conservative) 2 -The applied forces are functions of velocity, position and time.

Based on these assumptions the mechanical power of the applied forces is:

In the formula (32) Trqj represents the applied torque having the number s, s=1,2, ...,ni on body

i. ni may be different from body to body. w represents the vector of the angular velocity of the - I

body i on which the torque is acting on. The components of the torque and angular velocity are considered to be the projections on the body fixed reference system. In the contrary, the components of the applied linear forces are the projections on the axes of the global system of references. These assumptions will substantially simplify the resulting expressions and proofs. The power given by the torques and angular velocities terms are:

(33)

reorganizing the terms the result is:

Hence, the equivalence of mechanical power gives the equality,

40

Y 004102812004

-

or in a simplified form, ,=I( 4

-

j

p=l

Trqp are the generalized torque forces. The dependence of the angular velocities

av k ae, a n d xit - -aqPp, ' c--kjp,~+p aqp

on the components of the angular velocity on the mobile

p-1aqp system of reference of body i, ox,,wy1,o,, can be derived by using the equation (2) p-1

p=l

aqp

A similar procedure as in the case of the angular velocities and torques will be used for the linear

forces and velocities. The dot product between the velocities of a point on the line of action and the forces will give the mechanical power of the linear forces. This dot product should be defined in the fixed (g1obal)systemof reference. If there are t forces, , For,' t=1,2, ....,m m on each body i having the corresponding velocity v: the resulting mechanical power given by all these forces is:

The expression of the velocity can be derived by considering that the linear velocity of a point on the line of action is defined by the vector summation of the velocity of the body center of gravity and the relative velocity of the point with respect to the center of gravity. The velocity of the center of gravity in the global system of reference is given by the relation (4). The coordinates on the mobile system of reference of the body i of a point A: that is situated on the force's line of action are Xif.Yit,Za, . If the system has n bodies, k independent coordinates and each body has mm applied forces on it, the velocities of the points A,! are:

, where Ti, Ti, T,' are the partial derivatives of the Euler transformation matrix with respect to - - y / , , e l , p, . The Euler transformation matrix is given in the Appendix A relation (2A). Organizing the above expression with respect to the generalized velocities, the result is:

Y004/028/2004

41

L I=I

n

mm

i=l I=I

k

n

’qJ

mm

?Y’ =cCC. E-%, p=l ,=I ,=I

,‘I

Ti

-

line of action of each pplied force. Introducing the notation

1

, and making the dot product of these velocities with the vectors of the applied forces the total mechanical power of the linear forces is:

Taking into account relations (32),(35) and (38)the expressions for the equivalencies of the power of the generalized forces and applied forces is:

+

‘$Qpqp = i ( T r q p + F o r ~ p vFor,”vL ~ p-1

+ For,rv:f:)ip, hence, the expressions of the

generalized

p=l

forces are given by: Qp = (Trqp+ForLpv: + ForTv; + FoqLPv:f:)p=1,2, ........,IC (39) The general equation of motion (31) is defined completely at this point. It can be set up and formulated for any system for which the transmission functions can be found. The equation (31) is represented by a system of second order differential equations. The system (3 1) can also be used to predict the dynamic behavior of nonlinear systems such as the articulated mechanisms. As an application, the simulation of a two dimensional two degrees of freedom mechanism is discussed here. The purpose of this simulation is to find out if this formulation gives valid results, how involved are the computations required by numerical integration, how does index zero formulation reacts to different methods of numerical integration and to put in perspective how can this formulation be used to improve the computation efficiency toward real time simulation.

42

Y004/028/2004

3.

Application.

Figure 3. Five-bar mechanism

Figure 4. Planefary System

So far this formulation refers to articulated mechanisms. It can be extended to other systems if the transmission functions can be determined. The majority of the articulated mechanisms in the practical life have one or two degrees of freedom. The manipulators make the exception by having more than two degrees of freedom.

As an application and example the Five- bar Mechanism shown in the Figure 3 will be treated here. This is a two dimensional system having two degrees of freedom k = 2,2 ; it has five bodies n = 2,2, 5 with link number one as the ground. That means that in particularizing the equations (3 1) n=2,3,4,5; j = 2,2; I = 2,2; and r = 1,2. When applying a three dimensional formulation to a two dimensional mechanical system the plane that is parallel to the mechanism's motion has to be defined. For this case the choice is the fixed plane XOZ with the transformation having the nodal axis identical to the fixed X axis. Thus, the mobile system is rotated with 90 degrees so that the mobile Z is perpendicular to the fixed XOZ plane. The angle p, defines the position between the fixed X axis (nodal axis) and the mobile Xi axes. Such an example is p5 in Figure 3. The Euler angles yl = 0 , 6,= a12 and the Y direction coordinates remain constant. Hence, the values of the respective velocities and energy are zero.

...,

Taking into consideration equations (21),(29), the equation (3 I) that characterizes the dynamic behavior of the system from Figure 3 takes the form,

Y004/028/2004

43

where

Equation (40) represents the general form of the equation of motion for two dimensional two degrees of freedom systems for which the transmission functions can be found. In the case of the gear trains and planetary system represented in the Figure 4,the first order transmission functions are constant, hence, the second order transmission functions are equal zero. As a consequence in the equation (40) U =O and W=O.This simplifies the form of this equation. However, it should be stressed out that the functional form of Q, and Q2contribute to the computational efficiency of simulations that are using the equation (40). On Figure 4 pl0,p2,,,p3,,,p4,,and pCo represent the initial conditions for the gear angles. The mechanical data pertinent to the system from Figure 3 is given in Table 1. In this table the inertia moments for link 2 and link 3 contain the moment of inertia of the power train and the electrical motors. The simulation of the system from Figure 3 is a dynamic simulation. The dynamic behavior of the system is given by the action of the constant driving torques that are applied on links 2 and 3. First and second order transmission functions for this system are given in Appendix A, equations (3A). Considering these transmission hnctions and the data from Table 1 the functions F,!, and the variables U and W are computed after Velocity

Applied forces

conditions -1. [Nm]

I .025 N

I

Table 1. Data for thefive-bar Mechar sm each successful time step. Then the explicit system of DE (Differential Equations ) (40) is numerically integrated. Two methods are used for numerical integration. The first method is an explicit 4-th order Runge Kutta (RK)method and an implicit multi-step variable order BDF (Gear)

44

algorithm having the order 1 to 6 for which was forced a low order. That was done to make sure that the index zero DE works well with low order BDF methods. Caution should be taken in calibrating the error control so that both methods can be compared. The error control in the Gear algorithm is more involved than in a simple RK method. Llnk 5 CG coordlnares dlsplacomonts 1.5OE-01

1

I

I

I

I

1.WE-01

f

$i E 5.M1E-02

O.M)EtM)

Tlme (am)

-Link

5 X CG -Link

5 Y CG

Figure 6. Simulation results link 5 CG displacements

Figure 5 . Simulation multi-frame display

The simulation time is four seconds real time. The results of the simulation have shown that the two methods give identical results. The low order BDF methods give good results however their productivity is lower than the high order BDF methods because the time step is smaller. All orders of the BDF methods worked faster than RK. This is because of the higher number of function evaluation and the BDF time step variability. From the dynamic point of view the interesting result is that the driving torques on the cranks (links) 2 and 3 are smaller than the toraues aodied on rockers (links) 4 and 5 . \

1.

I

Angular Acceleratlons

Angular Velocltles I

I

I

I.,

5

I

150

1W

j

o

50

P

-5

-50

0

-lw -150 0

Tlm(sm) I-Unk4

-Link5

Figure 7 . Simulation results Link 4 and link 5 angular velocity

1

2

3

4

nm (OW) -Link4

-Link5

Figure 8. Simulation results link 4 and link 5 angular accelerations.

This means that the applied torques on the rockers and the gravity help drive the mechanism for some portions of the displacement accumulating energy into inertia that is given back when the torques become reactive or the rocker motion reverses. As it can be seen from Figures 6-8 the motion of the rockers has an oscillatory character. A multi- frame display and the path of the center of mass of link 5 are shown on Figure 5 . The CPU times for these simulations on a 2.4 GHz Pentium 4 machine were between 3.1 and 3.9 sec. These CPU times were achieved without writing any output. As far as the hardware is concerned these times can already be improved by

Y004/028/2004

45

running the simulation on the higher frequency machine. That is not the concern of this paper. The concern of this paper is how to improve the performance by means of the formulation of the equation of motion and software. 4.

Discussions and conclusions

Using the transmission functions one can always formulate the equation of motion in a explicit second order differential equations form as represented by the equation [31]. The geometric and kinematic constraints are taken implicitly into consideration, hence, no LaGrange multipliers are needed to define a given problem. The number of equations [31] is identical with k that represents the number of the degrees of freedom that is the minimum number of equations that can describe the dynamic behavior of the system. It was verified that equations [31] can be integrated numerically by any explicit or implicit, one step or multi-step numerical integration method guaranteeing good results when the process is convergent. In contrast with index three formulation the order of the integration method can be variable order and time step without deterioration of the results. The first and second order transmission functions used in this formulation are functions only on the positions of the mechanism. Hence all the functions F;, and

DJ,that depend on the transmission functions also depend only on the positions of the mechanism . This triggers the idea that by knowing the range of motion of the generalized coordinates, then the transmission functions and the general functions of the moments of inertia and masses can be generated before the effective integration takes place and tabulated in look up tables for speeding up the Computation. As far as the generation of the look up tables is concerned for a single degree of freedom system the procedure is straight forward. For multi degrees of freedom the problem is more complicated. For the systems from Figures 3 and 4, for each step of the first input a series of steps defined by the range of motion of the second input should be taken. In this case the range of motion for both inputs is 360'. Thus the generation of the look up tables for multi degrees of freedom systems requires large files. Although this formulation is good for any method of numerical integration, without the generation of the look up tables and pre-computation of invariants it seems more involved than the original ADAMS. Also from a user point of view it does require more engineering and mechanics science background.

46

References:

5.

I. 2. 3.

4. 5.

Gear, C. W. Differential-algebraic equation index transformation. S U M J. Sci. Siut Comp., 1988,9,39-47 Orlandea, N. V. and Coddington, R. Reduced index sparse tableau formulation for improved error control of the original ADAMS program. Mechanics in Des., 1996, I, 2 19-228, Toronto. Orlandea, N. V. A study of effects of the lower index methods on ADAMS sparse tableau formulation for the computational dynamics of multi-body mechanical systems. Journal of Multi-body Dynamics, Proc. Instn Mech Engrs. Vol. 213 Part K, 1999, London. Maros D. and Orlandea N. Contributions to the determination of the equations of motion for multi degrees of freedom systems. Journal of Engineering for Industry, Trans. ASME, Vol. 93 Series B February 1971, New-York Maros, D. and Orlandea N. The kinematics of the multi degrees of freedom planar mechanisms. Mathematica, Vol. 9 (32), 1, 1967,91-100, Cluj-Napoca

YOO4/028/2004

47

Appendix A Transformations of squared summations.

;:I

The Euler transformation matrix E=[;;;

::

'32

where a,, = cosp cosy - sin p sin y cos 0 ; a,, = -sin cosy - cosp sin y cos 0 uI3= sin Bsin y ; uZ1= cospsin y +sin p c o s y cos6 ; uZ2= -sin psin y

+ cospcos y cos6 ;

uZ3=-sinecosy; (3'4) u31=-sinesinp; u32=-sinecosp; u33

=-case.

The transmission function of first order for the five-bar mechanism from Figure 3. 3P4 - 12 Sin(% - PS) . ap2 14 s i n ( ~ ,-, ps)'

& = -f2. WP3 -R)

ap3 14 s i n ( -PA ~~ sin(% P4) . 394 = 13 M P 3 - P4) = ap2 15 sin(q -p4) ' c%p2 15 sin(p5 -q4) and the transmission functions of the second order,

*

48

'

Y004/028/2004

+

a 95

-

+

12 * cos(p,, - p5) 15(-)aP cos(p5 - p4) 15(-)8% aP2 dP2

Y004/028/2004

2

49

Marionette posture analysis by particle swarm optimization algorithm M OKUMA

Department of Mechanical and Aerospace Engineering, Tokyo Institute of Technology, Japan G M GERMAIN Department of Mechanical and Aerospace Engineering, Tokyo Institute of Technology, Japan (visiting student from Ecole Nationale Superieure d’Arts et Metriers, France)

1-Introduction The purpose of this paper is to investigate the applicability of the Particle Swarm Optimization Algorithm (PSOA in brief below) which was proposed by Kennedy and Eberhart* in 1995, to the static posture analysis of marionettes that are controlled by strings prior to its dynamic analysis. Marionette may be a kind of interesting multi-body system. A general marionette is controlled by about 10 strings and has some degrees of redundant freedom. In order to realize easy computer control system of marionettes, it is necessary to develop easy and quick methods for analyzing its static and dynamic posture. This paper presents a study on the static posture analysis of a marionette having closed link loops by the particle swarm optimization algorithm. Kennedy and Eberhart mentioned in their paper that the algorithm was originally developed based on long and careful investigation about the action of animals’ swarms such as bird swarms and fish schooling by zoologists. The authors of this paper investigate the applicability of the algorithm into the analysis of multi-body systems. This paper is the first report presenting a very easy and quick way of solving static posture analysis of a marionette having redundant freedom using the algorithm. This study can suggest more applicability of the algorithm for various analyses of multi-body dynamics. There are some kinds of marionettes. The ones are manipulated from the top. The others are from the bottom. In the former category, there are ones entirely manipulated with strings and others manipulated arms and legs under the condition of its body being held using metal rods. In the latter category, there are ones that are manipulated by a manipulator’s hand in its body and others that are done using rods. The type of marionettes in this paper is one manipulated using strings from the top as shown in Fig. 1. The final objective of this research project is to develop a computer controlled marionette system that can be easily operated with any story of play.

Y004/041/2004

Fig.1 Marionette

51

2-Marionette’s posture analysis As can understand in Fig. 1, a marionette is controlled generally using about ten strings. It has some degrees of redundant freedom. The marionette in Fig.1 has legs that can be considered as a closed loop link with leg parts and a control string. It is desired at first to know the relation between the posture of marionette’s parts and the position of strings’ control points. It is realized by static analysis. However, the analysis is not so easy due to some degrees of redundant freedom and nonharity. Figure 2 shows a leg of the above-shown marionette as a basic 2-dimensional example. The number of degrees of freedom is two. The control string is put through a hole, denoted by A, in the femur part from the top and tied to the lower end, denoted by C, of the tibia. So, at first, ones primitively want to know the posture of the leg dependent on the position of the string’s control point denoted by M in Fig.2. The concept of mechanics for the solution has no discussion. That is, it is the posture that has the minimum potential energy. So, this is a nonlinear optimization problem to find the posture having the minimum potential energy among the infinite number of geometrically feasible postures. In general, Newtonian methods of optimization can work efficiently if it starts at an initial point in the design region that forms the parabola phase in which the objective function is convex for the bottom or the top with respect to the optimum point. However, they may have difsculty for the solution of strong nonlinear Fig.2 A leg of the marionette problems having many local minima and the problems that cannot be expressed by mathematically differentiable equations. PSOA may be applied to such problems. 3-Particle Swarm Optimization Algorithm As abovementioned, Particle Swarm Optimization Algorithm is a quite new algorithm developed by R. C. Eberhart and J. Kennedy in 1995[1]. According to reference [l] and others, PSOA was developed based on long and careful investigation about social behaviors of some kinds of animals such as bird blocking and fish schools by zoologists. The algorithm is as follows. At first, suppose the following scenario. There is a group of animals, such as a bird flock and a fish school. The individuals are called “particles”. The particles in a group are searching food within an area. Each particle realizes its position in a coordinate system. The position is considered as its design variables. The position of food and the area are considered as the optimum position and the feasible design area respectively in PSOA. No particle knows where the food is. But they can know how far they are from the food in each time. Each time means each iterative process in PSOA. An effective strategy to find the food is basically to follow the particle being nearest to the food together with using recollectednearest positionswhere each particle has ever reached.

52

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Now, the algorithm of PSOA is reviewed as follows: Suppose the number of particles in a group as N. All particles are numbered No.1 to N. The position of particle No.k at discrete time i is denoted by p i . The optimum position is the place of food. The best ever position of the group and the best ever position of each particle No.k until the discrete time i are denoted by 6, and k $ l , respectively. Then, particle No.k moves to a new position according to the algorithm expressed by

. .(2) *

where represents the displacement to update the position at the current iteration, and CI and c2 are both constant parameters called “social scaling factors” such that cl=c2=2. The parameters 9 and r2 are uniform random numbers called “stochastic factor coefficient”. The parameter w, is called ‘’ininertiacoefficient” and is normally selected as a value between 0.8 and 1.4. 4-Marionette posture analysis by PSOA

As abovementioned, the purpose of this paper is to investigate the applicability of PSOA. Then, here is presented the posture analysis of 2-dimensional marionette model in the vertical plane as shown in Fig.3. The model consists of a head, an upper body, a lower body under waist, an upper arm,a lower arm, a hand, an upper leg, a lower leg as shown in Fig.3. It has 9 degrees of freedom and is suspended by three strings. Let us assume the string for the head is fixed at the origin of the coordinate system, and the length is constant. Another two control strings can change the lengths and their control points are assumed to be on Y-axis. Then, the length and the coordinates of the two control strings determine t k posture of the marionette model. The problem to solve is the posture of the model under the following parameters given: the mass, mass center, size, connecting points with neighbor parts with respect to each model part, the lengths of the three strings, and the Y-coordinates of two control strings. With respect to the connectivity from point 0 to Yl(O,yl), two algebraic equations are formulated as

where the first equation expresses the connectivity about Xcoordinate and the second is about Y-coordinate. With respect to the connectivity from point 0 to Y2(O,y2), two equations are formulated as ...(4)

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Fig3 Two-dimensional model of marionette for PSOA where the first equation expresses the connectivity about Xcoordinate and the second is about Y-coordinate. Then, the number of the variables to define the posture of the model is eleven. The number of equation about the connectivity is four. This leads the number of the degrees of redundant freedom is seven. If only the connectivity is considered, we can get infinite number of feasible solutions but no unique solution. However, the target to find is a solution that gives the marionette model minimum potential energy among the feasible ones. Seven variables in Eq.(3) and (4) are adopted as the position of each particle in PSOA. Substituting random numbers to seven variables in the four equations in Eq.(3) and (4), other four uriables are uniquely obtained because of variable dependency in Eq.(3) and (4). A number of particles are then created. Each particle keeps its potential energy by Eq.(S)

where tn, and xG, denote the mass of part N0.j (j=1-9), for instance No. 1 is the head, of the

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Fig.4 Initial distribution of particles

Fig.5 Converged result by PSOA

marionette model and the scoordinate of its mass center, respectively. In this paper, PSOA starts using 50 particles. “Social scaling factors” C I and cz are both set 2. At first, PSOA is executed by setting values between 0.8 and 1.4 for the inertia coefficient w, in accordance with the information from references. However, good and fast convergence of the particles cannot be obtained. Therefore, we set smaller values such as 0.015 for the inertia coefficient in PSOA, and then obtained good and fast convergence of the particles. The convergence is obtained within less than 20 iterations at high probability. Figure 4 shows initial postures of 50 particles. They are created using random numbers as the abovementioned way. Figure 5 shows the result of the particles’ convergence after some number of iteration. It is the resultant posture of the marionette model. It is found that appropriate values for the inertia coefficient are not always between 0.8 and 1.4 and depend on the characteristics of problem to be solved. PSOA gives optimum solutions, provided appropriate values are set for the inertia coeficient and social scaling factors in accordance with the dynamic characteristics of the problems to be solved. 5-Conclusions In this paper, the static posture analysis of a 2-dimesbnal model of a marionette having 8 parts is executed as an optimization problem of multkbody systems by PSOA (Particle Swarm Optimization Algorithm). Then, the following remarks are obtained

(1) PSOA can make this sort of problems easy to be tackled. In addition, this study can suggest the applicability of PSOA to dynamic analysis of multi-body systems. (2) Appropriate values for “Ssocial scaling factors” and ”inertia ccefficient” in PSOA may be dominantly depend on the characteristics of problems to be solved. Using appropriate values for

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the parameters, PSOA will be able to work properly with high probability of convergence for even very strong norrlinear optimizationproblems.

References (1) J. Kennedy and R.C. Eberhart, ‘‘Particle swarm optimiition”, Proceedings of the 1995 IEEE International conference on Neural Networks, volume 4,pp. 1942-1948, 1995. (2) A. A. Shabana, “Dynamics of Multibody System”, Second Edition, Cambridge University Press, 1998. (3) A. A. Groenwold and J. farkas, “Selecting representative objective functions: a case study using swarm intelligence”, Short paper of the 5th World Congress of Structural and Multidisciplinary Optimization, pp.19-23,2003.

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Visualization of dynamic multi-body simulation data A SIEMERS and D FRITZSON SKF, Sweden

Abstract

The purpose of this paper is to describe what is needed to create a complete multibody visualization system. The complete visualization procesa includes everything from data storage to image rendering, and what is needed for a meaningful user-to-data interaction. Normally the simulation output data has a large number of time steps, in the order of lo3 to 10'. In order to handle this large amount of data all possible bottlenecks need to be removed. This includes data storage, data processing, system modeling, and image rendering. An object oriented approach is used for the multibody model, its basic simulation data structures, and for the visualization system. This gives well structured models and supports both efficient computation and visualization without additional transformations. Simulation data can be classified into three classes, scalar-data, vector-data, and surfacedata. This paper focuses on time-varying vector data. The huge amount of data and time steps require data compression. Vectors are compressed using an algorithm specially designed for time-varying vector data. Selective data access is required during visualization. A block based streaming technique is created to achieve fast selective data access. These visualization techniques are used in a full-scale industrial system and has proven its usefulness.

1

Introduction

The purpose of this document is to describe what is needed to create a complete multibody visualization system. The reader should understand the complete visualization process, from data storage to image rendering. What is needed for a meaningful user-to-data interaction? Besides, some background information on multibody simulation will be given. The following topics are covered in this paper: Modeling of m u l t i b o d y systems. How to model the hierarchical structure of a multibody system for simulation and visualization?

R e q u i r e m e n t s on the visualization system. What are the requirements regarding performance and usability on the visualization system? ~~

Keywords: Visualization, Multibody, Simulation, Dynamic, Modeling, Contact, Surface, Vector, Compression, high formance

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Surface representations. Mechanical parts, or bodies, are defined by their boundary surfaces. How are surface represented during visualization? Classification of simulation data. There are different types of simulation data, e.g., scalar or vector data. The data used for this work needs to be classified. D a t a storage and access. A lot of time varying data is produced by a dynamic multibody simulation program. Compression is needed to reduce data size and fast data access needs to be granted during visualization. Visualization techniques for different t y p e s of simulation data. How to transfer different types of simulation data into visual representations understood by engineers and scientists? Graphics and visualization libraries. Different 3D libraries and toolkits are available for 3D image rendering and visualization. How useful are they for visualization of the simulation data used here? User interfaces for effective usage. What is needed for a meaningful and effective interaction with the simulation data? H a r d w a r e R e q u i r e m e n t s for Visualization. Large size data visualization is memory consuming and computation intensive. Is special graphics hardware needed for visualization? Some of these topics have well defined solutions while other need more attention and work. However, this paper describes a complete and working visualization system for multibody simulation data, called Beauty, based on these topics.

1.1

Multibody Simulation

There are many aspects of mechanical systems that need to be investigated. This is especially true for their dynamic behavior. Multibody systems are used to model mechanical systems in which several bodies interact with each other. A rolling bearing is an example of such a system, see Figure 1. Dynamic multibody simulations are conducted to investigate the dynamic behavior of these systems. In general, such simulations produce a large amount of data and different visualization techniques are needed for the analysis.

Figure 1: A ball bearing is a typical example of a multibody system. It consists of several bodies, i.e., an inner ring, an outer ring, rolling elements, and a cage. Two dimensional plots are commonly used to investigate time varying (dynamic) scalar data and vector components. These plots are commonly understood and therefore offer a

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good communication basis. However, they are not optimal for understanding of the dynamic behavior of a complete three dimensional multibody system. Thus a three dimensional visualization system is needed.

1.2

Overview of the Visualization Process

The main goal of multibody Visualization is to visualize 1. The mechanical system

2. Movement of the bodies in the system 3. Data from contacts between bodies

The simulation data used for this work is provided by a multibody simulation program called BEAST [13][24]. BEAST is specialized on contact problems of rolling bearings. This paper describes a visualization system for BEAST data, and thus focuses on the visualization of rolling bearing data. However, many of the described techniques solve general visualization problems. There are two different types of data supplied by the simulation program, static model data and dynamic simulation data. Static model data describes the initial state of the multibody system, this is surface geometry and orientation of bodies. Dynamic simulation data is time varying data and describes multibody system dynamics. This implies changes in surface geometry, movements of bodies, and contact related data, e.g., contact forces and moments. Memory

File

Simulation Data

Figure 2: Data flow from file t o the Screen. Figure 2 describes the data-flow during visualization. Throughout the visualization process the data appears in different representations. Body surfaces, for example, are defined by a composition of different static and dynamic functions. Dynamic simulation data is superimposed on the initial static surface geometry. Each surface is then transformed into a representation understood by the graphics renderer. However, the visualization process can be divided into two parts; visualization of the multibody system and visualization of dynamic simulation data.

1.3

Requirements on the Visualization System

A MBS model containing many contacts will generate lot of information during impacts and less in between impacts. Fixed time steps will either miss the important contact information or will generate enormous amount of unnecessary information in between impacts. A compact data format is very important in this kind of data. The number of time steps and data variables depend on the structure of the system and the simulation. BEAST simulations run in parallel, typically on a 40-node Linux cluster and take between 1 hour and 1 week wall-clock time. Typical simulations produce between lo3

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and loe time steps of dynamic data with 5,000 t o 25,000 variables per time step. This results in 500MB to 8GB of compressed data. The large number of time steps, data, and variables puts special demands on the visualization system. While simulations run on parallel machines, the visualization system should run on common workstations and PC’s with additional 3D graphics hardware. Scientists and engineers should be able to run the software in their office without the need for a high-end graphics workstations. Based on this, the following requirements for the visualization system are defined: Effective data compression for timevarying scalar and vector data is needed to reduce the size of the data to a minimum. The effectiveness of a compression algorithm is very much data dependent. Fast Data access is required for animation or visualization of time varying data. Models do not need to move in real-time but reasonably fast. A minimum frame-rate of 1 frame per second is acceptable for this work. Meaningful data interaction is required to give users the possibility to interact with the data and to understand it. Complex models, consisting of many bodies, cannot be investigated at a whole. Viewpoint adjustment and selection of model parts is required. Since the large amount of data that is created by the simulation program cannot be visualized all at onced data selection capabilities for variables and time-steps are required. Fast data transformations are needed to create a visible representation of the data. Hardware requirements need to be defined. What system should the visualization application run on?

2 2.1

Related Work General Visualization Systems

General visualization systems such as for instance AVS, Amira, and OpenDX are powerful tools for 3D data exploration. They work at a high level of abstraction, meaning that you have to prepare your data in a way understood by the tool. Alternatively you can write your own import module or filter. Thus there is still a lot of work needed to prepare your data. This is especially true for large dynamic data sets of multiple domains. Such data can often be found in the field of multibody dynamics, e.g., geometric data (surface geometry), surface related data (wear, pressure, etc.), and vector data (forces, speeds, etc.). These requirements do not exclude the usage of a general visualization system, but the choice of a appropriate visualization library or tool for the final image rendering process is just one part of the whole visualization process. Efficient access and storage of large data sets can be a problem for general visualization systems.

2.2

Visualization of Large Data Sets

Different techniques have been developed in the past to handle large data sets. When processing large data sets some form of smart memory management is needed. Operating systems have build in virtual memory management and many application relay on this. Another technique is called out-of-core visualization [ll]and is based on external memory algorithms [l]. These approaches implement their own memory management. Cox and Ellsworth [7] for example propose a general framework for application controlled virtual memory management.

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Most of these techniques address the problem of visualizing large scale volumetric data sets, mainly static data sets, and are thus not suitable for the time-varying data with thousands of variables used here. Streaming techniques are used to process large sets of time varying data, and allow data to be loaded in streams of segments or blocks. Visualization of time dependent data sets is a typical application for data streaming. Streaming techniques can be classified into two main classes, time continuous media streaming and large scale data streaming. The first is mainly used for audio and video streams [20] where continuous playback is often more important than quality. The latter is often used for large static data sets [2] [19] [17] which are processed in smaller blocks of data, to reduce memory load, or gain speed. In Section 6 , a streaming technique for time-varying vector and scalar data is presented. In contrast to media streaming techniques which process each time-frame separately a time-block based approach is used.

2.3

Visualization of CFD and FEA Data

Two other common areas where computer simulations are conducted are computational fluid dynamics (CFD) and finite element analysis (FEA) [3]. Most CFD simulations use the finite volume method while FEA simulations use the finite element method. The output from these simulations is typically a 2D or 3D mesh [5] [25] of nodes containing simulation data, most often a 2D or 3D vector field. Visualization systems for FEA and CFD data are mesh based and often work with a fixed mesh topology. They are therefore less suitable for the continuous surfaces geometry required by contact mechanics. These tools have predefined data formats which might be a limitation for simulation with large number of time steps and varying time step length.

2.4

MSC.ADAMS

- Multibody Simulation tool

MSC.ADAMS is the most popular and widely used multibody simulation system in industry. It is scalable, i.e., it can be used for both very large and small systems. MSC.ADAMS is a modular system, meaning that the base simulation package can be extended by several plug-ins t o fulfill the users needs, for instance:

ADAMS/Car is used by automotive engineers t o build models of entire vehicles. ADAMS/Flex allows import of flexible parts into MSC.ADAMS. ADAMS/Exchange can import CAD models into MSC.ADAMS. ADAMS/PostProcessor is an advanced post processor supporting 2D/3D plotting and 3D animation. ADAMS/View is the standard pre-processor used by MSC.ADAMS. It also has simple post-processing capabilities t o animate multibody dynamics. However, for advanced postprocessing, i.e., animations combined with 2D plots, ADAMS/PostProcessor is used. ADAMS/PostProcessor is a combined 3D animation, plotting, and documentation tool. Full 3D view of the complete model and animation of the system dynamics is supported. Multiple 3D animation views and 2D plots can be activated simultaneously. All views are synchronized during animation and the current animation step is marked within 2D plots. Force elements are visualized as animated vectors. Light sources, material colors, and camera can be adjusted. Cameras can be set t o follow any point in the model. Tkansparency is used to uncover hidden parts. Animations can be stored as AVI movies.

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The ADAMS/PostProcessor is a powerful visualization system for multibody simulation data but there are some differences to the requirements. The post-processor has limited capabilities for 2D data sets (2D contour plots are possible with the ADAMS/Flex extension only). In simulations with focus on detailed contact analysis 2D data is very important.

3

Object Oriented Modeling of Multibody Systems

Knowledge about the structure of the mechanical system is needed for simulation and visualization. Thus, an internal representation or model of the system is needed. Object oriented modeling [21] is often used to model physical systems. A bearing for example is a composition of bodies, i.e., rings, cage and several balls. Each body is a composition of surface segments, see Figure 3. Even segment contacts can be modeled as objects. This design has many advantages for multibody simulations and visualization because the hierarchical structure of the mechanical system is reflected in the object model. Thus many tasks are delegated directly to each object which simplifies implementation and data handling.

i *Generalization

kgregate

bornposition

.............................................................................

j

Figure 3: A simpltjied object oriented model of a ball bearing.

A model of the mechanical system is required for simulation and visualization. A common base design is therefore used for both programs, see Figure 4. Adjustments t o the model, the file 1/0 routines, and other basic components are necessary only once and changes, testing of new algorithms, and maintenance, are significantly simplified. Object oriented design has been used throughout the implementation and gives an integrated tool that supports natural model structure, efficient computation, and interactive visualization.

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Basic-Model

t Simulation-Tool

Vinualizaion-Tool

Figure 4: Simulation and uisualixation programs are based on the same class design.

4

Surface Representations for Multibody Systems

A geometric description of all bodies is needed for simulation and visualization. Since bodies are described by its surfaces, different surface representations need to be analyzed. There are two main classes of surface representations, continuous and discrete surfaces representations.

4.1

Continuous Surface Representations

Continuous surface representations are either parametric or non-parametric equations and play an important role in geometric modeling [12] [15]. However, the most useful ones are parametric equations. Parametric surfaces are defined by three two-parametric functions, x = f ( u , v ) , y = f(u,v ) , and t = f(u, v ) , where u and v are the parameters. The parametric form of a surface has many advantages in modeling of geometric shapes [15],e.g., the shape is independent from any coordinate system. Splines [8] (121 [15] are common in geometric modeling. A spline is a smooth piecewise polynomial function which is controlled by a set of spatially discrete points. Spline are commonly baaed on parametric equations. The most common splines are B-Splines and

NURBS. B-Splines or Basis-Splines provide local shape control and independence between number of control points and degree of polynomial function. They are generalizations of BQsier curves [12]. N U R B S are Non Uniform Rational B-Splines [MI.Only rational functions can represent a conic curve, e.g., a circle or an ellipse. Since a standard B-Spline curve is not capable of representing a conic curve, NURBS where invented. NURBS are generalizations of B-Splines and play an important role in “Computer Aided Design” (CAD).

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4.2

Discrete Surface Representations

Surfaces are sometimes represented by a set of spatially discrete points. This form of a surface is for instance used for fast graphics rendering where the points are connected t o polygons to build a piecewise linear surface. A piecewise linear surface is called a polygon-mesh. A mesh [5] [25] is a discretization of a geometric domain into small simple shapes, e.g., triangles or quadrilaterals. A polygon mesh is the piecewise linear version of the mesh. Many graphics renderers, e.g., OpenGL [26], support fast triangle and quadrilateral drawing and triangle and quadrilateral meshes are therefore commonly used for surface rendering. There are two major groups of meshes, unstructured and structured meshes. Unstructured meshed are commonly used in FEA [4] [14] [6] and surface reconstruction 1161, most commonly arbitrary triangle meshes based on the Delaunay criterion [9]. Unstructured meshes adapt very good to different surface topologies and are therefore widely used. However, unstructured meshes require significantly more memory than structured meshes and are very computationally intensive. Structured meshes [25] on the other hand are simple and efficient to calculate, and are less memory consuming.

4.3

Surface Representations Used in this W o r k

In the work presented here different surface representations are used for visualization. Surfaces are expressed in parametric form F(U,V)

= ?b(UIv)

+ ‘%(U,v)(AS(U,U) + hgeom(U,V))+ AFflex(utv)

Where Ft,is the base geometry function which defines the basic shape of the surface. The vector 51,is the normal of the base geometry in any surface point. The scalar As is the deviation from the base surface. All these functions are static functions and are calculated from user defined input parameters. The scalar hgeom and the vector AFfl,, are dynamic functions and are calculated by the simulation program. The scalar hgeomis the material removal function and the vector AFflexis the structural deformation of the body. The scalar function As is a composition of different parametric functions, e.g., splines and wave functions, and is used to describe surface imperfections. In order to create a visible representation of the surface, a quadrilateral polygon mesh is created and send to the renderer. The surface mesh needs to be recalculated during animation due to flexibility and material removal. This is a crucial part in the visualization process because mesh calculation is time consuming, especially for multibody systems with many bodies. Surface meshing is done in parametric space because it simplifies the meshing process significantly. A structured quadrilateral mesh is used because structured meshes are faster to compute. Furthermore is adaptive mesh resolution used, meaning that mesh resolution depends on different factors, i.e., overall shape of the surface, ratio between size of the surface and size of the view-port, and rendering performance.

5

Classification of Simulation Data

Simulation data is the time-varying data which is calculated by the simulation program and stored in the simulation output-file. Figure 5 shows a contact between one ball and the inner ring. As a result of the contact we will receive contact forces acting on both bodies. Many things happen within the contact and a large amount of data is produced by the simulation program. The data used here can be classified into three groups: Contact data is data which belongs to the contact itself, e.g., contact forces and moments. It does not belong to one single body but the resulting contact of two bodies. There

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Figure 5 : Contact between ball and inner ring. are two types of contact data, scalar and vector data (vectors of dimension three), e.g., a force vector P. B o d y related data is data which belongs to a certain body, e.g., velocity or acceleration. But it can also be the total force of all contacts acting on a body. As for contact data there are two types of body related data, scalar and vector data. Surface related data results from contact areas between two bodies. Two bodies do not contact in a single contact point but in a contact area. The contact area results in a pressure distributions and other distributed surface data. Such data is called “surface data”. Surface data values are either scalars, e.g., pressure, or vectors, e.g., velocity. Thus another possible classification of the data is scalar-data, vector-data, and surfacedata. Storage and visualization of scalar-data and vector-data is quite similar. Surfacedata on the other hand is much more complicated to store and visualize, and is therefore presented in a separate paper [23].The main focus here is on storage and visualization of vector-data.

6

Data Storage and Access

The amount of data produced by a simulation program depends on many factors, e.g., structure of the multibody system, simulation length, and applied loads. However, typical simulations produce a large amount of dynamic data. For each time step the simulation program calculates thousands of values. These values are written to a file for later analyzes. There are two main issues to overcome here, data size and data access: The size of the data should be reduced to a minimum. Therefore an effective and fast compression algorithm is required. Fast selective data access is required during the visualization. Dynamic or time varying data put special demands on the file format. It has to support fast access t o any variable at any time step.

6.1

Compression of Time-Varying Scalar and Vector Data

Data compression is used in many different areas and many different compression algorithms have been designed. The efficiency of a compression algorithm depends very much on the structure of the data you want to compress. Thus, it is useful to characterize the data first and try to find a compression algorithm which suits best. A compression algorithm for high-volume numerical data has been designed by V. Engelson et.al. [lo]. This algorithm is specially designed for the time varying vector data used here. It is based on delta compression where values are approximated from earlier time steps.

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Table 1: Compression of dynamic vector data for different multibody systems. Model Simulation Compression Steps Ratio Roller Bearing Roller Bearing 1002 2.6 Grinding Machine 1002 39.7

Several tests have been conducted to investigate the effectiveness of the compression algorithm. Table 1gives three examples of short but representative simulations. The compression ratio varies for different simulations. The high compression ratio for the grinding machine is to trace back to slow changes in the data which are approximated very well and therefore can be compressed very effectively.

6.2

Storage of Scalar and Vector Data for Fast and Selective Access

Data compression is only one aspect of the data storage issue. The compressed data needs to be stored for future analyzes. Fast selective data access is required. Delta compression complicates selective access since it requires historical data. One approach is to divided the data into blocks and compress block by block. During visualization the selected block is read an decompressed. The technique of processing data in several blocks rather than as a whole is called data streaming [20][2] [19][17]. A time continuous streaming technique is used in this work to store vector data, e.g., force and motion vectors. A fixed size memory block is allocated during simulation. This block is continuously filled with data. Full blocks are compressed and written to the file. A block header is attached to each block, see Figure 6. The memory block is cleared after writing and the compression algorithm reinitialized. The block header keeps information about vector offset in the block and time period. To allow fast arbitrary vector access the visualization program reads all block headers on startup. During animation block data is read sequentially and decompressed one by one.

Vector 1

vector 2 vector 1

YoELor n

Haader Block-3

Figure 6: Dynamic vectors are stored in data blocks.

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7 Visualization techniques for different types of data “Informally, visualization is the transformation of data or information into pictures” [22]. A large effort has been done to find proper ways to visualize data. Data visualization is based on transfer functions or transformation algorithms which describe how the data is transformed into it a graphical representation. Some of the most common algorithms are color-mapping, contour-mapping, vector-fields, and glyphs [22]. This paper explains visualization of vector-data. Visualization of surface-data is covered in [23]. Vectors are generally visualized as directed lines or glyphs. In this work a combination of both is used, a directed line with different glyphs at one end. Vector data can be divided into two categories; positional and non positional vectors. Positional vectors can define position and orientation of bodies and are used for transformations only. Non positional vectors are represented by a directed line with either a single arrow at the end, e.g., speeds and forces, or two arrows a t the end, e.g., moments. However, the essential part of vector visualization is positioning and scaling of vectors. Vectors can be of any unit, e.g., Nm or m/s, and therefore need to be transformed into model coordinates to be visible at full length. While positioning is fairly simple, its more a matter of choice, e.g., the origin of the bodies coordinate system, scaling needs some special attention. The following is taken into account for vector scaling: A reference length in model coordinates is needed to align vectors to, e.g., half model size or view-port size. For comparison of vectors of the same unit a common reference length is useful t o scale these vectors. Vector length needs to be adjusted if view-port size changes, e.g., the user enlarges the model. Fast changing values during animation might lead to jumping vectors. This can be avoided by a slowly adjusting scaling algorithm. The application of this is discussed below.

8

The Visualization System

The techniques described in this paper have been implemented in a visualization system called Beauty, see Figure 7. Many of the techniques described here have been integrated into this system as a base for further research and development. The main window allows fast access to all the common settings and actions, e.g., model visibility, animation settings, and many more. Visualization of the complete multibody system is the goal. A 3D scene with lights, coordinate systems, and the multibody system itself is created, see Figure 7. Mouse based rotation, translation, and scaling controls are used to interact with the model. An exploded view and transparency is used to view hidden parts. Besides each part the scene can be disabled (made invisible) or enabled.

8.1

Body and Surface Rendering

As mentioned earlier bodies are defined by its surfaces, and a boundary representation is used for visualization of bodies. Default surface colors are defined for different material types. These can be adjusted by the user. A colorize mode allows recoloring of all bodies for better distinguishing.

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Figure 7: The Beaut9 main window uisualizing a ball bearing in ezploded view. Surfaces of real-world objects are not perfect. Imperfect surface geometry can be described by the user with different parametrized functions, e.g., B-Splines. High contact forces might lead to material removal which destroys perfect surfacea as well. User defined magnification is used to visualize surface imperfections which are often on micrometer level, see Figure 8(a).

(a) Magnified inner ring gmrnetw.

(b) Colorized boll h r i n g model.

Figure 8: A ball bearing model once with the imperfect inner ring raceway magnified about 1000 times and once in wlorize mode.

8.2

Visualizing Multibody Dynamics

Animation is used to investigate the dynamic behavior of the model. Bodies move, rotate, and get in contact with each other. Movements are often very small because of the minimal gap between bodies. To be able to see small movements they can be magnified, see Figure 9. Thus users can for instance see how a single ball moves within the cage pocket. To improve this even more a locking mechanism for bodies is used. The user can lock the movement of any body in the system and the rest of the multibody system moves relative to this body.

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Figure 9: Radial motion of the balls magnified by factor 1000.

8.3

Vector Data (Forces and Motions)

Different kind of vector data is produced by the simulation program, e.g., forces, moments, and speed vectors. These vectors are of dimension three and are visualized as a directed line with an arrow at one end. Rotation and moment vectors are an exception, they are according to standard notation drawn with two arrows, see Figure 10. Vectors belonging to a certain body, e.g., speed vectors, are located in the origin of the bodies coordinate system. Contact vectors, e.g., total force in a contact, are located in the origin of the coordinate system of one of the contacting bodies.

Figure 10: Different vectors have different colors. The vectors legend indicates the valve of

the unit vector. Vectors are auto-scaled to fit into the view-port of the model. To achieve smooth autoscaling and avoid “jumping” of vectors a slowly adapting scaling algorithm is used. Old scale factors are weighted higher than new scale factors. To be able to compare vectors of the same type, e.g., forces, they are scaled by the same scale factor. Additionally, the user can set fixed scale factors or magnify the auto-scale factors. Vector values (length per unit) are displayed in an vector-legend beside the model, see Figure 10. Different colors and line styles indicate different vectors.

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8.4

Presenting Visualization Results

Engineers conduct simulations to improve a system or find problems. However, the results are often presented to customers and other interested parties. One way is to use the visualization system for the presentation. Some commercial visualization systems offer a free viewer for predefined animations. But often it is more useful to create an animated film which can be included into a standard presentation. The system described here generates animated films by capturing all rendered images and writing them as an image sequence into a file. This file can be converted into different computer video formats, e.g., MPEG-1, which can be imported into standard presentations.

8.5

Hardware Requirements for the Visualization System

The application is meant to run on standard workstations and PC’s. It is Motif and XWindows based and has been tested on various platforms, e.g., Solaris, Linux,and MicrosoftWindows (using an X-Server). Multibody systems with many bodies are memory consuming and at least 512MB is needed to be able to work with large models. The current version uses OpenGL to render the 3D scene. Since the whole system is three dimensional a 3D graphics card with OpenGL support is needed.

9

Conclusion

We have described what is needed to create a complete multibody visualization system. The complete visualization process includes everything from data storage to image rendering, and what is needed for a meaningful user-tedata interaction. Normally the simulation output data has a large number of time steps, in the order of lo3 to lo6. In order to handle this large amount of data all possible bottlenecks need to be removed, this includes data storage, data processing, system modeling, and image rendering. An object oriented approach is used for the multibody model, its basic simulation data structures, and for the visualization system. This gives well structured models and supports both efficient computation and visualization without additional transformations. Parametric surfaces are discretized during the visualization process. The final surface representation is a structured quadrilateral mesh. It gives fast re-meshing and interacts with surface data structures without additional transformations. Simulation data can be classified into three classes, scalar-data, vector-data, and surfacedata. The large amount of data and time steps require data compression. Vectors are stored as a stream of blocks each of which is compressed using an efficient algorithm specially designed for time-varying vector data. The block structures allows fast access to a certain time step during visualization. The visualization techniques have been implemented in a full-scale industrial system called BEAST by SKF, and has proven its usefulness. The system supports natural interaction with simulation data, e.g., selection of vectors and motion magnification. This is an important aspect in data visualization.

Acknowledgments The authors wish to thank the SKF Group Senior Vice President SKF Group Technology Development, Dr. Henning Wittmeyer, for his financial support and permission to publish this paper. This work was supported by SKF Group Technology Development, the KK-stiftelsens

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foretagsforskarskola in Linkoping, and the ECSEL (Excellence Centre for Computer Science and Systems Engineering in Linkoping) Graduate School.

References [l] J. Abello and J. Vitter. External Memory Algorithms. American Mathematical SOC., 1999. [2] J. Ahrens, K. Brislawn, K. Martin, B. Geveci, C.C. Law, and M. Papka. Large-ScaleData visualization Using Parallel Data Streaming. IEEE Computer Graphics and Application, 21(4):34, July/August 2001. [3] K. Bathe. Finite Element Procedures. Prentice-Hall, Inc., Upper Saddle River, New Jersey 07458, 1996.

[4] Bern and Eppstein. Mesh Generation and Optimal Triangulation. In Computing in Euclidean Geometry, Edited b y Ding-Zhu Du and R a n k Hwang, World Scientific, Lecture Notes Series on Computing - Vol. 1. World Scientific, 1992. [5] Marshall Bern and Paul Plassmann. Mesh Generation. In Jorg Sack and Jorge Urrutia, editors, Handbook of Computational Geometry, Elsevier Science, to appear. [6] Y. K. Cheung and M. F. Yeo. A Practical Introduction to Finite Element Analysis. Pitman Publishing Lmt., New York, 1979.

[7] Michael B. Cox and David Ellsworth. Application-controlled demand paging for out-ofcore visualization. In IEEE Visualization ’97, 1997. [8] C. deBoor. A Pmctical Guide to Splines. Springer-Verlag, New York, 1978. [9] Boris N. Delaunay. Sur la Sphere. Zzuestia Akademia Nauk SSSR, V I I Seria, Otdelenie Matematicheski i Estestvennyka Nauk, 7:793-800, Nov/Dec 2000.

[lo] V. Engelson, D. Fritzson, and P. Fritzson. Lossless Compression of High-Volume Numerical Data from Simulations. In Proc. of The 2000 IEEE Data Compression Conference, Snowbird, Utah, March 28-30 2000. [ll] R. Farias and C. T. Silva. Out-Of-Core Rendering of Large, Unstructured Grids. ZEEE Computer Graphics and Application, 21(4):42, july/August 2001.

1121 G. E. Farin. Curves and Surfaces f o r Computer Aided Geometric Design, A Practical Guide. Academic Press, Inc., 1250 Sixth Avenue, San Diego, CA 92101, USA, 1988.

[13] P. Fritzson and P. Nordling. Adaptive Scheduling Strategy Optimizer for Parallel Rolling Bearing Simulation. HPCN Europe ‘99, Amsterdam, April 1999. [14] H. Kardestuncer, D. Norrie, and F. Brezzi. Finite Element Handbook. McGraw-Hill Publ., New York, 1987. [15] M. E. Mortenson. Geometric Modeling. John Wiley & Sons, Inc., New York, 1985. [16] H. Mueller. Surface Reconstruction - An Introduction. In Proceedings of Dagstuhl’S7 Scientific Visualization Conference, 1997.

[17]R. Pajarola and J. Rossignac. Compressed compressive meshes. I E E E Thnsactions on Visualization nand Computer Graphics, 6(2), 2000.

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[18] L. Piegl. On NURBS: A Survey. IEEE Computer Graphics and Applications, 11(1):5% 71, January 1991. [19] C. Prince. Progressive Meshes for Large Models of Arbitrary Topology. M.S. Dissertation, University of Washington, 2000. [20] R. Ramanujan, J. Newhouse, M. Kaddoura, A. Ahamad, E. Chartier, and K. Thurber. Adaptive Streaming of MPEG Video over IP Networks. In Proceedings of the ZEEE Conference on Local Computer Networks, 1997. [21] J. Rumbaugh, M. Blaha, W. Premerlani, F. Eddy, and W. Lorensen. Object-Oriented Modeling and Design. Prentice-Hall, Inc., 1991. [22] W. Schroeder, K. Martin, and B. Lorensen. The Visualixation Toolkit. Prentice-Hall Inc., 2nd edition, 1998. [23] A. Siemers and D. Fritzson. Representation and Visualization of Surface Related Multibody-Simulation Data. To be published, 2003. [24] L-E. Stacke, D. Fritzson, and P. Nordling. BEAST-a rolling bearing simulation tool. Proc. Znstn Mech. Engrs, part K, Journal of Multi-body Dynamics, 213~6.3-71, 1999. [25] J.F. Thompson, Z.U.A. Warsi, and C.W. Mastin. Numerical Grid Generation: Foundations and Applications. Elsevier, New York, 1985. [26] M. Woo, J. Neider, and T. Davis. OpenGL Programming Guide. Addisin Wesley, 2nd edition, 1998.

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Fatigue analysis on a virtual test rig based on multi-body simulation S DIET2 and A EICHBERGER

lntec GmbH, WeRling, Germany

sYN0PsIs The contribution presents a new method of interfacing multi-body dynamics with FEM- and fatigue codes for durability and lifetime analysis. SIMPACK’s new interface to the durability software FEMFAT-MAX in connection with ANSYS was designed by INTEC in co-operation with ECS Steyr and MAN Nutzfahrzeuge AG. The whole simulation process, which includes multibody dynamics, finite element analysis and durability analysis, is easily applicable to long simulation periods even in the case of detailed finite element models and multi-bodysystem models, respectively. Moreover, compared to a transient finite element analysis computation times are significantly reduced.

NOTATION scalars

mu

modal co-ordinate excitation frequency residual transformation factor modal mass

Pk

component of attachment load vector

9,

R

vectors U

u,

deformation mode

4

eigenmode

u;

particular mode

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tJ tJI

tJ:

stress modal stress modal stress (eigenmode)

4

modal stress (particular mode)

P

4

attachment load residual mode

r

vector of free vibrations

matrices H L K

M UP Uh U'

I

P

Hooke's law strain displacement relation stiffness matrix mass matrix particular mode matrix eigenmode matrix residual mode matrix residual transformation matrix identity matrix mass density ( p = p I), with the 3x3 identity matrix

1 INTRODUCTION Classical approaches in the field of coupled finite element and multi-body-system analysis use modal stress calculation that require a huge number of modes and modal co-ordinates which have to be calculated in a multi-body code for later stress calculations. Most of these modes have no influence on the dynamic behaviour of the multi-body system and lead to high computation times. The unique feature of the new approach is, that only modes contributing to the relevant dynamic effects have to be taken into account. In general these are a small number of eigenmodes combined with some frequency response modes that represent structural deformations due to the interface loads within applied forces, constraints and joints. The result output of the multi-body simulation for a later stress calculation are a combination of interface loads and modal co-ordinates which describe the free oscillations of the structure. The small number of modes allows the use of efficient models in the multi-body software for dynamic load data generation. The new approach was applied to a spare wheel carrier of a truck. The computed stresses of the virtual test rig showed a very good correspondence to the measured values of the physical test. Critical areas of the design could be identified and were improved based on the results of the new method. 2 THEORETICAL BACKGROUND 2.1 Modal Stress Calculation In multi-body-systems flexible body deformation u is frequently represented by a linear combination of mode vectors u, and the time dependent modal co-ordinates q1

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n

u =c u , q j . j=l

Based on flexible body deformation stresses a c a n be calculated assuming linear material behaviour Hand a linear strain displacement relation L B = HLu.

(2)

The linear structural behaviour enables us to calculate modal stress vectors cr, corresponding to the modes u, oJ= HLu,

(3)

and to superimpose modal stress vectors using the modal co-ordinates of equation (1) in order to obtain the stress cr within the structure

,=I

Modal stress calculation can help to avoid time consuming transient finite element analysis, if the number of modes n is small compared to the number of degrees of fieedom of the finite element model. However, modal stress calculation based on a small number of eigenmode vectors uh yields poor results [2][3][6].Results can be improved not until the eigenmodes are extended by a set of particular modes u p , in which each particular mode u: represents the influence of a single unit attachment load pkon flexible body deformation. As shown in [2] so called inertia relief modes and frequency response modes

are suitable to be used as particular modes in this context. The calculation of frequency response modes requires the stiffness matrix K , the mass matrix M of the finite element model and the specification of an excitation frequency R , by the user. Frequency response modes can be calculated for floating structures, which are the frequent case in the field of multi-body dynamics, whereas constraints preventing rigid body motion have to be applied in the case of static correction modes. Thus, static correction modes cannot properly represent the behaviour of free floating bodies. 2.2 Modified Modal Stress Calculations in LOADS Durability Flexible bodies are frequently embedded within multi-body-systems by a multitude of connections such as force elements and constraints. Each connection can transmit forces or moments in certain directions, which are subsequently denoted as interface degrees of freedom. Precise modal stress calculation requires the consideration of one particular mode for each interface degree of freedom. Since most of these modes have no influence on the dynamic behaviour of the multi-body-system, LOADS Durability uses a modified approach.

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Equation ( 5 ) shows, that frequency response modes u: linearly depend on their corresponding unit attachment load pk. Instead of using the modal co-ordinates qk LOADS Durability directly uses the actually calculated attachment forces pkto superimpose modal stresses akp

u = -&q,

+Fu:pk

j=l

J=I

This enables SIMPACK to use only those modes that actually affect the dynamic behaviour of the multi-body-system. The complete set of particular modes is only considered during the post-processing in LOADS Durability. Due to this modification of the usual modal stress calculation procedure SIMPACK models can have a small number of degrees of freedom even in the case of a coupled FEM-MBS durability calculation. However, the process requires some further considerations. Each particular mode can be represented by a linear combination that uses the complete set of eigenmodes

tu:a,,

u: =

_. j=l

siguficant cigarnodes

+

-

,

i=nh+I

(7)

neglectedeigenmodes

withn, the number of significant eigenmodes that affect the dynamics of the multi-bodysystem, n the number of degrees of freedom of the finite element model and a set of unknown linear factors a,,and a,k.The linear combination of neglected eigenmodes

can be expressed by a residual displacement vector u; , which is the difference between the particular mode and a linear combination of significant eigenmodes u;, that optimally approximate the particular mode ukp

The same consideration applied to the stresses

shows that modal stress calculations as proposed in equation ( 6 )

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1

o = To;q,+ [ t o ; a J k +a; p , P I

,=I

must be performed using the residual stress vector cr; instead of particular stress vector$. Otherwise the influence of eigenmodes is represented by the factor q, + a,,p, , which leads to erroneous results.

2.3 Considering Residual Stresses Residual particular modes are linear combinations of those eigenmodes which are neglegted for multi-body-system analysis and modal stress calculations. Since all eigenmodes are orthogonal with respect to each other, each residual particular mode u; must be orthogonal to any significant eigenmode u: . Thus, the orthogonality relation

0 = VjUyp(u: - a , , u p v

(12)

and the definitions for the modal mass

mu = Ju:'pu:dV V

yields the residual transformation factors

a,, =-.mJk

mii

The residual transformation (9) can be rewritten in matrix form

with the modal matrices [uh

I U~]'[Uf I

[u* U']= [u:

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..* u;, UII,

I up I u;

... u;J, u;,],

77

the identity matrix I and matrix A, that contains the residual transformation factors

2.4 Preparation of Modal Stress Calculations with LOADS Durability After the time integration and measurements SIMPACK's results for modal stress caclulations are the modal co-ordinates of the eigenmodes q: collected in the vector qhand the attachment

forces pkcollected in the vector p . LOADS Durability considers the residual transformation

for later stress calculations by transforming the result vector

{};

{I*[

=

}:

instead of the modal stress matrix

LOADS Durability writes vector rand the attachment forces p a s result into a file interface for later durability analysis. Furhter work would show that vector r in equation (21) represents the modal co-ordinates of the eigenmodes qhminus the impact of static loading on the modal co-ordinates qt,ol,,of the eigenmodes

In other words, vector r describes the free oscillations of the structure that is embedded within the multi-body-system.

3 THE PROCESS In most cases the process starts with the component which is to be incorporated as flexible body into the SIMPACK model. As mentioned above this requires only those eigenmodes and frequency response modes which affect the dynamic behaviour of the multibody system. In the case of a very stiff component, the body may be modelled as rigid body and the LOADS Durability post-processing will exclusively be based on the attachment loads and the complete set of particular modes. After the time integration LOADS Durability is started and the user is requested to select the time interval for the durability analysis. Furthermore, the user may select between frequency response modes or inertia relief modes to be used for stress calculation. Then, the residual transformation coefficients are calculated.

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LOADS Durability continues with the measurements for each communication point within the selected time interval and calculates the vector of attachment forces p and then the resonance vector r by the residual transformation (21) of the modal co-ordinates 9;. of the eigenmodes. The vector of attachment forces and the resonance vector are written as time series into ASCII files, see Fig. 1.

+ FEA input files

time series of loads and

files containing unit stresses I

1-

I

FEMFAT input file xsiqnment of unitstresses and timeseries

Fig. 1 comparison of the test rig with the simulation Moreover, based on the information about the interface degrees of freedom and the modes which were used in SIMPACK, LOADS durability automatically generates input files for a finite element analysis that yields the complete set of modal stress vectors. The modal stress calculation and the durability analysis is performed in the durability software. In order to obtain the stresses, the modal stress vectors must assigned to their corresponding time series. Each particular stress vector 0: must be assigned to its corresponding attachment force pk and each eigenmode stress vector 0: must be assigned to its corresponding component r, of the resonance vector r . The required data for an automatic assignment are generated by LOADS Durability.

4 THE APPLICATION In the case under consideration, a test rig, investigating the frame add-on components, was simulated with particular attention paid to the fatigue life of the spare wheel carrier. A section of the test rig with the spare wheel carrier as central component was modelled in SIMPACK, see Fig. 2.

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Fig. 2 The Truck and its MBS model of the spare wheel mount The loads due to accelerations, which were measured directly along with the integrated state variables, were applied to the multi-body-system and a time integration was performed. In the multi-body-system model the elastic behaviour of the spare wheel carrier was described by a minimum number of eigenmodes and frequency response modes which guaranteed accurate dynamic behaviour during time integration. Fig. 3 shows the computed accelerations that are well correlated with the measured accelerations up to frequencies of 22 Hz. The measurement points were located at the spare wheel, see Fig. 2. FFT accelerallon of wheel (left)

FFT accelerallonof wheel (rlght)

\ 5

15

10

20

frewenw

25

30

35

40

5

10

[Hzl

15

20 25 frequency [MI

30

35

40

dlslrRUtlonacceleratlon of wheel (rlght)

dlslrlbidlon accelerationof wheel (left)

mmasur~m*nl slmuYlcn

-

I

E

B0

_-c--

1 1

e-.// I

10

100

number of cycles

loo0

loooo

4 1

I 10

100

1000

10000

rmmber 01 cycles

Fig. 3 comparison of the dynamic behaviour of simulation and measurement Here, the intertia relief approach [ 1][2] was used to compute the particular stress vectors with ANSYS. As mentioned above the ANSYS input files, including the unit load cases for all interface degrees of freedom and information about the used eigenmodes, were automatically generated by LOADS Durability. ANSYS completed its job after all the unit stresses were written into ASCII files. Finally FEMFAT-MAX [5] was used to read the assignment data, generated by LOADS Durability,

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the unit stresses and the time series. Cumulative frequency distributions of the stresses at the two critical spots (see Fig. 5) of the spare wheel carrier were compared with corresponding measurement results. The correlation of computed and measured stresses has the same quality, as the correlation of the computed and measured acceleration, compare Fig. 3 and Fig.4. dlstrlbutlon of stress at locatlon 1

8

dlstrlbution of stress at location 2

-50

R -100 -150 -200 -250

Fig. 4 Comparison of the cumulative stress distributions obtained by simulation and measurements. The results of the next task are shown in Fig.5, which shows a contour plot of the fatigue of spare wheel carrier, with two critical spots identifiable by the red coloured regions. Also this result corresponds with the results obtained by the test rig. However, the absolute lifetimes obtained by computation and the test rig were completely different. They were 27 hours for the test rig and 13 hours for the simulation.

Fig. 5 Damage contour plot with critical spots (red coloured areas) Nevertheless, LOADS Durability together with ANSYS and FEMFAT-MAX was a valuable tool even in the case of this pilot project, because a well directed redesign of the spare wheel carrier was assisted by the simulation results. Fig. 6 shows the damage contour plot after the redesign with no critical spots in the case of the current simulation scenario.

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Fig. 6 Improvement of the fatigue life due to a modified welding seam Finally, the simulation results were confirmed by the test rig. 5 SUMMARY

Due to an improved modal stress calculation procedure LOADS Durability saves both, a huge amount of computation times and hardware resources when compared with a transient finite element analysis. This enables the user to perform durability analysis over longer simulation periods even in the case of detailed finite element models and multi-body-system models, respectively. The inclusion of fatigue analysis in a MES simulation offers new perspectives in the CAE and development process. Virtual prototypes can be tested on a SIMPACK test rig or an entire vehicle model before going into production. Since the process can be performed in short computation times, it will allow optimisation of components of mechanical systems regarding fatigue life. The current focus, which is set to the relative fatigue life enables durability calculations, whose results reflect the improvements of component or the vehicle systems, respectively. At present the calculation of absolute fatigue life remains difficult. Nevertheless the future goal is to achieve better approximation of the absolute fatigue life time. 6 REFERENCES

ANSYS, inc.: Theory Reference Version 8.0,2004. Dietz, S.: Vibration and Fatigue Analysis of Vehicle Systems Using Component Modes, Fortschritt-Berichte VDI Reihe 12, Nr 401. VDI Verlag Dusseldorf. Dietz, S . Knothe, K.: Reduktion der Anzahl der Freiheitsgrade in Finite-ElementSubstrukturen, Bericht aus dem Institut ftir Luft-und Raumfahrt der Technischen Universittit Berlin, No. 315, 1997. Gasch, R. Knothe, K.: Strukturdynamik, Band 2 - Kontinua und ihre Diskretisierung, Springer Verlag, Berlin 1989. Magna Steyr: FEMFAT User Manual Version 4.3,2002. Melzer, Frank.: Symbolisch numerische Modellierung elastischer Mehrkorpersysteme mit Anwendung auf rechnerische Lebensdaueworhersagen,Fortschrittberichte VDI, Reihe 20 Rechnerunterstutzte Verfahren, No. 139, 1994.

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Representation and visualization of surface related multi-body simulation data A SIEMERS and D FRITZSON SKF, Sweden

Abstract

The demands on a multibody simulation tool focusing on contact analysis represents a special challenge in the field of scientific visualization [IZ]. This is especially true for multidimensional time-varying data, e.g., two dimensionalsurface related data and vector data. Typical simulations produce between IO3 and 10' time-steps of data, this results in 500MB to 8GB of compressed surface data. A surface data structure is presented which: supports the object oriented multibody modeling approach, represents both instantaneous (contact pressure) and irreversible (wear and life) physical phenomena, allows efficient computation and visualization, allocates memory on demand while still having fast data access, has effective data compression, has additional attributes for data reduction and other features to support computation, fits the multibody surface description and eliminates time consuming transformations, and supports interactive visualization. The visualization techniques have been implemented in a full-scale industrial system called BEAST, and has proven its usefulness.

1

Introduction

The purpose of this paper is to cover the problems of time-varying surface related data in multibody visualization. Surfaces play an important role in multibody simulations because contacts occur between surfaces. Large amounts of surface data are produced by a multibody simulation program that includes contact analysis. It should be understood what surface data is, how it is represented in the different stages of the visualization process, and why smart storage techniques are needed. Furthermore, solutions to the problem are presented and discussed. Results will be presented in form of surface visualization examples. In [12]a general description on multibody visualization techniques has been given where the following topics regarding multibody visualization have been identified: Keywords: Visualization, Multibody, Simulation, Surface, Contacts, 2D Data, Compression

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Modeling of multibody systems. Requirements on the visualization system. Surface representations. Classification of simulation data. Data storage and access. Visualization techniques for different types of simulation data. Graphics and visualization libraries. User interfaces for effective usage. Hardware Requirements for Visualization. This paper follows up [12] and focuses on surface data representation and visualization.

1.1

Multibody Simulations and Simulation Data

Multibody systems are used t o model and investigate the behavior of mechanical systems where several bodies interact with each other. A rolling bearing is a typical multibody system, see Figure 1.

Figure 1: A ball bearing is a typical ezample of 4 multibody system. It consists of several bodies, i.e., an inner ring, an outer ring, rolling elements, and a cage. The simulation data used for this work is provided by a multibody simulation program called BEAST [4] [13]. BEAST is specialized on contact problems of rolling-bearings but is capable of simulating other multibody systems as well. This paper describes techniques used in a visualization system for BEAST data. However, some of the described techniques solve general visualization problems. Visualization of multibody simulation data can be divided into two major steps, visualization of the multibody system itself and visualization of simulation data - this paper focuses on surface related simulation data only. Figure 2 shows the model of a grinding machine consisting of a grinding wheel and the ring which to grind. The multibody system is modeled in a object oriented way, see Figure 2(b). The model is a composition of bodies and each body is a composition of surface segments. Segment contacts, which are the essential part of the simulation program, are also modeled as objects. Contact calculations are delegated directly to the contact objects. Bodies consist of one or several surface segments. In this work each surface segment is an independent parametric surface. Parametric surfaces are defined by three two-parametric

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(a) Grinding wheel (GW) and ring.

(b) The object oriented model.

Figure 2: A simplified grinding machine model. The model consists of bodies which consists of surface segments. Contacts are defined between bodies surfaces. functions, x = f(u,v ) , y = f(u,v ) , and z = f(u,v ) . The parameter space of each surface is a rectangular subset, built by the two parameters u and v , of the 2D space. Surface boundaries are defined within this sub-space. The parametric form of a surface has many advantages in modeling of geometric shapes [9] [3], e.g., the shape is independent from any coordinate system. Two contacting bodies have contact on at least two surfaces, one surface of each body. The simple grinding machine in Figure 2 has exactly one contact between two surfaces. The high contact force between these surfaces and the high speed of the grinding wheel will result in material removal on the ring's surface. The contact also results in pressure distributions on both surfaces. Material removal and pressure distribution are two examples of "surface data". Several types of surface data are generated by the simulation program. Surfaces are described by continuous functions. Surface-data functions, e.g., pressure or temperature distribution, are thus continuous functions as well. Discretization is typically used for storage of continuous data. Surfaces are therefore discretized by the simulation program into equidistant two dimensional grids and surface-data is stored in the grid vertices. This is sometimes referred to as attribute-data [ll]or grid attributes.

1.2

Problem Definition

Each contacting surface in a multibody system creates surface data. Some multibody systems have many contacting surfaces, e.g., the ball bearing in Figure 1, and can thus produce high quantities of surface data. However, the number of time steps and variables depends on the structure of the system and the simulation. Typical simulations produce between lo3 and lo6 time-steps of data, this results in 500MB to 8GB of compressed surface. All this data needs to be read, visualized, and animated. The amount of time-steps, data, and variables put special demands on the visualization system. Data size on disk and in memory is a critical factor when processing large data sets. The size of the data also increases file access and reading time. Complex data structures slow down data access and transformations during visualization. The critical factors can be concluded as: 1. Disk and memory space.

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2. Access time on disk and in memory.

3. Time spent on data transformations. There is always a contradiction between time and space when large data sets are involved. Thus, it is important to find a good balance between the two factors. A data structure is described in this paper which has been designed to achieve this balance.

1.3

Related Work

Regular grids are often (but not always) used in terrain modeling and visualization. Here the grid data describes the height above a ground plane and is therefore often called a height-field. The advantage of regular grids over other mesh representations is its simplicity. They are simple to implement and visualize. However, many applications use very large terrain models and many techniques have been developed for efficient terrain rendering. These are divided into two main classes, simplification methods and level-of-detail (LOD) methods. Simplification methods [6] [l] transform the grid into other surface representations, for instance by removing information, e.g., grid rows or faces. LOD methods [8] [5] store the grid in different resolutions. They are often based on tree structures, where each level in the tree defines a level of detail. Depending on performance or other preferences the mesh can be displayed in different resolutions. Different LOD data structures allow different refinement techniques, e.g., local mesh refinement or progressive mesh refinement. Surface-data is similar to terrain-models because it is stored as a regular grid as well. However, one important difference is that surface-data is time-varying in contrast to most terrain models. Simplification and LOD methods are time consuming and cannot be applied to time-varying data if smooth animation is needed. These methods also destroy the simplicity of height-fields and the results cannot be stored and compressed as efficient as the 2D heightfield data. Especially if thousands of time-steps needs to be stored. These techniques are therefore not appropriate for surface-data. In image processing, streaming techniques are commonly used to process large sets of time-varying 2D data. Streaming techniques allow data to be loaded in streams of segments or blocks. A typical example are video streaming techniques [lo] where continuous playback of many time continuous images is needed. Many image and video compressions have been developed in the past. Most of them, however, require the image to be in a certain color space. Conversion of floating point numbers to color values is not trivial, especially if efficient compression is needed, and it might also lead to unwanted data loss. These techniques are therefore inefficient for this work.

2

A Data Structure for Surface Data Storage

The goal is to create a data structure which allows fast data access and low memory consumption during animation of surface data. A data structure which can be compressed, decompressed, and visualized efficiently. This structure will build the basis for all surface data storage and visualization. Several data-sets from the simulation have been investigated to find common surface-data properties. Two common properties have been identified for many of the data sets. First, many surface-data grids have many more rows than columns, where a row is defined t o follow the short side of the surface, see also Figure 3(b). Secondly, surface data is often active in small regions of the surface only. The contact between a roller and a ring for example is defined by a very small contact area. Thus, there is pressure in a few surface sections only, see Figure 3. The problem of rectangular grids with many zero entries is similar t o sparse matrices. Sparse matrices typically store each non-zero value separately together with its matrix indexes.

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fa) Normal-pressure distribution from ;oiler ring contacts mapped onto the rings segment.

(b) The same data displayed height-field.

88

a color-mapped

Figure 3: Contact surface data visualization. Data is applied in small regions of the surface only. This does not allow efficient access because an index search is needed each time. A different approach is presented here which also takes into account that there are often many more rows than columns in the grid. The solution presented here is derived from virtual memory management [7] where the memory pages, here rows, are accessed through a page-table. It is based on the idea to store non zero-rows only and assume all other data to be zero, see Figure 4. On initialization a row-table with a pointer for each row and a single zero-row is allocated. All pointers are set to point t o the zero-row. Memory is allocated for non zero-rows only and the row pointers set to point to the right row. On grid re-initialization all pointers are set to point to the zero-row again. Once allocated memory is not released on re-initialization but kept to avoid memory reallocations. Too avoid to many memory allocations it can be useful to allocate several rows at once. This results in unused memory, but avoids slow memory allocations. mitiallzed

Set I A l l o c a t e ) 3 ROWS

&-initialize

set

1 T-owe

Figure 4: Surface data storage during visualization. This structure allows very fast data access and optimized memory usage. It thus solves all problems related to memory access and space. Table 1 shows the number of active (non zero) rows compared to the grid size for different multibody simulations. Other tests have shown that the size for each surface-data grid is typically reduced by a factor 1:3 to 1:30. But this technique can also improve disk storage and visualization which is discussed in the next two sections.

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Table 1: Surface data comvression and active arid size. The data is token from simulation surface data. Values are 32bit floating point scaiors. results for different rpes grid size (bytes) compression size complete active rows complete I active rows size (bytes) ratio 1.340.160 I 36.400 1.715 I 22.39 60x698 60x20 27;49 577.920 21.025 1.340.160 60x301 60x698 82x1341 82x205 I 3.518.784 I 537.920 I 31.726 I 16,96 16,96 548.416 34.522 82x1341 82x209 3.518.784 81x44 4.525.632 114.048 4.093 27,86 11.999 21.16 4.525.632 254.016 81x1746 81x98

oi

E-

3

I

I

I

I

I

Compression of Surface Related Data

Data compression is needed for file storage because of the large number of time-steps. The two main issues of data compression are compression ratio and speed. Especially for visualization, fast decompression is needed. The compression takes advantage of the row based data structure. Only the rows containing data need to be compressed. Additional row data need to be stored to restore row information during decompression. Reduction of zero-rows does not increase compression ratio significantly but speeds up compression and decompression. A compression algorithm for high-volume numerical data [2]is used to compress all surface data. The algorithm is designed for one dimensional time dependent simulation data. Therefore all the non zero-rows are virtually connected to a one dimensional data stream. Some compression results are shown in Table 1.

4

Implementation in the Visualization System

Storage and compression of surface related data has been discussed so far. The described techniques have been implemented in a visualization tool for multibody simulation data. Some of the most common visualization techniques for 2D data are color-mapping, contourmapping, height-fields, and vector-fields [ll]. The first three techniques are scalar techniques. Meaning that each scalar of the 2D array is mapped either to a color (color-mapping and contour mapping) or used as an altitude to the XY plane (height-field). Vector-fields is a vector technique where each vector is displayed as an oriented line. Lines start in a point associated with the vector and are oriented and scaled according to the vector components (2,y,2). In the visualization system presented here color mapping and height-fields are used to visualize surface related data. Figure 5 for instance shows the pressure distribution in the cage pockets of the ball bearing, coming from the contact with the balls. Information from the row based data structure is used to gain visualization performance. Color calculation time is reduced by calculating colors for non zero-rows only. The color for zero-rows is calculated only once. Furthermore, the resolution of the height-field is higher in areas with data applied. Height-field creation and rendering is much faster this way.

5

Conclusion

The demands on a multibody simulation tool focusing on contact analysis represents a special challenge in the field of scientific visualization [12].This is especially true for multidimensional time-varying data, e.g., two dimensional surface related data, vector data.

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(a) Cage and cage pocket. Pressure distribution mapped onto different cage segments.

(b) Pressure distribution aa colored heightfield. The same data is used a8 height-field and color-map.

Figure 5: Surface data visuolization. The large amount of surface variables, time-steps, and resulting visualization data has been identified as the main problem for this work. Typical simulations produce between lo3 and lo6 time-steps of data, this results in 500MB to 8GB of compressed surface data. An surface data structure has been created. 0 0

0

0

0 0

It supports the object oriented multibody modeling approach. hstantaneous (contact pressure) and irreversible (wear and life) physical phenomena can be handled. To handle irreversible data accumulating and state buffering features have been implemented. The data structure supports fast data access for efficient computation and efficient visualization. To be able to handle reasonable model size on normal computers, the surface data structure allocates memory on demand while still having fast data access. Effective data compression gives data reduction rates of about 1:20. Additional attributes are used for data reduction and to support computation, i.e., shape of regions and active regions. The data structure fits the multibody surface description including grid resolution, which eliminates time consuming transformations.

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Interactive visualization of the data has been implemented based on color mapping and height-fields, including 4D plots, on surface mapping, view-point adjustment, and data selection.

The visualization techniques have been implemented in a full-scale industrial system called BEAST, and has proven its usefulness.

Acknowledgments The authors wish t o thank the SKF Group Senior Vice President SKF Group Technology Development, Dr. Henning Wittmeyer, for his financial support and permission to publish this paper. This work was supported by SKF Group Technology Development, the KK-stiftelsens fdretagsforskarskola in Linkoping, and the ECSEL (Excellence Centre for Computer Science and Systems Engineering in Linkoping) Graduate School.

References [l] P. Cignoni, C. Montani, and R. Scopigno. A Comparison of Mesh Simplification Algorithms. Computers and Graphics, 22~37-54, 1998. [Z] V. Engelson, D. Fritzson, and P. Fritzson. Lossless Compression of High-Volume Numerical Data from Simulations. In Proc. of The 2000 IEEE Data Compression Conference, Snowbird, Utah, March 28-30 2000.

[3] G. E. Farin. Curves and Surfaces for Computer Aided Geometric Design, A Practical Guide. Academic Press, Inc., 1250 Sixth Avenue, San Diego, CA 92101, USA, 1988. [4] P. Fritzson and P. Nordling. Adaptive Scheduling Strategy Optimizer for Parallel Rolling Bearing Simulation. HPCN Europe '99, Amsterdam, April 1999. [5] M. H. Gross, R. Gatti, and 0. Staadt. Fast multiresolution surface meshing. In Proceedings of the IEEE Visualization '95, pages 135-142. IEEE Computer Society Press, 1995. [6] P. S. Heckbert and M. Garland. Survey of Polygonal Surface Simplification Algorithms. Multiresolution Surface Modeling Course SIGGRAPH '97, May 1997. Multiresolution Surface Modeling Course. [7] B. Jacob and T. Mudge. Virtual Memory: Issues of Implementation. (IEEE) Computer, pages 33-43, June 1998. [8] P. Lindstrom, D. Koller, W. Ribarsky, L.F. Hodges, N. Faust, and G.A. Turner. Realtime continous level of detail rendering of height fields. ACM SIGGRAPH, 1996. [9] M. E. Mortenson. Geometric Modeling. John Wiley &. Sons, Inc., New York, 1985.

[lo] R. Ramanujan, J. Newhouse, M. Kaddoura, A. Ahamad, E. Chartier, and K. Thurber. Adaptive Streaming of MPEG Video over IP Networks. In Proceedings of the IEEE Conference on Local Computer Networks, 1997. [ll] W. Schroeder, K. Martin, and B. Lorensen. The Visualization Toolkit. Prentice-Hall Inc., 2nd edition, 1998.

[12] A. Siemers and D. Fritzson. Visualization of Dynamic Multibody-Simulation Data. To be published, 2003. [13]

LE.Stacke, D. Fritzson, and P. Nordling. BEAST-a rolling bearing simulation tool. Proc. Instn Mech. Engrs, part K, Journal of Multi-body Dylamics,

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3:63-71, 1999.

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Sub-surface visualization and parallel simulation A SIEMERS and D FRITZSON

SKF, Sweden

Abstract Multibody simulation includes the interaction of bodies [9]. Contact stresses between two surfaces penetrate the material underneath the surface. These stresses need to be stored during simulation and visualized during animation. We classify these stresses as sub-surfaces, thus a thin layer volume underneath the surface. Other sub-surface phenomena can be contact temperature fields, and material damage. A sub-surface data structure has been created. It has all the good properties of the surface data structure [8] and additional capabilities for visualization of volumes. Parallel Simulation puts special demands on the surface data and subsurface data structures. Data has to be packed and distributed to the different simulation nodes efficiently, in order to achieve good speed-up. Special attributes of the data are used to minimize the data to be packed. To achieve fast data transmission, all data is packed into one buffer. These techniques are used in a full-scale industrial parallel multibody system called BEAST [5] [lo]. The (sub-)surface data structure are designed for efficient visualization, storage, packing, and computation.

1

Introduction

T h e amount of data produced by a multibody simulation program entails special requirements o n t h e visualization and parallel computation process. T h i s is especially t r u e for multidimensional time-varying data its for example 2D a n d 3D scalar fields produced by contact calculation of two contacting surfaces. In [8] general problems within t h e a r e a of s u r f a c e d a t a visualization have been discussed. T h i s paper continues t h e discussion with two additional surface-data related issues: visualization of sub-surface d a t a and s u r f a c e d a t a packing for parallel simulation. Keywords: Visualization, Multibody, Simulation, Surface, Stresses, Subsurface, Volume, Packing, Parallel

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In [9] a general description on multibody visualization techniques has been given where the following topics regarding multibody visualization have been identified 0

Modeling of multibody systems.

0

Requirements on the visualization system.

0

Surface representations.

0

Classification of simulation data.

0

Data storage and access.

0

Visualization techniques for different types of simulation data.

0

Graphics and visualization libraries.

0

User interfaces for effective usage.

0

Hardware requirements for visualization.

This paper follows up [9] and [8]and focuses on sub-surface data visualization plus issues of surface-data packing in parallel simulation.

1.1

Sub-surface Data

Contact stresses between two surfaces affect the material underneath the surface. These stresses need to be stored during simulation and visualized during animation. This work classifies these stresses as sub-surfaces, thus a thin layer volume underneath the surface of the body. Techniques for storage and visualization of such sub-surface data have been created. Contact stresses is only one example of sub-surface data. Other sub-surface phenomena can be contact temperature fields, and material damage. The subsurface structure is a useful tool within the field of contact dynamics. In contrast to other volumetric data, subsurfaces belong to a certain surface segment and thus can store surface related volume data.

1.2

Parallel Simulation

In many application fields the simulation process is computationally intensive and fast computers, e.g., parallel computers or workstation clusters, are needed to obtain results in reasonable time. The simulation data used for this work is provided by a parallel multibody simulation program called BEAST [5] [lo]. BEAST is specialized in contact force calculations in rolling bearing applica, tions. Contact force calculations are very time consuming and therefore done in parallel in BEAST. A master-slave configuration is used for parallel computation. One process steers the computations, the so called master. The master distributes the work to the other processes, the so called slaves. A typical setup uses one processor for the master and one for each of the slaves. The processors can be on the same machine (multiprocessor computer), on different machines (cluster of workstations), or both.

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Table 1: Number of surface variables and m a . grid size for three different multibody systems. Model Maximum Number of Grid Size Surface Variables Ball Bearing 95x1225 Roller Bearing 204x1673 4896 Grinding Machine 1703x748 1080

One crucial aspect in parallel simulation is performance loss due to network traffic. Several factors determine the data transfer time, one of them is datasize. Large data sets, as for instance surface related data, need to be transfered between master and slaves. Sub-surface data is an extension to surface data, which are explained in more detail in [SI,and take up even more space. The influence of surface data and sub-surface data on the network traffic during parallel simulation need to be investigated.

1.3

Requirements

Simulations run in parallel, typically on a 40-node Linux cluster and take between l hour and l week of wall clock time. Common simulations produce between lo6 to loe time-steps of data, this results in 500MB to 8GB of compressed surface data. Surface and sub-surface data variables are the most memory consuming variables in BEAST. Table 1shows three BEAST models and the number of surface variables in each model. To achieve fast processing and visualizatioln of sub-surfaces and to minimize network load during parallel simulation, one has to focus on efficient processing of sub-surface data during visualization, and efficient packing of surface related data during simulation.

2 2.1

Related Work Visualization of Large Data Sets

Different techniques have been developed in the past to handle large data sets. When processing large data sets some form of smart memory management is needed. Operating systems have build in virtual memory management and many application relay on this. Another technique is called out-of-core visualization [4] and is based on external memory algorithms [l].These approaches implement their own memory management. Cox and Ellsworth [2], for example, propose a general framework for application controlled virtual memory management. Most of these techniques address the problem of visualizing large scale volumetric data sets, mainly static data sets. Each sub-surface, on the contrary, is quite small compared to other volume data sets. Instead, the large number of sub-surfaces and time steps makes the total large. This work focuses on compression and storage of time varying data sets for visualization.

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2.2

Volume Visualization

Sub-surfaces are volumes and thus volume visualization techniques are of interest here. The most common volume visualization algorithms can be classified into two classes, direct volume rendering and surface fitting algorithms [3]. Direct volume algorithms directly map the volume data into screen space. There are three classes of direct volume algorithms, image-order, object-order, and hybrids of both [q. Surface fitting algorithms [3] [6] on the other hand use surface primitives, e.g., triangles or quadrilaterals, to represent the data. They extract surface meshes from volume data to represent certain information. There are many techniques for volume visualization which can be used for the data presented here. There is no intention to develop new visualization algorithms, but different surface fitting algorithms are used in this work, see Section 4.

A Data Structure for Sub-surface Data

3

A sub-surface is a thin layer volume underneath the surface. Volume data is typically stored as a three dimensional equidistant grid. For sub-surface data, e.g., surface stresses, one is often interested in the data at a certain depth underneath the surface and thus smaller layer distance is needed at this point. A data structure which supports storage of multiple layers with varying depth underneath the surface is presented here. Similar to surface-data does each layer use a two dimensional equidistant grid for data storage. V

>

Figure 1: Surface volume data is stored in layers of equidistant grids. Since sub-surface data is closely related to surface data, see [8],it inherits all its properties, long elongated surfaces with a few small regions of data. Thus, the sub-surface data structure is an extension of the surface data structure explained in [8]. This structure is optimized for memory usage and data access during visualization. It is based on the idea to store active-rows (non-zero) only and assume all other data to be zero, see Figure 2. Memory is allocated for active-rows and a single row is allocated for zero-data. The main properties of this structure can be concluded as: 0

Storage of values in an equidistant grid.

0

Marking of used grid rows to improve grid compression and memory usage. Adaptive memory pool. Memory is allocated on demand for used grid rows only.

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0

Dynamic row pointers. Once allocated memory is reused to avoid reallocation during data animation.

Initialized

set IAllOcate) 3 Rows

Re-initialize

sot 2

COVB

Figure 2: Surface data storage during visualization. The data structure has been extended for this work to support multiple layers of surface data, see Figure 1. Each layer has its own memory pool. The distance between the layers is variable (not equidistant) but not changeable during a simulation. An additional z parameter is used for the depth of the layers. Thus, values are calculated from three parameters a = f(u,v,z). Interpolation is used to calculate values between the layers. One of the main advantages with this approach is that each sub-surface layer can be treated like a standalone surface data grid. In this way all functions for file-I/O, network-packing, and visualization can be reused. Only small changes are necessary to handle the additional layer information.

4

Visualization of Sub-surfaces

The first approaches for sub-surface visualization have been implemented using OpenGL [ll]. More sophisticated visualizations are currently investigated using the “Visualization Toolkit” (VTK) [7]. Sub-surfaces are stored as several layers, each at a certain depth underneath the surface. Since values at certain depths are often of interest, visualization of the separate layers has been implemented, see Figure 3. These techniques are based on surface data visualization techniques which are explained in [SI. Sub-surfaces are thin layer volumes and thus volume visualization techniques can be used to visualize the data. So far a view of three axial aligned cross sections, see Figure 4, has been implemented. Different direct volume rendering and surface fitting techniques, see also [7] [3], need to be tested on the data used here.

5 5.1

Surface Data Packaging for Parallel Simulat ion Packing Technique

A multibody system in BEAST is a hierarchical system [9] where the top level is called the model. In parallel simulation the model hierarchy is traversed and each component, i.e., bodies and segments, stores its data in a memory buffer. This is called down packing. The buffer is sent to the slaves, and back to the master where the data is restored. This is called up packing. Each component in

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(a) A single subsurface layer mapped onto the surface.

(b) A height-field of a single subsurface layer.

(c) All subsurface layers a8 heightfields in a single view.

Figure 3: Sub-surface layers visualized as texture-map and height-fields. the model is responsible for packing its own data, thus surfaces are responsible to pack their own data as well. Static surface-data properties, e.g., size and geometry, are sent only once at simulation startup to initialize the slaves. Grid values (meaning values stored in the grid) on the other hand are packed and sent multiple times between the master and the slaves. One can take advantage of the row based data structure for data packing. To reduce data size only active rows are packed during simulation. Additional information about active rows need to be packed as well, to tell the master and slaves how many rows are transmitted, see Figure 5.

6

Conclusion

In multibody simulation there are many phenomenas that happen under the surface of the contacting bodies, i.e., sub-surface contact stresses, contact temperature fields, and material damage. This is here classified as sub-surface data, i.e., a thin layer volume underneath the surface.

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Figure 4: A volume view of a sub-surface using three adjustable projection planes.

0 0

0 0 0 1

I O

0 0 0 0

0 0 0 0 '

Mastar

Slav.

Figure 5: Only active rows of each surface grid and sub-surface layer are sent between master and slave.

A sub-surface data structure has been created. It has all the good properties of the surface data structure [8] and additional capabilities for visualization of volumes. Different visual representations of the sub-surfaces have been implemented in the visualization system, i.e., height-fields and texture mapping, and 4D data, i.e., cross section planes. In order to achieve practical computation times for multibody simulations with many contacts, parallel computation is needed. Parallel simulation puts special demands on the surface and sub-surface data structures. Data has to be packed and distributed to the different simulation nodes efficiently in order to achieve good speed-up. Special attributes of the data are used to minimize the data to be packed. To achieve fast data transmission all data is packed into one buffer. These techniques are used in a full-scale industrial parallel multibody system called BEAST [5] [lo]. The (sub-)surface data structure are designed for efficient visualization, storage, packing, and computation.

Acknowledgments The authors wish to thank the SKF Group Senior Vice President SKF Group Technology Development, Dr. Henning Wittmeyer, for his financial support and permission to publish this paper. This work was supported by SKF Group Tech-

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nology Development, the KK-stiftelsens foretagsforskarskola in Linkoping, and the ECSEL (Excellence Centre for Computer Science and Systems Engineering in Linkoping) Graduate School.

References [l] J. Abello and J. Vitter. External Memory Algorithms. American Mathematical SOC.,1999. [2] Michael B. Cox and David Ellsworth. Application-controlled demand paging for out-of-core visualization. In IEEE Visualazation ’97, 1997. [3] T. Todd Elvins. A Survey of Algorithms for Volume Visualization. C o m puter Graphics, 26(3):194-201, 1992. [4] R. Farias and C. T. Silva. Out-Of-Core Rendering of Large, Unstructured Grids. IEEE Computer Graphics and Application, 21(4):42, july/August 2001. [5] P. Fritzson and P. Nordling. Adaptive Scheduling Strategy Optimizer for Parallel Rolling Bearing Simulation. HPCN Europe ‘99, Amsterdam, April 1999. [6] Marc Levoy. Display of Surfaces from Volume Data. I E E E Computer Graphics and Applications, 8(3):29-37, 1988. [7] W. Schroeder, K. Martin, and B. Lorensen. Prentice-Hall Inc., 2nd edition, 1998.

The Visualization Toolkit.

[8] A. Siemers and D. Fritzson. Representation and Visualization of Surface Related Multibody-Simulation Data. To be published, 2003. [9] A. Siemers and D. Fritzson. Visualization of Dynamic MultibodySimulation Data. To be published, 2003.

[lo] L-E. Stacke, D. Fritzson, and P. Nordling. BEAST-a rolling bearing simulation tool. Proc. Instn Mech. Engrs, part K, Journal of Multi-body Dynamics, 213:63-71, 1999. [ll] M. Woo, J. Neider, and T. Davis. OpenGL Programming Guide. Addisin Wesley, 2nd edition, 1998.

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Influence of modelling and numerical parameters on the performance of a flexible MBS formulation J CUADRADO and R GUTIERREZ

Escuela Politecnica Superior, Universidad de La CoruAa, Ferrol, Spain

SYNOPSIS Recently, the authors have developed an efficient, robust, accurate and easy-to-implement method for the real-time analysis of rigid-flexible multibody systems. The flexible bodies are modelled by means of the floating frame of reference formulation, along with modal superposition of both static and dynamic modes. The dynamic modes to be considered for each flexible body must be decided by the analyst. On the other hand, the co-rotational approach used to derive the inertia terms of the dynamic equations motivates that such terms depend on the discretization of the underlying finite element mesh. Therefore, the discretization size of the finite element model is another parameter to be selected by the analyst. Furthermore, the value of two other parameters must be chosen: the penalty factor for the dynamic equations, and the time-step size for the fixed single step numerical integrator. This paper studies the influence of the four parameters on the accuracy and efficiency of the abovementioned method, along with their relative dependence. To this end, a sweeping of the space generated by the parameters is carried out for a flexible system, and the corresponding results are analyzed in terms of accuracy and efficiency. In order to have a reference for comparison, the system is also solved through the nonlinear module of a finite element analysis commercial code. NOMENCLATURE pl, p2: points (natural coordinates) used to model the pinned-free beam. p2,, p2, : x- and y-coordinate of point p2. v l , v2: unit vectors (natural coordinates) used to model the pinned-free beam.

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VI,, VI, : x- andy-component of unit vector vl. v2,, v2, : x- and y-component of unit vector v2. CY static bending mode of the pinned-free beam. 7:amplitude of static bending mode @. n: number of the first dynamic bending modes considered. "1, Yz, ..., Y,,: n first dynamic bending modes of the pinned-free beam. c,, & , ..., 5" : amplitudes of the n first dynamic bending modes Y1, Y2, ...,Y,,. q: vector of problem variables. m:number of finite elements used for the discretization of the pinned-free beam. z, : history of a certain magnitude. 2,. : history of a certain magnitude for the reference simulation. lzlrnax : maximum absolute value of a certain magnitude during the simulation. cr: penalty factor for the augmented Lagrangian dynamic formulation. Ar: fixed time-step selected for the numerical integration. 1 INTRODUCTION

During the last years, the authors have developed an efficient, robust, accurate and easy-toimplement method for the real-time analysis of rigid-flexible multibody systemsl.2. The method employs natural coordinates for the modelling3, applies the co-rotational approach4 to derive the inertia terms of the flexible bodies, establishes the equations of motion through an index-3 augmented Lagrangian formulation with projections in velocities and accelerations', and carries out the numerical integration by means of the implicit, single step trapezoidal rule6. The kinematics of the flexible bodies is introduced through the floating frame of reference approach', along with modal superposition to describe the corresponding local deformations*, carried out by means of both static and dynamic modes defined with respect to a tangent frameg. When a certain multibody system containing flexible bodies is to be studied through the described method, four kinds of parameters are left to the analyst decision: a) The dynamic modes to be considered for each flexible body. Once the modelling in natural coordinates of the whole multibody system has been carried out, the static modes for each flexible body are automatically established' (some of them can be neglected, if desired, by imposing the corresponding constraint equation of null amplitude). However, the dynamic modes, which have the role of improving the representation of the deformation field given by the static modes, can arbitrarily be included in the model. Decision about how many and which dynamic modes to consider must be taken by the analyst. As demonstrated in previous works10Jl, both the accuracy and the efficiency of the simulation will be strongly influenced by this choice. b) In a general approach, a finite element (FE) model of each flexible body is also prepared. Such model serves, in a pre-processing stage, to obtain the static and dynamic modes, as well as the mass and stiffness matrices of the finite element method and, in a post-processing stage carried out at each time-step, to work out the values of elastic strains, stresses, displacements and efforts. Hence, the way in which the body is discretized becomes relevant, since it is

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expected to affect both to the accuracy and the efficiency of the simulation. For flexible bodies of simple geometry, like straight and uniform beams, the analytical form of the modes is available and, therefore, the described pre- and post-processing stages are not needed, and the simulation behaviour no longer depends on the FE mesh. However, if the authors’ method is used, the FE model still appears in the formulation. The reason is that, when the corotational approach is introduced, the inertia terms of the dynamic equations for each flexible body are obtained as products of several matrices which depend on the FE model!.*. Therefore, either if the analytical modes are available or not, the adopted FE discretization affects to the performance of the simulation. This is the second decision left to the analyst. c) In the proposed method, the equations of motion are established by means of an index-3 augmented Lagrangian formulation, which requires a penalty factor to amplify the internal forces caused by constraint violations. Such factor is crucial for the simulation stability, and constitutes the third decision to be taken by the analyst. d) Since the described method is targeted to achieve real-time performance, the fixed single step trapezoidal rule is used. Therefore, the fixed time-step for the numerical integration must be selected: this is the fourth decision for the analyst. Choices (a) and (b) can be referred to as the modelling parameters, since they deal with the modelling of each flexible body, while (c) and (d) may be called the numerical parameters, as they are related to the dynamic formulation and integration procedure of the whole multibody system. This paper aims to study the influence that the four mentioned parameters have on both the efficiency and the accuracy of the proposed method, and to search for relationships among such four parameters. To achieve these objectives, a sweeping of the space generated by the two modelling parameters is carried out for a flexible system, and the two numerical parameters are adjusted for each combination. The results are analyzed in terms of accuracy and efficiency. In order to have a reference for comparison, the example is also solved through the nonlinear module of a finite element analysis (FEA) commercial code. The remaining of the paper is organized as follows: Section 2 shows the flexible system to be analyzed, along with its modelling with both the proposed and the FE method; Section 3 explains the characteristics of the motion undergone by the system, the criteria to generate the multiple simulations executed, the magnitudes to be recorded, and the way to determine the error incurred by each simulation; Section 4 presents the results obtained for all the simulations, which are discussed in Section 5; finally, Section 6 summarizes the conclusions of the work. 2 THE EXAMPLE

The flexible system to be analyzed, shown in Figure la, consists of a beam pinned at one end to the ground, which starts from the rest and undergoes the bang-bang torque depicted in Figure 1 b. Gravity effects are neglected. Physical properties of the beam are: mass density 8000 Kg/m3, modulus of elasticity 2x1011 N/m2, length 1.5 m, cross-sectional area 10-4 m2, moment of inertia 10-10 m4.

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Yt

I -

0.25

-1.0

0.5

t (s)

-----

Fig. 1 a) pinned-free beam; b) bang-bang torque. Following the method developed by the authors, the modelling of the beam has been carried out as illustrated in Figure 2. At the pinned end, point p l and unit vectors v l and v2 have been defined, thus constituting the local reference frame of the body. In this case, point p l is fixed. At the free end, point p2 has been defined. The local displacement of point p2 in v2direction activates static bending mode a.Its local displacement in vl-direction has been prevented through a constraint equation, so as to avoid the appearance of the corresponding axial mode, not relevant in this example. To better represent the deformed configuration of the beam, as many dynamic modes as desired can be considered: they are the natural modes of vibration of the beam with fixed boundaries (points p l and p2, and unit vectors v l and v2), which means that, for their calculation, left end must be clamped and right end must be pinned. Figure 2 shows the two first dynamic modes, Y1 and "2.

Fig. 2 Modelling of the flexible pinned-free beam with the authors' method.

Then, if a certain number n of dynamic modes is chosen for the modelling, the vector of problem variables results,

where q is the amplitude of the static deformation mode a,and 5,. &, ..., 5, are the amplitudes of the dynamic modes considered Yl, "2, ..., Y,,. Therefore, the total number of variables is 7+n, with only 2+n independent. The analytical forms of both the static and dynamic modes have been used. For the underlying FE model of the beam, a mesh of m two-dimensional beam elements (BEAMZD) has been generated. All the elements are identical, with nodes of three degrees of freedom: two displacements in the plane of the beam and the corresponding slope. As the

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node placed at the pinned end of the beam can only experiment rotation, the total number of variables rises to 3m+l. 3 THE ANALYSIS

The motion of the flexible beam undergoing the described torque is simulated for 2 s. Simulations are run with a number of dynamic modes n going from 0 to 4, and a number of beam elements m ranging from 2' to 26. The penalty factor is initially adjusted to 109, and increased only when bad results are obtained. The time-step is set to 1 ms; in case that the simulation fails, the time-step is reduced until good behaviour is achieved. For each simulation, the following results are recorded: a) CPU-time required; b) history of the y-coordinate of the free end of the beam; c) history of the bending moment at the middle section of the beam. In order to have a reference for comparison, so as to evaluate the quality of the solution obtained at each simulation, the problem has also been solved through the nonlinear module of FEA commercial code COSMOS/M 2.8, using a discretization of 26=64 elements.

Two error values have been obtained for each simulation: a displacement error and a bending moment error. In both cases, the error has been calculated as follows. The history of the corresponding magnitude has been recorded at every 1 cs for both the simulation of reference and the simulation being evaluated. Then, the error is obtained as,

where 201 is the number of values considered (steps of 1 cs during 2 s of simulation), zi represents the history of the corresponding magnitude (y-coordinate of the free end of the beam, or bending moment at the middle section of the beam) for the current simulation, z,* is the same for the reference simulation, and lzlmaxis the maximum absolute value of the magnitude during the simulation. The resulting errors have the form of percentages. 4 RESULTS

Table 1 shows the obtained results for all the simulations performed. Remember that n is the number of dynamic modes, m is the number of beam elements, LT is the penalty factor, and dt is the fixed time-step selected for the numerical integration. The symbol "---" means that the simulation failed with such a combination of dynamic modes and discretization size. The simulation which produces the most accurate results has been boldfaced. The CPU-times reported have been obtained on a Pentium I11 @ 900 MHz.

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Table 1 CPU-time and errors for all the simulations performed.

1

7

1

I

3

I

I

---

I

---

I

---

I

---

I

__-

10

4 8 16

109 109 109

10-3 10-3 10-3

0.22 0.29 0.83

11.17 7.48 3.63

11

32

109

10-3

2.40

2.10

19.95

12

64

109

103

8.32

2.24

20.32

8 9

I

61.94 48.62 25.59

In order to provide the reader with a more visual presentation of the results, the CPU-time and errors for all the simulations performed are also given in Figure 3. CPU-times of 100, and error values of 20 for displacements and 100 for bending moments have been assigned to those simulations which failed (symbol "---" in Table I), so that plots are not distorted.

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Fig. 3 Results: a) CPU-time; b) displacement error; c) bending moment error.

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Both errors (displacement and bending moment) have been reduced to single figures, according to Equation (2), in an attempt of condensing the information and making easier its interpretation. In order to show the correlation between the error values presented in Table 1, and the actual discrepancies of the simulations with respect to the reference, plots comparing the histories of displacement and bending moment for simulations 1, 11,23 and 30 along with the corresponding histories for the reference simulation are depicted in Figure 4.

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Fig. 4: a) Errors in simul. #1 (n=O, m=2): displ. (8.86Y0)~bend. moment (97.68%); b) Errors in simul. #11 (n=l, m=32): displ. (2.10%), bend. moment (19.95%); c) Errors in simul. #23 (n=3, m=32): displ. (8.91%), bend. moment (51.43%); d) Errors in simul. #30 (n=4, m=64): displ. (5.22%), bend. moment (28.76%). From the presented plots, it can be seen that, if the error is determined as proposed in Equation (2), an error in displacement of 8% is a great error, while a value of 2% means very good accuracy. On the other hand, an error in bending moment of 100% represents a large error, and 20% indicates excellent agreement with the reference. Although the scales of both errors are different, as are the mean values of each kind of results, their trend is, in general, the same. 5 DISCUSSION

At the view of the results presented in the previous Section, it is clear that consideration of more dynamic modes does not necessarily leads to more accurate results, but always to less efficient simulations. In the example, the most accurate results are obtained with only one dynamic mode, and the corresponding efficiency is high in the context of all the executed simulations. This means that an optimum number of dynamic modes exists for a certain analysis. Regarding the discretization, it can be affirmed that a maximum mesh size cannot be exceeded based on the highest dynamic mode considered. Once under such maximum, more accurate results are obtained for more refined meshes until a certain value; further refinement does not lead to any improvement. On the other hand, the efficiency decreases as the mesh is refined. Therefore, it comes out that the required discretization depends on the number of dynamic modes, and that there is also an optimum size for the underlying FE mesh. It must be pointed out that the exponential increment in CPU-time reported in Table 1 as the number of finite elements rises, is due to products of matrices whose size depends on the mesh size, needed to build up the inertia terms. Such products can be done more efficiently if the sparse structure of the arrays is accounted for, so attenuating the pronounced growth of the CPUtimes with respect to the discretization size. Going now to the numerical parameters of the method under study, it seems that an increment of the penalty value can be worthy only for cases of insufficient FE discretization. On the

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other hand, the time-step size needed to perform the numerical integration shows to be related to the highest dynamic mode included in the modelling: higher dynamic modes imply smaller time-step sizes and, consequently, lower efficiency. From what has been said, it can be concluded that only one parameter is independent: the number of dynamic modes. The other three parameters discretization size, penalty factor and time-step size- can be established as functions of the number of dynamic modes. Moreover, there is an optimum value of the number of dynamic modes for a certain problem. The reason is that the motion of the body is properly captured with such optimum number of dynamic modes, so that the inclusion of additional modes only leads to the appearance of higher frequencies in the solution, which in turn hinder the numerical integration process, thus producing higher errors. Therefore, a method to determine how many and which dynamic modes must be considered for a certain analysis is crucial to develop models which can be run on real-time with the proposed formulation. Of course, the iterative process will always be available, and can be an option for some applications. In the field of structural dynamics, the optimum number of dynamic modes depends on the physical properties of the body and the frequency content of the applied forces, and both of them can be analyzed before the simulation is carried out. However, in flexible multibody dynamics, joint and inertia forces, which cannot be analyzed a priori, are of key relevance; they depend on the motion undergone by the body, which is unknown until a simulation is performed. Therefore, it seems that making an initial estimation of the optimum number of dynamic modes in flexible multibody dynamics won’t be an easy task, and that, at least, a previous rigid-body simulation will be required in order to get some insight into the form of both the joint and inertia forces.

I

Table 2 MPFs for the example. #dyn.mode I MPF(%)

1

This is exactly what is proposed in the modal participation factor (MPF) method, which has been successfully employed recentlyll.12 to estimate the dynamic modes that must be included in the model of a flexible multibody system. Such method has been applied in the paper to the studied example, so as to correlate it with the obtained results. To this end, rigid-body simulation of the system has been carried out, and the most critical position identified. The forces acting upon the body in such position have been recorded, and a static analysis of the body, now considered as a structure, has been conducted. For this purpose, the FE model of 64 elements, which had been served for comparison so far, has been used, and the first 16 dynamic modes have been obtained. Both the stiffness matrix and the applied forces have been projected to the modal space, and the corresponding modal amplitudes derived from the static equilibrium equation. Table 2 shows the MPFs obtained for the first 7 dynamic modes. The MPFs of the remaining modes were even smaller.

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From the results presented in Table 2,it is clear that the first dynamic mode prevails, and that only the second is above the commonly accepted limit of 1% share. Therefore, the MPF method indicates that the optimal selection consists in just taking the first dynamic mode, or perhaps the first and the second ones, which is in good agreement with the results previously obtained in the paper. Consequently, the MPF method can be considered as a good candidate to provide an initial estimation of the optimum number of dynamic modes for each flexible body. Once such decision is taken, the automatic tuning of the other three parameters to their optimal values (discretization size for each flexible body, penalty factor and time-step size), seems to be relatively easier; the development of some method for this purpose will be addressed in the future. 6 CONCLUSIONS

Based on the previously exposed results and discussion, the conclusions can be drawn as follows: a) The authors have recently proposed an efficient, robust, accurate and easy-to-implement method for the real-time dynamics of rigid-flexible multibody systems, based on the floating frame of reference formulation, with both static and dynamic modes. b) When applying such method, the analyst must decide on the value of four parameters: two modelling parameters -number of dynamic modes and discretization size of the underlying FE mesh for each flexible body-, and two numerical parameters -penalty factor for the dynamic equations and fixed time-step size of the numerical integration-. c) The four mentioned parameters are not independent: a certain value of the number of dynamic modes will ask for corresponding optimum values of the other three parameters. d) An optimum number of dynamic modes exists for a certain problem, which leads to the best results in terms of accuracy. e) The modal participation factor method can be used to provide an initial estimation of such optimum number of dynamic modes. f) A method to automatically obtain the optimum values of the remaining three parameters once the number of dynamic modes has been decided is left for future development.

ACKNOWLEDGMENTS This research has been sponsored by the Spanish CICYT (Grant No. DPI2000-0379)and the Galician SGID (Grant No. PGIDTOlPXI16601PN).

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REFERENCES

J. Cuadrado, R. Gutierrez, M.A. Naya and P. Morer, “A Comparison in Terms of Accuracy and Efficiency between a MBS Dynamic Formulation with Stress Analvsis and a Non-linear FEA Code”, Int. J. for Numerical Methods in Engineering, 51 (6),10331052 (2001). J. Cuadrado, R. Gutierrez, M.A. Naya and M. Gonzalez, “Experimental Validation of a Flexible MBS Dynamic Formulation through Comparison between Measured and Calculated Stresses on a Prototype Car”, Multibody System Dynamics, to appear. J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems -The Real-Time Challenge-, Springer-Verlag (1994). M. Geradin and A. Cardona, Flexible Multibody Dynamics -A Finite Element Approach,John Wiley and Sons (2001). E. Bay0 and R. Ledesma, “Augmented Lagrangian and Mass-Orthogonal Projection Methods for Constrained Multibody Dynamics”, Nonlinear Dynamics, 9, 1 13-130 (1996). L.F. Shampine, Numerical Solution of Ordinary Direrential Equations, Chapman & Hall (1994). A.A. Shabana, Dynamics of Multibody Systems, 2”d edition, Cambridge University Press (1998). L. Meirovitch and M.K. Kwak, “Convergence of the Classical Rayleigh-Ritz Method and the Finite Element Method”, AIM Journal, 28 (8), 1509-1516 (1990). R. Schwertassek, 0. Wallrapp and A.A. Shabana, “Flexible Multibody Simulation and Choice of Shape Functions”,-NonlinearDynamics, 20,361-380 (1999). [lo] R. Schwertassek, S.V. Dombrowsky and 0. Wallrapp, “Modal Representation of Stress in Flexible Multibody Simulation”, Nonlinear Dynamics, 20,381-399 (1999). [11]0. Wallrapp and S. Wiedemann, “Simulation of Deployment of a Flexible Solar Array”, Multibody System Dynamics, 7 , 101-125 (2002). [ 1210. Wallrapp and S. Wiedemann, “Flexible Multibody System Applications using Nodal and Modal Coordinates”, Proceedings of ASME 2003 DETC, Paper VIB-48305, Chicago, Illinois, USA, September 2-6 (2003).

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The multi-particle system (MPS) model as a tool for simulation of mechanisms with rigid and elastic bodies D TALABA and Cs ANTONYA

Product Design and Robotics Department, University Transilvania of Brasov, Romania

Abstract The paper presents an approach for the simulation of mechanisms with rigid and flexible bodies using the multi-particle system ( M P S ) dynamic formulation. According to this model, the bodies are represented through inertially equivalent systems of mass-points separated by constant distances (for rigid bodies) or spring-damper elements (for flexible bodies). This model provides the possibility to represent bodies with different number of particles, according to level of detail desired in each case. The flexible bodies are modeled as massspring systems, similar with the physically based models applied in haptic rendering applications. Each rigid body can be modeled with a minimum number of particles (at least 4 particles) separated by constant distances and the joints are modeled as constraints between particles, formulated as algebraic equations. The stiffness problem of DAE system is solved through Euler implicit integration scheme. Although the number of equations is larger than in multi-body formulation, the matrices are sparser and the computation efficiency is not very much affected. The constraints, dynamic formulation and integration for this approach are outlined, including a sample mechanism for which the simulation is presented. 1. Introduction

In recent years, multibody analysis computer packages became a usual tool in industry, research and development areas. The commercially available codes include nowadays a large range of facilities allowing simulation of sophisticated experiments with virtual prototypes of mechanical systems (mechanisms). The cutting edge research in this field is aiming towards developing new modelling and simulation facilities related on one hand to including into the analytical formalisms complex non-linearity like flexibility of the bodies, friction modelling etc and on the other hand to the increasing of the computing speed in order to enable the real time simulation. As resulting from the literature [4,5,6,7,8,9], three main representations have been assumed for the development of various methods and dynamic formalisms following the type

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of coordinates used: relative coordinates [6,9], Cartesian coordinates [4] and natural coordinates [ 5 ] . In this paper a new formulation is presented, namely the multi-particle system (MPS) according to which the mechanical system is represented as a collection of interconnected particles [7,8]. The method is an extension of the physically based modelling particle approach used in haptic rendering for single flexible body applications. 2. The multi-particle system (MPS) model

This model considers the mechanism as a collection of particles subject to a set of absolute and relative constraints. Some elements of this model are included in the representations utilized in their work by Alexandru et a1 [ 11, Jalon - Bay0 [5]. However, the main difference appears in the overall dynamic model that is a purely particle system, in which no body reference is included. Once the bodies have been replaced by particles that conserve integrally their inertial properties, the whole system can be treated as system of interconnected particle. The system mass matrix and external force vector are therefore computed for the entire system set of particles without relation to any Body Reference Frames. In this way, the body moments of inertia are no more relevant and the mass matrix is very simple (a diagonal matrix). The mechanism representation includes a particle based model for the rigid body and point contact models for each type ofjoint. The body model consists in a set of particles separated by constant distances, each particle being associated with a concentrated mass according to the inertial equivalence with the real object. For a body model in 3D space (Le. able to integrally conserve the mass properties of the original solid) minimum 4 particles are needed and 3 particles for the planar case. Once the particles location is established in the body frame, the concentrated masses can be easily obtained from the inertial equivalence conditions. These must ensure that for the particle system representing the body, the centroid position, the cumulated mass of the particles and the axialhentrifugal moments of inertia are the same as for the original body: m, y : +m,z: + m , y i + m , z i +m,y: +m,z: + m 4 y t +m4z: = J , m, x: MI

f

m,y :

+ m,xi

Y , f "2X2Y2

f

-Im2y i -I-m,x: f

",XSY,

f

m,y:

f

m4xi f m4y i = J ,

M4X4Y4 = J ,

m1 YI ZI m2y2z2 M3Y3-73 f "4Y4Z4 = J , (1) m , x , z , +m,x,z2 +m,x,z, + m 4 x 4 z ,= J , f

m, x,

f

+ m2x2-tm,x, + m4x4= 0

m1 Yl fm2Y2 +%Y, fM4Y4 = 0 m,z, +m2z2+m,z, +m4z4= O m,+m,+m,+m, = m According to these equations, 10 unknowns can be computed for each set of particle associated to a body that is at least 4 mass points (characterized by Cartesian 12 coordinates) are necessary to fully represent the inertial properties of a body. Higher number of points can be utilized, and in this case the rest of unknown coordinates must be numerically adopted by the user.

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For the planar mechanisms, only 6 equivalence equations could be formulated and therefore a planar body could be fully represented by minimum 2 mass points. 3. Joints and constraints in a MPS model

In the tri-dimensional space a particle have 3 degrees of freedom (f maximum three types of constraints can be imposed (fig. 1):

=

3), therefore

(i)Contact (coincidence) with a fixed point or another particle 3 f = 0, c = 3 (f is the degrees of freedom between the two particles

Fig. 1

and c is the number of dof restricted by this type of constraint) (ii) Contact with a 3D curve f = 1, c = 2. (iii) Contact with a 3D surface 9 f = 2, c = 1. The MPS body model includes the associated particles and a set of constant distances constraints between them, which represent the ideal rigid conditions, according to the usual definition. For example, for a body represented by 4 particles involving 4x3=12 generalized coordinates, a number of 6 distance constraints have to be imposed and finally 6 independent coordinates remain independent and fully characterise the body position and orientation in space(fig. 2):

+

~

'(Xfi-XP2)2+CYq

(xs *

-xP>)2+(YpI

-YP2)*+(Zq

-zPJ2=P,pz2

-YP,)2+(zP,

-zP3)z=~lp32

(XP* - x P 3 ) z + ( Y P *

-YPl)2+(zP2

-zP3)2=p2p32

(xfi

-YPJz+(zq

-Zp,l2

= s p 4 2

-YP4)2+(zP2

-zP4)2

=p2p42

-xpJ2+CYfi

(xp2 -Xp,)'+(Yp,

The kinematic joint model is defined as a combination of constraints between the various particles belonging to the adjacent bodies. The point type contact model defined in fig.1 allows the definition of practically any type of joint. The models of the most usual joints are detailed in table I. With these models defined for body and joint, a new criterion can be formulated for the mechanism mobility as:

M = S p - 21, (3) in which p is the number of the particles included in the model, S is the space dimension (S=3 for 3D space and S=2 for 2D space) and ci is the number of constraints defined between the particles of the system. It includes both the joint constraints and the constant distance constraints imposed to represent the rigid body conditions. The vector of the generalized coordinates has 3n, elements (np-number of particles) and has the form: rq1= [X/ Yl 21 x2 Y2 22 x3 y3 23 ... x p Y p zpl: (4)

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sua1joints representatioi Particle model Spherical joint

&@..

-77 P

0,

Cylindrical joint

Translation joint

Revolute joint

@, _/-

em t o

Plane joint

@ -9

Q,

The degree of freedom of the system can be computed with a relation similar with the Gruebler formula for multi-body systems, Le.: M = 3n - Zci, in which n is the number of mass points, Cci is the total number of constraints defined between them and M is the mechanism mobility . The vector of the generalized coordinates [q] can be obtained by numerical solving of the system of M + Zci algebraic equations corresponding to the M driving motions and Cci joint constraints. It must be noted that in most cases, for the M P S model, the joint equations can take a limited number of forms (as far as usual joints from the table I are involved) - that is only four type of equations: - Distance equation (xpl +(YPl -YPl)2+(zpl -ZP2)* =p,p22 (5) - Coincidence equation xPl=xQI,yPI=yQl,ZPl=zQl> (6) - Co-linearity equations '9

- 'PI - y9 - YPI - '9 - 'Qi

'h -'QL

-

114

~ P - I~ P I 'PI

(7)

-'@I

Co-planarity equation

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This provides a very simple mathematical background, facilitating the computer implementation. In the next step, through successive differentiation, velocity and acceleration equations can be easily derived obtaining thus the set of kinematic equations of the general form cp(q,r) = 0, b(q,t)= 0, &(q,t) = 0.

For the dynamic simulation, the equations have the same general form as for MBS model relation, i.e. mq - J T 1 = Q , (9) in which m is the mass matrix m=diag[ m, m, m, m2 m2 m2 m, m3 m,, ... mp m, m,], (10) J is the jacobian matrix, h is the Lagrange multiplier vector and Qex is the vector of the generalized forces. The number of Cci Lagrange multipliers include the joint reaction forces (that are purely forces since no torques are involved as the particle rotation is not relevant) and also the constant distance reaction forces between the particles of the same body. Although it generates a larger number of equations than in the multi-body formulation, the multi-particle formulation gives access to more level of detail in the analysis process. Thus, further to the applicability to the rigid body systems, it allows the simulation of systems with elastic bodies, by replacing the constant distance constraints with simple internal forces F, corresponding to spring-damper elements: [ F ]= 461+ c[8]. (1 1) in which k is the spring stiffness, c is the damping coefficient, [SI is the deformation of the spring and [8] is the deformation speed. In general, the number of points utilized for body models is larger than the minimum required by the inertial equivalence, since the definition of the various joints requires often more than one point. Of course, the number of mass points necessary to represent an elastic body is usually further larger than for a rigid body according to the body shape, the accuracy desired for the simulation etc. 4. The formulation for a mechanism with rigid parts

For the sample planar mechanism modelled as in figure 3, the number of mass points per body is 2, except bodies 3, which is defined with three particles. The total number of particles corresponding to the mobile bodies is p = 6 (AI, BI, B2, C2, C3, D3), that is S.p=2x6=12 generalized coordinates (two Cartesian coordinates for each particle):

...

(12) As constraints, there are 3 rigid body constant distances (AB, BC, CD) and 8 joint constraints, yielding Cci=l 1 constraints, that is M=S.p-Xci=12-11=1. The constraint equations set is:

4 =[-AI

YAl

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XBl

YBI

xB2

YBZ

xD3

YD3]?

115

In case of kinematic simulation, the 12Ihequation corresponds to the driving motion. The velocity and acceleration equations are derived by differentiation of the position equations J(j=Cl

sq=r' where the 12x 12 Jacobian matrix J has also a very simple expression not given here, for space reasons. For the dynamic analysis, one has to take into account that the particles are acted by external, reaction and inertia forces. Each force is applied to Fig.3 a particle (only forces, no torque can be applied to a point), which is an important simplification. The general matrix form of the differential equations is given also by (9), in which the mass matrix is 12x12 = diag[ lllA, mA, mAl mB, mB, mBl mB, mBz "' mQ mq m q l . For the simple slider-crank mechanism considered, with symmetrical shapes, the concentrated masses would result also symmetrical, such as m ~ 1 = m ~ 1 = 0 S mrn~z=mcz=0.5mz, l, mc3=m~3=0.5m3. The 11x12 Jacobian matrix corresponds to the constant distances & kinematic joint constraints and the Lagrange multiplier vector h has also 11 components. Thus, the DAE system has 23 equations (1 1 algebraic equations of constraints and 12 differential equations with Lagrange multipliers) with 23 unknown: 12 generalized coordinates and 11 Lagrange multipliers.

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5. Numerical example for an elastic mechanism The case of the planar slider crank mechanism with elastic coupler has been considered for dynamic simulation (fig.4). For this purpose, the coupler BC has been modelled with a larger number of points, i.e. 11 points and the material considered is rubber, such as spring damper elements are imposed between points instead of constant distances. The other bodies of the mechanism are modelled as rigid bodies using constant distance constraints between points. The body 1 is modelled with four points and body 3 is modelled with three points. The geometric shape of the elements being symmetrical, the masses of the points have been chosen m2.. .m5=mrank/4, m6.. .m16=mcouoieJ11 and m17.. .m19=mslider/3. For the Fig. 4 purpose of this simulation the values of spring-damper parameters have been adopted such as to approximate the rubber usual elastic characteristics. The multi-particle system (h4PS) obtained is thus containing 18 mobile particles interconnected through 4 joints (3 revolute joints and 1 translational) and 8 constant distances constraints (5 for the crank and 3 for the slider) that is 16 dof are restricted. The system degree of freedom is therefore M = 2x18 -16=20 dof. For the simulation purpose, the system is considered loaded with a constant force F acting on the particle no. 18, such as to force the slider to pass on the other side with respect to the revolute joint of the crank. To allow this, the coupler must bent elastically since in the rigid case, the slider could not change the position on the other side (see fig.4). The dynamic equation set includes 16 algebraic equations for the constraints and 36 differential equations resulting in a DAE system of 52 equations. Although the number of equations is higher than usually for a multi-body model of the same system, the matrices are sparser and therefore the computational efficiency is not much affected for this reason. However, the system of equations is stiff and implicit integration methods must be utilized in order to keep this the efficiency in reasonable limits. For the application illustrated in this paper the Euler backward integration scheme was utilized. The results of the simulation include the motion parameters for each particle. Starting from this, the graphic simulation could be achieved using appropriate techniques, i.e attaching the shape of the rigid bodies or approximating the deformed shape of the flexible bodies according to the number of mass points utilized in the model. Of course, any other post-processing scheme could be utilized for motion or forces diagrams etc. In this example, for each particle a sphere was associated and the graphic simulation of the sphere system was performed (fig. 5)

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I

I

Fig. 5

Conclusions and further research The multi-particle model is a general approach for the dynamic simulation of articulated mechanical systems. A general formulation has been presented including specific formulas for mobility computation joint constraint algebraic equations and the differential algebraic equations formulation. The M P S formulation was symbolically illustrated for a simple slider crank mechanism and a numerical example has been presented for the slider-crank mechanism with a flexible coupler. It has been showed that the M P S model has the same generality and usability as MBS formulation. Despite the larger number of equations than in multi-body formulations, the M P S model provides several interesting features: - The representation of forces and inertial mass properties is significantly simplified. No more torque is involved and the mass matrix is a diagonal matrix. - The constraints and the corresponding algebraic equations are of a small variety. This is simplifying both constraint and Jacobian matrix formulation. - Although the matrices are of larger dimensions, for the M P S formulation they are sparser than in the case of multi-body model and therefore the computational efficiency is not very much affected. - The M P S model allows the treatment of flexible multibody systems by replacing the distance equations with the flexibility principles for each body. The differential equations are stiff in this case and therefore stable integration methods have been utilized, like Euler backward implicit integration method.

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A first matter that has to be solved in the use of this formulation for the simulation of mechanisms with flexible parts is the equivalence between the spring-damper elements and the real flexible body. The most important aspect in this matter is to achieve a spring-damper meshing of the body and methods for determining the equivalent stiffness and damping coefficients has to be conceived. Another issue is related to the geometric representation of the deformed shapes of the flexible bodies on the basis of particles position, which are subject to change between the various frames of the simulation. These are subjects of further research in the next stage. However, whatever the difficulty and computational efficiency of these computations, it is important that they are made in the “pre-processing” stage of the simulation and the computational efficiency in the “processing” stage is not affected. This aspect is very important for the real time simulation applications.

6. References 1. Alexandru, P., Visa, I., Talaba, D., Utilisation of the Cartesian coordinates for the linkages study (in Romanian). The Romanian Symposium on Mechanisms and Machine Theory MTM ‘88, Politehnic Institute of Cluj-Napoca, vol. I, 1988, pp. 1-10. 2. Antonya, Cs. Dynamic transmissibility of car suspension mechanisms. PhD thesis, University Transilvania of Bragov, 2002. 3. Duditza F, Diaconescu D., Structural Optimisation of the Mechanisms. Ed. Tehnica, Bucharest, 1987. 4. Haug, J.E., Computers Aided Kinematics and Dynamics of Mechanical System, vol. I. Ed. Allyn and Bacon, 1989. 5. Jalon, J.G. and Bayo, E., Kinematic and Dynamic Simulation of Multibody Systems, Springer-Verlag, New York, 1994. 6. Schiehlen, W.O., Multibody Systems Handbook, Springer Verlag, Berlin-New York, 1990. 7. Talaba, D., Articulated mechanisms - Computer Aided Design. Transilvania Univ. Press, 200 1. 8. Talaba, D., A particle model for mechanical system simulation. Proceedings of NATO Advanced Study Institute ,,Nonlinear Virtual Multi-body Systems”, Prague, 2002. 9. Wittemburg, J., Analytical Methods in the Dynamics of Multibody Systems. Proc. IUTAM/ ISIMM Symposium on Modem Developments in Analytical Mechanics, Turin, 1984, pp. 835-858.

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St ructuraI Dy narnics

Stability and Chaotic Response of Elastic plate with large deflection L DAI and Q HAN Industrial Systems Engineering, University of Regina, Regina, Saskatchewan, Canada A LIU Environmental Systems Engineering, University of Regina, Regina, Saskatchewan, Canada

SYNOPSIS This study tends to investigate the stability and bifurcation of a nonlinear elastic rectangular plate of large deflection under various loading conditions and system parameters. An analytical mode is established with the Galerkin principle. The chaotic response of the plate is investigated theoretically and numerically. The results of theoretical bifurcation analysis are numerically verified with the consideration of vibration stability of the plate. The stability of the trivial and nontrivial solutions of the system is investigated theoretically and numerically and the conditions for the stability are provided.

NOMENCLATURE a

-

a/

-

a2

b

-

A

-

B

-

D E h N

-

Po PT -

5 -

e-

length of plate in the x direction; amplitude in expression of q, ; amplitude in expression of q2 ; width of plate in the y direction; constant of plate geometry; constant of plate geometry; flexural rigidity of plate; modulus of elasticity; thickness of plate; harmonic excitation; loading constant of harmonic excitation N, loading constant of harmonic excitation N, loading constant; loading constant;

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time function in expression of ; time function in expression of T2; coefficient in expression of T, ; first time function in transverse displacement expression; second time function in transverse displacement expression; lateral deflection; first location function in transverse displacement expression; second location function in transverse displacement expression; damping coefficient; stress function; constant of geometric and material characteristic of plate; constant of geometric and material characteristic of plate; constant of geometric and material characteristic of plate; characteristic value; Poisson’s ratio; phase angle in expression of q/ ; phase angle in expression of q2; material density; time function; radian frequency of harmonic excitation N, non-dimensional frequency. 1. INTRODUCTION

In the theoretical analyses of the dynamics of elastic plates, the effects of systematic nonlinearities must be taken into consideration especially for the cases of large deflections. The nonlinearities of vibration systems mainly due to the three aspects: (1) the physical nonlinearity, (2) the geometric nonlinearity and (3) the nonlinearity of boundary conditions [l-41. With the continuous development in nonlinear dynamics, the research fields of nonlinear science becomes great abundant. Significant contributions have been made in understanding the vibratory behavior of a single particle or particles as well as simple beams [2-71. However, there are few archival publications related to the chaotic motion and bifurcation behavior of plates or shells. The forced response of a nearly square plate, the nonlinear dynamics of a shallow arch, and the chaotic motion of a circular plate and the cylindrical shell are examples of some of the recent studies on the nonlinear aspects of mechanical and structural systems [S-131. The present research studies on the nonlinear dynamic behavior of an elastic plate subjected to a harmonic excitation. With the nonlinear effect taken into account, the nonlinear dynamic equation is derived. Using the Galerkin principle, results are presented for the double mode model of a plate instead of the widely used single mode model. The bifurcation behavior of the plate is examined in detail with the consideration of vibration stability of the plate. The plate governed by the averaged differential equations is then examined and the bifurcation behavior of the plate is determined. Demonstrative examples are presented and discussed on

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the basis of the governing equations with the assistance of phase portraits, time history and power spectrum corresponding to the dynamic response of the plate. The chaotic vibration of the elastic plate is investigated; and the comparison between the results generated by the single and double mode models is performed. Through the theoretical analysis and numerical computation of the present research, it is observed that the plate exhibits an extreme complex behavior over the range of the loading parameters. The condition under which the single mode method is valid and applicable is also determined. The results of the present research demonstrate that the single mode model can be used for studying the elastic structure’s nonlinear behavior for certain range of system parameter values. For the other values of system parameters, the single mode method usually used in nonlinear vibration analysis is less accurate in comparison with that of double mode and may lead to incorrect conclusions. Thus, higher order modes should be used in these cases. 2. DEVELOPMENT GOVERNING EQUATIONS

Consider the deformation of a plate with h as its thickness, and a and b as its length and width in x and y directions respectively. The plate is subjected to a simple harmonic excitation hJ, which has the following expression.

N = Po+ P, coswt

(1)

where PO,PT and w are loading constants describing the characteristics of the excitation acting on the plate. The lateral dynamic equation for the plate so defined can be given in the following form

I

DV‘W

+ ph-+d2W dr’

av

d’W

~7~---- hL(p,W) + N =0 dt dXZ

where

l4

Eh’ 12(1 - p 2 )

d2W + -.d2v, d2W - 2- d2v, .- d2W L(v,,W) = -.g2v, -

v

=-+2d4 dx4

dX2 d y 2

dP dx2dy2

d y 2 dX2

dxdy dxdy

(3)

+- d 4

By4

In Eq. (3) and Eq. (4), p denotes the Poisson ratio of the material, 8, the damping coefficient, p the material density, E the elastic constant, W the lateral deflection, and v, the stress function. When the plate is simply supported, the boundary conditions can be d2W d2W written as follows: W=-=O as x = O and x = a ; and W = - T = O as y = O and 23c2

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125

y = b , The transverse displacement of the plate is approximated in the following form with employment of the double mode model: m n y 2ny W ( x , t ) = T,(t) sin-sin+ T2(t)sin-2m sina b a b

(4)

Using Eq. (2), the stress function can be obtained.

where A=

2

[(%I +(”:)’;

B=

2

[;I +($I]’

In order to derive the equation in terms of the time-dependent variables, and Galerkin procedure is applied. DV4W + ph

at2

rn + So-- hL(cp,W )+ N -W;ds = 0

,’wl a2w1

dt

dX2

d2W rn DV4W + pb-+ 6,-- bL(p,W )+ N 7 W ; ~ =S 0

at2

where m sin-, n y W,’ = sin a b

at

(7)

BX

.

2 m . 2ny W, = sin-sina b

(8)

Substituting Eq. (8) and Eq. (4)into Eq. (7) and performing the integration, the following simultaneous nonlinear differential equations are obtained

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4

+ EP( A + B ) ( a )

(%) 4

T2T2= O

In order to convert the above equations in non-dimensional form, the following variables are introduced.

Considering the - buckling of the q, = q, = q, = q2 = P, = 0 , we have

plate

subjected

to

a

static

load,

q,[E-4 -4qfl=O .

let (13)

E

< 3.1, It is not difficult to find that Eq. (13) has a single unique solution q,$= 0, when corresponding to the pre-buckling state. If > 4, Eq. (13) has three solutions as indicated below.

E

1

4, = o

q,

=*iF

(14)

Consider the post-buckling state of the plate subjected to the static load, q, is defined as

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Making use of Eq. (1 5), the equations in Eq. (1 1) can be rewritten as

where

c-

The time-dependent variables are assumed to be of the form

q, =u,cos - o z + e ,

1

q2 = a 2cos(for+e2)

,

.

(1 8)

Using the method of averaging [ 141, a set of autonomous equations can be derived.

, ---[-4 w

a

I

2wu, + ~ . , u ps: i n ( ~ -, 28, )- 2 ~ 0s ,i n ~ ~ ,

1 ~ , 8=,-[4a,a,

+ ~A.,u,u: +&up: cos(28, - 2 4 ) + 34.: - 2 E . q C O S ~ Q , ]

4w

u, =-[-2Gu2 1

4w

(194

+A.,u,a: sin(28, -2Q,)-85a2 sin28,]

= a , cose,,

x 2 = a2cos0,

y, = a, sine,,

y, = a, sine,

XI

(19b)

(19c)

,

the set of equations in Eq. (1 9) can be rewritten as

x1 --[-

40

2Gx, - 2(2q

+ &(xi - Y:)X

128

+6)y, -2

- 2%V2Y,

4 :

+ y:)y,

- %(x? + Y?)Y,]

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1

1, = -[-

2wx2 - 4(a2 + 2 q ) y , + h,(x:

- y:)y2 - 2h,x,x2y, (22c)

4w

3. STABILITY AND BIFURCATION Using Eqs. (19a-d) one can obtain a trivial solution, corresponding to an equilibrium state of the plate; as well as non-trivial solutions, corresponding to the steady state motions under the effects of the nonlinearity. (1) The trivial solution is

a, =o,

a,

=o

(2) The non-trivial solutions is (when a, # 0, and a, = 0)

.

-

w

sin28, =-=-

PT (3) The non-trivial solutions is (when a2 # 0, and a, = 0 )

By linearizing the equations in Eq. (22) at the trivial solution (x,, y,, x,, y,) = (0, O,O,O),one obtains the corresponding characteristic equation and characteristic values.

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According to the equations (26) and (27), none of the characteristic values of the system is a pair of imaginary roots. Thus, it is impossible for the Hopf bifurcation to take place. Moreover, the trivial steady state motion becomes unstable only when any characteristic value is zero. When the axial excitation is small, the trivial solution is stable. With the increasing of the axial load 6 ,the characteristic value may be zero; in this case the trivial solution will become unstable. Thus, we obtain the following conclusions, if the following equation holds, 1 1 -+-[4a: 4 4w2

-F]=O

,

for the dynamic system(22) the bifurcation takes place in plane ( x i ,yl). Similar to this case, if -+-[CT:-~?]=O 1 1 4 w2

,

the bifurcation takes place in plane ( x 2 , y 2 ). Using the above equations, we can obtain the following necessary condition under which the bifurcation takes place.

If pTl= ?,, two characteristic values become zero, and the bifircation takes place at the same time in the plane ( x , , y , ) and ( x 2 , y 2 ). If the condition (28) or (29) is satisfied, the trivial solution becomes unstable and the bifurcation takes place in the plane (xl,yl) or ( x 2 , y 2 ) , which makes the non-trivial solution occur. Let us consider the following cases for stability of the non-trivial solutions. (a) a, #O, a2 = O The stability of this motion depends on the characteristic values of Eq. (22) at the point (a;, Si'). The corresponding characteristic equation in this case is

,I2 + A + D , = O where

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If u; in this equation is small, the motion of the single mode is then stable as a, # 0, a, = 0 . When D, = 0, the motion of the single mode becomes unstable, and the motion of the double mode occurs with bifurcation. The physical meaning of this is that the double mode motion starts as D,= O with initiation of bifurcation from the single mode motion. The corresponding energy of the system starts to transmit from that of single mode to the double mode. (b) a, =0, a, Z O The stability of the motion in this case depends on the characteristic values of Eq. (22) at the point ( a i , 6;). The corresponding characteristic equation is expressible as

a2+a+D,=o

(33)

where D, -- 4L

-[“)’.:

+ [w ~ +225& . T ~ ~4w

-(Lq] 2z

+&a;’~cos28;

. (34)

If ai in Eq. (34) is small, the single mode motion is stable as a, = 0, a, # 0. When D, = 0, the single mode motion becomes unstable, and the double mode motion occurs with bifurcation. Figure 1 shows the demonstrative examples of the mode bifurcation described above.

Figure 1 The bifurcation diagram In Figure l(a), when 6 reaches the corresponding value at point 0 as indicated in the figure, Eq. (28) holds. Thus, the single mode motion OA, as a, f 0, a, = 0 , takes place with the bifurcation from the trivial solution as illustrated in the figure. When 6 increases its value till that corresponding to the point A, we have D, = 0 . As D, = 0 , the double mode motion

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AB, under the conditions a, # 0, a, # 0, takes place through the bifurcation from the single mode motion OA. Figure l(b) shows the cases in which the bifhcation processes from the trivial solution to the single mode motion OA, as a, = 0, a, # 0, and then to the double mode motion AB. Figure l(c) exhibits that the bifurcation may take place directly from the trivial solution to the double mode motion.

4. CASE STUDIES To support the theoretical assertions, the P-T approach [ 151 with variable step was employed, where the following parameters of the plate were fixed as constants:

h=0.01m, a=b=0.3nl, p=O.3, p=2.78x1d(kglm3), E = 69.7Gpa9 8, =0.01

(35)

The initial conditions of the numerical simulations for the single mode model are x = 0.0, and x =O.O as t = O . For the double mode model, the initial conditions are x, =O.O, x, =O.O, x, =O.OOOOl, x, =O.O as t = O . With the numerical simulations, the relationship between the non-dimensional amplitudes of and the non-dimensional frequency W are determined and the dynamic loads (%,, graphically demonstrated in Figure 2. As can be seen from Figure 2, there are two characteristic values as zero at the same time corresponding to the point M, and at this point

F,)

w=w',pT -- j i 'T '

Figure 2 The

F,,e.,--W

curves

In Figure 2, the whole plane can be divided into four areas, A , , A,, A, and A d . The trivial solution is stable in area A, , the non-trivial solutions corresponding to the motions of two single modes are stable in area A, and A , , respectively. And the double mode motion exists in area 4. Moreover, along the curves (1) and ( 2 ) in Figure 2, following cases can be found. In the cases that the non-dimensional frequency satisfies the condition of 0 < -We, the motion of the single mode will take place as the external excitation varies

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along curve (l), i.e. a, # 0, a2 = 0. If G = &and = the motion of the double modes will take place, i.e.

5 q,

a, *o, a, ZO.

If Z> G*, the motion of another single mode will take place as the external excitation 6 varies along curve (21, i.e. a, # 0, a, = 0.

a) the q 2

0.5

- tcurve

0 -

-0.5O :

-1 -0.5

0

b) The phase portrait( q 2

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0.5

-

q 2)

133

00

O

t

2

3

4

5

6

(c) The power spectrum Figure 3 Wave, phase and power spectrum diagrams as pT = 100000,0= 10’. The phase portraits, time-displacement history diagrams and power spectrum diagrams generated with the numerical results can be developed. One set of the diagrams is shown in Figure 3. The numerical analysis with utilization of the models established, it is found that neither of the two modes has 1/2 subharmonic component as a, = 0 , u2 = 0 ; only one mode has 1/2 subharmonic component, the trivial solution is unstable and the single mode motion takes place with bifurcation as a, # 0 and u2 = 0 ; and both of the two modes have 1/2 subharmonic components as a, # 0 , u2 # 0 . The numerical results denote a bifurcation process from the trivial solution to the non-trivial solution of the single mode, and then to the non-trivial solution of the double modes. 5. CONCLUSIVE REMARKS

In the present research, the nonlinear dynamic behavior of an elastic plate of large deflection subjected to a harmonic excitation is performed is studied. Systematical studies on such a plate exerted by harmonic loadings with considerations of single and double modes are not found. From the present study, the behavior of the nonlinear elastic plate of large deflection is found very complex. Various vibratory behaviors are found in the present research and chaos of the elastic plate under the harmonic loading is evident. The governing equations for the plate are developed in terms of the time-dependent variables with the Galerkin principle and a double mode model. The trivial solution corresponding to an equilibrium state; and the non-trivial solutions corresponding to the steady state motions under the effects of the nonlinearity can be conveniently obtained. Furthermore, the stability and bifurcation of the vibration of the plate are analyzed in detail based on the governing equations developed. Numerical simulation is performed with employment of the newly developed numerical method, the P-T method. The results of theoretical bifurcation analysis are numerically verified through the P-T method. The relationship between the amplitudes of the dynamic loads and the frequency are determined. The nonlinear behavior of the plate is analyzed in detail for both the single and double mode models established.

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REFERENCES [l]. F.C. Moon, and S.W. Shaw, Chaotic Vibration of a Beam with Nonlinear boundary Conditions, Non-linear Mech., 1983, 18,230-240. [2]. P.D. Baran, Mathematical Models Used in Studying the Chaotic Vibration of Buckled Beam, Mechanics Research Communications, 1994,21, 189-1 96. [3]. P. Holms, and J. Marsden, A Partial Differential Equation with Infinitely Many Periodic Orbits: Chaotic Oscillation of a Forced Beam, Arch. Rat. Mech. and Analysis, 1981, 76, 135-165. [4]. V. Keragiozov, and D. Keoagiozova, Chaotic Phenomena in the Dynamic Buckling of an Elastic-Plastic Column under an Impact, Nonlinear Dynamics, 1995,13, 1-16. [ 5 ] . L. Dai and M.C. Singh, Periodic, Quasiperiodic and Chaotic Behavior of a Driven Frode Pendulum, Int. J. Non-Linear Mechanics, 1998,33,947-965. [6]. R. Han and A. Luo, Comments on the Subharmonic Resonance and Criteria for Escape and Chaos in a Driven Oscillator, J. Sound Vib., 1996, 196,237-242. [7]. A.H. Sheikh and M. Mukhopadhyay, Linear and Nonlinear Tranisent vibration Analysis of Stiffened Plate Structures, Finite Elements in Analysis and Design, 2002, 38, 477-502. [8]. P.C. Dumir and G.P. Dube, Geometrically Non-Linear Analysis of a Thick Annular Plate with Elastically Constrained Edge Using Galerkin’s Method, J. Sound Vib., 2001, 246,556-565. [9]. X.L. Yang and P.R. Sethna, Nonlinear Forced Vibrations of a Nearly Square Plate-Antisymmetric Case, 1. Sound Vib., 1990, 155,413-441. [IO]. W. Tien, N. Sri Namachchivaya and N. Malhotra, Non-linear Dynamics of a Shallow Arch Under Periodic Excitation-11.1 :1 Internal Resonance, Int. J. Non-Linear Mech., 1994,29,367-385. [ l l ] . Q. Han, H.Y. Hu and G.T. Yang, Chaotic Motion of an elastic Circular Plate, Transactions of Nanjing University of Aeronautics & Astronautics, 1998, 15,206-210. [12]. Q. Han, H.Y. Hu and G.T. Yang, A Study of Chaotic Motion in Elastic Cylindrical Shells, Eur. J. Mech. NSolids, 1999, 18, 351-360. [13]. L. Dai, Q. Han and M. Dong, A Single and Double Mode Approach to Chaotic Vibrations of a Cylindrical Shell with Large Deflection, to appear in the journal of Shock and Vibration. [14]. A.H. Nayfeh, Problems ofperturbation, 1993, John Wiley & Sons, New York. [15]. L. Dai and M.C. Singh, A New Approach to Approximate and Numerical Solutions of Oscillatory Problems, J. Sound Vib., 2003,263,535-548.

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Dynamic simulation of civil engineering structures in virtual reality environment Cs ANTONYA and D TALABA

Product Design and Robotics Department, University Transilvania of Brasov, Romania

1. Introduction. IRMA YR s o m a r e The civil engineering metallic structures are usually conceived as mechanical systems with zero degree of freedom, although mobile joints like in usual mechanisms are involved (revolute, spherical, etc). The degree of freedom is usually zero in order to precisely predict the force flows in the various branches of the metallic structure. In the first stage of design, the structure is conceived like a mechanism able to transmit the vertical and horizontal forces. In the next stage, the structure mobility is canceled by articulating further bodies @racings) into the system. In this way, the effort flows are well established from the very beginning and the structural role of each element is decided. This paper presents the model and simulation of a civil engineering metallic structure in an earthquake situation integrated in a virtual reality application. The structure taken into consideration is part of a real life building and it was modeled as a multi-body system (MBS) using ADAMS, as well as a multi-particle system (MPS). The two approaches are discussed from the results and efficiency viewpoint. The simulation results are displayed in an interactive manner via the JRMA virtual reality software developed in the framework of the FP5 funded project the IRMA “A reconfigurable virtual reality system for multipurpose manufacturing applications”. The VR software overall aim is to facilitate the control and monitoring tasks, training of personnel and failure modes of plants and plants components. In this context, the typical VR facilities available in IRMA software are providing added value to the MBS/MPS simulations in the post-processing stage, e.g. walkthrough facilities, taking measurements of the damages, comparing different designs, immersion etc.

2. M B W P S simulation module for civil engineering structures The dynamic simulation module developed for the IRMA package was adapted particularly for applications in civil engineering. The application chosen for the demonstration is the simulation of industrial building structure behaviour during earthquake. The software developed includes three modules, devoted to the three main stages in model definition and simulation: pre-processing, processing and post-processing (Fig. 1).

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For the processing module, as fig. 1 illustrates, the work included two parallel approaches: (i) the utilisation of commercial software and (ii) the development of a new software, based on the multi-particle formulation approach. For both approaches, pre-processing software modules and a common post processing software module have been developed, in order to allow integration and communication with the IRMA core software. 3. The multi-body systems approach For the multi-body systems approach the software ADAMS has been used, which represents the mechanical system as a collection of interconnected bodies and formulates the motion equations as

{

[Jl[iil =[VI, [ml[iil - - [ J I T [ 4 = [Q,,],

(1)

in which the kinematic constraints corresponding to the joints are represented by the algebraic equations of the generalized accelerations [q] and the internal forces are included in the generalized forces vector, [Qex]. The internal forces introduced by the flexible beams are modelled as ,,beams” entities which introduces spring-damping forces on all six degrees of freedom of the 3D space of the form

Fig. 2. The MBS model of the metallic structure

138

Fig. 3. Detail of comer

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[FI = [kl VI + [CI [&I (2) in which [k] is the stiffness matrix, [6] is the deformation vector, [c] is the damping matrix, [8] is the deformation speed of the beam and [F]=[Fx F, F, M, My MzlTis the vector of the spring-damping force. The stiffness and damping constants from (2) are determined with relations that depend on shape and dimensions of the real beam (in section and length), as well as materials used. Thus, the geometric input data of the beam model for the MBS software simulation is limited to the description of the profile and the connection points with the adjacent bodies. In principle, modelling a flexible beam consists in its division in a number of rigid segments, linked by elastic beams. In order to illustrate the modelling steps and feasibility of the MBS simulation approach, a simplified metallic structure was selected from a real life project developed at the Canam Steel Romania Company, the civil engineering partner company in Romania. ?

The metallic structure selected is part of an industrial plant and includes 6 columns, 6 beams and 4 bracings. These elements are made in steel laminated profiles of I shape of 205x210 mm for the columns (marked with l), 107x403 mm for beams (2) and 105x107 mm for bracings (3). The dimensions of the metallic structure simulated are: 20 m length, 10 m depth and 14 m high (fig. 2). In the comers of the structure the different kind of elements are linked with revolute joints. In order to avoid the structure to have mobility, bracings are articulated in different perpendicular frames of the structure. The discretisation of the constructive elements into rigid parts is Fig. 4. The structure in a deformed stage made such as to obtain segments of 2 m length. In this way, the columns have been divided into 7 rigid parts each, the beams in 5 parts and the bracings have been divided in 9 rigid bodies. The building foundation is modelled as a rigid body to which the ground motion is applied (the earthquake ground motion). The columns (i.e. the firsts segments of the columns) are linked by spherical joints to the ground. The first and the last rigid segment of each beam are articulated to the columns by revolute joints. The axes of these joints are perpendicular on the plane of the adjacent bodies, i.e. the beam rigid segment and the column rigid segment. The bracings are linked to the columns through revolute joints in a similar way as the beams. For example, in the comer A from fig. 3 three revolute joints are illustrated: two between the beam segments (2) and column segment (l), respectively the revolute joint between the bracing segment (3) and the column (1). The model obtained has 109 rigid ungrounded bodies, 20 revolute joints, 6 spherical joints and 530 DOF. Out of the foundation body and the end segments of the beams, columns and bracings, all the other segments are not kinematically constrained but only the internal forces (flexible links). The weight of 150N/m2, corresponding to the roof has been taken into account by charging each beam with a supplementary mass of 10kg. For the purpose of the simulation the “earthquake” has been introduced as a simple harmonic excitation force on the foundation of the building (this body was linked to the ground by a planar joint in horizontal plane) with the direction corresponding to the diagonal of the foundation rectangle, amplitude of the 15kN and frequency of 2Hz.

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Another building modeled at the request of the civil engineering company and simulated in the virtual reality environment with the IRMA software was the model of a full warehouse. This steel-structure building has a larger number of elemenst and it was modeled as follows: 393 moving parts (not including the grounded body), 133 revolute joints, 22 spherical joints and other 10 fixed joints for the roof elements. The number of the degrees of freedom of the entire model was 1566. The excitation taken into consideration was a real earthquake of magnitude of 6.5 degrees on Richter scale, so displacement of the base is imposed. In this case the peak values of the acceleration were: 0.31 m/s2 (for up-down direction), 0.77 m/s2 and 0.52 m/s2 in the horizontal plane and for the displacement 1.6cm (up-down), 5.4 cm and 2.5 cm (two directions in the horizontal plane). The simulation proved that the structure was supporting easily the earthquake, the most important displacements being recorded at the roof level. The building state in a simulation frame as resulted from D A M S is shown in fig. 5.

Fig. 5. The simulation of the SPES building appliction For the virtual reality simulation of this building’s behavior the IRMA-VR software has been used. The core integration module is the Delfoi IntegratorTM, a PC based message broker used to link a virtual environment and a production/failure database scenario, to different simulations (QUEST and ARENA material flow simulation, IGRIP robot kinematics simulation) or to programmable logic controls using their OPC interface. In order to use the IRMA VR system this software was used to link an Excel database (containing the precomputed data history of the displacement and rotation of every solid element within the modeled steel structure) with VRML (virtual reality modeling language) model of the building in order to obtain the VR simulation. Due to the large number of entities, communication of all needed information with IRMA, via the Integrator sofiware model takes approx. 4 min. The IRMA module allows a set of VR facilities for simulation manipulation, as freezing the system for measurements of earthquake effects at certain moments of time, zooming, setting up the observer location and attachment (local position when the observer is located on the building floor and observes the earthquake or global position when the observer is not static located, out of the earthquake), loading the transparent model of the unaffected building and comparing with the current simulation position (fig. 6) - very useful in designing the path of pipings to ensure maximum safety.

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Fie. 6. Comoarina the simulation with the unaffected uosition 4. The multi-particle system approach Within the lRMA Project a new simulation software module has been added and developed. The objective of this was to estimate the industrial steel structures behaviour during earthquakes using the multi-particle approach and post-processing the results using the facilities offered by virtual reality simulations. Within this formulation, the rigid bodies are modelled as sub-systems of mass particles that are conserving entirely the original mass properties of the body. This condition requires a minimum number of 4 equivalent mass points in 3D space for each rigid part. The beams, columns and bracings has been divided in

Fig. 7.The equivalent spring mass system for a beam modelled by 2 rigid parts the same way as for multi-body approach, and furthermore each rigid part was modelled with a set of 6 mass points in order to facilitate the definition of the mobile and flexible connections between the various bodies. The 6 mass points associated with each rigid part are located at the extremities, in order to facilitate the definition of the springs elements between the two adjacent parts (figure 7). The multi-particle model considers the mechanical system as a collection ofparticles subject to a set of absolute and relative constraints. The main difference with respect to the conventional multi-body models consists in the overall dynamic model which contains no bodies because they are replaced by point masses. Therefore no body reference is included and thus the rotational motion which usually characterizes the 3D bodies becomes irrelevant for the case of point masses. The mass matrix and external force vector are computed for particle without relation to any Body Reference Frames. In this way, the body moments of inertia are no more relevant and the mass matrix is very simple. The mechanism representation includes a particle based model for each rigid body and point

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contact models for each type of joint. The rigid body model consists in a set of particles separated by constant distances, each particle being associated with a concentrated mass according to the inertial equivalence with the real object. Once the position of the particles is established in the body frame, the point masses can be easily obtained from the inertial equivalence conditions: m,y,2+m,z,2 +rn2y2’ +m2z22 +m3y32 +m3z32 +m4y42 +m4z42 + m 4 y 42 +mgzg 2 +mgyg 2 +m.+g m l x I 2+ m l z , 2 + m 2 x 2 2+ m 2 z 2 ’ + m 3 x j 2 + r n , ~ + , ~m 4 x 4 2+ m 4 z 4 2+ m 4 x 4 2 + m6262 + m6x62 + m6262 m l x 1 2+ m l y 1 2+ m 2 x 2 2+ m 2 y 2 2+ m j x 3 2 mlxlyl fm2x2y2 +m3x3Y3

+m4x4Y4

=J,

=J,

+ m 4 x 4 2 + m 4 y d 2+ m 4 x 4 2 +m6yb2 +m6x62 + m 6 y 6 2 = J ,

+m5x5y5 +m6x6Y6

= J.xy

(3)

mlylzl + m2y2z2 + m3Y3z3 + m4y4z4 m5y5z5 + m6y6z6 = J p m l z , x , + m 2 z 2 x 2+ m,z,x, + m 4 z 4 x 4+ m 5 z 5 x j+ m6z6x6 = J , mlxl + m2x2 + m,x, + m4x4 + m5x5 + m6x6 = 0

mlyl + m2Y2 + m3Y3 m4Y4 + m5y5 + m6Y6 = m,z, i m2z2 + m3z3+m4z4 +m5z5+m6z6 =0 ml + m 2 + m , + m 4 + m 5 + m 6 = M

The generalized co-ordinate’s vector representing the system state space has the form: r91 = [XI Yl ZI x2 yz z2 x3 y3 z3 ... x p yp zpl: For dynamic simulation, the equations have the same general form as for MJ3S model

(4)

mq - JT;l = Q,,, in which the mass matrix is a diagonal matrix m=diag[ m, m, m, m2 m2 m2 m3 m3 m3 ... mp mp mp1. (6) The Lagrange multipliers include the joint reaction forces (including no torques as rotation of point masses is no relevant) and also the constant distance reaction forces between the particles of the rigid bodies. As the starting point for simulation is the conventional geometric model of the system the authors have developed software modules for preparing the input data (VR-PRE module), solving the differential algebraic system of equations (DAE) and presenting the simulation results using the IRMA facilities and Delfoi Integrator. The pre-processing module has two main roles: a. to provide the MPS geometric model of the structure and facilitate conversion from the multi-body model in a format ready for input into the processing module. b. to generate an ASCII file with all the data about the multi-particle system used in the simulation, the connections between particles and the boundary conditions in which the entire structure is analyzed (i.e. the boundary conditions associated with the earthquake external loads). This file includes all necessary input data for the mathematical solver module based on the multi-particle system formulation. The application taken into consideration is the same as for multi-body approach (the structural cell of a building, fig. 2). The input data to generate the VRML file includes only the number of mass-points in which the entire metallic structure has been divided. The software used in the pre-processing module is developed using Borland Delphi, C++ and AutoLISP programming languages associated with the geometric modeller AutoCAD. The user tasks in the pre-processing stage, facilitated by the VR-PRE module are: a. Geometric modelling of the mechanical structure - this is performed using the standard capabilities for 3D modelling of AutoCAD,

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Discretisation of each element of the structure in smaller rigid parts and association of a set of 6 equivalent mass points to each rigid part - his is performed automatically by a program in Auto LISP able to exploit the AutoCAD geometric model, c. Calculation of equivalent spring-mass system of each element of the structure, d. Generation of the input datafile for the M P S based solver module. The multi-particle model resulted for the structural cell contains the following elements: 745 mass-points corresponding to 113 solid elements, 576 translational springs and dampers, 20 revolute joints, 6 spherical joints, 1 planar joint, 1557 constant distance constraints. For the virtual reality simulation of the results a graphical user interface has been developed (figure 8) to link the M P S solver and the with IRMA VR software and to enables various tools for comparing different designs and evaluate the simulation results. The GUI allows the same facilities for simulation manipulation as the one presented earlier for the multi-body approach. For comparing different designs, the input data for the solver can be changed with the interface and, after re-computing the structure’s behavior, the new result can be put side by side with the old ones in the virtual reality environment. Also magnification factors has been introduced in order to scale building of different dimensions to be able to compare different designs. In figure 9 a VR simulation frame of the structure is shown as displayed by the post-processing module. The spheres corresponding to the point-masses are coloured according to the magnitude of the forces acting on them and show the level of local stress. b.

Fig.8. The GUI for the VR simulation 5. Comparison between MBS and MPS models Despite of the larger number of equations, the MPS model provides several features with relevance to the non-linear mechanical system simulation: - The representation of forces and inertial mass properties is significantly simplified.

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- The constraints and the corresponding algebraic equations are of small variety. This is simplifying both constraint and Jacobian matrix formulation. The M P S model allows the extension for the treatment of flexible multi-body systems by replacing the distance equations with the flexibility principles for each body. The computation time for the two approaches for a 1 second simulation in 100 steps and the same external load (amplitude of the 15kN and frequency of 2Hz) are the followings: ADAMS, 1.02 min (Adams-Bashforth-Moulton integration method) C++ application, 18.7 min (backwards Euler integration method) MATLAB-version of the M P S solver, 20.77min (backwards Euler integration method) The MPS formulation is obviously computationally much slower, since the number of equations is much higher than the MBS formulation. In the presented case a body is represented by 6 mass points which generate 6x3=18 differential equations plus 12 algebraic constant distance equations to model the rigid body conditions, that is 18+12=30 DAE equations per body, comparatively with 6 differential equations per body in MBS formulation. Although from a general perspective the M P S approach has important advantages and from the IRMA software viewpoint, has disadvantage in the increased number of entities describing the model (for each body at least four points are required, while in MBS approach one body=one entity). This is requiring longer times, in the post-processing stage as well, for

Fig.9. M P S simulation of the structure communication via the network the results and up-dated geometry.

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Conclusions IRMA VR MBS/h4PS software module is focused on the simulation of articulated mechanical structures used in civil engineering in earthquake situations. Two approaches have been considered for the simulation: the multi-body approach and multi-particle approach. The post-processor achieved for IRMA, VR-Post allows to present the results in both approaches and allows the user to freeze the system in a desired state, take measurements of the displacements, to display the strengths in the beam segments, to walk into the structure and ,,see" in more detail the behaviour of the respective structure part during the earthquake. This set of facilities is offering VR capabilities to the user, to evaluate the behaviour of the building during earthquake. Comparatively, although simpler, the MPS formulation involves a large number of inertial entities, therefore the simulation is computational less efficient than in the MBS case. In turn, offers interesting discretisation possibilities and convenient way of simulation of flexible systems. References 1. Antonya, Cs. Dynamics of the double wishbone car suspension mechanism. SMAT Conference 200 1, vol. 1, Craiova, 2001, p. 173-176. 2. Antonya, Cs. Dynamic transmissibility of car suspension mechanisms. PhD thesis, University Transilvania of Brasov, 2002. 3. Talaba, D. A particle model for mechanical system simulation. NATO Advanced Study Institute Series, Praga, 2002. 4. Talaba, D. Articulated mechanisms - Computer Aided Design. Transilvania Univ. Press, 2001. 5. ***ADAMS -User's reference Manual, Mechanical Dynamics Inc, USA, 2001.

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Wind force time-history generation by discrete Fourier transform (DFT) P J MURTAGH, B BASU, and E M ERODERICK Department of Civil, Structural, and Environment Engineering,Trinity College Dublin, Ireland

ABSTRACT This paper presents an approach to artificially generate wind force time-histories for use in the time domain response prediction of a flexible line-like structure, such as a wind turbine tower. Although such analyses are usually carried out in the frequency domain, a time domain approach is sometimes favourable, as it allows for the inclusion of behavioural non-linearity and response coupling. The method presented here expresses the frequency dependent longitudinal turbulence within a wind flow in terms of a fluctuating velocity or drag force, by employing the well known Discrete Fourier Transform (DFT). The velocity or drag force time-history is created from spectral energy contributions summed over a discretized continuous frequency band. The structure may be discretized, with its mass lumped into several nodes, ultimately creating a multi-degree of freedom (MDOF) entity. Fluctuating nodal wind velocity time-histories are first generated using this DFT approach in conjunction with a turbulence spectrum such as those suggested by Kaimal, Von-Karman or Hams. A subsequent calculation will then easily yield the corresponding nodal wind force timehistories. Nodal force spatial correlation may then be incorporated into the system by including coherence information. This information is used to obtain the modal force timehistories for the required number of degrees of freedom in a MDOF model of the structure. This representation facilitates estimation of the structural response using a mode superposition technique such as mode displacement or mode acceleration.

INTRODUCTION Although dynamic analysis is traditionally employed using a frequency based inputoutput approach, a time domain solution may sometimes be favourable. Thus, this paper is interested in the temporal variation of wind forces acting on a flexible line-like structure. A flexible structure subjected to dynamic loading will experience inertia and damping forces, as well as elastic forces. Its response is thus made up of the addition of a series of weighted mode shapes. By discretizing the inherent properties of the structure into several lumped nodes, the structure becomes a multi degree-of-freedom (MDOF) entity. Each mode will have a modal force time-history associated with it. 1.

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If recorded time-histories are not available, it is necessary to generate the artificial data using widely disseminated spectral data. Three methods are generally employed, the first is based on the Fourier Transform, the second on the Wavelet Transform and the third on an auto regressive-moving average (ARMA) based time-series approach. Kumar and Stathopoulos [l] simulated wind pressure time histories by using a Fast Fourier transform (FFT) based algorithm. Li and Kareem [2] simulated a multivariate non-stationary random process by use of spectral decomposition, which also used a FFT based algorithm. Kitagawa and Nomura [3] used wavelet theory to generate wind velocity time-histories. Minh et a1 [4] investigated the time-domain buffeting of long-span bridges by simulating force timehistories using the ARMA method. The time-history simulation method presented in this paper makes use of the fact that any random signal with a varying frequency content may be represented by a Discrete Fourier Transform (DFT) encompassing that frequency content. The Fourier coefficients associated with this DFT are obtained as randomly generated numbers with zero mean and a specific standard deviation. This standard deviation is related to the energy content contained within a discretised power spectral density (PSD) function. Thus, in order to randomly generate a timehistory, a spectrum must be available. A turbulent wind flow may be visualised as an array of swirling vortices of air of different size and strength imparting kinetic energy into any structure with which it comes in contact with. It is convenient to quantify this energy in terms of frequency to form a PSD function. Several models of wind velocity power spectra have been proposed over the past sixty years, by V o n - K h B n [SI, Davenport [6], Harris [7] and Kaimal [8]. The spectrum proposed by Kaimal [8] is used in this paper. Spatial correlation, or coherence relates the similarity of signals measured over a spatial distance within a random field. Earthquake engineers have studied the relationship between ground accelerations at different points on the earth's surface; publications in this regard include Hao et a1 [9] and Harichandran and Vanmarcke [lo]. Coherence is also of great importance to the wind engineer, especially if gust eddies are smaller than the height of a structure. Some of the earliest investigations into the spatial correlation of wind forces were carried out by Panofsky and Singer [ 1 11 and Davenport [ 121 and later augmented by Vickery [13] and Brook [14]. The coherence model proposed by Davenport is adopted for use in this paper. Recent publications involving lateral coherence in wind engineering include Hnrjstrup [15] and Snrrensen et a1 [16]. Minh et a1 [4] studied the buffeting of long-span bridges, accounting for the spatial correlation of wind turbulence along the length of the bridge. This paper illustrates a method of artificially generating modal wind drag force timehistories for use in the response estimation of a MDOF line-like structure. The spatial correlation of forces is included in the generation of the time-histories. These time-histories may ultimately be used to estimate the response of the structure to realistic wind loading, using a mode superposition technique.

THEORETICAL CONSIDERATIONS When a bluff body is immersed within a wind flow, the body will experience pressures distributed over its surface. These pressures result in a net force on the body, the along wind component of which is known as the drag force and the across wind component is known as the lift force [17]. The total drag force, F(t), experienced by a point like body is expressed as 2.

F(t) = 0.5CDpb2V2(t) = 0.5CDpb2[V + v'(t)I2

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where CD is the coefficient of drag, p is the density of air, b is the width or height of the body, V(t) is the total velocity of the wind flow, Vis the mean component of V(t) and v'(t) is the fluctuating component of V(t). Thus, the total drag force at node 'n', F,(t), is composed of a mean and a fluctuating component as

F,, (t) = T,, in which the mean nodal drag force,

+ fh (t)

(2)

T,, is expressed as

Tn = 0.5CDb2pV2

(3)

and the fluctuating nodal drag force, f'(t) is fn'(t)= C,b*pVvb(t) For a MDOF structure, the mean modal drag force, ,? orthogonality as

(4)

may be obtained by virtue of modal

where [@I is the mode shape matrix and superscript 'T' denotes matrix transpose. Nigam and Narayanan [ 181 presented an expression for the modal fluctuating drag force power spectrum, Sm, for a continuous line-like structure, which may be discretized into a MDOF system to yield

where f denotes frequency (Hz), k and 1 are spatial nodes, svkvI(f) is the velocity auto power spectral density (PSD) function when k=l and the cross PSD function when k+l, V kand VI are the mean wind velocities at nodes k and 1 respectively, and@,(k)and gj(1) are the node k and 1 components of the 'jth' mode shape. The auto and cross PSD terms may be evaluated as

with SVkk and Svll being the velocity PSD functions at nodes k and 1 respectively and coh(k,l;f) is the spatial coherence function between nodes k and 1. SVkk and Svll are derived by the expression offered by Kaimal et a1 [8] as

where z is the elevation (m), S,,(z,f) is the PSD function of the fluctuating wind velocity as a function of elevation and frequency, v. is the friction velocity, and n is known as the Monin coordinate. The latter two terms may be obtained from the expressions

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-

1

v(z) = -v, k,

ln-

Z

(9)

zo

with V(z) being the mean wind velocity at height z, is von KhrmBn's constant (typically around 0.4), and is the roughness length. Figure 1 illustrates the distribution of the spectral energy with frequency, obtained using equation (8). Also included in figure 1 are the PSD distributions suggested by von K h h [SI,Davenport [6]and Harris [7]. This paper makes use of the coherence function suggested by Daveport [12], coh(k,l;f), which relates the frequency dependent spatial correlation between nodes k an 1, and is represented coh(k, 1; f) = ex(

-

$!!)

10'

1o2

3 io1

-

w-

E

10"

1 0.'

1o'2 1 O'l Frequency (Hz)

1oo

10'

Figure 1 Comparison of spectral energy distribution of various proposed turbulent wind velocity spectra where Ik-11 is the spatial separation and LS is a length scale given by

where

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? = 0.5(Fk + 7 , ) and C is a decay constant. The fluctuating component of the modal force, as in equation (4) may be obtained by virtue of the fact that any arbitrary fluctuating force signal f’(t), with zero mean, may be represented by a DFT with a discretized version of a continuous frequency content, as

where ak and bk are the Fourier coefficients, ok is the k’ discretized circular frequency (o= 2nf, f is frequency in Hz) and t is the time instant. The PSD function, obtained by evaluating Sa in equation (6) is conceptually divided into ‘n’ frequency bands of size df. The area under the PSD function between the limits of fi and fi + df is equal to the variance of the signal $, at the discrete frequency fi. The Fourier coefficients in equation (14) are obtained as normally distributed random numbers, generated with zero mean and standard deviation o i . The fluctuating modal drag force is hence composed of a number of contributions from a discretized form of a continuous frequency band. The mean nodal wind velocity is assumed to vary with height according to a logarithmic law, values of which may be obtained using equation (9). These values are inserted into equation (3) to obtain mean nodal drag forces, which are in turn subsequently employed with equation (5) to obtain mean modal drag forces. The total modal drag force time-history is simply the sum of the mean and the fluctuating components.

3. NUMERICAL EXAMPLE A numerical example comprising of a flexible steel tower of height 60m is presented here. The tower is discretized into 8 DOF, and a time-varying nodal wind force time-history acts at each of the nodes. These nodal force time-histories, along with their spatial coherence information are converted into modal force time-histories. The parameters considered were Cd = 2, b = 2.65m, p = 1.225 Kgm”, = 0.4, v, = 0.16 d s , zo = 0.08m, C = 9. Three modal force time-histories for the first mode of the tower are presented in figure 2, each one representing a mean wind velocity of IOds, 2 0 d s and 3 0 d s at the top of the tower. As expected, the modal drag force with the mean wind velocity of 3 0 d s produced the highest force magnitudes. This is due not only to the presence of the highest mean wind velocity, but also because of the highest spectral energy content within the wind force PSD.The modal drag force time-history with the mean wind velocity of 2 0 d s contained the next highest force magnitudes, followed by that of 1O d s . CONCLUSIONS This paper presents a method of generating a modal drag force time-history acting on a MDOF line-like structure. The derived modal drag force time-histories include coherence information regarding the vertical spatial correlation of forces at each of the nodes. Once the modal drag force time-history is generated, it may be used to predict the response of the structure when used in conjunction with an algorithm to obtain the modal (generalised) coordinate and a mode superposition technique, such as the mode displacement or mode acceleration method. The method presented here may also be extended for use in the

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prediction of the response of a number of separate structures having specific lateral coherence relationships, such as in a wind farm. A series of modal drag force time-histories, each acting of a different wind turbine tower, could be generated due to ambient wind turbulence coupled with localised turbulence from a wake of another wind turbine tower. A specific lateral coherence model, as investigated by Hrajstrup [151 could be employed in this regard. lo4

3.5

0.5

-

-v = lOm/s

0

0

Figure 2 Modal drag force time-histories for three varying mean wind velocities

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8.

152

Suresh Kumar K., Stathopoulos T., ‘Computer simulation of fluctuating wind pressures on low building roofs’, J. Wind Eng. Ind. Aerodyn. 69-71 (1997) 485-495. Li Y., Kareem A., ‘Simulation of multivariate nonstationary random processes by FFT’, J. ofEngrg. Mech., ASCE 117(5) 1037-1058. Kitagawa T., Nomura T., ‘A wavelet-based method to generate artificial wind fluctuation data’, J. Wind Eng. Ind. Aerodyn. 91(2003) 943-964. Minh N. N., Miyata T., Yamada H., Sanada Y., ‘Numerical simulation of wind turbulence and buffeting analysis of long-span bridges’, J. Wind Eng. Ind. Aerodyn. 83 (1999) 301-315. V o n - K h b , T., ‘Progress in the statistical theory of turbulence’, Proc. Nat. Acad. Sc., Washington D.C., (1948) 530-539. Davenport A. G., ‘The spectrum of horizontal gustiness near the ground in high winds’, J. Royal Meteorsol. SOC.,87 (1961) 194-211 Harris R. I., ‘The nature of wind, in the modern design of wind sensitive structures’, Construction Industry Research and Information Association, London, 1971. Kaimal J. C., Wyngaard J. C., Izumi Y., Cote 0. R., ‘Spectral characteristics of surface-layer turbulence’, J of Royal Meteorol. SOC.,98 (1972), 563-589

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9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Hao H., Oliveira C. S., Penzien J., ‘Multiple-station ground motion processing and simulation based on smart-1array data’, Nuclear Engrg. and Design, 111,293-310. Harichandran R. S., Vanmarcke E. H., ‘Stochastic variation of earthquake ground motion in space and time’, Journal of Engineering Mechanics ASCE 112(2) (1986) 154-174. Panofsky H. A., Singer I. A., ‘Vertical structure of turbulence’, J. Royal Meteorol. SOC.,91 (1965) 339-344. Davenport A. G., ‘The dependence of wind load upon meteorological parameters’, in Proceedings of the International Research Seminar on Wind Effects on Buildings and Structures, University of Toronto Press, Toronto, 1968, pp. 19-82. Vickery B. J., ‘On the reliability of gust loading factors’, in Proceedings of the technical meeting concerning wind loads on buildings and structures, National Bureau of Standards, Buildings Science Series 30, Washington D.C., 1970, pp. 93-104. Brook R. R., ‘A note on vertical coherence of wind measured in an urban boundary layer’, Bound. Layer Meteorol., 9 (1975) 247. H~jstrup.J., ‘Spectral coherence in wind turbine wakes’, J. Wind Eng. Ind. Aerodyn. 80( 1999) 137-146. Smensen P., Hansen A. D., Andre P., Rosas C., ‘Wind models for simulation of power fluctuations from wind farms’, J. Wind Eng. Ind. Aerodyn. 90 (2002) 1381-1402. Simiu, E., Scanlan R., ‘Wind effects on structures’, John Wiley & Sons, New York, 1996. Nigam N. C., Narayanan S., ‘Applications of Random Vibrations’, Springer-Verlag, New York, 1994.

Fourier coefficient Widthheight of body Fourier coefficient Coefficient of drag Decay constant Frequency Frequency at interval i Total drag force Total nodal drag Force Nodal mean drag force Nodal fluctuating drag force Modal mean drag force von Karmans constant Length scale Monin coordinate Modal force power spectral density function Velocity power spectral density functions time mean wind velocity at node k mean wind velocity at node 1 mean wind velocity fluctuating wind velocity

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m

Hz Hz N N N N

m Ns m’s-l S

ms” mi’ mi‘ ms-l

153

P

[@I

154

Total wind velocity friction velocity mean velocity elevation roughness length density of air Modal matrix j'hmode shape component at node k jthmode shape component at node 1 Varianc'e at interval i

mi' mi' mi' m m Kgm-3

m2

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FRFs for wind turbine lattice towers subjected to rotor mass inbalance P J MURTAGH, B BASU, and B M BRODERICK Department of Civil, Structural, and Environment Engineering, Trinity College Dublin, Ireland

ABSTRACT This paper investigates the frequency response functions (FRFs) for wind turbine lattice towers subjected to rotor imbalance loadings. FRFs or transfer functions are damping dependent representations of structural response as a function of frequency and are usually obtained analytically by employing a mathematical transformation technique such as the Laplace or Fourier Transform. The knowledge of several parameters is necessary to analytically obtain a transfer function, namely the mass, stiffness and damping characteristics of the model. However, these parameters may be difficult to obtain especially if the model is geometrically complex. In this study, a finite element analysis of a lattice tower has been employed to develop FRFs for the nodal displacements and member forces in the tower. The model under consideration is a wind turbine tower consisting of a 3D lattice tower supporting a concentrated nacelle and rotor mass. The finite element code ANSYS has been employed for this purpose. FRFs are obtained (with varying nacelle masses) for the nodal displacement and member forces caused by imbalance effects of the rotating blades, which create a lateral loading at the top of the tower. These FRFs may then be used in subsequent frequency domain stochastic analyses to estimate the response of the structure. 1. INTRODUCTION

Although the majority of the large-scale wind turbine towers worldwide make use of a tapered tubular steel tower, designers have also opted for steel lattice tower designs. The lattice tower, in fact the most widely used assembly for small scale turbines in private use. Although common among the first generation of wind turbines, the lattice tower has subsequently seen a demise in its popularity. This appears to have arisen largely on aesthetic grounds, Le. a greater visual impact than the tubular tower, though this is clearly debateable. The lattice tower actually has many advantages over the tubular tower, such as a decrease in material usage (typically half the material needed as in a tubular tower) and ease and practicality of erection. The lattice tower wind turbine tower is the assembly of interest in this paper.

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Structures may be dynamically analysed in both the time and frequency domain, with each approach having its individual merits. Analysis through the frequency domain employs the well known systems approach, which also provides a statistical framework for response estimation. This approach is composed of three separate components, the characterisation of the loading (input), the characterisation of the structure (frequency response) and the characterisation of the response (output). The latter component is usually the unknown and is of most interest to design engineers. It is formed from the product of the loading and structural characteristics. Transfer functions as so named because they transfer energy input into the structural system into energy output by the structural system. They are fundamentally formed by gathering knowledge of the structure’s individual modes of vibration and presenting that information in terms of frequency. Mathematically, they may be formed by employing the Laplace Transform technique [I] or the Fourier Transform technique [2] to convert a differential expression into an algebraic expression which is easier to solve. An Inverse Transform then converts the expression back to its original differential form. Dyke et a1 [3] stated that the experimental determination of transfer functions falls into two categories, the swept-sine method and the fast Fourier transform method. While both produce accurate representations of a structure, the latter is faster and more convenient to use as it estimates the transfer function over a concurrent band of frequencies. The transfer function is also the basis of modern systems identification. Fukuwa et a1 [4] used experimentally-derived transfer functions in estimating the natural frequency and damping properties for a steel framed building. Trowbridge et a1 [ 5 ] used the transfer function capability of the software code NASTRAN to predict the transient structural deformation and force within rod and plate structures. Kareem & Kline [6] recently used the systems approach in the study of structural control, developing transfer functions which incorporated the characteristics of multiple and tuned mass dampers to estimate structural response due to wind and seismic excitations. Dyke et a1 [3] also developed transfer functions by state space realization for use in seismic testing. A wind turbine tower assembly consists of several rotor blades (usually three) connected to a nacelle casing (which houses the mechanical and electrical equipment) which in turn sits on a supporting tower. This paper investigates the effects of rotor imbalance loading coupled with a large nacelle mass acting at the top of the tower. Changes in the tower’s displacement and member force transfer functions, due to horizontal loading from the rotors and gravity loading from the nacelle mass, are determined. A description of the effects of rotor imbalance loading is given in section 3.

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2. THE MODEL

Member 16

Member 22

Figure 1 3D Lattice tower as viewed in ANSYS The 3D wind turbine tower consists of a lattice tower with a concentrated mass at the top, representing the nacelle and rotors, as in figure 1. The main vertical members are steel angle sections of dimension 20Ox2OOL24. The horizontal and oblique members are also steel angle sections, of dimensions 10Ox1OOL12. The mass ratio (mr) is defined as the ratio of combined nacelle and rotor mass to the tower mass, expressed as a percentage. A finite element model is created using the software code ANSYS [7].In order to create a pin jointed structure, a truss element LINK8 was employed. LINK8 is a 3D truss element that does not have bending or shear stiffness and has only three degrees-of-freedom at each node. Structural members have the material properties of steel and real constants are sourced from the Steel Design Guide to BS 5950 [8]. In order to simulate the concentrated mass at the top of the tower, the density of the four uppermost horizontal members are raised relative to the other members. ANSYS has a harmonic analysis capability in which the user can harmonically vary a load on a model over a user specified frequency range and obtain the steady state displacement response. ANSYS harmonic analysis has three solution methods, full, reduced and mode superposition. The full method is used to obtain the member force transfer functions and is the most computationally complex. The solution algorithm for this method may be viewed in the ANSYS Theory Reference [lo]. The mode superposition method is used to obtain the displacement transfer functions and is the least computationally complex. A modal analysis must first be carried out to obtain the fkee vibration characteristics, and then the program uses the solution algorithm explained in the ANSYS Theory Reference [ l l ] .

3. THEORETICAL CONSIDERATIONS 3.1 Rotor Mass Imbalance inducing Asymmetric Centrifugal Force The centrifugal force experienced by a rotating wind turbine blade is designated as the thrust force and is ultimately transferred into the hub and drive train. Harrison et a1 [9] class the thrust force as being symmetric about the rotor axis Le. for ‘n’ number of blades rotating, there is a conservation of force magnitude within the system, which in theory is of zero

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magnitude. However, this is true only if the centripetal force characteristics of each blade are the same. When a mass is rotating in a circle, it has an angular velocity component and an orthogonal tangential velocity component. Although the magnitude of the velocity may be constant, the direction is constantly changing and this gives rise to the well known centrifugal acceleration phenomenon. Centrifugal acceleration, A, acts always in the direction towards the centre of rotation and is represented by A,

V2

= -= r o 2

r

(1)

where v is the tangential velocity, r is the distance from the mass to centre of rotation, and o is the angular velocity. By virtue of Newton's Second Law, the centrifugal force experienced by the rotating mass is given by Fc = (WAC

(2)

where M is the mass of the rotating body. The centrifugal force that a rotating prismatic beam of length r experiences may be obtained using equation

with being the mass per unit length of the beam and v(r) is the tangential velocity as a function of r. Mass imbalance effects arise when blades in a series of one or more possess different mass variations or geometries. When this occurs, a resultant force is imparted to the top of the tower. This force has both horizontal and vertical component, though only the former is considered in this paper. In order to obtain the resulting displacement or member force FRF, the resultant of equation (3) is set to unity, and applied to the model as a horizontal force. 4. RESULTS 4.1 Displacement FRFs

This section presents the displacement FRFs obtained by executing an ANSYS mode superposition harmonic analysis at the top of a tower of height 60m, base width 6m and top width 3m. Figures 2, 3, 4, 5 represent the FRFs obtained by harmonically varying a unit horizontal force at the top of the tower with mass ratios (mr) of 0%, lo%, 20% and 30% respectively. The tower was prescribed structural damping of 1% of critical. In figures 2 - 5 , the notation IH(f)I denotes the modulus (amplitude) of the FRF as a function of frequency.

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Figure 2 Displacement FRF for 0% mass ratio I\I\LFys

Figure 3 Displacement FRF for 10% mass ratio I\I\Lsys

"X

0

2

I

D

.

6 6

0

7

1

e

0

1

L

L

Z

'

I

S

.

I

I

Forclog Prqurncy (ICs)

Figure 4 Displacement FRF for 20% mass ratio

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159

"X

Figure 5 Displacement FRF for 30% mass ratio

4.2 Member Force FRFs This section presents the member force transfer functions obtained by executing an ANSYS full harmonic analysis at the top of a tower of height 60m, base width 6m and top width 3m. Two members were chosen, and may be viewed as members 16 and 22 as in figure 1. Figure 6 represents the transfer function obtained for member 22, and figure 7 shows the transfer function for member 16. The tower has the same geometry as that investigated in section 5.1, with a mass ratio of 30%, and was prescribed a structural damping ratio of 1% of critical. In figures 6 and 7, the notation lH(f)l denotes the modulus (amplitude) of the FRF as a function of frequency.

mass ratio

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Figure 7 Member force FRF for member 16 for tower with 30% mass ratio 5. DISCUSSION Comparing figures 2, 3, 4 and 5, it is evident that the largest displacement occurs at distinct loading frequencies associated with the natural frequencies of the tower. The mass ratio greatly affects the form of the displacement transfer function. Increasing the mass ratio shifts the response peaks to lower frequencies, while at higher mass ratios, the effects of the higher modes reduce, as the magnitude of their response peaks decrease significantly. The member force transfer functions also show maximum forces around the natural frequencies of the tower. Comparing figures 6 and 7, the transfer functions for members 22 and 16 differ significantly. At the fundamental frequency, the magnitude of the forces in member 22 are approximately double that of member 16. At higher frequencies, the magnitude of the forces in both members are similar.

6. CONCLUSIONS The harmonic analysis capability of the finite element software code ANSYS has been employed to derive both displacement and member force transfer functions for a wind turbine lattice tower. The chief merit of this approach lies in the program’s ability to obtain the stiffness characteristics of the model, which if done by hand could be difficult, especially for a geometrically intricate model. Frequency response functions for tower nodal displacement were obtained using a mode superposition method which was very computationally efficient. Member force transfer functions may only be obtained using the full harmonic method. Results for two lattice members are included in this paper, though any of the eighty members contained in the model could have been presented as easily. The transfer functions derived by ANSYS could subsequently be used to estimate the response of the tower to rotor imbalance forces, using the systems approach for frequency domain analysis.

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REFERENCES 1. 2. 3.

4. 5. 6. 7. 8. 9. 10.

11.

Jeary A. ‘Designers guide to the dynamic response of structures’, E & FN Spon, Great Britain, 1997 Clough R., Penzien J., ‘Dynamics of Structures’, McGraw-Hill, Inc., New York. 1993. Dyke, S. J., Spencer B. F., Quast P., Sain M. K., Kaspari M. K., Soong T. T., ‘Acceleration feedback control of MDOF structures’, J. Engrg. Mech. ASCE (1996) 122(9) 907-917 Fukuwa, N., Nishizaka R., Yagi S., Tanaka K., Tamura Y., ‘Field measurement of damping and natural frequency of an actual steel-framed building over a wide range of amplitudes’, J. Wind Eng. and Indust. Aerod, 59(2-3) 1996 325-347 Trowbridge, D.A., Grady J. E., Aiello R. A., ‘Low velocity impact analysis with NASTRAN’, Computers and Structures, 40(4) 1991 977-984 Kareem, A., Kline S., ‘Performance of multiple mass dampers under random loading’, J. Struct. Engrg. ASCE (1995) 121(2) 348-361 ANSYS Corporation, ‘ANSYS online manuals’, Release 5.6.1, ANSYS Corporation, 1999. The Steel Construction Institute ‘Steel Design Guide to BS 5950: Part l’, Volume 1 Section properties and member capacities 5‘hEdition, Great Britain, 1990. Harrison, R., Hau E., Snel H., Large Wind Turbines, Design and Economics, John Wiley & sons, England (2000). ANSYS Corporation, ‘ANSYS Theory Reference’, Release 5.6, ANSYS Corporation, section 17.4.2, pgs 1101-1103, (1999). ANSYS Corporation, ‘ANSYS Theory Reference’, Release 5.6, ANSYS Corporation, section 17.4.5, pgs 1106-1 107, (1999).

List of Notation A, Centrifugal acceleration Fc,bearn Centrifugal force on beam FRF Frequency response function Modulus of frequency response function IH(f)l Mass of rotating body M mass per unit length of beam mb mr mass ratio co-ordinate along the length of beam r tangential velocity as function of r v(r) 0 angular frequency

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Periodic motions in a periodically forced, piecewise linear system ACJLUO

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, USA

ABSTRACT In this paper, periodic motions for a three-piecewise linear system under a periodic excitation are predicted analytically from the mapping structures for specified periodic motions. The symmetry for the stable asymmetrical periodic motions of such a system is observed. The presented methodology is applicable to other non-smooth systems. Keywords: non-smooth dynamical systems, periodic motion, mapping structure. NOMENCLATURE excitation amplitude spring stiffness d damping coefficient Der(.)determinant of matrix Jacobian matrix of Poincare mapping Jacobian matrices of basic mappings discontinuous force discontinuous displacement Imaginary component of complex

a c

spring force function mapping basic mappings Real component of complex time

switching time Trace of matrix displacement solution vector = (t,y ) switching solution vector switching displacement velocity = f switching velocity eigenvalue of matrix switching sets natural frequency excitation frequency critical excitation frequency

1 INTRODUCTION In 1983, Shaw and Holmes [I] investigated a piecewise linear system with a single

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discontinuity through the Poincare mapping and numerically predicted chaotic motion. In 1991 Nordmark [2] used the mapping technique to investigate non-periodic motion caused by the grazing bifurcation. In 1992 Kleczka et al [3] investigated the periodic motion and bifurcations of piecewise linear oscillator motion, and observed the grazing motion. Foale [4] used the Nordmark’s idea about the grazing bifurcation to analytically determine the bifurcation in the impact oscillator in 1994. To determine complex periodic motions, in 1995 Luo [5] initialized the concept of mapping dynamics for discontinuous systems and applied to impacting oscillators and a ball bouncing on a vibrating table (also see, Han et a1 [6],Luo and Han [7]). In 2002, Luo [8] discovered the two asymmetric period-I motions by introduction of a time interval between two impacts, and one of the two asymmetric motions for such an impact system were observed through a numerical investigation in [9]. In 2004, Luo and Menon [IO] used the concepts of mapping dynamics to investigate chaotic motions in such an oscillator. In addition, without the mapping techniques, the following contributions on the piecewise linear system should be mentioned. The early study of a piecewise linear system without damping was completed by Hartog and Mikina [ 113 in 1932 and a closed-form solution for symmetric and periodic motion was obtained. Timoshenko [ 121 discussed undamped piecewise linear systems in 1937. In 1989 Natsiavas [ 131 identified the responses of a system with tri-linear springs with a time-incremental method, and by use of a similar approach, the dynamics of oscillators strongly nonlinear asymmetric damping was investigated [ 141. In 2000, Theodossiades and Natsiavas [ 151 discussed the modeling of gearpair vibration as a piecewise linear problem, and the periodic solutions and stability for such a system were discussed. In this paper, periodic motions in a three-piecewise linear system under a periodic excitation will be investigated. The analytical prediction of all stable and unstable periodic motions will be given by specified mapping structures. The local stability and bifurcation will be obtained through eignenvalue analysis. Numerical simulations of periodic motions are presented. 2 SWITCHING SETS AND GENERIC MAPPINGS

Consider a periodically excited, piecewise linear system as f + 2&+ k(x) = ucos Rr, where x = dx/dt. The parameters respectively. The restoring force is

(n and cx-e, cx+e,

a ) are excitation frequency and amplitude, for for for

x 2 E; -E<x<E; x I -E;

wirh E = e/.. In the foregoing system, there are three linear regions of the restoring force (Region I: x 2 E , Region 11: -E 5 x < E and Region 111: x 5 - E ) , The solution for each region can be easily obtained [Ill. For description of motion in Eq.(l), two switching sections (or sets) are defined as

The two sets are decomposed into C+ = C: UZ’

164

u {t,,E,O} and C-

= C;

uC: u {t,,-E,O},

(4)

Y004/015/2004

where four subsets are defined as

1

z:=((t,,x,~y,)l~,=E,X,=y,>O and C ~ = ( ( r , , x , , y , ) l x , = E , X , = y , < O ) ; C; =((r,,x,.y,)lx,=-E,x, = y , >O) and

XI =[(t,,x,,yi)lx,=-E,% = Y , < O

I.

Fig.1 Switching sections and generic mappings in phase plane The points {ti,E,O} and {ti,-E,O} strongly dependent on the external force direction are singular. From four subsets, six basic mappings are:

4 : c:

-b

z:,

p2 :C'

-b

XI, 4 : CI + z;, P6 :E; +cs.

(7)

p, :c; +XI, 4 :X: +E;,

In Fig.1, the switching planes and basic mappings are sketched. The mapping 4 :(ti,E,yi)+ (ti+,,E, y f + ] ) indicates that the initial and final states are ( t , ~ , . k ) , ~=, ,(t,,E, , ~ , y,) and ( t , ~ , f )=~(t,+l,E,yi+l) ~~, in Region I, respectively. For y, > 0

and y,+, < 0, two governing equations for mapping

4 are obtained from the reference [lo]:

[C, (tf)cosw(tl+l- r, ) + C, (t,,y,)sin ~ ( t ,-t,+ )]e-d(f~+l-tJ ~ +a(D, c o s ~ t , , ,+ sin at,,,) = 0, Y,,, + { [c,(I,) d - c, (ti,y, ) w ] cos w(t,+,- t , )+ [C, (t,) + C, ( t , ,y, ) d ] sin w(t,+,- t, )}

e-d('l*l+J

+an(D, sin Rt,,, - D2cos RZt,,,) = 0.

I

(8)

Similarly, the mapping pZ : ( t , , E , y , )+ (t,+,,-E,yi+,)gives the initial and final states (t,, E, y , ) and (t,+l,-E,y,+l) in Region 11, respectively. The corresponding governing equations for the mapping p2 at x, = E for yi < 0 and y,,, < 0 are obtained in [IO]:

c (t 3

l>Yl )e-2d(h-h)

y i+l +2dC3 (,19

Y ,

]

+C,(t,,y,)+a(D3cosRt,+,+D4sinRr,+,)+2E=0,

)e-2d(f~+'-r')+uR(D3sinRt,+, -D4cosRt,+l)=0.

(9)

The motion of system enters Region 111 after time t i , and returns back to the boundary of

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Regions I1 and I11 until time f,+, . Such a motion is measured through the mapping4 , Note that the governing equations for y, c 0 and y,+l> 0 is the same as in Eq.(8). The mapping p4 brings the motion from the boundary of Regions 111 and I1 at time t, to the boundary of Regions I1 and I at time t,,, . The governing equations for such a mapping in [IO] are: C 3 ( ~ , , y i ) e - 2 d ( ' * , ' - ' ' ) + C 4 ( t , , y , ) + a ( +D4sinRt,+,)-2E ~3co~Rt,+, =0, if1

+2dC3 ( 1,,Yi ) e - 2 d ( f , + l - f t )

+uR(D3sinQt,,, -D4cosRt,,,)=O.

]

(10)

The mapping 4 brings the motion from the boundary of Regions I and I1 to the boundary of Region I and I1 at time f,,, . Similarly, the governing equations for such a mapping are:

c3 ( ti*Y,

]

(t,,y,)+a(~~co~ +04 ~ t s, i+n~R t , + , ) = ~ ,

)e-2d('l+l-'J+~4

y,+l+ 2 d ~ 3 ( t , , y , ) e ~ 2 d ( ' " ' - ' ' ) + u R ( ~ 3 -sD i n4Rc toi~+R , t,+,)=0.

(11)

The mapping P, maps the motion from the boundary of Regions I1 and I11 to the boundary of Regions I1 and I11 at time I,+,, The governing equations for such a mapping are the same as in Eq.(ll) with y, > O and y,+, c 0 . From the above mapping definitions, mappings <,4,4 and P, are termed the local mapping, and mappings p2 and p4 are termed the global mapping.

3 MAPPING STRUCTURES For simplicity, the following notation for mapping is introduced as

p,,n,..'", = P", 4, O

where P,~

O.'.

O

p, .

{ q l j 1,2 = ,...,6} and ni ={1,2,...,6}. Note that the rotation of the mapping of

e,.,

periodic motion in order gives the same motion (Le., .,,", ,P,*.,,",", Pn,,,,,,,_, ), and only the selected Poincare mapping section is different. The motion of the m-time repeating of mapping P, ", is defined as ,..e,

To extend this concept to the local mapping, define m-acts

m-sN

For the special combination of global and local mapping, introduce a mapping structure

m-rets

From the definition, the motion for Eq.(l) can be very easily labeled through mapping

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structure accordingly. Consider a mapping structure

<(36),232(15),,1.

When k, = k,, the mapping

structures gives the motion possessing a symmetrical mapping. For k, # k,, the motion having an asymmetrical mapping structure is obtained. The physical model is given in Fig.2.

Fig3 A global periodic motion with k2-cycle (left) and k,-cycle (right) local motions The more generalized mapping structure for the complex, periodic motion is expressed by <(36)h"32(1)),," 1...4(36~132(1)),,11

=p4 O 4 2 "

o<"p2°4:'. O

4

0-oP,~421 "PZ O<

O 4 ?

"4

40 4 = P4o P 3 ~ P 6 ~ ~ 0 .4 ~ 04 o < oP,aP, oP,o...o4 0 I

k,.

I

- se1 (laeal)

I

4..

set floul)

I

(16)

From mapping structures of periodic motions, the switching sets for a specific regular motion can be determined through solving a set of nonlinear equations. For instance, for a periodic motion with a mapping structure ~(36)""32t1~)h"l.,,4(36)~'3z(1)),,', (i.e.9 < ( 3 6 ~ 3 2 ( l ) ) , , " l , , , 4 ( 3 6 ~ h 1 3 2 ( I J )= , , l x), IX

where

f(n') =

a set of vector equations is as

(A(fl'),fp)ris

relative

to

governing

equations

of

mapping

Pn,( n, ~ { 1 , 2..., , 6 ) ) and x, = ( t , , i , ) r =(t,,y,)'. For the period-1 motion perN-periods, the periodicity condition requires

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Solving Eqs.( 17) and (18) generates the switching sets for periodic motions, 4 STABILITY AND BIFURCATION

The periodic motion can be obtained through a specific Poincare mapping given by one of the combined mappings developed in the previous section. In other words, the period-] motion is determined by the fixed point of the Poincar6 mapping for a given period T = 27r//R (Le., %(36)""32(IJ)h"l 4(36)"2~32(15)"lx~ =

2 ~ ~ " * h "where ~ x, =(r,,y,)'). Once the I.,

initial condition (r,',E,y:) for the periodic motion is obtained, the switching times and velocities for all the switching planes are determined accordingly. The stability and bifurcation for period-1 motion can be determined through the corresponding Jacobian matrix of the Poincark mapping. From Eq.(16), the Jacobian matrix is computed by the chain rule, i.e.,

where II represents the series multiplication. From the six mappings, two unknowns (f,+l,y,+l) can be expressed through the initial conditions, i.e., I,,, = rt+l(rl,yl) and y,tl = yi+,(f,,y1). The linearization of mapping P, and 8 in the neighborhood of the solution of periodic motion (t:,y,') generates the Jacobian matrices:

The linearization of mappings Jacobian matrices for

I68

4 4,

P5 and& in the neighborhood of (t,',y,') generates the

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2dy, + a cos Qt, (e2d(f1*'-fo - 1)

1 -e2d($+l-4)

1

e 2 d ( f , ~ ~ - f ~ ) ~ + , a- c~ous cn toi s+ ~, ~ ,acosnt,,, +, - Z',+,e2d(f,+t-fJ (I; .Y; , d l .Y;*,) (21)

where N , =ucos(Q~,)-&, and Z', =acos(Rti)-2dy,.

(22)

The eigenvalues of a fixed point for the periodic motion mapping is expressed through the trace ( T r ( D P ) )and determinant ( D e t ( D P ) )of the Jacobian matrix DP,Le.,

If [Tr(DP)I2< 4 D e t ( D P ) , equation (23) is expressed by A,2 = Re@) f j Im(A), with j = f i

.

(24)

where Re (A)= Tr (DP)/2, and Irn (A) = J4Det (DP)- [Tr (DP)]2 /2. If two eigenvalues lie inside the unit circle, then the period-I motion pertaining to the fixed point of the Poincare mapping is stable, while if one of them lie outside the unit circle, the periodic motion is unstable. Namely, the stable periodic motion require the eigenvalues be Ia,l
(25)

When this condition is not satisfied, the periodic motion is unstable. For complex eigenvalues /A,21l, the periodic motion becomes an unstable focus. If IA,a,21 = 1 with complex numbers, namely,

Det(DP) = I ,

(26)

the Neimark bifurcation occurs. For two eigenvalues being real, the stable-node periodic motion requires maxlA,,(< I , i =1,2 and the unstable node (or saddle) one requires

[A, I > 1,

i = 1 or 2 . The saddle-node of the first kind is as Ai > 1, i = 1 or 2, and the saddle-node

of the second kind needs A, <-1 , i = 1 or 2 . If one of the two eigenvalues is -1 (i.e., A,(or2) = -1) and the other one is inside the unit cycle, the period doubling bifurcation occurs. Further, the period-doubling bifurcation condition is Tr (DP)+ Det (DP)+ 1 = 0.

(27)

If one of the two eigenvalues is +1 (Le., &,r2) = + I ) and the second one is inside the unit cycle, the first saddle-node bifurcation occurs. Similarly, the corresponding bifurcation

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169

condition is Det (DP)+ 1 = Tr (DP). For the grazing of periodic motion, one of the eigenvalues becomes infinity since the Jacobian matrix is singular.

0.0

a)

1.6

1.7

1.8

1.9

Excitation Frequency Cl

0.0

2.0

b)

Excitation Frequency fl

1.5 C)

1.5

1.0

0.5

2.0

1.5

Excitation Frequency

1.5

d)

1.0

0.5

1.6

1.7

1.8

I

0

n

1.9

2.0

Excitation Frquency fl

Fig.3 Switching phases and velocities varying with excitation frequency for ~ ~ 1 6 ) 1 2 ~ i 5 ~ , periodic motion:(a) switching phases and (b) switching velocities; (c) zoomed switching phases and (d) zoomed switching velocities. The circular symbols denote bifurcation points.(d=0.5, c=IOO, E=1.0, a = 2 0 , N = l ) . 5 ILLUSTRATIONS

Consider the switching phases and velocities of the motion of p4(36)32(,J)I for the entire range of excitation frequency as illutration as shown in Fig.3 with parameter ( d = 0 . 5 , c=lOO, E = 1 .O, a = 20, N = 1). The circular symbols denote bihrcation points. The thin and thick solid curves represent the stable symmetrical and asymmetric motions, respectively. The dashed and dash-dot curves denote the unstable symmetrical and asymmetrical motions. The corresponding eigenvalue analysis for the motions is presented in Fig.4 for local stability and

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bihrcation conditions. The solution structure for this motion is similar to the motion of pd321. For the symmetrical motion of p4(36)32(15)1,the critical values for disappearance gazing are Rc+)

= 1.889 and Rc,+)= 0.176 The saddle-node bifurcation points of the first kind are I

Rc,2(,)= 1.88 and R,,,(.) = 1.706. For the asymmetrical motion of p4(36)32(15)1,the critical values for the disappearance grazing are Rc,I(o)= 1.65 and Qm+) = 0.788, and the saddlenode bifurcation points of the first and second kinds are Rc,2(o)= 1.614 and RC+)= 1.592, respectively.

2

-s

4-

-----

k-

-

aI

1

5

.-wM

ei o 3

OS nul’

0.0

t t

0

.3

1.0

a

a -0.5 -1

-2

0.0 a)

0.2.5 1.6

1.7 1.8 1.9 2.0

Excitation Frequency R

0.0

b)

d

I a

k e

C)

Excitation Frequency R

Excitation Frequency R

1.0 -

O5

-

0.0

-

5 a

-0.5-

,$ A,

-1.0 -

t

d)

O.ii.5 1.6 1.7 1.8 1.9 2.0

m0-0

-1.5 -1.5 -1.0 -0.5 0.0

0.

0.5 1.0

1.5

Real Part of Eigenvalues (A,,,)

Fig.4 Eigenvalues for periodic motion relative to mapping P4(36+2(,5), :(a) real parts and (b) imaginary parts; (c) absolute values and (d) complex plane. The hollow and solid circular symbols denote symmetrical and asymmetrical periodic motions. ( d = 0 . 5 , ~ = 1 0 0 ,E=1.0, a = 2 0 , N = l ) . 6 NUMERICAL SIMULATIONS

The switching velocity and phases at switching planes are analytically predicted. The motions can be simulated numerically through Eq.( 1) with appropriate initial conditions. The analytically determined initial conditions are selected from the switching plane ( t , , y , )E Z:

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171

given analytically. For comparison, a set of parameters ( d = 0.5, c = 100, E = 1 .O, a = 20) is used again. The computation precision at switching planes islO-lo. In Fig.5, the motions relative to P,,,, are illustrated. The stable right-asymmetric and left-asymmetric motions are in FigS(a)-(b), respectively. The phase trajectory of the right-asymmetric motion turning 180 degrees clockwise is identical to the phase trajectory of the left-asymmetric periodic motion. In FigS(c)-(d), the phase trajectories for P,32(15)1 and <(36)321 are illustrated. The stable rightasymmetric and left-asymmetric motions for p4(36)32(15)1and

<(36)232(,5)21

are given in Fig.6(a)-

(b) and (c)-(d), respectively. The skew-symmetry for asymmetrical motions is observed. 7 CONCLUSIONS

The analytical prediction of stable and unstable periodic motions in a three-piecewise linear system under a periodic excitation is given from specified mapping structures. The symmetry for the stable asymmetrical periodic motions of such a system is observed. The methodology presented in this paper is applicable to other non-smooth systems such as friction-induced vibration, impact oscillator and power control systems. REFERENCES 1. S.W. Shaw and P.J. Holmes, 1983, A Periodically Forced Piecewise Linear Oscillator. Journal ofSound and Vibration 90( l), 12 1 155. 2. A.B.Nordmark, 1991, Non-periodic motion caused by grazing incidence in an impact oscillator. Journal ojSound and Vibration 145,279-297. 3. M. Kleczka, E. Kreuzer and W. Schiehlen, 1992, Local and global stability of a piecewise linear oscillator. Philosophical Transactions of the Royal Society of London: Physical Sciences and Engineering, Nonlinear Dynamics of Engineering Systems 338,533-546. 4. S. Foale, 1994,. Analytical determination of bifurcations in an impact oscillators. Philosophical Transactions ofthe Royal Sociey ofLondon 347,353-364 5. A.C.J. Luo, 1995, Analytical modeling of bificrcations, chaos and multifactals in nonlinear dynamics,Ph.D. Dissertation, University of Manitoba, Manitoba, Canada. 6. R.P.S. Han, A.C.J. Luo and W. Deng, 1995, Chaotic motion of a horizontal impact pair. Journal of Sound and Vibration 181,23 1-250. 7. A.C.J. Luo and R.P.S. Han ,1996, Dynamics of a bouncing ball with a periodic vibrating table revisited. Nonlinear Dynamics 10,1-18. 8 . A.C.J. Luo, 2002, An unsymmetrical motion in a horizontal impact oscillator. ASME Journal of Vibrations and Acoustics 124,420-426. 9. G.X. Li, R.H. Rand and F.C. Moon, 1990, Bifurcation and Chaos in a forced zero-stiffness impact oscillator. International Journal of Nonlinear Mechanics 25(4), 414-432. 10. A.C.J. Luo and S.Menon, 2004, Global chaos in a periodically forced, piecewise linear system with a dead-zone restoring force, Chaos, Solitons and Fractals 19, 1 189-1 199 11. J.P.D. Hartog and S.J.Mikina, 1932, Forced vibrations with non-linear spring constants. ASME Journal ofApplied Mechanics 58, 157-164. 12. S. Timoshenko, 1937, Vibration Problems in Engineering, New York: D. Von Nostrand

-

co. 13. S.Natsiavas, 1989, Periodic response and stability of oscillators with symmetric trilinear restoring force. Journal of Sound and Vibration 134(2), 3 15-331.

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14. S.Natsiavas and G.Verros, 1999, Dynamics of oscillators with strongly nonlinear asymmetric damping. Nonlinear Dynamics 20,221 -246. 15. S. Theodossiades and S. Natsiavas, 2000, Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. Journal of Sound and Vibration 229(2), 287-3 10.

-3 a)

~

50 -

3

2

-10

-3

I

I I I

I I

I I

I

O

I

I

-1

0

0

1

2

3

1

>

I

I

-2

Displacement x

b)

I

-

-10

3

4r3

I

-5

-

%l (-Lj 1

0

-1

Displacement x

10

3a

-2

1

1

-5

1

1 1 1

I

1

1

-1 0

: (a) asymmetric at x, = E ( 0 = 3.2, a t , w 5.6830, Fig.5 Periodic motion for y, =5.5475),and (b)asyrnrnetricat x, =-E ( n = 3 . 2 , Q t , ~ 6 . 2 1 9 5 ,y, 115.7207); (c) periodic motion for P,,,(,,),( a= 2.8, Rt, c 5.3225, y, c 4.6288); (d) phase motionforp,(,,,,,, ( 0 = 2 . 8 , Ot,*6.2502, y , ~ 7 . 2 8 1 0 ) .( d = 0 . 5 , ~ = 1 0 0 ,E=l.O, a =200)

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173

10

I

I I I

i

A

e

I

I

I

I I I

I I I

5

5 -

2 >"

I

I I

,g

j 0

0-

-5

-

-5 I I

I I

-10 -3

-2

-1

0

1

2

3

Displacement x

b)

10,

I I

I

I

I I

I I

I

5A

2.

'8 -

0

5

-5 I

I

-10

-

-

-3

d)

Fig.6 Periodic motions of

<(36)32(15)1

-2

I I

I I

I

I

-1

0

1

2

3

Displacement x

: (a) asymmetric at x, = E ( R = 1.6, Rt, z 5.5191,

y, z 4.8633), (b) asymmetric at x, =-E (R = 1.6, Rt,

= 5.7805, y, = 5.4937);

: (c) asymmetric at x, = E ( R = 1 . I 15, Rt, i5: 5.4467, periodic motion ofF'(36)232(15)21 y, ~4.6830)and(d)asymmetricatx , = - E (R=1.115, Rt, ~ 5 . 6 0 1 7 y, , ~5.2969). ( d = 0.5, c = 100, E = 1.0, a = 200)

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YOO4/015/2004

Dynamic models for components considered in the design of a solar concentrator H R HAMIDZADEH and L C MOXEY Department of Mechanical and Manufacturing Engineering, Tennessee State University, Nashville, USA

ABSTRACT In this paper an overview of the mathematical models for different structural components proposed for a solar concentrator assembly are provided. The analysis includes dynamic responses of these structures. In particular, attention has been confined to study modal analysis of rigidized struts, thin film lens, and the inflated supporting torus components. The provided solutions for these components are based on theory of elasto-dynamics, The proposed analytical solution can provide numerical results for natural frequencies, modal damping, and mode shapes for each of the three structural components. The validity of the developed models was verified by comparing some of the computed results with those established for special cases. The developed analytical solutions can be expanded to consider the assembly of these parts in the overall design of the supporting structure for a solar concentrator by dynamic impedance matching among the components. NOTATIOIN

Radius of the cylinder or Radius of the generating circle of torus Angle between consecutive stations Modulus of elasticity Bessel function of the first kind Thickness of membrane Half of the axial wave length L N,, Ne, Nas Forces per unit length of the membrane N Circumferential wave number p Internal pressure R Radius of torus r , e , z Cylindrical coordinates u, v, w Meridional, circumferential, and normal displacements Velocity of propagation of dilatational waves v, a

d E J h

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175

Velocity of propagation of distortional waves Meridional position angle Ratio of two radii of the torus p Eao, Eo, Membrane strains Sa,So Membrane pre-stress forces p Mass density (pa2 / E p2) w2,dimensionless frequency R w Circular frequency 0 Circumferential position angle rotation of normal to the membrane $0, $00 w Circular Frequency w, Non-dimensionless factor v Poisson’s Ratio V2 Laplacian Operator v2 a

a la -+--+--+-

l a 2 a2

ar2 r a r r 2 ae2 az2 2,p Lame’s elastic constants y,t+,y Displacements along r, 8, and z directions 1.

INTRODUCTION

In view of many potential applications of lightweight structures, employing thin inflatable or rigidized membranes for use in the aerospace industry have received renewed emphasis in recent years. These structures have shown great promise for use in space due to their inherent lightweight, low packaging and launch volume, and relative simplicity of deployment. Among the vast variety of applications for these structural elements, one can mention solar sails, space solar power generation systems, solar thermal propulsion vehicles, large space telescopes, and inflatable communication antennas. In the design and development of these structural elements, the overarching criteria are low cost, lightweight and high reliability. In this paper, analytical procedures are proposed to determine dynamic characteristics of thinfilm inflated structural components considered in design of a solar concentrator. This power system typically consists of a large flexible fresnel lens that focuses solar radiation into a receiver where high intensity heat is collected. This heat is then used to generate mechanical power. The lens is surrounded by a supporting inflated torus that is connected by three rigidized struts to complete the solar concentrator assembly. Figure 1 illustrates a schematic of a satellite with Solar Thermal Upper Stage, which utilizes solar concentrator assemblies. Due to the complexity of this structure it is convenient to consider it as an assembly of substructures. The theoretical analysis of this large system can be conducted with higher efficiency if it is broken into its parts and analyzed separately and the whole assembly reconstituted in terms of the models of each individual components. The advantage of applying this technique is that each substructure can be modeled using different techniques: analytical method, numerical method, or from experimental modal analysis.

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Figure 1. A Solar Collector This paper presents analytical procedures to determine vibration and physical characteristics of thin film lenses considered in the design of a solar concentrator. The work presented here also includes the study of free vibration of inflated circular toroidal membranes with circular cross sections. The first step toward the analysis is to develop a more accurate solution to the governing equations of motion. This paper gives an outline of the general method starting from the equations of motion presented by Sander (1963) for uniform thin toroidal membranes. Based on the governing equations presented by Liepins (1965) a modified governing equation in the form of state equations is developed and solved by a more accurate numerical method. The accuracy of the solution is assessed by comparing some of the results with those of Liepins ( I 965) given for different toroidal membranes. The analysis provides natural frequencies and mode shapes for a toroidal membrane and specifically is employed to determine modal information for the torus used in the Shooting Star Experiments. Analysis is conducted to determine natural frequencies and mode shapes for a variety of modes for a simply supported cylinder. Further more this article addresses the study of free vibration of solid cylindrical foam struts, to introduce viscoelastic damping in the composite cylinder, the analysis allows complex shear modulus. As presented by Hamidzadeh and Chandler (1991), the problem is formulated in the form of a matrix of equations that relates displacements of a point in the medium to the boundary stresses. Invoking the zero boundary stresses leads to a frequency equation for vibration of the solid viscoelastic cylinder. Hamidzadeh and Sowaya (1993) presented an analytical procedure applicable to multilayer cylinders of arbitrary laminated configuration and any material damping for each layer. 2 MODAL ANALYSIS FOR THE RIGIDIZED STRUTS

In this section, the problem of propagation of harmonic waves along a solid circular cylinder within the framework of the three-dimensional theory of elasto-dynamics is considered. The Mathematical model for determining frequencies and mode shapes for a wide range of axial and circumferential wavelengths for the simply supported solid cylinders is developed based on Armenakas et. al(l969).

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177

Consider the isotropic elastic cylinder shown in Figure 2. The governing equations for threedimensional wave propagation in an isotropic elastic medium in invariant form can be written as: pvii + (a+ p)vv.ii= p6 (2.1) where ii is the displacement vector p is the density 1and p are Lame's constants V2 is the three-dimensional Laplace operator. The eigenvalue problem consists of finding solutions ii , which satisfy the governing equation and the boundary conditions: aii CJ, =a, =a, at r =a, and -=0 at r = O (2.2) dr

Figure 2. Cylindrical strut with its coordinates Using Lame's potentials 4and Hsolution to the problem may be given by ii = V # + V x H where the potentials 4 and fi should satisfy the following wave equations

(2.3)

v2,v24= 8

(2.4a) (2.4b) where v, and v2 are the velocity of propagation in an medium of dilatation and distortional wave and are given by v; =- (1+ (2.5a)

v22V2H =H

w

P

.;=E

(2.5b)

P

Solutions to the Lame's potentials are in the form 4 = f(r)cosnOcos{zexp(iwt) (2.6a) H,= g,(r)sinnOsin{zexp(iwt) (2.6b) He= g,(r)cosnOsin{zexp(iwf) (2.6~) H,= g,(r)sinnOcos{zexp(iwt) (2.6d) Considering that the cylinder is solid one solution to the resulting Bessel equation fieldsf; gr and g, may be written as f ( r ) = AIJ" (ar) (2.7a) (2.7b) g, ( r )= -go (11 = 4 J , + i (Pr) (2.7~) g,( r ) = 4J" (P-1

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(2.7d) The displacements and stresses are expected in terms of functionsf g , andgousing equations (2.7), (2.6), and (2.3). (2.8a) (2.8b) (2.8~) (2.9a)

r

(2.9b)

+(p2- ~ ' ) ] g , - ~ g , } x c o s n ~ s i n ~ z e x p ( i w t )

(2.9~)

where primes denote differentiation with respect to r. Satisfying the boundary conditions stated in equations (2.2) yields the system of homogeneous system of linear equations (2.10) c,A, = 0 ( i , j = l,3) The elements of matrix cg are given by CII = [2n(n - 1) - (p' - g')u']J"(aa)+ 2aaJ,,,(aa) (2.1 la) c12

= 2gp'pa'J,(Pa)

c13

= -2n(n

-

- 2 M n + l>Jn+,(Pa) (pa>+ 2npdn+l(pa>

(2.11b) (2.1 IC) (2.1 Id) (2.1 le)

c*, = 2n(n-I)J,(aa)-ZnaaJ,+,(aa) c22

= -4Pa2J,(Pa)+ 2(n + 1)5aJ,,,

c23

= - [2n(n - 1) - P 2 a 2J" ] (Pa)- 2 W " + ,(Pa)

(Pa)

(2.1 If)

c3,= -2n5dn(aa) + 24aaZJ,+,(aa) '32

=-npdn(pu)+(P2 -62>a2Jn+l(pa)

c33

= n w " (Pa)

(2.1 lg) (2.1 1h) (2.11i)

A nontrivial solution of equation (2. IO) requires the vanishing of the determinant IcU(.

Therefore c - 0 ( i , j = l ...,3) 101is the frequency equation. For given dimensions and elastic constants the above equation is and where L is the half wavelength and an implicit transcendental function of

5

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yws

179

2.1 CASE STUDY

Assuming material density of p = 32.04 kg/m3,Poisson’s ratio of 0.3, and elastic modulus of E = 60.97kPa. The computed natural frequencies for a specific circumferential wave number n are listed below in table 1. Table 1. Natural Frequencies for the first six lateral modes of a simply supported solid cylinder Mode 1 2 3 4

(Hz)for n = 0 1.9816 1.9948 1.9785 2.0001 1.9756 2.0066 1.9732 2.0140 1.9715 2.0223 1.9703 2.0313

6 Mode 1 2 3 4 5 6

lo5 x Natural Freauencies (Hz)for n = 1 0.0027 0.3633 0.5476 1.0441 1.2576 1.3912 0.0105 0.3713 0.5528 1.0441 1.2593 1.3929 0.0228 0.3838 0.5587 1.0442 1.2621 1.3958 0.0387 0.3999 0.5643 1.0444 1.2659 1.3999 0.0575 0.4187 0.5716 1.0449 1.2706 1.4051 0.0784 0.4392 0.5808 1.0458 1.2760 1.4116

Mode 1 2 3 4 5 6

0.4593 0.4581 0.4565 0.4549 0.4536 0.4532

lo3 x Natural Freauencies (Hz)for n = 2 0.5997 0.8601 1.3134 1.5451 1.8530 0.6044 0.8622 1.3139 1.5460 1.8538 0.6118 0.8658 1.3148 1.5474 1.8552 0.6215 0.8709 1.3160 1.5495 1.8571 0.6330 0.8774 1.3177 1.5521 1.8596 0.6459 0.8853 1.3197 1.5552 1.8625

0.7076 0.7069 0.7059 0.7049 0.7039 0.7033

x Natural Freauencies (Hz)for n = 3 0.8237 1.1784 1.5698 1.8129 2.2216 0.8266 1.1802 1.5704 1.8136 2.2210 0.8313 1.1831 1.5714 1.8147 2.2201 0.8375 1.1871 1.5728 1.8163 2.2191 0.8450 1.1923 1.5746 1.8184 2.2182 0.8537 1.2002 1.5769 1.8210 2.2174

0.9238 0.9236 0.9234 0.9231 0.9229 0.9229

lo5 x Natural Frequencies (Hz)for n = 4 1.0420 1.4954 1.8180 2.0769 2.4835 1.0438 1.4967 1.8186 2.0775 2.4837 1.0469 1.4989 1.8196 2.0786 2.4840 1.0511 1.5020 1.8210 2.0800 2.4843 1.0563 1.5059 1.8228 2.0818 2.4848 1.0624 1.5106 1.8251 2.0841 2.4855

5

Mode 1

2 3 4 5

6 Mode 1

2 3 4 5

6

I80

lo3 x Natural Freauencies 0.7446 0.7858 1.3741 0.7342 0.8005 1.3749 0.7238 0.8175 1.3761 0.7148 0.8362 1.3779 0.7073 0.8560 1.3803 0.7018 0.8768 1.3832

0.0413 0.0825 0.1236 0.1643 0.2045 0.2442

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The following illustrations show a few of the infinite modal shapes of vibration in a cylindrical strut.

w

_.---

*‘’

.____.I

(b) Second Axial Mode

J (

(g) Third lobar mode, n=4 No nodal circles with four nodal diameters

,.....

...... .--.. J‘ (c) Breathing Mode

(h) Fourth Lobar Mode, N=5 No nodal circle with five nodal diameters

c

(d) Rigid Body Mode No nodal circles with one nodal diameter

;+

c:+

..-_-,

(i) Radial mode One nodal circle with no nodal diameters

(0 _---c_

(e) First _lobar . _ _ _ _mode, -n=2

(0

No nodal circles with two nodal diameters

I

I

c>

(i)Torsional mode One nodal circle with no nodal diameters

‘.-/I

‘Ld.‘

(f) Second lobar mode, n=3

No nodal circles with three nodal diameters

(k)Axial mode One nodal circle with no nodal diameters

Figure 3. Modal Shapes for the vibrating cylindrical strut.

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\

\

'.--.'

(I) Axial mode Two nodal circles with no nodal diameters Figure 3 cont'd. Modal Shapes for the vibrating cylindrical strut. 3 MODAL ANALYSIS FOR THE SUPPORTING TORUS

Consider a thin-film pressurized circular membrane torus with generating circle and torus radii of a and R respectively and membrane thickness of h. The torus is inflated to the pressure of p and is freely vibrating. Figure 4 illustrates a typical torus with its geometrical notation. In order to make this paper self-contained a brief review of basic membrane equations of motion are presented here.

.......,..,...,.......... ........... ...... ......

Figure 4. Typical geometry and coordinate system for a toroidal membrane The linearized equations of motion in the case of circular torus in terms of stresses, strains, membrane pre-stress forces, displacements and rotations of the normal to the membrane as reported by Sanders (1963) . . are as follows: -(rNa)+--N,sina+rSa anh,

a

aa

ae

par@=+ phao'ru

182

.

-

=0

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aNa a -+-(rNaa)+ Noo sin a + r Sa(saa+@.,)+So ae aa

sin a + Oacosa

par@,,+ phaw'ru = O

1

1+

-par(.ca+ce)

(3.lb)

(3.1~)

with p as mass density of the material of the torus and w as the circular frequency. Clearly u, v, and w denote meridional, circumferential, and normal components of the displacements at any point on the medium. In addition to the equations of motion, the strain-displacement and rotation-displacementrelations are: Strain-Displacements aU

(3.2a)

a&, =-+w

*+

da

race =

ae

usin a - wcosa

avau

2ra.sa,,= r -+ -- vsin a aa

ae

(3.2b) (3.2~)

Rotation-Displacements ada

=-z &+U

aw

rade =-tvcosa

ao

au a ae aa (4

-2raqjae = -- -

(3.3a) (3.3b) (3.3c)

The material constitutive relations in the linear elastic case may be derived from the classical Hooke's law. EhE, = Nu -vNe (3.4a) EhE, = No-vNa (3.4b) EhCas = (1 - V ) Nus (3.4c) The stress resultants Sa and Se are function of the geometry and the internal pressure of the torus and they can be determined using static analysis. As it was shown by Liepins (1965) they can be written as: S, = p a ( 1- % E cosa)/(l - E cosa) (3.5a)

sa=%Pa

(3.5b)

The equations (3.1 through 3.4) complete the formulation of the problem. These equations can be simplified using separation of variables by introducing the Fourier components of stresses, strains, displacements, and rotations in circumferential direction. Series expansions of the stresses, strains, displacements, and rotations are:

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As Reported by Liepins (1965) by substituting equations (3.6) into equations (3.1) through (3.4) for each value of n they result into twelve coupled ordinary differential equations in terms of a.After simplifications these equations can be reduced to the following relation AZ" + BZ'+CZ = 0 (3.7) Where A, B, C are square periodic matrices and are given by Liepins (1965). Elements of these matrices are function of angle a,geometry of the torus, material properties, internal pressure, and dimensionless frequency of R. Z = {ti,,, v,,, #A*and the prime indicates differentiation with respect to a,.Equation (7) can be expressed in the state form as: {A'} = [ K ( 4 1 { 4 (3.8a) where

[

[ K ( a 1 f i ) l = A.;

(3.8b)

A.i;]

and { A } T = {Z,Z'}

(3.8~)

The geometry and pre-stresses are symmetrical about a = 0 and a = 7c. Dividing the range of a = 0 and a = 27c into 18I equally spaced station, then the spacing between stations is d = 2d180 and the position angle for the ith station can be written as ai=( i-1) d. Solution of the equation (3.8a) at the station i+l in terms of the solution in its previous station can be presented by: {AI,+, = {AI, (3.9a) where (3.9b) [Y(fi)], = exp @(a, 4 4 since the geometry of the torus requires {A}] = {A)/81 using equation (3.9a) for every station yields

[VN,

3

PI181

=[fi[v)l,]{A)l

={A),

(3. loa)

Solution of the above eigen value problem can provide the frequency equation in the form of: (3.10b) Mode shapes for the torus can be obtained using equation (3.9a) at R obtained from equation (lO.b). Mode shapes at the i" station is Zi and it can be obtained considering the following mode shapes at the first station where i =I.

I!= Z,

I84

forfhesymefricmodes

(3.1 la)

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and

2,=

ion

Where

for the untisymetric mod es

(3.11b)

is a hnction of elements of matrices A, B, C and n.

3.1 CASE STUDY Natural frequencies and mode shapes for different cases that reported by Liepins (1965) were computed with results obtained using the proposed technique and were compared. Comparison demonstrated an excellent agreement between these results. In the forthcoming section attention is confined to the determination of free response of a torus made from inflated Kapton polyimide thin-film material manufactured by DuPont. Properties of Kapton 200-HN is provided in the following table. Table 2. Static Properties of Kapton I Kauton200-HN Thickness - m 0.0000508

1

1419.99

Poisson’s ratio E - kPa

2620.007 MPa

Free vibration of an inflated torus made of Kapton 200-HN with generating radius of 0.3048 m, the ring radius of 2.896 m, and inflated pressure of 5.37931kPa has been determined. Based on the presented solution for linear model, natural frequencies and mode shapes are determined using an iterative method. The vibration modes considered here includes families of (a) flexural modes associated with low frequencies; (b) predominantly extensional deformation of the meridional curve; (c) predominantly extensional deformation in the circumferential direction; (d) purely dilatationless circumferential vibrations. The frequencies of the last three families are high compared to those of the first family. Natural frequencies and mode shapes of a torus under consideration were computed and results are summarized in Table 3. Ta Hz.

The study concludes that modes of vibration can be grouped into four families: a family of flexural modes, modes related to the extension of meridian curve, extensional modes in the circumferential direction, and dilatationless circumferential modes. The modes of the first family consist of the coupling of two types of vibrations: the modes of a torus whose meridian curve is not allowed to distort (ring vibration), and the modes of pre-

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stressed circular string (cross-sectional vibrations). The first type of modes is in-plane bending modes, out of plane bending mode, extensional modes, and torsional modes. The string modes are two divided into lower and upper families. It is the lower family of string modes, which couples up with the ring modes.

For a torus made of Kapton 200-HN with generating radius of 0.304 m, the ring radius of 2.8956m, and pressure of 5.3779 kPa natural frequencies and mode shapes for symmetric modes with circumferential wave numbers of n = I and n = 4 are given in figure 5 . w

.

n =0.7506

freq-82.58Hz

1 18

U

2

a

1 0.5 v

o -0.5 -1

Symmetric mode shapes for n = 1 w

n =16.31

frea'384.9Hz

1 U

18

2

1

1

0.5 v

B"

o -0.5

-1

0

0 -1

100

200

300

0

200

100

a

300

a

Symmetric mode shapes for n = 4 Figure 5 . Displacement mode shapes for n

I86

=I

and n = 4

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4

MODAL ANALYSES FOR THE CIRCULAR LENS

The classical mathematical model for transverse free vibration of the membrane can be represented by the following Helmholtz equation (4.1)

r17zw-phw=o N

where p is the mass density, h is the film thickness, and N is the radial uniform boundary tension per unit length. wn(r,8,t) = W,( r )cos n(8 -a)e'"' (4.2) Substituting equation (4.2) into the governing equation result in the following Bessel differential equation:

a2w, awn

r2-+r-+(A2r2 ar2 ar

-n')W, = O

(4.3)

phw2 where n are integer number and L2= N The solutions to the equation (1.3) are known as Bessel functions. For each value of the integer n there are two linearly independent solutions of Bessel's equation. W,(r) = AnJn(Ar)+Bnq(Ar) (4.4) Since W,, is always a finite value and is zero at the boundary of r = a, thus B, = 0 and J,(aa) = o (4.5) The above equation is the frequency equation and has infinite number of roots for each value of n. A few of these roots are listed in the following Table. Therefore, the natural frequencies of the membrane are given by (4.6) a Free vibration of a thin film circular lens made of CP1 supported by an aluminum ring is considered. Two different analytical models based on pre-strained membrane and plate theory are developed and the results are compared with experimental modal analysis. Comparing the results of experiment and membrane theory provides estimation for the prestrain in the film. CPI material has the following dimensions and properties. Film Thickness 0.0005 in Film Diameter 17.5 in Mass density 0.0001341 Ib s2/in4 Poisson's ratio 0.34 Modulus of Elasticity 300000 psi

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As is demonstrated in the above Table the mode shapes are in the following order.

0 0

m=O, n=O

m=3 n=O

m=3, n=l

m=l, n=O

m=2, n=O

cl3

m=l, n=l

m=2, n=l

m=3, n=l

m=l, n=2

m=O, n=3

m=2, n=3

m=3, n=3

@ m=3, n=2

m=O, n= 1

m=l, n=3

m=O, n=2

0

Strain components in the medium can be written by the following expressions

Where E' = E ( l + i q ) is the complex modulus of elasticity. Since the medium is stretched uniformly then NBB= N ,. Consequently, it can be concluded that E@ = E,, and the tension per length in the medium in terms of strain is

N=-E'he I-v Substituting the above equation into equation (4.6) it yields w

.2 mn

=

(ila)2,EE

a2p(l- v)

(1 + $1

Considering that zeros of first Bessel functions for non-negative orders are always real, from the above equation one can conclude that for linear viscoelastic material the modal loss factor for each mode is the same as the material loss factor. Computed natural frequencies for first sixteen modes for different values of strains are computed and presented in Figure 6 . The results illustrate as the pre strain increases natural frequencies for all modes increase too. Theses results are compared with those obtained from Finite Element analysis for the fundamental modes. The comparison establishes an excellent agreement between analytical and Finite Element results.

188

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2500

2000

2 8

1500

it8

h

13 IOoo Z

500

0

o

0.002 0.004 0.006 0 . 0 0 ~ 0.01 0.012 0.014 0.016

o.oie

0.02

FmStraln

Figure 6. Variation of natural frequencies verses pre-strain 5. CONCLUSION Vibration analysis for components of a solar concentrator system were performed by conducting modal analysis for long rigidized struts, inflated supporting torus, and a circular lens. The developed analytical solution provides modal parameters for each of the system components. In the case of the strut modes such as axial, torsional, transverse, lobar and their corresponding thickness modes are considered. Flexural and extensional modes for both symmetric and anti-symmetric modes are considered in the analysis of the torus. Classical analysis was conducted for the circular lens, which is made from thin-film material (Kapton 200-HN). The analysis was used to determine dynamic properties of the Kapton lens. 6. REFERENCES

Armenakas, A. E., Gazis, D. and Henmann, G. Free Vibrations of Circular Cylindrical Shells, Permagon Press Inc., 1969. Hamidzadeh, H. and Chandler, D. “Circumferential Vibrations of Three Layered Sandwich Cylinders”, Proceedings of the Thirteenth Biennial ASME Conference on Mechanical Vibrations and Noise, DE-Vol36,233-237, 1991. Hamidzadeh, H. and Sawaya N. N. “Free Vibration of Thick Multi-layered Cylinders with viscoelastic Cylinders”, Journal of Shock and Vibrations, Vol. 2, No. 5, 393-401, 1993. Liepins, A. A., “Free vibrations of me-stresses toroidal membrane,” NASA Report CR-296, 1965. Sanders, J. L., 1963, “Nonlinear theory for thin shells,” Quart. Appl. Mech. 21,21-36.

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A simple beam element for large rotation problems K KERKaNEN, J SOPANEN, and A MIKKOLA Department of Mechanical Engineering, Lappeenranta University of Technology, Finland

ABSTRACT

In this study, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The non-linear elastic forces of the beam element are obtained using a continuum mechanics approach. Linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The phenomenon known as shear locking is avoided through the adoption of selective integration within the numerical integration method. It is shown that by using the proposed element, accurate non-linear static deformations, as well as realistic dynamic behaviour including the capturing of the centrifugal stiffening effect, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements. 1 INTRODUCTION

The description of non-linear deformations is a challenging and active research topic in the area of multibody dynamics. The goal of these studies is to obtain a more realistic simulation models for applications such as belts and cables. Non-linear deformation in multibody dynamics can be treated using, for example, the absolute nodal coordinate formulation (1, 2) or the large rotation vector formulation (3). The absolute nodal coordinate formulation has many advantages, which include the exact description of an arbitrary rigid body motion and a constant mass matrix. Despite numerous investigations into the usability and accuracy of the absolute nodal coordinate formulation (4, 5), there is still a need to improve its accuracy and appropriateness for computer calculation.

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191

The objective of this investigation is to develop a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation. The most distinctive feature of the formulation is that slopes and displacements are used as the nodal coordinates instead of finite or infinitesimal rotations. The proposed beam element uses a linear displacement field and a reduced amount of slope coordinates in comparison to the previously introduced absolute nodal coordinate finite elements. It has been perceived that higher terms in description of the displacement field in the axial direction are purposeless, because the element have a feature of exhibiting linear bending behaviour when a continuum mechanics approach is used in the description of the elastic forces. The smaller number of nodal coordinates leads to a reduced degree of freedom in the finite element leading to computational advantages in structural analysis. 2 KINEMATICS OF THE LINEAR BEAM ELEMENT

Using the absolute nodal coordinate formulation, the global position vector, r, of an arbitrary point in a planar case can be written as

where S is the element shape function matrix, x and y are the local coordinates of the element and e is the vector of the nodal coordinates. The proposed beam element uses linear polynomials instead of cubic polynomials to interpolate both the transverse and longitudinal components of displacement. The use of linear polynomials leads to eight unknown polynomial coefficients and for this reason the slope coordinates ar' /& can be neglected. The assumed displacement field of the two-dimensional shear deformable element can be defined in a global coordinate system using the following linear polynomial expression:

.=[:I=[

a, + a,x + a,y + a,xy bo +b,x+b,y+b,xy

1

Four nodal coordinates can be chosen for each node of a two-noded beam element as follows: e, =[r;

5J.

(3)

where r, is the global position vector of node I and vector ar; / @ is the slope of node I defining the orientation of the height coordinates of the cross-section of the beam (5). The element shape function matrix, S, can be expressed by using the nodal coordinates and the interpolating polynomial of Eq. (2) as follows:

192

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S=[S,I S,I S,I S,I]

(4)

In Eq. (4) I is a 2 x 2 identity matrix and the element shape functions, SI.. 3 4 , can be written as

s,

SI= l - t ,

=

m - 57)

9

s, = 5 ,

s,

= 157

I

where 1 is the length of the element in the initial configuration and the non-dimensional quantities, 5 and v, are defined as

The shape functions contain only one quadratic term, xy, while the remaining shape functions are products of one-dimensional linear polynomials. 3 DESCRIPTION OF ELASTIC FORCES

The definition of the elastic forces for the absolute nodal coordinate beam element can be obtained by using a continuum mechanics approach (4, 6). In this investigation, a non-linear strain-displacement relationship is employed for the elastic forces. By utilizing the fact that vector r defines an arbitrary point on the element in the global coordinate system, the gradient of the deformation vector can be defined as

In Eq. (5), X and x are the vectors of the global and local element coordinates, respectively. The vectors of the nodal coordinates in the deformed and initial configuration are presented by e and e.Matrix J is the deformation gradient and matrix JOa constant transformation matrix. The Green Lagrange strain tensor, E"', can be written using the right Cauchy-Green deformation tensor as follows: = -1( D ~ D - I ) 2 The strain tensor of E"' is symmetric, and therefore, only three strain components are needed to identify it. These components can be written in vector form as E=[€;;

E;

2€$.

(7)

Using matrix E, which contains the elastic coefficients of the material, the expression of the strain energy can be written as follows:

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193

U=’j EI’EEdV 2,

(8)

Matrix E can be expressed for an isotropic homogenous material in terms of Lame’s constants, 1 and p, as follows:

E=

(9)

In Eq. (9), A = E v / [ ( l + v ) ( 1 - 2 ~ ) ]and p = E / [ 2 ( 1 + v ) ] ,where E is Young’s modulus of elasticity and v the Poisson’s ratio of the material. It is demonstrated in Refs. ( 5 , 7) that the use of Eq. (9) leads to erroneous results in bending. The transverse normal strain, E:, that is constant over the cross-section, is coupled via Poisson’s ratio with the strain ET; that varies linearly over the cross-section. This leads to linearly varying transverse normal stress over the cross-section that causes an overly stiff behaviour in bending. This locking phenomenon is known as Poisson’s locking or thickness locking. By neglecting the Poisson’s effect, the strain energy, U, can be written using Young’s modulus of elasticity, E, and shear modulus, G, as follows (8): 1

U =-

I(E&,’ + EE,’ + 4 k , G ~ , ’ )dV

2 ,

In order to obtain the correct shear strain energy, the shear correction factor, ks, is needed to minimize the error between the constant and the known true parabolic shear strain contributions. The vector of the elastic forces, Qe, can be defined as the derivative of the strain energy expression with respect to the element nodal coordinate vector as follows: dU

I’

Q, = -(de)

4 INTEGRATION OF ELASTIC FORCES

The shape functions of the proposed two-dimensional shear deformable beam element include only one quadratic term, xy. Therefore, the element is able to exhibit only a rectangular deformation shape. This characteristic results in parasitic shear strain under pure bending. As a result, the element stores excess shear strain energy that leads to a phenomenon called shear locking (9).

194

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To avoid the accompanying defects of spurious shear strain, selective and reduced integration is adopted within the numerical integration method. Firstly, integration over the cross-section (y, z) was performed using only one point. Furthermore, one Gauss was used to evaluate the contribution of shear strain and two Gauss points were used to evaluate the contribution of normal strains in the longitudinal direction (x). This method led to convergence problems when the number of elements in dynamic model was increased. When a small number of proposed elements was used, the model had a tendency to converge larger deformations in comparison to the other models. These results can be explained as a consequence of using reduced integration in the selective integration of the strain components of the element. Reduced integration has a soRening effect and may also introduce some spurious modes, such as zero-energy deformation modes or hourglass modes. The spurious modes incorporated by the stiffness matrix of the element can deactivate the resistance to nodal loads. As a result, spurious zero energy modes are activated in the element (9). The convergence problem was solved by increasing the number of Gauss points from one to two in integration over the cross-section. In final form, integration in closed form is used to evaluate the contribution of normal and shear strains over the cross-section of the element while numerical integration method with one Gauss point is used to evaluate the contribution of strains in longitudinal direction. 5 EQUATIONS OF THE MOTION

The mass matrix given by the absolute nodal coordinate formulation is constant and symmetric. Using the element shape function given by Eq. (4), the mass matrix, M, can be written as M= IpTSdV,

(12)

V

where p and Vare the material density and volume of the finite element, respectively. Using the constant mass matrix and the elastic force vector, which omits a non-linear description when a continuum mechanics is used, the equations of motion of the deformable finite element can be written as (1) Me = Q, - Q, , where Qk is the vector of the generalized external nodal forces. Since the mass matrix is a constant matrix, the vector of the accelerations e of Eq. (13) can be efficiently solved using numerical procedures on the following equation:

The kinematic constraints that depend on the nodal coordinates and possibly on time in the multibody system can be written in vector form as (10)

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195

(15)

C(e,t) = 0 ,

where C is the vector of linearly independent constraint equations, e the nodal coordinate vector and f time. The equation of motion that accounts the constraints can be defined using Lagrange's equation in matrix form by employing an augmented formulation as follows:

In Eq. (1 6), Cf is the Jacobian matrix that is the partial derivative of the constraint vector with respect to nodal coordinate vector and 1 is the vector of Lagrange multipliers. Note, that the quadratic velocity inertia forces are zero in the elements based on the absolute nodal coordinate formulation. The unknowns 1 and e of Eq. (16) can be determined by differentiating the constraints of Eq. (15) twice with respect to time:

and writing a system of differential and algebraic equations in matrix form as follows:

6 NUMERICAL RESULTS

In this section, the performance of the proposed shear deformable beam element is investigated in static and dynamic problems. The results of the examples for the proposed beam element are compared to those of the analytical solutions and/or to the solutions obtained using a commercial finite element code ANSYS as well as a two-dimensional shear deformable beam element proposed by Omar and Shabana (6). The strain energy of the proposed beam element is calculated using Eq. (10) by employing a shear correction factor k, = 5/6. Eq. (8) is used to determine the strain energy in the case of the element proposed by Omar and Shabana. For both element types, integration in closed form is used to evaluate the contribution of normal and shear strains over the cross-section of the element. Numerical integration method with one Gauss point is used for the proposed element to evaluate the contribution of strains in axial direction, while four Gauss points is needed for the element of Omar and Shabana. This is due the fact that the element proposed by Omar and Shabana uses third order polynomial expansion. In the first example, large non-linear deformations of the simple cantilever structure are considered and compared to the non-linear solution of the BEAM188 model in ANSYS (5, 11). The cross-section of the beam is rectangular and the length of the beam 2.0 m. The material of the structure is assumed to be isotropic, the Young's modulus of the material is 2.07.10" N/m2 and mass density 7850 kg/m3. The other end of the beam is clamped by boundary conditions that

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eliminate the horizontal and vertical displacement and slopes aril+ and ar213y of the first node. The vertical displacements of the endpoint are investigated using different numbers of elements for the two different cantilever models: In the first model (Model I), the beam has a 0.1-m-sided square cross-section and value of the Poisson’s ratio is 0.3, while in the second model (Model 2), the height h of the beam is increased from 0.1 m to 0.5 m while the Poisson’s ratio is 0.0. A vertical force, F = -5.0.108 .h3N, is applied to the free end of the cantilever. The results of the second problem are shown in Tables 1 and 2.

element of Omar and

beam element

As can be seen in Table 1, in the case of Model 1 the beam element of Omar and Shabana seems to suffer from residual transverse normal stresses leading to overly small displacements (5). The predicted displacements of the proposed model and the BEAM188 model are very similar with the exception of the case of two elements.

I

Number of elements

I

2 4 8 16 32 64

The ANCF 2D beam element of Omar and Shabana 1.86909, -0.64098 1.84841, -0.69436 1.84498, -0.70341 1.84407, -0.70573 1.84378, -0.70643 1.84367, -0.70667

I I

Tip Position (X, Y), [m] Proposed ANCF 2D beam element 1.87307 1.85001 1.84412 1.84271 1.84237 1.84228

-0.65134 -0.69591 -0.70709 -0.70970 -0.71034 -0.71050

I 1

ANSYS: BEAM188

I

1.86749, -0.67783 1.85551, -0.69700 1.85246, -0.70179 1.85169, -0.70299 1.85150, -0.70329 1.85145, -0.70337

According to Table 2, the results of the proposed elements and elements of Omar and Shabana are in good agreement, but the BEAM 188 model slightly underestimates displacements in comparison to the other models. It is important to note that, in this problem, computer times in iterations with the proposed element were two times shorter than with the element of Omar and Shabana. In the first dynamic problem, the dynamic behaviour of a simple planar pendulum, which consists of one beam, is investigated using different numbers of proposed two-node two-

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dimensional shear deformable beam elements. The pendulum is connected to the ground by a The revolute joint, and the only force acting on the system is gravity, which is equal to 9.81 ds2. cross-section of the beam is a 0.1-m-sided square, while the length of the beam is 2.0 m. The material of the structure is assumed to be isotropic and the Young's modulus of the material is 2.07. lo7 N/m2, the Poisson's ratio 0.0 and mass density 7850 kg/m3. The initial position of the beam is horizontal with zero initial velocity. The energy balance of the beam should remain constant due to the fact that the free-falling pendulum is a conservative system. This can be written as

2(Ti +

+ U'

= constant,

i

where n is the number of elements of the system, T' the kinetic energy, V' the potential energy and U' the strain energy of the element i (4).The energy components of the pendulum made up of 4 elements are shown in Fig. 1. It can be seen that the energy balance remains constant with excellent accuracy; in this case, the greatest exception of the sum from the constant is 0.62. lo4 J.

.2wo

1

0.5

(3

Tlms Io]

Figure 1. The energy components and energy balance of the falling flexible beam modelled using 4 elements. A comparison of the vertical displacement between the proposed element and that presented by Omar and Shabana is shown in Fig. 2. The results are obtained using 8 elements and good agreement can be observed between the models. Using the proposed beam element, a significant saving in computation time can be achieved in comparison to using the beam element presented by Omar and Shabana. This is due to the fact that less nodal coordinates and simpler polynomials are needed to identify the element, and the dimensions of the vectors and matrices in the

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calculation are smaller. A comparison of the performance of the elements in terms of relative computer time is shown in Table 3. Number of elements 4

CPU time for the element of Omar and Shabana 100 % 100 %

8

CPU time for the proposed element 65 % 68 %

rims ISI

Figure 2. A comparison of the vertical displacement between the proposed element and that presented by Omar and Shabana using 8 elements. In the second dynamic problem, the centrifugal stiffening effect in the spinning cantilever beam modelled with the proposed elements is under investigation. A proper definition of beam deformation for spinning beam demands a coupling of axial force with a bending moment. The capability of capturing this so-called geometrical or centrifugal stiffening effect is examined by modelling a rapidly spinning flexible beam using the parameters and angular displacements reported by Wu and Haug (12). The beam has a length of 8 m, width of 1.986,103m, height of 3.675.10-' m, modulus of elasticity of 6.895 .IO" N/m2 and density of 2766.67 kg/m3. The angular displacement is given as follows:

e= @,

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(I

-

$),

199

According the Eq. (20),the steady state angular velocity is reached after Tsseconds. The beam is modelled using three elements, an angular velocity w s of 2 rad/s, acceleration time T, of 15 s and above parameters. A useful indicator of the capability of capturing geometrical stiffening effect is the steady state axial extension of the beam. The exact solution for the beam axial extension ux can be written in following form: u, =z[T-l), tan(aZ)

where a = ,/$on.

In Eq. (22) wsis the steady state angular velocity (13). The analytical value of the axial extension of the beam at steady state phase in this case is 2.7386.10’ m. The difference between the global horizontal positions of the end points of the modelled beam and the straight shadow beam and the value of the axial extension of the beam at steady state phase are depicted in Fig. 3.

1

I

Figure 3. The difference of the end point horizontal displacements between the modelled beam and the straight shadow beam and the steady state extension of the rotating beam. The results shown in Fig. 3 are in good agreement with the results of Refs. (12, 13). There exist small vibrations during the steady state phase, which was expected by results of (14), where the centrifugal stiffening effect using the absolute nodal coordinate formulation is studied. As can be seen from Fig. 3, the steady state axial extension of the beam corresponds with the analytical value with good accuracy. On the basis these results, a capability of automatic accounting of centrifugal stiffening effect of a spinning beam can be reached by using the proposed elements.

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7 CONCLUSIONS It has been perceived that although the displacement field of the element proposed by Omar and Shabana includes a cubic interpolation polynomial in the axial direction of the displacement, the element exhibits linear bending behaviour when a continuum mechanics approach is used. For this reason, the advantage of the third-order polynomial expansion is debatable. The objective of this investigation was to develop a computationally efficient two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation. The beam element uses a linear displacement field neglecting higher-order terms and a reduced number of nodal coordinates. The expression of the elastic forces is non-linear. The accompanying defects of the phenomenon known as shear locking are avoided through the adoption of selective integration within the numerical integration method. Several numerical examples were used to demonstrate the functionality and usability of the proposed beam model. The results were compared to the results of a commercial finite element code ANSYS, the results of the previously published beam element model by Omar and Shabana and analytical results. Generally the results are in good agreement. The computing times of the iterations were two times faster using the proposed elements than using the elements of Omar and Shabana. For the proposed element, integration in closed form is used to evaluate the contribution of normal and shear strains over the cross-section of the element while numerical integration method with one Gauss point is used to evaluate the contribution of strains in axial direction. By using this combination, the element locking is eliminated and an accurate and fast convergence is possible to achieve. In addition, the use of complicated cross sections of the elements is not restricted.

In the case of a simple pendulum, the results of the proposed beam element demonstrate good functionality. The energy balance of the dynamic model remains exactly constant, and the results are in good agreement with the beam model of Omar and Shabana with less computational effort. The results of the spinning beam shows that a capability of automatic accounting of centrifugal stiffening effect of a spinning beam can be reached by using the proposed elements.

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NOMENCLATURE Symbols ao, ..., a3

A bo, ..., b3 b

C C, D e E E F G h

I I J Jo

ks

I M n QC

Qe Qk

r SI, ...,s4 S

t T TS UX

U V X

X

X Y

202

polynomial coefficients area of the beam cross-section polynomial coefficients width of the beam in initial configuration vector of linearly independent constraint equations Jacobian matrix gradient of the deformation vector vector of the nodal coordinates Young’s modulus matrix of elastic coefficients of the material force applied to a node shear modulus height of the beam in the initial configuration node I identity matrix deformation gradient constant transformation matrix shear correction factor length of the beam in the initial configuration mass matrix number of elements vector that arises by differentiating the constraint equations twice with respect to time vector of the elastic forces vector of the external generalized forces position vector in a global coordinate system shape functions element shape function matrix time kinetic energy acceleration time beam axial extension strain energy volume of the element or potential energy local coordinate vector of the local coordinates vector of the global coordinates local coordinate

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Greek Letters Green Lagrange strain tensor normal strain in x-direction normal strain in y-direction normal strain in xy-plane non-dimensional quantity angular displacement Lame’s constant vector of Lagrange multipliers Lame’s constant Poisson’s ratio non-dimensional quantity material density angular velocity

Ern

4 E;;

7

e a i P V

6 P w

REFERENCES Shabana, A. A., 1998, Dynamics of Multibody Systems, John Wiley & Sons, New York. Shabana, A. A., 1998, “Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics”, Nonlinear Dynamics, 16, pp. 293-306. Cardona, A., and Geradin, M., 1988, “A Beam Finite Element Non-Linear Theory with Finite Rotations”, International Journal for Numerical Methods in Engineering, 26, pp. 2403-2438. Berzeri, M., and Shabana, A. A., 2000, “Development of Simple Models for The Elastic Forces of The Absolute Nodal Coordinate Formulation”, Journal of Sound and Vibration, 235(4), pp. 539-565. Sopanen, J. T., and Mikkola, A. M., 2003, ”Studies on the stiffness properties of the absolute nodal coordinate formulation for three-dimensional beams”, in Proceedings of ASME 2003 Design Engineering Conferences and Information in Engineering Conference, DETC2003/VIB-48325. Omar, M. A., and Shabana, A. A., 2001, “A Two-Dimensional Shear Deformation Beam for Large Rotation and Deformation”, Journal ojSound and Vibration, 243(3), pp. 565576. Rhim, J., and Lee, S. W., 1998, “A Vectorial Approach to Computational Modeling of Beams Undergoing Finite Rotations”, International Journal for Numerical Methods in Engineering, 41, pp. 527-540. Sharf, I., 1999, “Nonlinear Strain Measures, Shape Functions and Beam Elements for Dynamics of Flexible Beams”, Multibody System Dynamics, 3, pp. 189-205. Cook, R. D., 1981, Concepts and applications ofjnite element analysis, Wiley, New York. Shabana, A. A., 1994, Computational dynamics, Wiley, New York. ANSYS User’s Manual, 2001, Theory, Twelfth Edition, SAS IP,Inc. 0.

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[I21 [13] [ 141

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Wu, S.-C., and Haug, E. J., 1988, “Geometric non-linear substructuring for dynamic of flexible mechanical systems”, International Journal for Numerical Methods in Engineering, 26, pp. 221 1-2226. Simo, J. C., and Vu-Quoc, L., 1986, “On the Dynamics of Flexible Beam Under Large Overall Motions - The Plane Case: Part II”, Journal of Applied Mechanics, 53, pp.855863. Berzeri, M., and Shabana, A. A,, 2002, “Study of the Centrifugal Stiffening Effect Using the Finite Element Absolute Nodal Coordinate Formulation”, Multibody System Dynamics, 7 , pp. 357-387.

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Engine Dynamics

Advanced CAE simulation and prediction of drivetrain attributes H FOELLINGER Powertrain Applications, Ford Werke AG, Germany

KEYWORDS MBD, NVH, DOE, Vehicle drivetrain, Attributes. ABSTRACT A robust design methodology is required which can prevent the late and costly need for production palliatives in the auto drivetrain. This necessitates that the factors which cause failure or discomfort, and the failure mechanism itself, are fully understood. In complicated multi-body systems a local malfunction can cause a system reaction. For example, judder is excited at the clutch but this manifests as a system failure. Equally, a local disturbance may excite a modal response. For example, when a drivetrain mode coincides with an acoustic cavity mode to cause interior boom. Hence, it is clear that a multi body CAE approach is absolutely necessary to simulate and predict the dynamic conditions in a complete drivetrain when it is subjected to a range of user inputs. A multi body analysis would be essential to investigate complex interaction NVH (Noise, Vibration and Harshness) concerns, such as judder, whine, boom, rattle, and clonk, together with system investigations into driveability and shiftability attributes.

A fully verified multi body system simulation will allow "what-f I' design factor analysis before the need to commit the design to production tooling. It also allows the resolution of conflicting package and design requirements; cost optimisation, the cascading of design objectives from the system level, and the feasibility of design attributes.

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OVERVIEW AND PROBLEMS.

INTRODUCTION TO AUTOMOBILE

DRIVETRAIN

NVH

NVH problems (Noise, Vibration and Harshness) are complex. The systems and the components are frequently non-linear, and in addition, system behaviour is sensitive to component variation. The systems may also be subjected to complex excitation signals. See Fig 1.

Figure 1. Symbolic representation of driveline NVH. NVH concerns may be intermittent, transient, unpredictable, and they may lack the repeatability that is needed to support a controlled investigation. NVH concerns are sensitive to driving style, component variation, component assembly processes, environmental conditions, and many other factors. As a result, the dynamic vibration behaviour of a non-linear vehicle driveline system may be difficult to reproduce and predict, especially when it is also necessary to consider the associated noise radiation from an excited complex structure. Complex NVH system problems may be represented and simulated by the use of a suitable simulation code in a multibody analysis. Multibody analysis enables the simulation of complex NVH systems, the identification of root causal factors, the optimisation of system function, and the attributes that can provide robust performance.

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The following are some brief auto NVH definitions and descriptions related to the drivetrain:

Body Boom. Definition: Body boom is a low frequency audible noise, which may be excited throughout the engine speed range and in any gear. Boom will occur when body structural modes clash with internal acoustic cavity modes.

To simulate boom, it is necessary to predict internal sound pressure variation with respect to frequency and spatial location, by the use of a vibro acoustic coupling analysis between the body structure and the cavity. An example of a measured acoustic mode is shown in Figure 2.

Figure 2. 101 Hz acoustic mode excited at 3030 rpm. Transmission Gear Rattle. Transmission gear rattle may occur in neutral or in drive gear. Idle rattle, for example, may occur with the transmission in neutral and the engine at idle rpm.

This rattle is audible due to the low background engine noise levels. Gear rattle is sensitive at high temperatures, when the oil viscosity and gear drag torque is low. Engine speed variation from the clutch causes idle rattle and hence it may be 'switched off by clutch disengagement. Idle rattle may be readily resolved using a wide-angle low rate clutch characteristic. This will provide isolation between the flywheel and the transmission, but it may also aggravate the clonk condition.

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Clutch Squeal. Definition: Clutch squeal is a high frequency noise emanating from the clutch assembly, which may occur during high speed gear shifting and which generates a short-term resonant condition between pressure plate and clutch disc friction material. A replacement clutch disc normally resolves this problem. Gear Whine. Definition: Gear whine is a high frequency airborne and structure borne noise emanating from gear teeth contact and is caused by mesh error in transmission or axle. It is sensitive to the vibration path into the vehicle. Shuffle. Definition: Shuffle is a low frequency fore and aft motion of the vehicle, which may be excited by throttle tip-in with an engaged clutch. The torque pulse excites the first torsion eigenmode of the drivetrain, which generates the longitudinal response. The condition is maximised when the torque pulse time period is equivalent to the shuffle mode kequency. Clutch Whoop. Definition: Clutch whoop is a low frequency audible and tactile response from the clutch pedal, which is experienced during clutch pedal actuation. It is sometimes referred to as pedal growl. The source of excitation for this problem is engine crank bending coupled to the flywheel nodding or whirling mode.The problem is sensitive to the engine type, flywheel type, and to the vibration path between pedal and flywheel. Judder. Definition: Judder may be experienced during pullaway. It is related to the first torsion mode of the drivetrain system and is excited by the clutch friction material characteristics. It may be palliated by driveline damping. A negative friction coefficient gradient is the major causal factor for judder. The friction coefficient gradient is sensitive to wear, surface temperatures, clamp variation, friction material, and vehicle usage.

Driveline Clonk. Definition: Driveline clonk is a high frequency metallic noise emanating from the drivetrain and is caused when an impulsive torque is applied to the driveline, either by clutch engagement or by rapid throttle application after coast. Clonk occurs when impulse torque has passed through an accumulation of low resistance lash zones in the drivetrain. The colour map (see Figure 3) is a spectrogram of an impact and shows the intensity as a function of frequency and time.

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Figure 3. Spectogram of driveline impact. Clonk derives from high energy lubricated impacts in the driveline. A close examination of the local tribological conditions is required, since the impact energy will excite the high frequency response of neighbouring elastic structures and the onward radiation of unwanted noise. The following Figure 4 shows the structural modal shape of a driveshaft tube at 3256 Hz, which is excited by impact and is in the clonk frequency range.

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Figure 4. Structural modal shape of a driveshaft tube at 3256 Hz

COMMENT. All of the above-mentioned drivetrain NVH concerns share functional similarities although they each have different frequency domains, are perceptively different to the vehicle user, and may be influenced by different driving styles. In the past, an empirical approach to the solution of these NVH problems would be taken. This often involved a 'one factor at a time' development process and then subjectively rated for acceptability. It was not a convergent process, and it was time consuming and inefficient.

MJ3D may be very usefully employed in the resolution of NVH problems and the avoidance of palliation, which is both costly and time consuming. It is a logical process based on objective data, see below, and it allows the investigation of complex system dynamics and an examination of system factor interactions.

I

Identify the NVH

L problem.

212

Construct the MBD model. -+

Verify the model with supporting data.

Develop robust NVH

design metrics.

-

Implement the design actions and lessons learnt.

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THE APPLICATION OF MULTI BODY DYNAMICS TO DRIVETRAIN NVH. MBD simulations enable scenario building investigations to be conducted at much reduced cost and time when compared to comparable physical prototypes. Alternative designs may be evaluated, and the necessary time to bring a new product to the market may be shortened by the elimination of many iterations of laboratory testing and prototype fabrication. In the aircraft industry of course, this is an absolutely essential process. A vehicle drivetrain system is unlike a linear system dynamics problem. It is a complex nonlinear multi-body system, there are many interactions between the parts, and the number of factors affecting the design function may be significant. For such a system, it would not be cost or time effective to cany out 'what if scenarios on full size physical prototypes, since a large number of such prototypes might be required. Instead, system simulations are used for design optimisation studies. The investigation of complex vehicle noise, vibration and harshness (NVH) problems requires that analysis tools need to be sufficiently detailed to allow the end user to effectively and readily employ their capabilities. Virtual prototyping tools take the form of powerful computer-based software packages that provide modelling and simulation environments for their users. The tools must be user friendly. The model input parameters should represent real physical data such as geometrical dimensions, masses, moments of inertia, stiffness and damping coefficients. When parts are available the model may be verified. MULTI-BODY DYNAMICS. Multi-body dynamics (MBD) is the physics of interaction of an assembly of rigid or flexible inertial bodies or parts held together by some form of constraint or restraint such as joints, couplers, gears, bearings, bushings. Other forms of constraints arise from pre-specified motions, which ensure that inertial components in an assembly of parts follow a pre-defined type of motion. The overall behaviour of a multi-body system is therefore influenced by its individual inertial elements, which may exert forces upon each other when exposed to external force excitation. Since the combination of constraints, restraints, applied forces/ torques, and the inertia of the parts, govern the overall motion and response of the multi-body system, it is necessary to devise a method of formulation and solution in order to obtain and to understand the resulting dynamic behaviour. The methodology used to mathematically define and solve the physics of the multi-body system is based upon constrained Lagrangian dynamics, which employs the fundamentals of dynamics to simplify the formulation procedure. The method of formulation lends itself to the automatic generation of equations of motion for all individual parts in a multi-body system separately. The equations of motion for every part in the multi-body mechanism are then expressed in a single sparse matrix form (the matrix commonly known as the Jacobian matrix), which is often very large in size and consists of many zero entities. The solution procedure requires the simultaneous solution of these equations by applying numerical methods, which use simple but effective matrix manipulation techniques to obtain the required solution eigen vector.

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The dynamics of vehicle drivetrain systems are complex in nature. In particular, these systems often include non-linear characteristics, such as sources of compliance (stiffness, lash and damping). The assembly of parts or components having relative degrees of freedom with respect to one another introduces constraints that are also represented by complex non-linear functions. The equations of motion, the algebraic constraint functions and applied forces / torques, require matrix formulation for simultaneous solution. Thus, a suitable formulation and solution algorithm must be employed in order to obtain the required system response in incremental steps of time. The results obtained from multi-body dynamic analysis are usually in the form of displacements, velocities, accelerations and reaction forces / torques. These time-domain outputs can be converted into frequency signals to obtain the spectra of individual vibration of parts in the system. Based on this information, modifications may be made to multi-body system factors to guard against resonant conditions when prevailing operating conditions coincide with the fundamental natural frequencies of the overall system or its subsystem. The reaction force output from the system also allows for component redesign in terms of reducing stress levels, thus improving the fatigue life of parts under cyclic loading conditions. A typical MBD model of a vehicle drivetrain with 6 DOF (Degrees of Freedom) is shown in Figure 5 below.

Figure 5. MBD model of vehicle drivetrain.

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ADAMS is a general purpose modelling software that allows a wide range of dynamic multibody mechanisms to be simulated and analysed. It has the capability to predict the response of new designs, to evaluate the performance of existing designs or to provide extreme operating conditions to analyse the output of the mechanism, thus making the building of physical prototypes largely unnecessary. It provides the linkage between the CAD model and the ultimate use of DOE analysis. It allows the CAE models to be fully parameterised, thereby enabling Design of Experiments (DOE) to be performed on the system in order to improve and optimise the performance. An example of the application of A D A h 4 S is shown in Figure 6.

Figure 6. Application of ADAMS to drivetrain. DOE - DESIGN OF EXPERIMENTS. Design of Experiments is a systematic test approach that seeks to identify the contribution from a number of pre-assigned factors, to the yield of a complex process or function. DOE is most suitable when applied to multi-factored complex systems, when a robust system performance is required, when various factor contributions must be sorted into order, when optimisation is required, and when parameter and tolerance design studies are being performed.

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DOE is a controlled series of tests, which are conducted in line with a predetermined test matrix. This matrix identifies for each test run,the factors and the levels at which the factors are to be set. Factors may be best identified by the use of a so-called P diagram, which identifies by judgement, the significant control and noise factors, which affect the true functioning of the system. A very useful way of determining the most relevant system factors to include in DOE experimentation is to conduct a brainstorm, with contributions from all the appropriate technical specialists. Such a process uses a Cause and Effect fishbone structure. For example in the case of driveline clonk, the cause and effect fishbone was generated by an expert team, and is shown in Figure 7 as follows:

Figure 7. Cause and effect diagram for driveline clonk EXAMPLES OF MBD APPLIED TO THE RESOLUTION OF ACTUAL VEHICLE NVH PROBLEMS. Example 1. MBD and DOE applied to the study of driveline judder. A driveline model in ADAMS has been developed, which included the flywheel and clutch assembly, and the remainder of the drivetrain to the rear tyres. A pullaway simulation was developed, which illustrated how the clutch friction characteristic was the most significant contributor to the excitation of the judder mode. The model was used to show how a modification to the friction characteristic. The CAE work was backed up by acquisition of friction material coefficient data from the lab, and by !dl

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vehicle dynamometer verification, In this case the vehicle testing included an in depth DOE study to confirm the factor ranking. Example 2. MI3D and DOE applied to the problem of driveline clonk. Here, the excitation was a hard transient impact caused by impulsive torque traversals through the accumulation of backlash in the driveline. The impulsive torque would be generated by clutch or throttle actuation. The resultant high frequency impact energy caused neighbouring elastic structures to radiate noise.

This was a fairly complicated investigation since it required a coupled vibro acoustic analysis. The model extended from the flywheel to the road wheels, and it was subjected to a gear tooth impact that occurred as a result of an initial torque surge from a clutch engage. The extremely short duration impact excited the elastic driveshaft tubes into high frequency resonance. It was necessary to couple a high mesh density FE model of the tubes with an MBD model of the driveline system, in order to capture the modal detail after a driveline gear impact had been made. An FE mesh of the cavity airspace was included, in order to evaluate the coupling behaviour between shaft and cavity. It was shown how the frequency content of the impact, matched the natural panel modes in the driveline and also the radial acoustic cavity modes in the driveshaft. The speed of sound in the tubes at resonance was found to be supersonic. Component flexibility for driveshaft tubes was applied in the multi-body dynamic analysis, using the super-element finite element method in NASTRAN and the component mode synthesis in ADAMSFlex. The resulting solution combined a large displacement low frequency carrier wave motion (shuffle) with a superimposed small amplitude elastodynamic high frequency vibration (clonk) driven by an impulsive shock force as the lash was taken up. This drivetrain model was extended to include the acoustic cavity response, using the same super-element technique. This resulted in the creation of elasto-acoustic models of the thin hollow driveshaft tubes, which clearly indicated the coincidence between some of the structural modes excited by impulsive loading of the system, and the acoustic modes of the cavities. It has been shown that elasto-acoustic coupling takes place when frequency coincidence occurs. This results in travelling sound waves of a supersonic nature leading to high levels of sound wave propagation.

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Figure 8. MBD prediction of internal sound field of driveshaft at clonk frequency. Example 3. MBD and DOE applied to the problem of clutch whoop. This is also quite a difficult condition to model. Whoop occurs during clutch pedal actuation and is a combined problem of audible noise and tactile pedal vibration. It is always important to establish the boundary for all MBD investigations, to include all the source(s) and vibration path(s) and output(s). In the case of whoop, experimental work had earlier indicated that the likely boundary would include a simplified engine model, the clutch pedal and actuation link to the clutch assembly, the dynamic movements of the flywheel and clutch during clutch actuation, and the remaining drivetrain. The investigation uncovered many interesting dynamic modes including pressure plate wobble and flywheel whirling, which generated the force inputs to the system. This investigation established the dominant causal factors by a CAE DOE and also indicated that many of the factors were strongly coupled in the whoop mechanism. Example 4. MBD applied to the problem of body boom. Since the main body structural modes are excited at 2"dorder engine speed (firing frequency), we are interested in the frequency range up to approx 150 Hz.(since this corresponds to 4500 rpm). In a boom investigation, it is necessary to conduct a coupled structure - acoustic analysis, with particular regard to the many and various vibration paths into the body structure. It is also necessary to take full account of the body interior furniture and the passengers, and the subsequent effects on internal sound pressure distribution.

21 8

We find that longitudinal acoustic modes in a commercial vehicle are excited up to 100 Hz, and beyond this frequency we also start to excite diagonal and transverse sound pressure waves. Hence, it is important to understand this spatial sound pressure variation with frequency especially at passenger head positions, and the factors that influence the variation. Experimental work must be conducted in parallel with the model simulation in order to verify the assumptions made in the model including the model boundary, and the model accuracy. Finally, the MBD approach to this boom problem allows us to investigate the total system behaviour as well as the factor effects, using analytical DOE studies and 'what if scenarios.

SUMMARY. Some vehicle drivetrain NVH problems have been briefly introduced and their characteristics have been considered. The drivetrain system is excited by complex vibration signals. The internal and external restraint functions are also complex and non-linear. In addition we have normal piece-to-piece and vehicle-to-vehicle variation to consider. Due to the presence of damping, compliance, and free play in the drivetrain, the driven behaviour is always difficult to simulate. And yet an ability to predict drivetrain NVH is absolutely necessary in order to avoid and prevent unexpected dynamic behaviour. We have found that, due to the necessary need to minimise weight and cost in the drivetrain, these systems are lightly damped and resonant conditions may be readily excited over a wide frequency range. Palliation is not only time consuming and costly, but it may also have limited effectiveness. For all the reasons stated, and for the need to achieve challenging program timing requirements, it is absolutely necessary to be able to construct system failure conditions and to identify the root causal factors at a very early stage of the engineering program. Subsequently, MBD has been employed to great effect in the description and solution of these dynamic problems Subsequently, MBD has been employed to great effect in the description and solution of these problems, and several instances have been referenced to illustrate this approach. Once a fully representative and verifiable MBD model has been established, the system behaviour may be much better understood and the search for design robustness is quickly achieved through the use of experimentation and model investigation.

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Vi r tual test ing supPO rts reIiabiIity engineeri ng of engine prototypes G RAINER AVL List GmbH, Austria

KEYWORDS Reliability, MBD, Simulation, Development Process, Virtual Testing ABSTRACT Reliability, safety and quality are key issues for placing products successfully on the market. To achieve these goals Reliability Engineering has to be employed from the concept phase and throughout the lifetime of a product. During the product development phase it is important for the engineer to better understand the factors that cause components and systems to fail. This precondition should considerably lower the risk of delaying the SOP due to failure of prototypes. It is necessary to intensively use computer simulation tools and application methodologies during the design phase and the prototype development in order to make up-front investigations based on accurate digital models under real operation conditions. Targets for such calculations are to Analyse proposed design and evaluate the reliability potential, Ensure that all components, subsystems and systems in a design will behave as the designer anticipates and Prepare procedures for the later test runs on components, subsystems and systems. The prerequisites to successfblly develop and employ such sophisticated simulation methodologies are application platforms which consist of Various mathematical simulation tools, Testing tools corresponding to the mathematical simulation tools, Data and workflow management. The paper describes the application platforms for durability and NVH and how they are employed to optimise crankshaft design and to evaluate the stresses considering the cylinder head - cylinder block compound based on system simulation. Comparisons of simulation and experimental results are shown and an evaluation of the method in terms of maturity and limitations to reduce the risk of failures is outlined.

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INTRODUCTION The automotive industry’s marketing policy is increasingly imposing new challenges by the need for Faster product development cycles to meet the requirement of increased frequentation of new car models, A growing range of product variants based on a reduced number of platforms, Development of products with reduced life cycle costs, Increasing system complexity (function, comfort and luxury) and Increased safety. It is well known of course that the only way to succeed is to make the right design decisions during the concept phase. This makes it necessary that reliability is designed and built into products at the earliest possible stages of product development. In these early stages of the development process the costs of change are very low compared to changes during the development phase, while the locked-in life cycle costs are very high, Fig. 1.

Fig. 1: Locked in and occurring costs through different stages of the life cycle Therefore it is clearly visible that the up-front mathematical simulation has to be adopted to reduce reliability problems. This will reduce the product development cycle time and cost dramatically compared to experimental tests during the prototype stage as mathematical simulation helps to avoid the danger of encountering a problem close to the start of production. To apply mathematical simulation as a tool for reliability engineering during the concept and

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design phase it is an absolute must to have all information available concerning The allowable material stresses in the structure depending on the geometry, The loads on the structure under real operating conditions and To have multidimensional mathematical models available which offer high accuracy of the results by short calculation lead time. The results of such risk analysis have to make sure that the structure will not fail, if the loads on the structure do not exceed the allowable stresses. More precisely: the probability that a structure will fail is acceptably small, if the designer is lead by reliable calculations. The challenges to apply mathematical simulation as a “virtual testbench” to perform risk analysis are Integration of design, simulation and test systems, Implementation of data management and implementation of process management. All of the above mentioned items have to be treated as equivalent partners to reach the target. Process oriented activity ensures that the simulation fits into the process flow and is backed by appropriate tests. In parallel to that, the technology development guarantees accurate mathematical and physical models to simulate the operating conditions.

THE ENGINE DEVELOPMENT PROCESS Simulation and testing form an important part of the engine development process. In all phases of the engine development it is necessary that the following items progress in the order shown below to achieve an integrated development process: Design Virtual test Real test (real life conditions) Fig. 2 shows the influence of the different depth of simulation, virtual testing and real testing:

Fig. 2: Influence of virtual testing and real testing on the Engine Development Process

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The traditional development process, Fig. 2a, starts with system and assembly specification supported by simulations using simple simulation models calculating many variants within a short time to find an optimum solution. In a second step individual components are optimised using different physical models, e.g. FEM, CFD. During the prototype phase the product development is carried out by prototype testing supported by sophisticated simulations to finally optimise the entire system. The future approach, Fig. 2b, allows virtual testing based on mathematical simulation of the entire system starting during the concept phase. Components are tested as a part of the system, not individually. Thus a big effort has to be put into activities linking all the product shaping process steps, Fig. 3, in order to really benefit from them. Multidimensional modelling and simulation for powertrain and vehicle systems have become important areas of research for this reason. The data exchange between software tools of different domains, e.g. CAD data, multi-body system simulation data, control system simulation data or FEM data at any stage of the product development process has to be given the highest priority. By using experimental, test bed and road test data the correlation between the virtual and physical prototype can be improved. This is an indispensable prerequisite for further development of the mathematical simulation methods.

Fig. 3: Integrated Development Process The proportion to which each of the above items contributes to the course of the development process is changing. Fig. 4 shows the intensity of computer and test bed simulations in the traditional (b) and the future (a) engine development process.

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While in the traditional process the main development steps are done on the test bed by real testing, the future process relies on virtual testing during the design phase and uses testing only as a validation procedure.

Fig. 4: Impact of development based on virtual testing and on real testing on the Engine Development Process INTEGRATION OF THE VIRTUAL TEST INTO THE DEVELOPMENT PROCESS The applicability of the virtual test to reliability engineering and therefore the integration in the development process is still seen with some scepticism by engineers. Reasons for that are Missing expertise in interpreting results and therefore Difficulties to communicate results, Easy to use software (lack of training) and Links to different software to perform multidimensional simulation, Confidence in mathematical simulation. In order to establish virtual testing and - as a precondition - to overcome the fore mentioned scepticism it is essential to develop measures which form a “simulation environment” consisting of

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Software capable of predicting the behaviour of complex systems under real operating conditions with the highest accuracy. It must be ensured that software packages based on different physical or mathematical models are combined in the best suitable way via interfaces, co-simulation or simultaneous simulation. Workflow management hiding the complexity of the simulation system from the engineer. This includes model generation including load definition and material data acquisition and result presentation in a task oriented manner. Computer power to achieve sufficient turnaround times for the virtual testing to effectively drive the development process and Educating engineers to understand different engineering disciplines and to make use of complex interrelations of numerous design parameters. Apart from the technical aspect this human factor is the key to employ such sophisticated and extensive simulation procedures in a successful way. THE AVL PLATFORM CONCEPT Based on the definition of AVL’s engine development process and on the capabilities of AVL‘s and third party simulation software products a “platform concept” was worked out and turned into reality. Fig. 5a shows the components of the platforms and Fig 5b shows examples for various platforms which are developed to support the engine development process with simulation and test procedures. The following example of the “Durability and NVH Platform” emphasises the mathematical simulation part of the platform.

Fig. 5: A n ‘ s Platform Concept DURABILITY AND NVH PLATFORM APPLIED TO CRANK TRAIN DESIGN Traditionally the crank train layout was based on rough calculations, afterwards e.g. crankshaft and connecting rod were simulated with FEM (quasi- static) and optimised separately. Increasingly structures and systems of products become more and more complex. It appears that when various reliable components are combined into a system, the result is not necessarily a reliable system. Subsequently, the complexity of today’s powertrains does not

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allow component level simulation and component level test as the only scrutiny. It is also compulsory to do simulation and testing on the system level. Fig. 6 shows the modelling of the entire engine system and the displaced results for a part of the engine block. The increasing level of teamwork required in today’s development process, and the corresponding holistic thinking leads to the fact that engineers are engaged in several and differing levels of the process at the same time. As a consequence, they have to use different programs, which requires a graphical user-interface (GUI) which is as simple and intuitive to use as possible.

Fig. 6: System modelling and result evaluation for engine durability simulation The analysis of the results, apart from the specialised knowledge possessed by the engineers, demands visualisation software that enables fast interpretations of the results. In addition to the “static” result representations, video animations are created because they permit reliable result interpretations. This is especially important for structural and fluid flow simulation. Fig. 7 shows the workflow for the durability evaluation of the crankshaft. This new approach starts with the layout of the crank train using flexible MBS simulation under real operating conditions. The calculation model for the system is steadily refined, the durability of the components (crankshaft, con- rod, main bearing wall, etc.) are optimised within the system simulation. The project lead time is steered by the fact that the creation of the flexible MBS- model takes a considerable effort and also the calculation of the stresses has to be done externally with a commercial FE solver. As the generation of the flexible MBS crankshaft model (structured model) takes a lot of modelling time considerable effort was invested to reduce this part of the workflow by an automated procedure that at the same time maximises the reliability of the calculation model to avoid hard to detect errors.

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Fig. 7: Workflow for durability of crankshafts (structured model) In a first attempt the creation (generation) of the flexible MBS model ( crankshaft structured model) was automated as this is the most time intensive work, Fig. 8. The steps of the automated procedure are: Based on a CAD- STL file the crankshaft is separated by the AUtoSHAFT into simple units (cylindric parts like journals, crank pins) and complex units (webs). In the next step a FE meshing of the complex parts is done, a mass partitioning calculation is performed, followed by an elasticity evaluation using a static analysis with an integrated FE-solver with all necessary boundary conditions and load cases set up automatically. Finally the complex parts represented by the numerically derived mass and stiffness matrices and the analytically calculated pins and journals are assembled together using the tool Shaft Modelier to form the structured model of the crankshaft. AVL’s target solution is that also all post-processing steps of the crankshaft analysis (stress calculation and fatigue evaluation) is included to make this part of the analysis workflow easier and faster. This is enabled by strategic partnerships with the companies ABAQUS and Safe Technology Limited.

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For NVH applications the parameter identification for the engine mount model has been automated based on measured stiffness values. These measures reduced the modelling time for structured crankshaft models and engine mounts by 80%. Additional measures were taken to reduce the overall CPU processing time. A special task was to develop new models for considering the oilfilm in the bearings specifically for durability and NVH applications.

Fig. 8: Flexible crankshaft MBS model for 1D and 3D analysis automated derived from CAD-STL file This reduced the CPU time - depending on the application - between 20 and 85% compared to the use of overqualified, highly sophisticated elasto-hydrodynamic bearing models for these applications. Implementing these measures into the durability and NVH platform are preconditions to make this sophisticated multidimensional mathematical simulation a part of the standard engine development process. Finally the procedure had to be verified by real testing. The simulation results displayed in Fig. 9 show a sufficient correlation with the experimental results.

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Fig. 9: Comparison of results for the main journal torque of a 4- cylinder engine Now the virtual tests can be easily applied in a very early phase of the product development and lead to a reliable development status creating confidence in the quality of the product. So this leads to a decision making process which gives the right answers beforehand. An additional fact is that the virtual testing through multidimensional simulation can simulate the real operating conditions before hardware is available. Test equipment often has limited capabilities to simulate real operating conditions if only single components are tested. In our case the test could only consider the torsional behaviour while the simulation included also bending and longitudinal effects automatically. It is clearly visible that the replacement of testing time by up-front mathematical simulation will reduce the product development cycle time and cost dramatically. SUMMARY High precision mathematical simulation tools are already available to make detailed investigations of single components and can increasingly make investigations at system level. This is supporting the reliability engineering already during the concept and design phase. The effort to develop such tools which are powerful enough to predict product reliability in the early development stage is considerably high but helps to dramatically reduce the costs of change during the prototyping and pre-production phase. To further improve this way the demands on this process are: Reliable results for the entire systems provided on time. Simple handling of the simulation tools. Integration of the simulation tools into the workflow. Further steps, required for the integration of the computer simulation tools and tests, will be necessary to bring together the already existing hardware components of the prototype and the computer models to employ hardware in the loop (HIL) simulations.

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Periodic response and stability of reciprocating engines I GOUDAS, P METALLIDIS, I STAVRAKIS, and S NATSIAVAS Department of Mechanical Engineering, Aristotle University, Thessaloniki, Greece

ABSTRACT Periodic steady state response and stability of reciprocating engine models is captured directly, by applying appropriate numerical methodologies. First, small-scale models of single-cylinder engines are examined. The set of equations of motion derived is strongly nonlinear due to the rigid body rotation of the crank and the connecting rod, as well as due to the nonlinearities associated with the bearing action and the fact that both the driving and the resisting loads are expressed as a function of the crank rotation. Subsequently, the same methodologies are extended and applied to a much more complicated model of a four-cylinder in-line engine. Both the crankshaft and the engine block are discretized by finite elements, resulting in a large-scale system. In all cases, the influence of the system parameters on its long time response and stability can be studied in a systematic and effective way.

NOTATION radial clearance of the bearing br damping coefficient modelling piston motion mechanical energy losses CP cr

c, D

I; FP

4 F,9Fv

IC

I,, m,

rolling bearing damping coeEcent hydrodynamic bearing clearance hydrodynamic bearing diameter restoring part of radial force developed on bearings force exerted by the gas on the piston radial force developed on bearings forces developed at the support location of the crankshaft centroidal mass moment of inertia of the crankshaft functions of i , j generalized coordinates used to calculate system

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kinetic energy centroidal mass moment of inertia of the connecting rod crank normalized centroidal mass moment of inertia connecting rod normalized centroidal mass moment of inertia bearing equivelant stiffness crank length connecting rod length hydrodynamic bearing length crankshaft mass piston mass connecting rod mass resisting moment with respect to the Oz axis bearing constants total rolling elements in contact pressure mean effective value

yo

232

generalized coordinates vector system kinetic energy displacement component of the crankshaft along the Ox axis radial deformation of the bearing center displacement component of the crankshaft along the Oy axis coefficients of constant, linear and quadratically varying part of resisting moment angular position of the m -th rolling element diametrical clearance to crank length ratio system virtual work normalized rolling bearing damping coefficient angular rotation of the crank normalized spin speed of the crank average normalized spin speed of the crank crank length to rod length ratio crank mass center distance from bearing center to rod length ratio connecting rod mass center distance from crank-rod joint to rod length ratio hydrodynamic bearing oil viscosity coefficient connecting rod mass to crank mass ratio piston mass to crank mass ratio crankshaft angular velocity normalizing frequency normalized crank spin speed

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1 INTRODUCTION

Examining and understanding the dynamic response of reciprocating engines is of great importance for their economical and safe design. However, a systematic investigation of dynamics and vibration of such engines is difficult to perform due to their mechanical and geometric complexity and the inherent nonlinearities associated with their response. The great majority of previous research work refers to single-cylinder mechanisms, whose crankshaft is supported rigidly (1-3). Some recent work has also appeared on the dynamics of more involved and complete engine models (4-6). The main objective of the present study is to develop and apply a systematic numerical methodology, leading to a direct determination of periodic motions of internal combustion engines. First, a single-cylinder engine mechanism with rigid members but supported by either roller or oil journal bearings dynamic model is examined. The forcing is non-ideal, since it depends on the crank rotation. Apart from providing useful information on the engine dynamics, the original results provided a basis for checking the accuracy and effectiveness of the methodology before it was eventually applied to a more complicated engine model. The organisation of this paper is as follows. The equations of motion of slider-crank mechanisms with compliant supports and subjected to non-ideal forcing are first presented in the following section. Then, numerical results referring to periodic steady state response and stability are presented in section 3. Finally, results obtained by the same methodology when applied to a large-scale four-cylinder engine model are included in section 4. The paper concludes with a summary of the highlights of the work.

2 EQUATIONS OF MOTION FOR A SINGLE-CYLINDER MODEL The validity and accuracy of the numerical methodology developed was first applied to a single-cylinder engine mechanism, shown in fig 1. This mechanism involves rigid parts, but its crankshaft is supported on either roller or oil journal bearings. Therefore, if u(t) and v(t) represent the displacement components of the crankshaft along the axes Ox and Oy of the inertial reference frame Oxy, respectively, the dynamic configuration of the system examined here can be described by the generalized coordinates 6'(f), u(t) and v(t). The equations of motion of the system examined were derived by applying Lagrange's equations (7). For this, the kinetic energy is first expressed in the form

where the quantities I e q , mu,, mvv, m,, m,, and m,, represent complicated but known functions of the generalized coordinates 0 and v (8). Since all the mechanism members are rigid, the elastic energy is zero. Moreover, the virtual work of the system is expressed in the form

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where M , is the resisting moment developed with respect to the axis

02,

F, is the force

exerted by the gas on the piston, while c, is a damping coefficient modelling the mechanical losses of energy caused by the motion of the piston. For the purposes of the present study, it is assumed that the resisting moment appears in the form

M,(B) = a0+ poB+ y o @ ,

(3)

with coefficients depending on the nature of the engine load (6). On the other hand, the driving torque is generated by the gas force Fp . Evaluation of the pressure giving rise to this force involves the solution of coupled thermodynamic and dynamic laws. However, a simplified thermodynamic cycle is assumed here, yielding the pressure developed within a cylinder as a function of the corresponding crank angle (6). This pressure distribution is completely determined in terms of the mean effective pressure, p e N Likewise, . the quantities

Fu and F, in equation (2) represent the forces developed at the support location of the crankshaft. Employing relations (1)-(3) and applying Lagrange's equations yields after considerable algebraic manipulations the equations of motion of the system in the following matrix form

q(t) = (B(t) u(t) v(t))' . The coefficients appearing in matrix M and vector with complicated but known functions of the generalized coordinates (8).

are

3 NUMERICAL RESULTS FOR SINGLE-CYLINDER ENGINE The equations of motion (4)are strongly nonlinear. Consequently, the system dynamics can only be obtained numerically. For this, suitable numerical methodologies were developed, capturing periodic motions in a direct manner. These methodologies were also accompanied by an appropriate method yielding the stability properties of the located periodic motions (810). The results presented in this section were obtained for engine mechanisms with their crankshaft supported on flexible bearings. In particular, if u, represents the radial deformation of the bearing center, the radial force developed on bearings with rolling elements has the form

with restoring part expressed in the form

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In the last expression, b, represents the radial clearance of the bearing, a, is the angular position of the m -th rolling element (of the total N elements in contact), n is a constant, while the coefficient K, is determined from the bearing characteristics and loading conditions (1 1). First, the crank displacements are normalized by the crank length normalized with the frequency w, =

,/-,

e , . Moreover, the time is

where k, = ~,b:-’/2 and m, is the mass of the

crankshaft. Next, the following set of dimensionless parameters is also introduced

where I , and I, represent the centroidal mass moment of inertia of the connecting rod and the crankshaft. Finally, the default numerical values of these parameters were selected to be:

R = 0 . 3 , / 1 , = 0 , / 1 2 = 0 . 5 , p 2 = 0 . 1 , p 3 = 0 . 1 , J I = 1 0 0 , J 2 = 0 . 1 , p=O.OOl a n d c = 0 . 1 . Figure 2 presents some typical numerical results. First, fig 2(a) shows results obtained for the special case where the angular velocity w of the crankshaft is constant (ideal forcing). The amplitude of the crank lateral displacements is shown as a function of the frequency R = w / w o . Large amplitudes appear near the main and many secondary resonances. Moreover, the amplitude of the displacement u along the direction of the external forcing (denoted by the thicker curve) is in general larger than that of v along the transverse direction (thinner curve). Finally, the broken parts of the curves represent unstable periodic motions. Next, fig 2(b) shows the amplitude of the same displacements as a function of the effective gas pressure, for non-ideal forcing. Again, there appear several branches of unstable periodic motions. The loss of stability was found to take place either through saddle-node or via Hopf bifurcations (8). In particular, the saddle-node bifurcations are associated with the appearance of the Sommerfeld phenomenon, which is well-known and expected to arise in dynamical systems subjected to non-ideal forcing (10). On the other hand, Hopf bifurcations lead to appearance of quasiperiodic or chaotic response (8). The results of fig 2(c) present the dependence of the average crank spin speed on the effective gas pressure parameter, for three different combinations of the system parameters. The nominal case (not shown in the figure) and the special case with low damping examined in fig 2(b), give virtually the same results in terms of the amplitude of However, the case with smaller bearing damping (shown in fig 2(c) in place of the nominal) leads to several ranges of per where the located periodic motions are unstable, just like in fig 2(b).

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Finally, fig 2(d) depicts the trajectory of the crank support point in steady state conditions obtained for the nominal case, at pe8 = 10. The jagged part of the trajectory on the ( u , v ) plane indicates the existence of strong higher harmonic components in the response, which become dominant mostly in the interval just after the firing of the gas. When the crankshaft is supported by hydrodynamic journal bearings, the dynamics observed is quite different. Here, the oil journal action was modeled by employing the finite-length impedance method (12). In the following calculations, the journal bearing supporting the crankshaft was chosen to have a length-to-diameter ratio LID = 1, a radial clearance of the bearing C, such that C 7 / ! ,= 0.001 and an oil viscosity coefficient of p = 0.03Nsfrn’ . First, fig 3(a) presents frequency-response diagrams for the lateral displacements of the crank, obtained for typical values of parameter il under ideal forcing. No branch of unstable periodic motions was detected here. Comparison with fig 2(a) reveals that the displacement amplitudes are much smaller, while the form of the response diagrams is in general much more regular than those obtained for bearings with rolling elements. Finally, the displacement v exhibits again a lower amplitude than the corresponding amplitude of the displacement u . Next, fig 3(b) presents similar results but for non-ideal forcing. The amplitude of the lateral displacement of the crankshaft, obtained at steady state conditions, is presented for a typical range of the effective pressure parameter and the same values of the parameter 2 .The effect of this parameter is qualitatively different in the range with the smaller than in the range with the larger values of p e g .Likewise, fig 3(c) presents the average spin speed of the crankshaft as a function of the effective gas pressure for different values of A . Clearly, the larger the value of il the larger the resulting average spin speed 8,,, which follows qualitatively the same trends with those observed for bearings with rolling elements. Finally, fig 3(d) shows trajectories on the (u,v) plane corresponding to the periodic steady state obtained for three different values (namely a small, an intermediate and a large value) of the gas pressure parameter per. Direct comparison with the results presented in fig 2(d) for bearings with roller elements reveals that the trajectories of the crank are much more regular when oil bearings are used. This indicates a significant design advantage of the oil journal bearings over the bearings with rolling elements for the specific application.

4 NUMERICAL RESULTS FOR A LARGE-SCALE ENGINE MODEL The application of the numerical methodologies to the small-scale models examined in the previous two sections provided the means to check their accuracy and effectiveness. Next, these methodologies were modified and expanded appropriately and they eventually were applied to more complicated dynamical systems. Here, a small sample of characteristic results obtained for an involved finite element model of a four-cylinder in-line car engine model (fig 4) will be presented. The crankshaft interacts with the engine block through oil journal bearings, while the block is connected at three points to the car body through nonlinear

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bushings. Before application of these methodologies, the degrees of freedom of the resulting system were reduced substantially by applying a component mode synthesis method (9). First, fig 5(a) presents a typical diagram, showing the maximum and minimum values obtained for the axial displacement of one of the three points of the engine block connecting it to the car body, as a function of the fundamental frequency of the gas forcing. The thidthick curves correspond to results obtained for a model including linear modes of the shaft and the block up to 100 and 1000 Hz, respectively. Likewise, fig 5(b) shows the corresponding force developed between that point and the corresponding point on the car body. Next, fig 6(a) shows the displacement of a crankshaft point, which is in the area of the hydrodynamic bearing close to the flywheel. The thidthick curves were obtained for the fully nonlinear and the corresponding linearized model (which is often used in practice). The results indicate significant differences between these two models. Finally, fig 6(b) depicts frequency response diagrams obtained for the three displacement components of a point on the engine block outer surface. Since the whole response histories are available at each forcing frequency, such information is useful in performing dynamic packaging studies in a systematic way. 5 SUMMARY

Periodic steady state motions of reciprocating engines were located in a direct manner by applying appropriate numerical methodologies. The stability characteristics of the located periodic motions were also determined. First, these methodologies were applied to simplified models of single-cylinder engines, involving rigid members but compliant crank bearings, subjected to non-ideal forcing. The resulting equations of motion included strong nonlinearities. The study of these models provided useful insight into the system dynamics In particular, it was demonstrated that saddle-node bifurcations of a periodic motion are associated with the onset of Sommerfeld phenomena, while a Hopf bifurcation marks the beginning for the appearance of quasiperiodic or chaotic motions. Next, the same methodologies were extended and applied to a much more complicated model of a fourcylinder in-line engine. Both the crankshaft and the engine block were discretized by finite elements. This resulted in a large-scale system, whose dynamics was analysed after reducing substantially its dimensions by applying a suitable methodology. The methodologies developed help the efforts to predict the influence of parameters on the long time response and stability of large order nonlinear systems in a systematic manner. Acknowledgments: This research was funded by the Greek Ministry of Education and the European Union, through the Heraclitus research program. This support is gratefully acknowledged.

REFERENCES 1. Viscomi, B.V. and Aye, R.S., ‘Nonlinear dynamic response of elastic slider-crank mechanism’, ASME Journal of Engineering for Industry 93, 1971,251-262.

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2.

Hsieh, S.-R. and Shaw, S.W., ‘The dynamic stability and nonlinear resonance of a flexible connecting rod: single-mode response’, Journal of Sound and Vibration 170,

3.

Wauer, J. and Buhrle, P., ‘Dynamics of a flexible slider-crank mechanism driven by a non-ideal source of energy’, Nonlinear Dynamics 13, 1997,221-242. Okamura, H., Shinno, A., Yamanaka, T., Suzuki, A. and Sogabe, K., ‘Simple modeling and analysis for crankshaft three-dimensional vibrations, Parts 1 and 2 ’, ASME Journal of Vibration and Acoustics 117, 1995,70-86. Mourelatos, Z.P., ‘An efficient crankshaft dynamic analysis using substructuring with Ritz vectors’, Journal of Sound and Vibration238,2000,495-527. Metallidis, P. and Natsiavas, S., ‘Linear and nonlinear dynamics of reciprocating engines’, International Journal of Non-Linear Mechanics 38,2003,723-738. Greenwood, D.T., Principles of Dynamics, 2ndedition, Prentice Hall, Englewood Cliffs, New Jersey, 1988. Goudas, I., Stavrakis, I. and Natsiavas, S., ‘Dynamics of slider-crank mechanisms with flexible supports and non-ideal forcing’, Nonlinear Dynamics, (in press). Veros, G. and Natsiavas, S., ‘Ride dynamics of nonlinear vehicle models using component mode synthesis’, ASME Journal of Vibration and Acoustics 124, 2002, 427-

1994,25-49.

4.

5. 6. 7. 8. 9.

434. 10. Harris, T.A., Rolling Bearing Analysis, New York: John Wiley, 1966.

1 1. Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979. 12. Childs, D.,Turbomachinery Rotordynamics, J. Wiley & Sons, New York, 1993.

Fig. 1. Model of a single-cylinder engine with flexible crank support.

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0.003

Uma

"ma

17/

I n m

0 05

0

0.75

R

0.0015

0

1.5

0.18

0

10

Peff

0

u

20

0.0015

e,,,

V

0.09

0

0 0

10

Pett

20

.0.0015 -0.0015

0.0015

Fig. 2. Results for roller bearings: (a) Frequency-response diagrams for ideal forcing. (b) Response diagrams for non-ideal forcing. (c) Effect of length ratio A and mean effective pressure on the average crank spin speed. (d) Trajectory of the crank support on the (u,v) plane.

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0.001

h = 0.2 ........... h = 0.1 ......

urnax

%ax

vmax

0.0005

/

.................:- .............. -. ......

%

n 0

0.75

R

1.5

0.18

0.0002

.

.......... ..

:.-.

v

'

0

20

Per

10

0.001

.............

pe"' 2.94 ea,

U

0.09

0

n 0

10

Pen

20

.0.001 -0.001

0

v

o 101

Fig. 3. Results for oil journal bearings: (a) Frequency-response diagrams for ideal forcing. (b) Response diagrams for non-ideal forcing. (c) Effect of the length ratio ;1 and mean effective pressure on the average crank speed. (a) Trajectory of the crank support on the (u,v) plane for three different values of the mean effective pressure.

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Fig. 4. Finite element model of a four-cylinder in-line engine: (a) Crankshaft. (b) Engine block.

0.56 max x /min x

[mml

-130

-0.56

25

Q[HZ]

50

4 25

1

n[Hz]

50

Fig. 5. (a) Frequency-response diagram for the displacement of an engine connection point. (b) Corresponding force developed at the same point.

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241

- nonlinear /min x

0-

-11

1

2s

R [Hz]

50

Fig. 6. Frequency-response diagrams for: (a) the displacement of a crankshaft point, (b) the three displacement components of a point on the engine block outer surface.

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Coupled torsional and transverse vibration of engines A L GUUOMI, S J DREW, and B J STONE School of Mechanical Engineering, The University of Western Australia, Australia

ABSTRACT: Failures still occur in reciprocating engines and compressors as the result of torsional resonance problems. In practice torsional vibration measurements are still relatively rare and the measurement of the vibration of engines is normally made with accelerometers placed on the engine block. It is of some interest to investigate if torsional vibration is always manifested in transverse vibration of the engine block. Much recent research on torsional vibration of engines has assumed that the engine block is fixed to earth. This makes for far greater simplicity in modelling and is perhaps the main reason why the vibration of an engine on its mounts with coupled torsional and transverse vibration has received little attention. This paper describes the modelling of a single cylinder engine with such coupling of the vibration. The engine block was constrained to vibrate in the plane of motion of the reciprocating mechanism. An overview of the modelling approach is presented and some predictions made with the model are presented that indicate its validity. Some initial experimental results are also presented and, when compared to predictions made &om the model, indicate that the predictions are very sensitive to certain engine parameters that are not easy to measure experimentally.

NOMENCLATURE Acceleration , ~ ~of block ~ in Xdirection of block in Yu ~ , ~Acceleration ~ ~ , direction Coupling point of crank and A connecting rod. B Coupling point of connecting rod and piston, centre of mass of piston. C Centre of mass of crank. u

~

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eo eox eoY FAX

FAY

Block centre of mass distance &om 0 x component of block centre of

mass distance from 0 y component of block centre of mass distance from 0 Reaction force at point A in Xdirection. Reaction force at point A in Ydirection.

243

Reaction force at point B in Xdirection. Reaction force at point B in Ydirection. Horizontal crankshaft bearing force. Vertical crankshaft bearing force. Positive force dynamometer horizontal axis Positive force dynamometer vertical axis Acceleration due to gravity. Ratio of length OC to OA. Mass moment of inertia of block about its centre of mass Mass moment of inertia of crank about its centre of mass. Mass of block. Mass of crank. Mass of piston. Mass of connecting rod. Positive force dynamometer moment Centre of rotation of crank, as observed in X-Y plane. Piston loading force. Crank throw, OA. Horizontal mount reaction force (Left hand side) Vertical mount reaction force (Left hand side) Vertical mount reaction force (Right hand side) Side force acting on piston applied by cylinder wall. Crankshaft torque. Time. x coordinate (attached to block)

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Terms involving acceleration terms derived from X equation manipulation Transverse displacement of engine block in X-direction. Transverse acceleration of engine block in X-direction. y coordinate (attached to block) Terms involving acceleration terms derived from Y equation manipulation Transverse displacement of engine block in Y-direction. Transverse acceleration of engine block in Y-direction. Angular displacement of engine block. Terms involving acceleration terms derived from 6 equation manipulation Angular velocity of engine block. Angular acceleration of engine block. Angular displacement of crank (relative to block) Terms involving acceleration terms derived from 6 equation manipulation Angular acceleration of crank (relative to block) Absolute angular displacement of crankshaft Absolute angular acceleration of crankshaft Angular displacement of crankshaft centre (0)from block COM measured from x coordinate

1 INTRODUCTION Internal combustion engines are commonly used in society today in applications ranging from gardening equipment, such as chainsaws and lawnmowers, to large marine engines and automotive vehicles. Despite their wide-spread use over a long period of time there is currently very little research focused on the dynamic modelling of reciprocating mechanisms, particularly the modelling of crankshaft torsional vibration. Most of the current research in the automotive industry is directed towards minimising noise, vibration and harshness (NVH).This is because NVH performance is a common

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criterion that must be met in today's automobile industry market. It has been identified that root cause identification and solution of power train NVH calls for detailed modelling of the internal combustion engine using a multi-body dynamic approach (Anundale, Gupta & Rahnejat 2000, p. 207). Because crankshafts still fail as a result of torsional vibration, detailed theoretical models that can predict their torsional natural frequencies are considered extremely useful. Hesterman (1 992) used a receptance-based approach and simplifying assumptions to develop a model to predict the torsional natural frequencies of an engine. Hesterman's model was found to accurately predict the torsional natural frequencies of the crank for the experimental rig used. In her analysis the Free Body Diagram (FBD) of the piston (Figure la-2) and crankshaft (Figure la-3) identified three forces of interest. These were the piston side force S and the main bearing reaction forces Fox and Foy. Hesterman also noted the possibility for S to transmit high-frequency vibrations to the support structure through the mounting system.

Figure 1: a) Hesterman's representations of (1) reciprocating mechanism, (2) piston FBD and (3) crank FBD @esterman 1992) b) Predicted and experimental forces for (1) Piston side force S (Watling 2000) (2) Main bearing reaction Fox (3) Main bearing reaction FOY(Yarwood 2002)

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Watling (2000) furthered Hesterman's work, allowing prediction of the forces Fox, Foy and S. After constructing a test rig, Watling was able to experimentally measure the piston side force S and compare this with the predicted values. She found that there were noticeable differences between the two. The difference appeared to be a result of a missing factor of two, which she suggested may have arisen from an integration error in the analysis. Yarwood (2002) used the rig to further the work of Hesterman and Watling by measuring Fox and Fay, and found that, like S, there were large differences between the theoretical and measured values. However, unlike S, the values did not seem to differ by a simple factor of two. Yarwood thought that the error must have arisen from assumptions used in deriving the Hesterman model that were no longer valid when considering the experimental rig. He thought that the most likely contributor to these errors was the assumption that the engine block translational and angular accelerations were not important. Using rotating axes, Yarwood developed acceleration equations to account for these additional effects. However, because the engine block of the test rig was firmly held by the force dynamometer, it was found that there was little difference between the Hesterman model and the model developed with rotating axes. Yarwood performed simulations that demonstrated that for a block which experiences vibration there may be large differences between the two models. The initial aim of the investigation reported in this paper was to develop a time domain model that predicted the motion of a reciprocating engine block when mounted on flexible mounts and when torsional vibration of the crankshaft was present. To do this some of the work completed by the previous researchers (Hesterman 1992, Watling 2000, and Yarwood 2002) had to be used. It was believed that an attempt should be made to reduce the known errors between force predictions and measurements before performing the large extension intended. Thus a reanalysis of their works was performed while at the same time developing the model of an engine on flexible mounts. The reanalysis identified some errors that were rectified and, where appropriate, experimentally investigated. 2 ERRORS IN PREVIOUS WORK

Before demonstrating the errors it is necessary to briefly explain the experimental technique used by Yarwood and Watling to measure the forces S, Fox and FOY.To verify the engine forces predicted from theory, Watling developed a test rig that used a Villiers engine that had been stripped down to resemble the simplified piston, conrod and crank model (Figure la-I). The head, valve gear and flywheel were removed, so that the engine had to be motored. The engine was motored by a 1.7kW AC servo motor that could apply a constant rotational speed to the crank and/or provide torsional excitation through an oscillating angular velocity. To measure the engine forces a Kistler force dynamometer was used to firmly hold the engine block. The crankshaft angular velocity was measured using a Dantec Laser Torsional Vibrometer (LTV). A light sensor was set to trigger when the piston was at TDC. The signals measured by these four instruments were recorded with an HP 35670A digital signal analyser, from which time and frequency recordings were made. It was found that both Watling and Yarwood had incorrectly resolved the forces that acted on the force dynamometer. Both researchers believed that the force dynamometer measured the forces that acted on the engine components. However, since the engine

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block is mounted on the force dynamometer the measured forces correspond to those on the block. The component forces hence have negative signs. This explains why Yarwood's theoretical and experimental results for Fox and S appeared to be separated by a large phase difference. The apparent phase difference was a consequence of neglecting the negative signs, which caused the results to be inverted. If the negatives are included there is immediately greater correlation between the predicted and measured forces. In Hesterman's analysis a piston loading force Q(t) was identified. In operating combustion engines this force arises (mainly) from the combustion pressure. In normal operating engines the piston loading force Q(t) is complex since it depends on the gas forces acting both above and below the piston. In normal operating conditions the gas pressures above the piston are typically orders of magnitude more positive or negative than the pressure forces acting below the piston. This is due to many factors, including 1) breathing holes in many engines that minimise crankcase pressurisation and 2) the high positive gas pressures associated with combustion. Since the test engine had no cylinder head, the top surface of the piston was exposed to atmospheric conditions. Watling and Yarwood therefore assumed that the piston loading force was zero. However, in this study the validity of this assumption was questioned. Although the top of the piston is exposed to atmospheric conditions, the underside of the piston is exposed to the much smaller crankcase volume, which contains three small breathing holes. These holes are a result of removing engine parts. One hole is the crankcase breathing hole; the other two holes exist as a result of removing the valves. Although the holes allow air movement, they are not as free-flowing as the top side of the piston. Due to their small size and location it was thought that pressure differentials would occur across the piston top surface since the effectiveness of these breathing provisions would be reduced as piston speed increased. This being the case, the piston loading force Q(t) could not be set to zero. It was considered that this may explain some of the discrepancies between theory and experiment encountered by both Yarwood and Watling in the forces S, Fox and Fay. It was initially thought that the effect of crankcase pumping could be experimentally found using Watling's rig and the force dynamometer. One experiment would have been conducted having the holes blocked and another with more open. However, after finding the error related to the dynamometer readings and examination of the S, Fox and Foy equations, it was found that, although Q(t) affects each of these forces, the forces as seen by dynamometer resulted in Fx and Fy being unaffected by its presence. Since it is not possible to observe any pressure effects in Fx and FY simply using the force dynamometer, it was decided to fit a pressure sensor to the crankcase. From the measured forces and the corresponding pressure trace it was possible to determine the actual Fox and Foy values.

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Figure 2: a) FBD of engine b) positive dynamometer notation c) corresponding block FBD Modelling the actual process that occurs during motoring of the rig is complex. At low speeds the air can flow much more easily through the orifices since drag and turbulence effects are proportional to the square of velocity. The interaction of gas flow and gas forces is further complicated if the engine is run from stationary to running speed. This is because at low speeds the effect will be minimal, but would become more significant as speed increases. At some critical piston velocity, gas flow could become choked, meaning that a constant air-mass flow rate would be transferred through the orifices. Since air at low crank rotational speed can move in or out, it is unlikely that there is any noticeable difference in pressures acting above and below the piston. Hence the piston loading force Q(t) can be set to zero. However, as the rotational speed increases, because of the location of the holes and their relative size, the pressure acting on the underside of the piston will no longer be the same as that above. The true effect of these holes on the generated pressures inside the crankcase is complicated. It would be possible to accurately model this interaction using gas dynamics and a suitable computer package, but time restricted this possibility. 6 4

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Figure 3: Piston loading force Q(t) for rig as used by Watling and Yarwood The measured pressure force Q(t) is shown in Figure 3 and was assumed by Yarwood and Watling to be zero. However, since the forces that result from Q(t) are comparable to the forces Fox, Foy and S measured by Watling and Yarwood (Figure lb), the effect of crankcase pumping should have been included. It was found that including the effect of crankcase pumping did affect S but it failed to greatly improve the correlation between the measured and predicted forces. Thus the errors investigated, though significant, are not the only ones. This is an area for further work. The modelling of an engine on flexible mounts was pursued in parallel with the error analysis and is described in the next section. However it needs to be noted that the model is likely to contain the same unknown error(s) as for the earlier analyses.

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3 ENGINE ON FLEXIBLE MOUNTS This section shows how the model was developed to describe engine block motion when mounted on flexible mounts. Although real engines exist in 3-dimensional space and have six Degrees of Freedom @OF), the model developed only considers motion in the plane of the reciprocating mechanism. This was done because the reciprocating mechanism causes most of the motion to occur in this plane. The model was developed using a global reference frame attached to earth at 0 when the block is in its static equilibrium position (Figure 4a) (without the reciprocating mechanism). This global reference frame is denoted by capitalised (X,Y) and the local reference frame (attached to the block as shown) is denoted by lower case (x,y). It was decided to use the global reference frame to define the position of the crank since this allowed use of the acceleration equations derived by Yarwood and confirmed in this study. The equations in this study are derived in this general form to make the model applicable to many engine geometries. The FBD for the engine mounted on flexible mounts is shown in Figure 4b. The forces S, Fox and Foy are those derived in this study that account for crank and piston rotation as a consequence of block rotation. Since the engine block rotates due to vibration, these equations are applicable. The FBD shows a single horizontal spring force which represents resistance to motion in the X-direction. Including two horizontal reactions would cause four unknown mount reactions, which would make the system statically indeterminate. Thus only a single reaction was included to make static analysis of the FBD solvable. With horizontal stiffness included in the derivation the model developed is more representative of real engines. It is possible to set the X stiffness to zero if desired. Through use of Newton's Second Law the equations of motion for the block are represented by Eqs 1-3 (Guzzomi (2003)),

Fu'

Figure 4: Engine block on mounts: (a) Global reference frame and (b) corresponding FBD (c) crank FBD

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However, if the block Centre of Mass (COM) is not at 0 (the crank bearing centre), aBlodX and aBlmky do not simply correspond to X and Y . This is because X and Y denote the location of point 0 relative to the global coordinates. Thus, when the COM is not at 0 the translational accelerations of the block COM X a n d Y comprise the accelerations of all three of the engine degrees of freedom: X, Y and a . Using rotating axes (Meriam & Kraige 1998), the translation accelerations where found to be:

+ y)+ a 2.e, cos(a + y ) cos(a + y ) - .e, sin(a + y )

aBlockX = X + a.e, sin(a

(4)

aBlmky = Y - &.e,

(5)

Since the mechanism forces of Figure 4b result from rotation of the crankshaft it was necessary to define the crankshaft equations. From the FBD of the crank shown in Figure 4c it was possible to find the crank angular acceleration relative to the vertical.

~ , r ( ~ + ~ ) c o s y / + ~ , , r ( ~ + ~ ) s i n y + ~ - ~ , ~ ~ r c o s y / - ~= , ,f ~, r s~ i n y /(6) The model for a single plane has four degrees of freedom: three corresponding to block motion Eqs. 1-3, and a fourth to specify the crank motion Eq. 6. Although Eqs. 1-6 appear to be relatively simple, the complex nature of the component accelerations results in each of the four accelerations being coupled. That is, that accelerations of the other DOF appear in each of the equations. Because of this coupling, the easiest way to obtain expressions containing only the acceleration of interest was to write the set of equations in matrix form. The mathematical analysis performed is very complex, as it represents a general case and cannot be included in a short paper such as this. It is possible to rearrange Eqs. 1-6 into the form shown by Eqs. 7-10. The equations pertaining to each of the four variables were manipulated so that all accelerations appeared on the left-hand side and all the velocity and displacement terms appeared on the right. This process involved extensive equation manipulation. Eqs. 7-10 can be represented in matrix form (Le. Eq. 1 I), which can be represented as Eq. 12. Matrices M and C comprise velocity and displacement terms. This allows the four variable accelerations of matrix 2 to be found solely from the displacements and velocities of each variable through use of simple matrix operations (Eq. 13).

c, X.y, + Y . y , +&.y, + e . y , = c, x . x , + Y.x2 +a.x, + e x , =

X.a, + Y.a, +&.a,+ B.a, = C, 2.q+ Y.e, +&.e,+ e.8, = c,

xy ]

= [ ]

a

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To determine the displacements that result from these accelerations it was decided that numerical integration should be used together with a computer program. The numerical technique that was implemented was a Runge-Kutta 4th Order technique (R-K4). R-K4, although only a single step method, is accurate and efficient (O’Neil 1995), since the error associated with the method is proportional to the 4th power of the step size. For RK4 to be used, the initial conditions (at t=O) of the velocity and displacements must be known. The language used to write the program to iterate the equations was MATLAB.

3.1 Model validation and checking The complex nature of the model meant that it was difficult to verify the complete program. Since iterations in R-K4 required looping through the complete set of equations, it was not possible to check the code in separate subsections without performing a complete loop. Thus to verify the model, various simple cases were tested and the simulated results confirmed using vibration theory. The relevant equations of motion (Thomson 1998) were derived from the FBDs and solutions to these equations were found by assuming there was undamped free vibration at the natural frequency. In most cases the system had only 1 DOF and hence one corresponding natural frequency. Firstly, simulations were conducted using the engine block simplified to not include the reciprocating mechanism. This was done by setting the parameters of the reciprocating mechanism components to small values (since setting them equal to zero caused the matrix to become singular and hence unsolvable) and making the crank fixed. The block COM was also set to be located at 0 (Figure 4a). The block was then constrained to vibrate, vertically and then horizontally and the frequency of free vibration in each case was as expected from the mass of the block and the stiffness of the springs. The block was also caused to rotate on its mounts and the frequency of free vibration was again found to be as expected. To investigate the effect of coupling between the Y and Q terms the system was reduced to effectively that of a block on two vertical supports vibrating in a single plane. The two natural frequencies for this system were derived and free vibration of the model was as expected. By moving the centre of mass location the coupling became more noticeable. Different COM locations were tested and in all cases it was found that the results were consistent with the theoretical predictions. Secondly, the crank was allowed to rotate freely. When the block is fixed, there is no damping on the crankshaft and friction is not included. With the piston located at TDC the system contains potential energy. When the block is slightly tilted, gravity causes the piston to fall and convert the potential energy to kinetic energy. However, the energy of the system must be constant according to Work-Energy. Hence the piston rises until it converts all its kinetic energy back to potential energy. The piston then stops and the process is reversed, in a similar fashion to a pendulum. The simulated results for crank rotation and angular velocity were as expected. Finally, the model was validated W h e r by performing a simulation allowing free vibration of all DOF. For the case tested, t = 0, B = 0.25 rad (0.04rev) and all other initial conditions were zero. With X, Y, a and B unrestrained, and for realistic engine parameter values, the simulated displacement time characteristics are shown in Figure 5. It is apparent that all DOF experience vibration as expected. This is because all DOF are coupled and hence motion of one cannot occur without causing subsequent motion of other DOF. Also demonstrated is that each DOF is bound and does not increase, which fi.uther demonstrates the reliability of the model. Figure 5a shows that, like the

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verification for B alone, the crank rotates from the initial set value of 0.04 rev. to effectively the same piston location which results when B is approximately 0.96 rev. The X and a motions indicate a vibration about a mean not equal to zero. This is because the simulation had the engine block COM located on the right-hand-side of the crank. Hence, block static equilibrium occurs when the block is rotated CW (negative) an amount a which, with the effect of the reciprocating mechanism, causes the crank (and hence X) to displace. Also evident is that Y changes the most when the piston is near BDC. This is because the dynamic effect of the main bearing reaction force FOYis pushing down on the block when the piston is near this location. In practice, however, this scenario would not be possible, since friction in the engine is too high and the crank would not begin to rotate under the effect of gravity.

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3.2 Experiments To experimentally verify the model developed, the test rig built by Watling (2000) was modified to allow greater, observable block vibration, from which block accelerations could be measured using accelerometers. The developed rig had springs and used a flexible coupling to drive the engine; these are labelled in Figure 6a. A horizontal plate (X-plate) was used to restrain motion out of the X-Y plane. To measure the block accelerations in the X, Y and a directions four accelerometers were fitted to the engine block. For simulations to be performed various engine parameters had to be found. Since the forces S, Fox and FOYaccount for the gravity and inertia terms of the piston, conrod and crank it was necessary to find the mass, COM and inertia of the engine block alone. These were found experimentally. Since the model required the torque, stiffness and damping values to be known the values corresponding to the test engine had to be found. The stiffness and damping values were found from the accelerometer readings obtained when the block was displaced and then released. The sensitivity of the model to parameter values is clearly apparent. The location of the block COM and the horizontal stiffness values were not known exactly, thus various simulations were performed with variations in their values. The T1 results correspond to

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initial values determined experimentally. The T2 values result from a horizontal change to the block COM location by lOmm which was investigated since additional mass was added to the block after initially locating the COM. T3 corresponds to a vertical shift of the COM while maintaining the initial horizontal value. Simulations T4 and T5 result from a change in the value used for the horizontal plate stiffness by a factor of 10. This value had not been experimentally determined and therefore had to be inferred from the frequency of free vibrations. The range of the predicted values compared to the experimental is promising, but the form of the curves is at times significantly different.

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Figure 6: (a) Experimental rig (b) measured rotational speed (c) predicted block rotational acceleration ii (d) measured vertical acceleration Y (e) predicted vertical acceleration Y ( f ) measured horizontal acceleration 2 (e) predicted horizontal acceleration 2 When viewing the differences in the plots, the model complexity must also be considered. Currently it is not possible to use the measured running speed data in the

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model, hence only an average speed is used which is not ideal. This is because in the model the rotational speed is referenced to the bore however, the measured values were referenced to earth, hence the values differ by the block movement. The accelerometers used to measure 2 and Ywould also be effected by the angular velocity of the block which causes an acceleration towards the centre of rotation. It is also believed that some errors arise because of the coupling and the unknown madinertia effects of the combined assembly including the AC servo drive. Further testing and additional modelling is required to determine if the servo drive should be included in the system modelled and if so, how to account for it. It has been shown that the model is sensitive to the location of COM of the block and the stiffness of the X-plate. This stiffness has been assumed to be linear, however the plate is prone to buckling. It is highly probable that the model is sensitive to other parameters which may not be accurate.

4 CONCLUSION: A review of the works of Watling and Yanvood found many errors, requiring much analysis to be repeated from first principles. Two such errors were included in this paper. After correcting the errors it was considered to be justified to develop the model for an engine on flexible mounting. However, although the corrections reduced the differences they did not remove them so that unknown errors still exist. The complexity of the dynamic interactions of the system meant that the model was a far more complicated than first anticipated, requiring extensive equation manipulation. A better correlation between predicted and measured results could perhaps be obtained by fitting the parameter values and more modelling. Therefore hrther experimental work and dynamic modelling is required to achieve this.

5 REFERENCES: Arrundale, D., Gupta, S. & Rahnejat, H. 2000, 'Multi-body dynamics for the assessment of engine induced inertial imbalance and torsional-deflection vibration' in Multi-body Dynamics: Monitoring and Simulation Techniques - 11, eds. H. Rahnejat, M. Ebrahimi & R. Whalley, Professional Engineering Publishing, London. Guzzomi, A. L. 2003, Time Domain Modelling of Engine Forces and Vibration, Honours dissertation, University of Western Australia. Hesterman, D. C. 1992, Torsional Vibrations in Reciprocating Pumps and Engines, PhD Thesis, The University of Western Australia. Meriam, J. L. & Kraige, L. G. 1998, Engineering Mechanics: Dynamics, 4'h edn, John Wiley & Sons, Brisbane. O'Neil, P. 1995, Advanced Engineering Mathematics, 4'h edn, Brooks/Cole, Pacific Grove, CA. Thomson, W. T. 1993, Theory of Vibration with Applications, 4" edn, Stanley Thornes, Cheltenham. Watling, K. 2000, Torsional vibration - analysis of crank-assembly model for a reciprocating engine, Honours dissertation, University of Western Australia. Yarwood, D. 2002, Vibration of a Single Cylinder Engine, Honours dissertation, University of Western Australia.

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Quality and validation of cranktrain vibration predictions effect of hydrodynamic journal bearing models

-

G OFFNER and H H PRIEBSCH

Christian-Doppler Laboratory for Engine and Vehicle Acoustics, Technical University of Graz, Austria MTMA

Advanced Simulation Technolo ies, AVL UK Limited, Lidlington, U K U KARLSSON and A WIKSTR8M Engine Dynamics and Acoustics, Scania CV AB, Sweden B LOIBNEGGER

Advanced Simulation Technologies, AVL List GmbH, Graz, Austria

Abstract The dynamic behaviour of moving parts in a running internal combustion engine is significantly affected by the constraints in their supports. This is particularly true for the moving cranktrain parts, which are supported by hydrodynamically lubricated journal bearings. Therefore, a good mathematical model of these lubricated contacts between the moving parts is required since the quality of the bearing model will significantly impact the accuracy of simulation results. Although significant scientific work has been carried out by many researchers in the area of crankshaft and engine dynamics, there still remain many unresolved issues, e.g. the correct modelling of vibration transfer induced by bending vibrations of the crankshaft in the main bearings. In this paper, the theoretical framework for elastic MBD used for the moving crank train parts, e.g. connecting rods and crankshaft and for the engine block vibrations is outlined. Then, some approaches for modelling the hydrodynamic behaviour of journal bearings in the cranktrain assembly are described. The effect of the bearing models on the quality of simulation results, e.g. vibration and noise transfer prediction, is discussed. The bearing joint models used are validated by experiments through comparison of the calculated and measured accelerations at number of points on a large diesel engine.

1 Introduction Virtual design and prototyping in the development of new combustion engines and power units play a critical role in today’s automotive industry. Significant reduction of development time and cost can be achieved by high quality simulation results. At the same time, increasing power and speed of the vehicles, combined with demands for light and compact design, and resulting complex geometry of engine parts, require detailed and time consuming calculations in the process of engine design.

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To meet these challenging requirements, many sophisticated simulation tools have been developed, which are applicable in both concept and detailed analysis phases of engine development process. Results of numerical simulation of power units, as complex as they may be today, will no longer be satisfactory unless they allow conclusions to be drawn with respect to the specific stages of the development process. The requirements are to predict stress, durability, vibrations and specific vibro-acoustic phenomena. This is particularly true in regard to the cranks train as the central part of the engine. Due to increased efforts for detailed modelling of elastic multi-body systems and relevant non-linear body contacts, simulation models have become very complex and a high amount of computation time is necessary to calculate accurate results. Thus, another challenge is to deliver the results in the short time required for a fully integrated simulation solution in the different stages of the development process. Modem methods for simulation of engine dynamics consider the global movements, the coupled torsional and bending vibrations of the crank train parts and the hydrodynamic influence of the slider bearings under running engine conditions. Specific results are normally produced in both time and frequency domains, including the detection of possible resonance (e.g. of a flywheel) and the prediction of the strength of connected parts. The solution procedure commonly used in the crankshaft dynamic analysis is based on a combination of the multi-body dynamics and the Finite Element Method (FEM) [lo]. AVL have introduced the software EXCITE for this purpose, [I]. Due to the prediction requirements in the low frequency (stress) and the high frequency ranges (noise transfer), various detailed models are developed. Hence, requirement for high quality results on one hand and demands for less pre- and post-processing and calculation time on the other generate conflicting demands for the engineers. Therefore, many efforts have been made to automate the modelling process, e.g. for the complex design of the crankshaft [9, 111. Furthermore, the contact models have been developed for sliding contacts in order to obtain better results with reduced computing time [3, 81. In addition, an economical bearing model has been developed, which is able to capture the physical behaviour of journal bearings with less computing effort. The emphasis of this paper is on different approaches for modelling the non-linear behaviour of journal bearings in the cranktrain assembly. The advantages of the models are discussed and assessed by comparison of calculated and measured results. It is necessary and important to assess the effect of the various bearing models on the crankshaft vibrations and transfer of noise to the radiating surfaces of the engine. 2 Theoretical background

Because of the complexity of an internal combustion engine, the total mechanical system of the engine has to be divided into the coupled sub-systems in order to model its dynamic behaviour. The sub-systems represent the engine components, which have to be considered within the calculation model. For example, for the engine studied in the present work a power unit, a crankshaft and six connecting rods have to be considered.

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In addition, the chassis is represented by some rigid nodes, to which the power unit is connected by engine mount joints. The connecting joints, e.g. journal bearings, are modelled by non-linear contact forces acting between the components. 2.1 Body equations In modelling of the dynamics of engine components different types of motions are considered. These include global motion (e.g. crankshaft rotation, connecting rod movement) and local vibration motion.

The vibration model has been developed based on Newton's equation of momentum and Euler's equation of angular momentum as discussed in [2]. By transformation of these equations from the global coordinate system into the local component fixed coordinate system, the classical equation of motion for the linear systems can be obtained. It is given in the component fixed coordinate system and represents the dynamic behaviour of the total system of rigid partial masses, M.q+D.q+K.q=p' +fa'. (1) The bodies can be modelled either by beam-mass elements [9], or by three-dimensional (3D) volumes. Equation (1) is solved for the generalised displacements q, which contains translational and rotational degrees of freedom of all nodes.

Since equation (1) is derived from the component fixed coordinate system and the structural geometry can be considered to be linear due to small displacements, the structural matrices M , D and K are invariant in time domain and can be computed in a pre-processing step using commercial FE-tools such as ABAQUS or MSC.Nastran. at the right hand side of equation (1) is a sum The vector of external forces and moments f of exciting joint forces and moments f * and the external loads f a

f"'= f"+f' The f a is time dependant (forces, moments) including e.g. gas loads and the output torque. The non-linear excitation forces, f * , are given by joints, (e.g. contact models for the main bearings). Details of the joint models will be discussed in the next section. Governing equations for global motion of a component are derived based on the principles of momentum and angular momentum of discrete partial masses of the body. The equation of translational motion is written as follows,

where, the total body acceleration is,

I,,,, = x B + 2 . 4 .iB +(A, + 4). (x, + c,)

(4)

and the nodal acceleration reads.

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The equation of angular motion is expressed as,

A description of the variables used in equations (3) - (6) can be found in the Nomenclature part of this paper. 2.2 Joint equations

For the computation of non-linear contact forces and moments, f' , of each body, a set of joint models have been developed by AVL development engineers, and these have been implemented in AVL EXCITE multi-body dynamic software. The choice of the joints depends on the field of application. The model complexity and corresponding computational costs vary with the use of different types of joints. The joint models, which were used in the present study, are discussed in this section. The simplest joint connection of two bodies can be done by spring-damper functions,

f=f(kJointrdJ,i",,hr,~). (7) The coefficients k~,,, and d ~ ~ ~represent ,,, optionally non-linear stiffness and damping characteristics. TheAxandAtdenote the distance and its first derivative in time of the connected nodes. Depending on the application, the Ax has to be subdivided according to the used spring-damper arrangements. A simple node to node connection is sufficient for modelling a radial slider bearing whereas higher sophisticated spring-damper arrangements may be needed for frequency dependant applications (e.g., engine mounts) [ 5 ] . To practically model the journal bearings, a joint model based on the spring-damper function has been developed, which can connect one journal node to a number of shell nodes. This joint model is known as the NONL in AVL EXCITE software.

To accurately model the hydrodynamic behaviour of journal bearings, a modified Reynolds equation [4] is used,

Equation (8) is solved for the hydrodynamic pressure distribution P = P ( F , z , t ) of the oil film - in the lubricated region and the oil percentage, described as the fill ratio 0 =B(X,z,t) in the

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cavitation region between the bearing shell and the bearing journal. The excitation joint forces that will be applied to the discrete body nodes are obtained by integration of the hydrodynamic pressures. The joint based on the solution of the Reynolds equation is known as the elastohydrodynamic (EHD) joint. In AVL EXCITE, this is referred to as EHD2joint. Both the NOM, and EHD2 joint models have been used to model the main bearings of a large truck engine in this study.

2.3 General simulation procedure The general simulation procedure can be summarised as three main steps:

1) Pre-processing. For each (condensed) body of the multi-body system the table of degrees of freedom and geometry as well as stiffness, damping and mass properties have to be generated. In addition, the generation of external body loads (e.g. gas forces, output torque, etc) from measurement or pre-calculated data needs to be done. 2) Calculation. Due to the high non-linearity of the MBD system, the multi-body dynamic analysis is done in time domain. In each time step, the equilibrium has to be fulfilled for both bodies and joints and the entire system. In order to minimise numerical error, the implicit Newmark method is used [6]. 3) Post-processing. Within this step a data recovery of calculated acoustic results to the uncondensed set of degrees of freedom can be performed [7]. 3 Forced vibration analysis In this section, the general procedure for performing the forced vibration analysis with AVL EXCITE is described, with a particular reference to the work presented in this paper, where an inline 6-cylinder truck Diesel engine (Scania DC1201) is studied. 3.1 Analysis procedure In order to perform a forced vibration analysis, generally the following steps are necessary:

1) Creation of a 3D crankshaft assembly model including flywheel, damper primary/seismic mass; the crankshaft assembly can be represented either by a structured model [9, 111) or by a reduced 3D solid FEM model. 2) Modelling of connecting rods, representation of piston mass at connecting rod small end. 3) Modelling of the power unit structure including gear box and all add-on parts. 4) Calculation of natural frequencies and eigenvectors. 5) Defmition of boundary conditions, loads and coupling conditions between the different bodies. 6 ) Forced vibration analysis over 3 to 5 full engine cycles under ‘steady state’ condition. 7) Result evaluation. As the simulation has to be carried out in the time domain, large FE models have to be reduced to some representative nodes in order to reduce computing cost. Usually a combined static and dynamic condensation is performed. The static reduction is used for the nodes,

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where external loads have to be applied (e.g., nodes to apply gas pressure and piston side forces), and which are necessary to connect different parts (for example the nodes for the main bearings) or to mount the power unit. The dynamic behaviour of the remaining structure is considered by dynamic reduction (CMS), where the dynamic behaviour is described by vectors for the desired frequency range of up to 3000Hz. Figure 1 shows the entire FE model of the power unit of a 6-cylinder truck Diesel engine used in the present study. This model contains about 63000 elements and 75000 nodes. The model was condensed to 2600 master degrees of freedom, with 250 retained modes.

L

Figure 1: FE model of the power unit of an inline 6cylinder truck Diesel engine used in this study.

Figure 2: Springdamper arrangement for modelling engine mounts.

3.2 Boundary conditions In the present analysis, an inline 6-cylinder truck diesel engine manufactured by Scania was investigated. Further details of the engine will be presented in Section 4. In the multi-body model of the engine, the non-linear behaviour of the oil film in the main bearings is described by non-linear joints, which consist of a springdamper function (NONL) and detailed EHD model (EHDS), as described in section 2.2. Coupling conditions are defined in five sections of each main bearing to consider the effect of crankshaft journal misalignment. The spring characteristics in the NONL joint model are derived from the maximum gas force and the bearing clearance, and are scaled by the number of connected shell nodes as well as the number of the sections over the bearing width. The damping lies usually in the range of 1 to 50 Ndmm for this engine type. The spring and dampers are only active in direction of compression. The EHD joint needs bearing data such as diameter, width, clearance as well as oil viscosity. The main bearing clearance was estimated from measurements. In addition, pressure boundary conditions for the oil supply have to be defined. The main bearings have central oil grooves on the entire upper bearing shells.

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Some other special coupling joints are used to model the axial thrust bearing, the vibration damper and the guidance of the connecting rods in the cylinder liners. The engine is mounted at four points on the engine flange and at four points on the lower deck. At these points, the spring-damper joints are applied, as shown in Figure 2. If no measured data are available, the stiffness of these springs is chosen, so that the global movement at the mounting points is be below 0. I m m displacement in all directions. 3.3 Loads Measured cylinder pressures were used to calculate the external loads. The gas pressure is applied to the different connecting rods in the firing order 1 - 5 - 3 - 6 - 2 - 4. Additional to the gas pressure, a mean output torque is applied. This is assumed to be constant during the whole engine cycle, and it has to be tuned so that the speed stays nearly constant for the duration of the calculated engine cycles. The multi-body dynamic calculations were performed for the two different operating conditions, Le., 1100 rpm full load and 1900 rpm full load. 3.4 Creation of calculation model As soon as all the needed input data are ready, the calculation model can be created and simulation can be performed within AVL Workspace. Figure 3 shows the graphical representation of the calculation model of a 6-cylinder truck Diesel engine, where the main bearings are modelled as E m 2 joints.

Figure 3: GUI representation of an EXCITE MBD model of a Scania truck engine with main bearings being modelled by EHD2 joints

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4 Experimental analysis In this section, the set up of experimental tests to measure accelerations at some selected points on the engine is described. The measurements were conducted on a Scania DC1201 engine with the PDE-injection. The engine was mounted on a test rig, which contains the chassis with the transmission and wheels, but without the cab. The measurement and analysis of the measurement data was done using LMS Cada-X. Accelerometers of B&K 4394 were used to measure the acceleration on the bolts of main bearings 3 and 5. To do this a cube was screwed onto the bolt as shown in Figure 4. The accelerometers were screwed onto the cube (see Figure 5). In order to protect the accelerometers against oil contamination, the silicon was used.

Figure 4: The cube screwed on to the main bearing bolt.

Figure 5: The accelerometers screwed on to the cube.

The acceleration was measured in three directions relative to the engine block. It is noted that the x-direction is along the crankshaft rotational axis, while the z-direction is along the cylinder axis. The measurements were done on both bolts for main bearings 3 and 5. That gives a total of 12 accelerometers, as can be seen from Figure 6 and Figure 7. All measurements were performed in steady state from 900 to 2000 rpm with an increment of 100 rpm.

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Figure 6: The accelerometers on the bolts of main bearing 3.

Figure 7: The accelerometers on the bolts of main bearing 5.

5 Results and discussion

In this section, the calculated and measured results for a Scania truck engine are presented and discussed. 5.1 Comparison of measured and simulated accelerations It is essential that a simulation tool should be validated by experiments before it can be applied with confidence. There is no exception for AVL EXCITE. Although a variety of results can be extracted from AVL EXCITE simulations, in this paper only accelerations at some locations on the engine are discussed since these are available experimentally. Figure 8 compares the calculated and measured acceleration levels on the right bolt of main bearing 3 in the x-direction (denoted as MB3R:+x in Figure 6 ) for two engine speeds of 1100 and 1900 rpm over a whole engine cycle. As discussed earlier, in the calculations seven main bearings were modelled as NONL and EHD2 joints respectively. Figure 9 shows comparison of the calculated and measured acceleration levels on the right bolt of main bearing 5 in the y-direction (denoted as MBSR:+y in Figure 7) for two engine speeds of 1100 and 1900 rpm over a whole engine cycle. Figure 10 illustrates comparison of the calculated and measured acceleration levels on the right bolt of main bearing 5 in the z-direction (denoted as MB5R:+z in Figure 7) for two engine speeds of 1100 and 1900 rpm over a whole engine cycle.

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Figure 8: Measured and calculated accelerations in the x-direction on the right bold of main bearing number 3 over a complete engine cycle.

Figure 9: Measured and calculated accelerations in the y-direction on the right bold of main bearing number 5 over a complete engine cycle.

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Figure 10: Measured and calculated accelerations in the z-direction on the right bold of main bearing number 5 over a complete engine cycle. Generally, the predicted and measured accelerations are in good agreement in the domain. In order to further access the quality of calculated results, the time histories of accelerations depicted in Figure 8 to Figure 10 were transformed into frequency domain by Fast Fourier Transformation (FFT). The transformed results are presented in Figure 11 to Figure 13. In these figures, the acceleration level versus the 3rd (1/3) octave mean frequency is given in an acoustic relevant frequency range. The upper and lower bands of several measured engine cycles of the corresponding steady state condition are presented. Again, the calculations with both types of bearing joints correlate well with the measurements. Broadly, the acceleration levels predicted are slightly lower than those measured in the x- and y-directions, but higher than measured in the z-direction (vertical).

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Figure 11: Measured and calculated accelerations in the x-direction on the right bolt of main bearing 3 over the interested range of frequency domain.

Figure 12: Measured and calculated accelerations in the y-direction on the right bolt of main bearing 5 over the interested range of frequency domain.

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Figure 13: Measured and calculated accelerations in the z-direction on the right bolt of main bearing 5 over the interested range of frequency domain.

It is noted from these figures that in general, the Em2 joint type offers better predictions than the NONL joint type. This is to be expected. However, the computing cost associated with the EHD2 joint is very high. For the engine MJ3D system under investigation, the CPU times used for the simulation of five engine cycles at 1900 rpm are 31 hours and 1.4 hours respectively with the EHD2 and NONL joints on an Itanium 900 h4Hz HP workstation.

5.2 Predicted surface velocity levels In order to evaluate the effect of two types of bearing joints used on vibrations of the engine structure, some calculated surface velocity levels are presented in this section. Because a condensed model of the power unit has been used in the calculation of forced vibrations, a result data recovery was performed to obtain the velocities at the nodes on the outer surface of the power unit. The data recovery was done with MSC.Nastran. The recovered nodal velocities of the surface nodes are converted into velocity levels, VL by software AVL IMPRESS. The velocity level, VL, is a measure for structure borne noise and is calculated by,

[:I

V,=lOlog 7 =20log - , where,

vo = 5 x 10" (m/s), v is the nodal velocity in the direction normal to the surface. Figure 14 shows the distributions of the calculated surface velocity levels for 1 Wz 3rd octave band from the solutions with seven main bearings modeled by EHD2 and NONL joint types respectively. It should be noted that the absolute values are not given in the figure due to the confidentiality agreement. Nevertheless, it can be seen that the two distributions are fairly similar. The EHD2 joint produces modestly higher levels of vibration in the vicinity of

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the engine skirt. This is consistent with the acceleration levels calculated on the bolt of the main bearing number 5 for 1 kHz as illustrated in Figure 12 and Figure 13 earlier.

Figure 14: Surface normal velocity levels for 1 kHz 3rd octave band predicted with the EHD2 and NONL main bearing joints.

Another criterion for the evaluation of structural borne noise level is the mean velocity integral level. This is obtained by an averaged integration of the velocity levels for all nodes over the examined surface area for a defined frequency band. Figure 15 shows comparison of the mean velocity integral levels versus a number of 3rd octave centre frequencies, calculated over the complete oil pan surface area from the solutions with the EHD2 and NONL main bearing joints respectively. It is noted that agreement between the E m 2 and NONL joints is good despite there is about 5 dl3 difference at the lower frequencies of 630 and 800 Hz.

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Figure 15: Mean integral levels of normal velocities on the oil pan surface for 31d octave bands predicted with the EHD2 and NONL main bearing joints.

In summary, both the EHD2 and NONL bearing joint models are able to produce a reasonably accurate prediction of the vibration of the cranktrain studied. Generally, the quality of the results from EHD2 solutions is better than that from NONL solutions. However, this is on the expenses of high computing cost associated with the use of EHD2 joints. 6 Conclusions

In this paper, an approach to MBD dynamics of internal combustion engines has been presented. The joint models used for main journal bearings have been described. These include a so-called NONL joint based on the spring-damper function, and an EHD (known as EHD2) joint based on the solution of a sophisticated Reynolds equation. In addition, the analysis procedure for forced vibrations has been given with a particular reference to the engine studied. In order to validate the models used, experimental tests have been conducted on a Scania truck engine. Accelerations at a number of locations on the engine have been m measured. The calculated results have been compared with the measured ones. From the results presented, the following conclusions may be drawn: Calculated accelerations with both the NONL and EHD2 main bearing joints agree well with experimental measurements in both time and frequency domain. The quality of results from the solutions with the EHD2 joint model is better than that with the NONL model. However, the computing cost associated with the EHD2 joint model is markedly (about 20 times for the engine studied) higher than that with the NONL joint model. As a practical approach, the NONL joint model is sufficient for performing routine design analysis tasks.

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Nomenclature

270

Vector of the centre of gravity of the un-deformed body

Circumferential velocity of the shell body

Position vector of node i in the un-deformed structure (geometry)

Vector of the translatorial motion of the reference coordinate system given in the reference coordinate system of the body

Non-linear damping of a joint

Circumferential coordinate

External loads vector

Axial coordinate

External forces and moments vector

Sparse symmetric tensor of angular velocities

Force vector that is applied to node i given in relative coordinates

Sparse symmetric tensor of angular accelerations

Moment vector that is applied to node i given in relative coordinates

Sparse symmetric tensor of local rotations

Connecting forces and moments vector

Damping matrix of an elastic body in the bodies coordinate system

Clearance gap height

Inertia tensor of the un-deformed body

Non-linear stiffness of a joint

Mass moment of inertia of node i

Mass of node i

Stiffness matrix of an elastic body in the body-fixed coordinate system

Mass of the un-deformed elastic body

Mass matrix of an elastic body in the body-fixed coordinate system

Hydrodynamic oil film pressure

Local rotational displacement vector of node i

Vector of generalised displacements

Dynamic oil film viscosity

Time

Fill ratio, fraction of volume that is filled with oil

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u,

UJOUmOl

p*

Local translatorial displacement vector of node i

Wog

Average angular velocity of journal and shell

Circumferentialvelocity of the journal body

'

Vector of angular velocity of the body

Vector of non-linear inertia forces and moments resulting from the transformation of the equations of momentum and angular momentum to the body fixed coordinate system

References [ 11 AVL-EXCITE Reference Manual (Version 6.Q AVL LIST GmbH, Graz, 2003 [2] Bestle D.: Analyse und Optimierung von Mehrkiirpersystemen,Springer Verlag, Berlin Heidelberg, 1994 [3] Knaus 0.; Loibnegger B.; Herbst H.; Kreuzwirth G.: Influence of Structure Dynamics and Elasto-hydro-dynamic Contacts on Con-rod Design, MTZ 7,2002 [4] Krasser J.: ThermoelastohydrodynamischeAnalyse dynamisch belasteter Radialgleitlager, PhD Theses, Technical University Graz, 1996 [5] Loibnegger B.; Mikosch T.: Simulation and Correlation of Engine Mount Vibrations up to I kHz, JSAE, 2001 [6] Newmark N. M.: A Method of Computationfor Structural Dynamics, Journal of the Engineering Mechanics Division, 1959 [7] Offner G.: Mathematische Modellierung des Kolben - Zylinder - Kontakts in Verbrennungskraftma-schinen und numerische Simulation des durch mechanischen Kolbenschlag angeregten Koerperschalls (Mathematical Model of the Piston to Liner Contact in Combustion Engines and Numerical Simulation of the Structure Borne Noise Excited by Mechanical Piston Impact), PhD Theses, Technical University Graz, 2000 [8] Offner G.; Priebsch H. H.: A Numerical Model for the Simulation of Piston to Liner Contact Excitation considering Elasto-hydrodynamics, WTC Congress, 200 1 [9] Parikyan T.; Resch T.; Priebsch H. H.: Structured Model of Crankshaft in the Simulation of Engine Dynamics with AVL EXCITE, ASME Fall Technical Conference, Argonne, 2001 [ 101Priebsch H. H.; Krasser J.: Simulation of Vibrationand Structure Borne Noise of Engines - A Combined Technique of FEMand Multi Body Dynamics, CAD-FEM Users' Meeting, Bad Neuenahr - Ahrweiler, 1998 [ 111Rasser M.; Resch T.; Priebsch H. H.: Enhanced crankshaft stress calculation method and fatigue live evaluation, CIMAC Congress, Copenhagen, 1998

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Multi-body analysis and measurement of valve train motions M TEODORESCU, H RAHNEJAT,and S J ROTHBERG Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, U K

Abstract Valve train systems exhibit a number of complex motions, as the result of interplay between the large displacement dynamics of components, such as the valve, rocker arm and cam, and the small-scale interactions between the various load bearing surfaces. The frictional characteristics between the valve and the valve guide, the cam and tappet pair, the valve and the valve seat and the rocker and its retaining shaft play an important role in the deviations of inertial dynamics from the idealised required function, in the form of small amplitude vibration. These lead to losses described as frictional. At the same time, the non-ideal dynamics of the system lead to undesired contact conditions, such as separation in cam-tappet conjunction, valve spring surge and tilting motion of the valve. The study of the interplay between these phenomena, which are so widely separate on the physical scale, is a non-trivial problem, requiring combined numerical analysis and experimental effort. This paper concentrates on kinematics and kinetics of contact conjunctions. Keywords: multi-body dynamics, frictional contacts, torsional vibration, tappet spin

1- Introduction The ideal function of a valve train system is to synchronise the opening and closing of inlet and exhaust valves with the required thermodynamics of the combustion process. As such, a kinematic type mechanism is desired. However, timing requirements in the action of each valve and between any inlet-exhaust pair require the use of contacting pairs of suitable profiles. The existence of contact, particularly of lubricated nature, renders the problem one of complex non-linear dynamics [ 1,2]. This is further exacerbated by the translational imbalance of reciprocating elements such as the valve itself, which cannot easily be countered [3].

To reduce the inertial forces responsible for this, provisions are made to limit the mass of the moving parts, whilst maintaining a sufficient force to guard against loss of contact at the cam-

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tappet interface. This is usually achieved by means of suitably preloaded valve springs. Due to the high pressures generated in the combustion chamber, the coil springs must be of sufficient stiffness, which necessitates the use of fairly significant coil thickness constructions. The mass of such springs is not negligible, contributing to the inertial dynamics of the system [1,2,4]. Furthermore, due to significant variation in the applied combustion force with its sharp rise and fall rates, the retaining spring is subjected to conditions resembling shock loading, making it necessary to use a construction with irregular coil spacing to withstand the range of conditions encountered. Under certain operating conditions, however the spring becomes unloaded and undergoes large displacements, a phenomenon referred to as the valve spring surge effect [ 1,2,4]. This phenomenon is accompanied by loss of contact in the cam-tappet pair (known as the separation effect) and subsequent rebound of the tappet onto the cam surface. The motions described above are, therefore, quite complex, requiring combined detailed analysis and intricate measurement techniques under controlled conditions to gain a fimdamental understanding of use in the design process. Because of this, the current paper focuses on some specific and important aspects of this problem, namely the interactions between the inertial dynamics of the system, frictional interactions in the tappet assembly and the prevailing conditions in a spinning tappet-to-cam contact. This is a critical part of valve train behaviour, because these contacts account for a major proportion of the frictional losses in the system. In the case of a spinning tappet, the sliding motion of the contact line affects the ideal entraining motion of the lubricant and can lead to shear thinning. However, if the spinning action was to be eliminated, any cessation of entraining motion could lead to depletion of the lubricant film and the consequent repeated asperity contact at that location could lead to severe scuffing of the tappet [5]. The tilting secondary dynamics of the tappet causes it to impact with its guide [6]. This misaligns the contact, and thus truncates it, effectively further complicating the kinematics of contact by a reciprocating motion which affects the mechanism of fluid film lubrication.

2- Experimental investigation 2.1- Experimental set-up Tappet spin occurs as the result of the moment imbalance generated by the impact and frictional forces between the tappet and tappet bore and fluid film traction in the cam-totappet conjunction [SI. The first step in the investigation is to accurately measure tappet spin as the valve goes through its cyclic translational motion. This is quite difficult due to the complexity of the aforementioned interplay between the various motions of the valve train system. In a controlled experiment, where the effects of other sources of vibration, such as camshaft elasticity are to be minimised, it is preferable to motorise rather than fire the engine. Furthermore, an engine with lower inertial imbalance and lighter loaded contacts is more suited to such experiments. Thus, a small 5 hp single cylinder Honda IC engine is mounted onto a rigid frame and driven by a DC servomotor via a suitable coupling. A torque meter mounted between the servomotor and the engine measures the frictional losses in the valve train, since the piston assembly which usually accounts for the major portion of frictional losses in any IC engine and the connecting rod are removed. The experimental set up is depicted in figure 1 .

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With this motorised arrangement, it is also possible to cut away observation windows in the engine casing, exposing the cam-tappet pair. Through these small openings, it is also possible to drip feed the cam-tappet contact with small quantities of lubricant, in order to guard against wear. Figure 2 shows a view of the inlet and exhaust cam-tappet contacts.

Figure 1: The experimental set up

Figure 2: A view of inlet and exhaust cam-tappet pairs Figure 3 shows the laser beams incident on the target surface. The two parallel laser beams from the laser rotational vibrometer are able to resolve the spinning of the tappet from any tilting (Le. pitching) motion.

Figure 3: Laser beams target onto a flat tappet In the ideal configuration, the two laser beams, which are 8 mm apart, are directed onto the target surface of the spinning tappet in a plane that is perpendicular to its axis of rotation. The measurement can be adversely affected if the translational motion of the tappet causes the

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beams to fall beyond its edges. To guard against this problem and to insure the laser beams are perpendicular on the target during the experiment, a very light sleeve is made to fit over the tappet (see figure 2). The very small error resulted from the tappet tilt in its bore (smaller then 0.06': see figure 5) was integrated into the general error of the measuring system.

2.2- Experimental findings Figure 4 shows the cam lift, taking place over the event angle. It corresponds to opening of the valve, reaching the fully open position at the maximum cam lift at the position of its nose after 0.05 s. The valve commences closure thereafter, reaching fully closed position at 0.1 s. Due to the eccentric position of the line of contact between the cam and tappet with respect to the vertical axis of symmetry of the latter, a moment is applied to it, which tends to spin the tappet, which is resisted by a torque acting upon the tappet, caused by interactions between the tappet and tappet bore. These points are described in more detail later. 6.E-03 1

l.E-03

-

5

Expehental spin

O.E+OO

"

--

-

0.00

0.02

0.04 0.06 Time <s>

0.08

-20

-25 0.10

Figure 4:Cam lift and the corresponding tappet spin In addition, the figure shows both experimentally measured and numerically predicted tappet spin velocity. It can be observed that highest spin velocity occurs at the maximum lift, where the largest drag torque is introduced as the cam nose comes into contact with the tappet, whilst the tappet assumes the ideal vertical orientation with respect to its bore (see figure 5, obtained numerically), thereby reducing the resistive torque by reduced contact friction between tappet and tappet bore. Prior to and after the time of maximum spin velocity (i.e during the opening and closing part of the cycle), the driving torque is reduced due to lower cam-tappet contact force, whilst the tilted attitude of the tappet with respect to its bore makes for increased friction due to a larger boundary friction contribution and a reduced or diminished fluid film there. The slight oscillations on the experimental trace are surmised to be due to stick-slipping in the misaligned contact of tappet-to-tappet bore.

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0.06 0.04 ~

I

0.02

cn

al

7J

-cn

0.00

a

-0.02

c

-0.04

’

-0.06 0.0

L -20 0.0

0.0

0.1

0.1

0.1

Time <s Figure 5: Cam lift and the corresponding tappet tilt

It is clear that a fundamental understanding can only be achieved through use of a numerical model, described below. Such a model would be most useful when made as simple as possible. One complicating factor would be any significant inertial effect. Other complications may also arise as a result of cam-tappet contact separation, valve spring surge and camshaft wind up and wind down [ 1,7]. These are the precise reasons for the use of a low power motorised engine, with reduced loading and low inertial imbalance.

3- Dynamic analysis This paper is confined to the analysis of cam-tappet motion, which is a combination of rotational motion of the cam relative to the tappet, which is usually considered in the tribological studies of this contact. As the cam goes through its cycle from its base circle contacting the tappet onto its flank,followed by its nose and back to the base circle, the tappet goes through a combined primary translational and secondary tilting motion with respect to its bore, as shown in figure 6. It is clear that as the cam nose comes into a loaded contact with the tappet surface (as described above), two important events take place. Firstly, the contact force increases, whilst the speed of entraining motion decreases. This depletes the lubricant film thickness, leading to the promotion of, or dominance of, boundary lubrication and increased friction. As a result, there is a tendency for direct contact of the contiguous surfaces and their scuffing by repeated cyclic action along a given line of contact. To reduce this problem the centre of gravity of the tappet, G in figure 7 (a), is offset from the centre of the axis of symmetry. This causes a drift in the line of contact due to the introduced sliding motion (described below), thus reducing a defined and evolving wear scar. However, the sliding motion can itself contribute to the thinning of the lubricant film and increased frictional losses. In figure 7(a), the line of contact is instantaneously represented by the horizontal line, indicated on the figure. The centre of contact is at a radius d from the centre of the tappet, the eccentric location of which from the lines of symmetry are shown by the quantities: e and 6 . The tappet spin velocity is shown on the figure, with the drive torque due to cam contact and resistive torque.

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Figure 6: Cam-Tappet motion The free-body diagram is complete with tappet -tappet bore contact forces (see Figure 7(b)). The forces to be determined are wb/, wb7. and P, from moment equilibrium condition around point "A", and from the equilibrium of forces in X and Y directions. Thus: D

~ , c o s ( y ) [ d c o s ( y ) + e , ] + -sin(y)-W,, ~,, 2

[L/

" I

cos(y)--sin(y) 2

- F/ ( L - . L / ) -

-~,sin(p)l,cos(y)+~,cos(p)[e,+ ( l j -~,)sin(y)]-y~~= O

(1)

W,,+ P , s i n ( y ) + F 8 c o s ( y ) - W , , - P , s i n ( p ) = 0 p. cos(y)- Flc sin(y) - Fj - rnK - P, cos(p) = 0 where:

Since the tilt angle y is very small as can be observed in the results of figure 5 , the following set of simplifying assumptions have been made (refer to figure 7(b)):

=

sin(y) 0 d+egzd

and and

cos(y) a 1 l,-lg+eg~l,-lg

Thus, the contact forces become:

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Cam. Tappet middle conlad llns

Figure 7: The eccentric position of line of contact with respect to the tappet centre and the kinetics of the system 4- Simulation results and comparisons with measurements

The numerical predictions for the tilting motion of the tappet have already been shown in figure 5 above. This causes friction forces and resisting torque with the tappet guide as already described in the case of figure 4, together with the experimental findings under identical conditions. The engine speed can be varied to see the effect of the competing kinetics upon tappet spin characteristics. Figures 8(a) and (b) show the results of measurements and predictions at different engine speeds respectively. 0-

4-

B s

E

;-12 0

z

0 -18 -

-24

0

0.1

0.2

0.3

0.4

I 0.5

Time cs>

Figure 8(a): Measured tappet spin velocity at different engine speeds

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.

300 rpm

500rpm

-24

0.0

0.1

0.3

0.2

0.4

0.5

Time

Figure 8(b): Predicted tappet spin velocity at different engine speeds Good agreement can be observed between numerical predictions and measurements, both qualitatively and quantitatively. The percentage error does not exceed 15%. The small differences are due to a number of factors. Firstly, the model of tappet-tappet bore interactions is based on an approximate pair of short bearings, which is described in detail by Teodorescu [5] and Zhu [8]. In reality the contact is more complex. Secondly, there are some problems with sampling from the laser torsional vibrometer output to precisely capture the tappet spin cycle, although this is a relatively minor problem. It is interesting to note the rather fast nature of tappet spin velocity for relatively low engine speeds. This is of the order of 180-200 rpm for an engine speed of 500 rpm. However, this is in accord with other experimental investigations in the case of higher powered engines, where spin velocities of the order of 1200 rpm have been observed for engine speeds of similar value [9]. The variation of spin velocity with engine rpm can be described physically in terns of the contact conditions between the cam-tappet and tappet-to-tappet guide. At low speeds the lubricant film thickness is diminished in both case. However, in the case of the cam-tappet contact the tappet spin is not the only velocity that promotes entraining motion of the lubricant, whilst in the case of tappet-to-tappet guide contact this is usually the case. Figure 9 shows that the speed of entraining motion in the cam-tappet contact has two velocity components; one due to tappet spin which alters along the line of contact as a function of distance from the axis of symmetry, and the other being due to the relative rotation of the cam with respect to the tappet being the same for all contact localities, and which is the dominant contributory factor to lubricant entrainment. Thus, the increased friction in the case of the tappet-to-tappet guide contact dominates, and the resistive torque here is greater than the driving torque of the cam contact, thus a reduced spin velocity is observed. As the engine speed is increased the fluid film lubrication in the tappet-to-tappet guide improves, reducing the resistive friction torque and the spin velocity is increased.

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The center of the

Cam n tappet contact line

Figure 9: Kinematics of cam-tappet contact The kinematics of the contact, as shown in figure 9, reveal the interesting point, in that the speed of entraining motion is not merely due to rolling motion of the contiguous solids, but also as a result of the variations at each point of contact due to spin of the tappet. This is not usually taken into account in the tribological studies reported in literature, and should be considered. However, it makes for much longer analysis of contact conditions. To find the local entraining velocity, the surface velocities of the cam and the tappet at the sliding point of contact are obtained (in the same direction), without regard to the spinning motion. The additional spin velocity of the tappet relative to the cam is obtained, clearly proportional to any contact locality's eccentric position relative to the spin axis. Then, it is clear that lubricant entrainment takes place as the average surface velocities of the contiguous solids in contact, or:

1 u =-(v, 2

+V,"

+ vficosa)

(4)

The figure also shows that the sliding velocity V, between the cam and tappet at any location can be obtained as the vectorial deduction of V, and Y,. The friction force acts in the opposite sense to this velocity and produces the driving torque over the moment arm d, in figure 9. Clearly, the important issue with cam-tappet contact is to enhance the formation of a coherent lubricant film, and reduce any contribution from boundary lubrication, thus improve engine efficiency. Prediction of film thickness between cam and tappet is, therefore, very important. This, however, normally requires transient analysis of lubricated contact, which has been reported in reference [lo]. Here a quasi-static analysis can be carried out in small intervals of time with an extrapolated lubricant film thickness equation for finite line contact geometries such as the one reported in reference [1,6], and including the effect of squeeze film motion (for normally approaching and separating surfaces), in addition to the above mentioned contact kinematics. In this manner rapid prediction of lubricant film thickness can be made throughout the cam-cycle. This is shown in figure 10 below for the engine speed of 500 rpm, and the corresponding tappet spin velocity variation in figure 4 or 8(b).

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Figure 10: Lubricant film thickness time history throughout cam cycle Note that the film thickness increases from an approximate thickness of 70 nm when the tappet is in contact with the cam base circle at the beginning and the end of this figure. As the contact moves onto the cam flanks the effective radius of curvature increases and the speed of entraining motion is enhanced because of both cam surface velocity and tappet spin. The film thickness rises to nearly 0.3 pm. As the event angle leading to the cam nose commences, the effective radius of curvature begins to reduce. Additionally, the tilting motion of the tappet also increases the resistive torque from tappet-to-tappet bore contact, thus slowing down of tappet spin occurs in a dramatic manner. This combination of events diminishes the effective speed of entraining motion, thus reducing the film thickness. Immediately prior to the cam nose contact, the relative motion between the surfaces without tappet spin changes direction. This momentary cessation of entraining motion removes the most effective mechanism of lubrication (Le. the wedge effect). In such circumstances, and in practice, squeeze film action through mutual approach of the contiguous surfaces acts in a very modest manner to entrap lubricant to separate the surfaces. However, the effectiveness of this is very limited indeed, thus the reason to introduce off-set tappets and encourage spin action to enhance lubrication. Figure 10 shows the greatly reduced film in these regions, prior to, and immediately after the cam nose contact, referred to as film thickness minima. Note that even with spin, the very small film of a few nanometres is not improved, indicating that for a very short period of time boundary lubrication is dominant and the tappet spins through boundary friction torque. Although from here to the cam nose the lubricant film thickness is enhanced due to increased entraining action, the film thickness is quite low (approximately 40-50 nm) and the driving spin torque dominates the resistive tappet-to-tappet guide torque. Just after the cam nose contact the next half-cycle (Le. the closing part of the cycle) is almost identical to the previous, except that as the lift diminishes and the next minima is approached, loss of contact occurs, resulting in rapid oscillatory behaviour in the film due to jounce and rebound, as depicted in the figure.

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5- Conclusion The paper has described fundamental physics of valve train behaviour for cam-tappet and tappet-to-tappet guide contacts, including the complex prevailing kinematics of rolling, squeezing and spinning motion of the former and tilting and sliding of the latter. Very good agreement is also obtained between the simplified tribo-multi-body dynamics model and experimental results. Acknowledgements Authors wish to express their gratitude to SKF Engineering Research Centre for financial support extended to this research project. Nomenclature: apparent contact area minimum separation film for tappet-tappet bore contacts radial clearance between the tappet and the bore tappet diameter boundary friction component of the cam-tappet friction force total friction forces between the tappet and the tappet bore friction forces on the tappet-tappet bore interface cam-tappet friction force tangential tappet-tappet bore friction force viscous friction component of the cam-tappet friction force inertia moment of the tappet around the vertical axis cam width tappet section length tappet length normal force on the cam-tappet contact force between pushrod and tappet cam instantaneous radius of curvature cam base circle radius cam instantaneous radius of curvature spinning torque applied on the tappet tappet-tappet bore resistant torque equivalent surface velocity at the tappet-tappet bore interface translation component of the cam-tappet contact point velocity on the tappet rotation component of the cam-tappet contact point velocity on the tappet cam-tappet contact point velocity on the tappet sliding velocity tappet velocity along tappet bore axis; velocity of the cam-tappet contact point on the normal forces on the tappet-tappet bore interface distance from the cam-tappet contact line middle point to tappet bore axis distance between the friction force direction and the tappet centre eccentricities for the tappet sections centre of mass eccentricity from tappet bore axis

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oil film thickness tappet lift distance from the pushrod application point to cam tappet interface load carried by the asperities tappet angular velocity around tappet bore axis tilting angle distance between the centre of the cam tappet contact line and X-axis dimensionless eccentricities for the tappet sections oil viscosity oil viscosity in the tappet-tappet bore interface separation film for cam-tappet contact separation film for tappet-tappet bore contacts boundary friction coefficient Eyring shear stress is the rate of change of the shear stress with pressure camshaft angular velocity References:

[l] Kushwaha, M., Rahnejat, H., and Jin, Z. M. (2000)Valve-train dynamics: a simplijied tribo-elasto-multi-bodyanalysis. Proc. Instn. Mech. Engrs. 214 Part K: 95-110 [2]Chen, F. Y.(1982)Mechanics and Design of Cam Mechanisms Pergamont Press [3] Rahnejat, H.(1998)Multybody Dynamics: Vehicles, Machines and Mechanisms, July 1998 (Professional Engineering Publications and Society of Automotive Engineers). [4]Kim, W.J., Jeon, H. S., and Park, Y. S., Contact Force Prediction and Experimental Verijication on an OHC Finger -follower - type Cam - Valve System, Experimental Mechanics, June 1991,pp. 150-156 [5] Teodorescu, M. (2003) Modular Approach for Valve Train Dynamics and Friction Simulation with in situ Experimental Validation. PhD thesis, Wayne State University, Detroit, MI [6] M. Teodorescu, M. Kushwaha, H. Rahnejat, D. Taraza: Transient Analysis o f a Line of Valves with Camshaft Flexibility and Valve-Spring Surge Eflects Proc. 2003 ASME International Mechanical Engineering Congress and RD&D Expo November 15 - 21, 2003, Washington, D.C., USA [7] Koster, M. P. (1975)Effect offlexibility of driving shaft on the dynamic behaviour of a cam mechanism. Trans. ASME, J. Engng for Industry, 595-602 [8] Zhu, G.and C.M. Taylor, Tribological Analysis and Design of Modern Automobile Cam and Follower. Engine Research Series, ed. D. Dowson. 2001,London and Bury St. Edmunds, UK: Professional Engineering Publishing Limited. [9] Schmidt, A. (1997). A Contribution to Cam Tappet System Tribology. MTZ Motortechniche Zeitschrift 58(1): 20 - 27. [ 101 Kushwaha, M. and H. Rahnejat, (2002)Transient elastohydrodynamic lubrication of finite line conjunction of cam tofollower concentrated contact. Journal of Physics D: Applied Physics, 2002. Vol. 35(No. 3): p. pp. 2872 - 2890.

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Drivetrain Dyn amics

The torsional vibration of gear boxes with backlash M L COLTRONA and B J STONE School of Mechanical Engineering, The University of Western Australia, Australia

Abstract: The study of vibration with backlash in geared systems has been a popular topic for academic research in recent years. A subset of this research deals with the subharmonic response of gear pairs with backlash. A review of this prior research reveals that it may be complemented in two ways. Firstly, there is a need for experimental work that allows external excitation of the gear pair at a frequency other than their own meshing frequency. Secondly, an analysis of the type of damping employed by the physical model (viscous and/or impact) is required, since including impact damping seems more appropriate and yet is not evident in prior work in this area. Mathematical models for both the impact damping and viscous damping cases have been programmed into Matlab’s SIMULINK environment. Both models allow for external excitation at an arbitrary frequency and can (optionally) include the effects of time varying mesh stiffness of the gears on the system. Stable subharmonic motions were found to exist for both models; thus, the viscous damping model was favoured due to its simplicity -justifying its use in prior research. The simulation results show that subharmonic motions only develop if the system is given a certain initial energy input. An experimental rig has been constructed, but conclusive evidence of subharmonic motions is yet to be detected in it - probably because precise (and unknown) combinations of initial conditions are required for such motions to occur. These findings have implications for the designers of geared systems that may have backlash; subharmonic motions can lead to different-than-expected responses if linear models are employed that ignore backlash. NOMENCLATURE C Viscous damping coefficient [Ns/m2] or impact damping ratio d Half of the total backlash amount [m] Reduced external excitation function [N] F(t) Moment of inertia of the ilh gear [kg m2] Ji Time varying mesh stiffness function [N/m] k(0 External moment applied to the i* gear [N/m] M,(t) n Number of cycles of output completed during one input cycle

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t to

To tI (2

T,

xo xo

5 p,(t)

w oi(5)

wn XI X2

Time [SI Time at which gear teeth pass through the centreline [SI Time at which teeth separate in steady state [SI Time at which gear teeth touch in steady state [SI Time taken until teeth pass through centreline after separating [SI Time at which maximum displacement occurs [m] Amplitude of external excitation [m] Initial displacement of the mass [m] Viscous damping ratio Angular coordinate of the i“ gear [rad] Forcing frequency [rads] Constant mean angular velocity of the ithgear [rad/s] Natural frequency [rads] Displacement, velocity and acceleration of the mass Displacement, velocity and acceleration of the frame

INTRODUCTION The dynamics of nonlinear vibrating systems has been a popular topic for academic research, possibly owing to the complexities and unusual phenomena that such systems exhibit. The backlash (or ‘clearance’) nonlinearity has been an area of particular interest, and was investigated as early as 1933 by Buchold. Since that time, there have been refinements in both experimental techniques and mathematical modelling owing to technological advancements. Backlash is present in almost all real systems - it may be incorporated purposely to prevent binding, or it may be the result of loose couplings, wear, inaccurate assembly or poor design. Whichever the case, the prevalence of backlash in dynamical systems justifies a detailed analysis of its effects. One such effect is the development of subharmonic motions. Subharmonic motions arise in an oscillating system when the output of the mechanical oscillator vibrates at a frequency usually equal to an odd fraction of the forcing frequency. The analysis of subharmonic motions is important for a number of reasons; firstly, from a vibration control viewpoint, understanding subharmonic motions can lead to a better understanding of what would otherwise be an unpredictable vibration response, and secondly, for safety reasons. Take for instance a subharmonic motion oscillating at one-third of the excitation frequency. When the excitation is at three times a resonant frequency the system will be vibrating at resonance (see Stone and Jones (1964)). This is significant because resonance can lead to catastrophic failure, at the worst, or increased levels of noise, wear, and vibration, at best. A review of prior work reveals that whilst vibration with backlash has been investigated from a number of perspectives, the study of subharmonic motions has received less attention. Furthermore, key limitations evident in prior work leave considerable room for improvement. These limitations include the choice of damping model and frequency of external excitation. An analysis of the initial conditions leading to the development of stable subharmonic motions is also absent from the literature. Thus, the main objectives of this paper are: to investigate the effect of backlash on the vibratory response of a nonlinear dynamical system, to investigate the type of damping model which is most suitable, to investigate experimentally the phenomenon of subharmonic motions (at a range of excitation frequencies), and to examine the initial conditions required for subharmonic motions to develop and become stable.

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PREVIOUS WORK One of the first analyses of vibration with backlash was conducted by Stone and Jones (1964) who modelled and observed subharmonic motions (having a period equal to odd multiples of the forcing period) in a single degree of freedom axially vibrating system with symmetric backlash. However, analysis of the subharmonic motions was made tedious by the computer power available at the time, and damping was ignored. Their experiments would only yield subharmonic motions when the mass was given some additional impulse, suggesting that subharmonic motions required some input of energy before they could become stable. Since the late 197Os, the investigation of backlash in geared systems has become increasingly popular. Significant contributions to the literature have been made by: Azar and Crossley (1977); Singh, Xie and Comparin (1989); Kahraman and Singh (1990); El-Saeidy (1991); Kahraman and Singh (1991); Padmanabhan and Singh (1992); Padmanabhan and Singh (1995); Kahraman and Blankenship (1997); Theodossiades and Natsiavas (2000); Wang, Zhao and Manoj (2002) and Litak and Friswell (2003). Some of these papers were able to demonstrate the existence of subharmonic motions with a period of up to nine times the excitation period. Despite all this research, a series of common limitations underlies these papers. In the subset of these papers where the analysis of subharmonic motions was a focus, experimental evidence only included excitation at the gear meshing frequency (or GMF). This is a serious limitation because in a real gearbox, torsional excitation can arise from more sources than simply the meshing of the gears of interest. For instance, there may be another pair of gears meshing at a different frequency in the same gearbox, or, for example, there could be vibration from an engine travelling through the drive train. These papers also reported experimental results obtained from a gearbox test rig with a mean preload applied to the gears; a situation which is unrepresentative of situations where there are rattling gears under no load. Additionally, none of these papers accounted for backlash in the damping. Thus, despite the considerable attention already given to the torsional vibration of backlash gear pairs, there is an apparent need for experimental results on an unloaded gear pair with a periodic external excitation at a frequency other than that of the gear meshing frequency. This has applications for understanding neutral gear rattle and nonlinear vibration in general. Also, it is possible that modelling of the system may be improved by incorporating impact damping, rather than linear viscous damping (as used by Kahraman and Blankenship (1997), and other authors in the area), and this should be investigated. Whilst the concept of impact damping is not new and has been applied to nonlinear vibrating systems before - as stated by Shaw and Holmes, (1983) - it does not appear to have been used in studying the vibration of backlash gear pairs. A closer look at the energy input required for the development of stable subharmonic motions is also warranted, so that systems that are at high risk of developing them can be identified. Conversely, this type of analysis will identify situations when subharmonic motions are less likely to occur, and may therefore be disregarded.

THEORY The complete theoretical derivations developed are presented in the final year honours thesis of Coltrona (2003). A number of subtly different mathematical models were used in order to examine the effect of different assumptions on the accuracy of model predictions. This also

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facilitated a comparison of the effectiveness of the various models employed. All of the models are based on a similar representation of the backlash gear pair, as illustrated in Figure 1.

Figure I . Model of backlash gear pair. The total amount of backlash in the system is given by 2d in linear form. (Adaptedfrom Blankenship and Kahraman, 1995). The backlash gear pair may be modelled as two circular inertias fixed to rotate about a point in space connected to each other by a linear spring (with backlash) representing the mesh stifhess. By modelling the damping as either impact and/or viscous damping, and by modelling the mesh stiffness as either constant, or as a function of time, six variants of the mathematical model are possible: 1

Viscous damping model with constant mesh stiffness;

2

Viscous damping model with time varying mesh stiffness;

3

Impact damping model with constant mesh stifmess;

4

Impact damping model with time varying mesh stiffness;

5

Impact and viscous damping model with constant mesh stiffness; and

6

Impact and viscous damping model with time varying mesh stiffness.

Since the focus was on quantifying the difference between the viscous and impact damping models, the last two models, though intuitively appealing, are inappropriate. As such, only the first four models have been analysed. Furthermore, a number of different analyses of a given model are possible. These include estimation of the system natural frequency, time domain response analysis and steady state response analysis in the frequency domain.

Mesh stiffness The linear spring in Figure 1 is used to represent the stiffness of the meshing gear teeth in the physical system. Under the action of an applied load, each gear tooth will deflect elastically, because it is made of a material with finite stiffness. The total deformation of a single pair of meshing teeth consists of a combination of bending and shear deflection, axial compression, and hertzian deformation at the point of contact (see, for instance, El-Saeidy, 1991). In practice, it is much simpler to infer the mesh stifhess from experimental procedures, than to derive it from material properties. Notwithstanding this, for a detailed derivation of the mesh stiffness from first principles see El-Saeidy (1 991).

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The somewhat complicated system of Figure 1 can be shown (Coltrona (2003)) to yield the same general equations of motion as the system shown in Figure 2. This is the same model as investigated by Stone and Jones (1964); hence similar results were expected.

Figure 2 Simplijied mathematical model. The reduced equation of motion actually represents the equation of motion of this much simpler axial oscillator.

Backlash Backlash in the (simplified) model is accounted for as a gap of total distance 2d at either ends of the mass. The combination of a linear spring and a linear amount of backlash greatly reduces the complexity of the model without compromising its accuracy appreciably. Obviously, this gap would tend to zero for a system without backlash. However, this must be applied with some caution; decreasing d to zero does not mean that both springs are in contact with the mass the whole time - it simply means that one of the springs is in contact with the mass at all times.

Viscous damping For a nonzero value of ‘cyin Figure 1, a viscous damping model is assumed. In this model, dashpots with viscous damping coefficient ‘cy are used to represent equivalent viscous damping from the bearings as well as the meshing gears. The damping force is modelled as being proportional to velocity, and thus is not affected by whether or not the gears are in mesh or between the backlash. Furthermore, this model does not incorporate any energy loss upon impact of the gear teeth, as may intuitively be expected to be the case in the physical system. The advantage of the viscous damping model is that it is relatively simple to implement in simulations.

Impact damping For the pure impact damping model, the dashpots shown in Figures 1 and 2 would be removed and energy loss in the system would be accounted for as an instantaneous reduction in (angular or linear) velocity upon impact. This means that in the pure impact damping model, there is no energy loss whilst the teeth are between the backlash distance, and after they have initially impacted. The proportion of angularllinear velocity remaining after impact is essentially a material property - for materials with higher energy dissipation, this proportion will be lower.

General mathematical model The model in Figure 2 may be considered a general model in the sense that ‘c’ may be altered between zero and a non-zero value (to represent the absence or presence of viscous damping)

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and k(t) may represent either a constant mesh stiffness or time varying mesh stiffness. As noted earlier, impact damping may be modelled as an instantaneous fractional velocity reduction upon impact. In this model, the so called ‘static transmission error’ is ignored as it is assumed that the gears are made perfectly (since the focus of this study is backlash and not gear profile imperfections). Setting F(S, = 0 whilst making Xo a nonzero number (see Figure 2) changes the system only slightly; instead of a force input on the mass (i.e. a moment input between the gear teeth) there is a positional excitation of the frame (i.e. an angular excitation of the input gear shaft). The only effect this has on the equation of motion is that the forcing term is replaced by the positional excitation term. This was implemented so that more direct comparisons may be made with the work of Stone and Jones (1964).

Time domain simulations The nonlinear term stiffness results in a so called ‘stiff nonlinear differential equation; that is, the system involves rapidly changing components together with slowly changing ones (Chapra and Canale, 1998). In general, sophisticated mathematical techniques such as the harmonic balance method are required to solve such equations. These methods generally impose certain frequency criteria on the solution and involve tedious Fourier transforms and matrix algebra. Alternatively, approximate solutions may be found by numerical integration techniques specially designed for such systems that are available in computer packages such as Matlab.

soop.

Delay

Anlmrtion

B

x out

Figure Figure 3. Impact damping, time time varying varying mesh mesh stiffness stiffness model. model. The The overall overall model is shown shown at the the top, and the the time time varying varying mesh mesh stiffness stiffnesssubsystem appears below. below.

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The equations of motion for both the viscous and impact damping cases, assuming both constant and time varying mesh stiffness were programmed into the Matlab SIMULINK environment in block diagram form. The most complicated model is obviously the impact damping model with time varying mesh stiffness. The block diagram for this model is shown in Figure 3. The Matlab fimction ‘Impact Damping’ is a simple logical test that reduces the velocity to a fraction ‘c’ of its former value if an impact has just occurred. Each of the SIMULINK models required a Matlab m-file to control the inputs to them. In particular, a special m-file was created that automated the frequency response calculation by repeatedly running a given model for a certain length of time at a number of frequencies and recording the output at each stage.

Steady state analysis Stone and Jones (1964) presented a steady state analysis of a single degree of freedom system with symmetric backlash allowing calculation of the ftequency response. The analysis could be extended to include subharmonic motions in their axially vibrating system. If it is assumed that the position of the input gear is known exactly, (Le. it is assumed that the input shaft is a rigid abutment vibrating sinusoidally) and if damping is ignored, their analysis may be applied to a backlash gear pair, with some adaptation required. The analysis is based on matching conditions at different points throughout the meshing cycle, as depicted in Figure 4 (along with the results), and requires the assumption of constant mesh stiffness. x2 (output disolacement)

t

Backlash centreline 6

Teeth separate Y

’

t=t

t=tn

-Stone&

5

Jo1~s(1964)

?lKOretical

4 A

$ 3 2

Stone & Jones (1964) Expe*ntal C o h n n (2003) Simulated

1

0 0

0.5

1

1.5

2

2.5

3

3.5

4

FracIbnofnaId hqwncy(whm)

Figure 4. Steady state motion (halfof one cycle) and response curves (Afrer Stone and Jones (1964)). n = 3 indicates a sub-harmonic motion.

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At time t = to the gear tooth passes through the centreline of the motion with constant angular velocity, hitting the other gear at time t = tl. After this point, the mesh stiffness defines the motion of the gear until the teeth separate, with the same speed as before impact. The full derivation of this model is quite complicated, and draws heavily on trigonometric identities. The results of this model (as calculated by Stone and Jones (1964)) are discussed below and compared with the time domain simulation results graphically in Figure 4 (b).

EXPERIMENTAL RIG An experimental rig was designed and built with the aim of investigating subharmonic motions in a physical system. After considering a number of alternatives, the rig shown in Figure 5 was chosen.

Plastic gear The bearings on this top shaft can be moved up and down to adjust the cen distance between the shafts. Output (measured with laser vibrometer) Input (sinusoidal - from torsional exciter)

c)

c

e

int of gear contact with

-tachable

steel inertia

Fixed bearings

Figure 5. Experimental rig (a) General view; (b) elongated holes allow for adjustment

of the backlash distance; (c) Schematic view Plastic gears were used in order to reduce the mesh stiffness (and thus the system natural frequency) owing to limitations in the range of excitation frequencies that can be excited by the torsional exciter. Although most gears in industrial and automotive use are metallic, the results gained from this rig are still applicable elsewhere because the model under investigation takes material properties into account. Detachable inertias were added at the end of each shaft to increase the effective inertias of each gear and therefore provide an additional reduction to the natural frequency of the system. By making the inertias detachable it was

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possible to test the theoretical relationship between natural frequency and mesh stiffness (see Coltrona, 2003) by running tests with different combinations of inertias attached. The shaft centre distance may be changed by moving the top set of flange bearings in their elongated holes (Figure 5 (b)). This of course alters the amount of backlash between the gear teeth, allowing different configurations to be tested. The shaft centre distance was set using a vernier calliper to measure the distance, before the bolts on the flange bearings were tightened. Input to the lower shaft comes from a ‘torsional exciter,’ which is essentially an electric motor with a controller programmed to allow both a constant average angular velocity plus a harmonically varying component. The torsional exciter allows excitation of the meshing gears at a frequency other than their GMF. The vibratory output of the system is measured with a laser vibrometer, which can be pointed at different locations on the rig if necessary.

RESULTS Extensive simulations were performed to investigate the effects of time varying mesh stiffness and also the most appropriate form of damping. The main conclusion was that all of the models except the impact damping with time varying mesh stiffhess model provide an acceptable description of the harmonic frequency response function of the physical system. Figure 4 (b) shows the generally good agreement between the SIMULINK simulations, the steady state approach of Stone and Jones (1964) and the experimental results of Stone and Jones (1964). It proved very difficult to simulate a subharmonic motion as the existence of such harmonics depend critically on the initial conditions. However, such motions were observed and Figure 6 (a) shows a close up view of stable subharmonic motion which developed in the viscous damping, constant mesh stiffness model after around 140 seconds. The impact damping model gave similar results, as shown in Figure 6 (b), but the subharmonic motion took longer to stabilise (300 seconds). By converting the initial displacement and velocity of the hypothetical mass into potential and kinetic energy terms, Coltrona (2003) showed that systems with small amounts of backlash require less energy input to develop stable subharmonics - highlighting the dangers of ignoring subharmonic motions in systems with seemingly innocuous amounts of backlash.

Figure 6 (a) Viscous damping model and (b) impact damping model simulated steady state subharmonic response (n = 3)

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Since both models resulted in similar subharmonic motions, and since the viscous damping model is quicker to analyse and run, the viscous damping model with constant mesh stiffness is the focus of the remainder of this paper. This is the same conclusion reached when looking at the frequency responses of the different models, and further justifies the use of the viscous damping, constant mesh stiffness model. Attempts were made to find such subharmonics on the experimental rig. It was considered that by finding the first resonance using steady state excitation that an excitation at just less than three times this frequency might, with an appropriate impact, result in the desired subharmonic. However it proved extremely difficult to obtain a consistent steady state response. This was found to be the result of a developing failure in the torsional coupling between the exciter and the rig when it failed catastrophically. A new coupling also failed in a fairly short time so that experimental confirmation of the sub-harmonic motion was not achieved. CONCLUSIONS The investigation reported in this paper has confirmed that the modelling of vibration with backlash indicates the possible presence of subharmonic motion. Simulations based on a viscous damping, constant mesh stiffness model have demonstrated the dangers of ignoring backlash in gearbox vibration, since subharmonics may arise unexpectedly. However, attempts to find such subharmonics experimentally were unsuccessful because of failures in couplings. As these couplings had not failed in many hours of testing on other rigs it is suggested that backlash can indeed have very serious consequences. REFERENCES Azar, R.C. and Crossley, F.R.E. 1977. Digital simulation of impact phenomenon in spur gear systems. Journal of Engineering for Industry 99(3), pp. 792-798. Buchold, W. 1933. Forced vibrations with a clearance between mass and spring, Doctors Thesis, University of Darmstadt, Referenced in Stone and Jones (1964), (see below). Coltrona, M.L. 2003. Vibration with backlash. Honours Thesis, University of Western Australia, School of Mechanical Engineering. Chapra, S. and Canale, R. 1998. Numerical methods for engineers with programming and software applications. WCB McGraw-Hill, Sydney. El-Saeidy. 1991. Effect of tooth backlash and bull bearing deadband clearance on vibration spectrum in spur gear boxes. Journal of the Acoustical Society of America 89(6), pp. 2166-2773. Kahraman and Singh, 1990. Non-linear dynamics of a spur-gear pair. Journal of Sound and Vibration 142, pp. 49-75. Kahrarnan and Sing, 1991. Interactions between time varying mesh stiyness and clearance nonlinearities in a geared system. Journal of Sound and Vibration 146, pp. 135-156. Kahrarnan, A. and Blankenship, G. 1997. Experiments on nonlinear dynamic behaviour of an oscillator with clearance and periodically time-varying parameters. Journal of Applied Mechanics 64, pp. 217-227. Litak, G. and Friswell, M. 2003. Vibration in gear systems. Chaos, Solitons and Fractals 16, pp. 795-800. Padmanabhan, C. and Singh, R. 1992. Spectral coupling issues in a two-degree-of-fleedom system with clearance non-linearities. Journal of Sound and Vibration 155(2), pp. 209230. Padmanabhan, C. and Singh, R. 1995. Dynamics of apiecewise non-linear system subject to dual harmonic excitation using parametric continuation. Journal of Sound and Vibration 184(5), pp. 767-799.

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Shaw, S. and Holmes P. 1983. A periodically forced piecewise linear oscillator. Journal of Sound and Vibration 90(1), pp. 129-155. Singh, R., Xie, H. and Comparin, R. 1989. Analysis of automotive neutral gear rattle. Journal of Sound and Vibration 131(2), pp. 177-196. Stone, B.J. and Jones, M.G. 1964. Vibrations with non-linearities. Honours Thesis, Department of Mechanical Engineering, University of Bristol. Theodossiades, S. and Natsiavas, S. 2000. Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. Journal of Sound and Vibration 229(2), pp. 287-310. Wang, M., Zhao, W. and Manoj, R. 2002. Numerical modelling and analysis of automotive transmission rattle. Journal of Vibration and Control 8, pp. 921-943.

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Coupled torsional and transverse vi bration of gearboxes M SARGEANT and B J STONE School of Mechanical Engineering, The University of Western Australia, Austraila

Abstract: The majority of torsional vibration models developed for geared systems ignore the effects of coupling between torsional and transverse vibration. Although this is valid for very stiff bearings placed close to the supported gears, coupled models are important because most geared applications use rolling element bearings with significant flexibility. To analyse the effect of torsional/transverse coupling on the natural frequencies and torsional modal damping levels in geared systems, receptance theory was utilised in the frequency domain to model both the torsional and transverse response. A two shaft system connected via a gear pair model was developed, which demonstrated that coupled torsional natural frequencies and torsional modal damping levels are directly related to bearing properties, and that torsional damping in some modes can be increased by a factor of 400% via coupling effects. The coupled model was then extended to investigate the torsional vibration response of a torque regenerative (back-to-back) gearbox system. The model developed predicted the first 1 1 natural frequencies to within 8%. The coupled model more accurately predicted the dynamic response compared to an uncoupled torsional model. The latter predicted only 9 natural frequencies over the relevant frequency range since it was unable to predict coupled transverse natural frequencies. These findings validate the necessity of including coupling effects into geared system vibration models to accurately model the system’s vibration response.

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NOMENCLATURE General receptance General receptance locations Transverse force General excitation at location c in direction n Moment General receptance directions Rotational angle Radius Torque General response at location b in direction m Tilt angle General co-ordinates

1 INTRODUCTION Geared systems are important for power transmission due to their ability to change rotational speeds and amplify torque. Unfortunately, equipment failure due to torsional vibration remains a problem for industry. This is because excitation sources exist which superimpose oscillating stresses onto the system that can lead to instability problems or premature failure due to fatigue. The latter problems arise because torsional damping present in geared systems is generally low and such oscillating stresses can be high, particularly when the system is excited close to a natural frequency. Since many of these excitation sources are inherent with rotating systems, resonance avoidance is an important issue for geared system designers. This is achieved through analysing the natural frequencies of the system in torsion and ensuring the system's natural frequencies are sufficiently removed from any excitation frequencies. This analysis is repeated for transverse vibration. Such vibration modes are analysed independently given that no simple method is available to analyse them while they are coupled together. However, torsional and transverse vibration modes are coupled in gearboxes and this coupling can cause the estimated uncoupled natural frequencies to have an error up to 15% as modelled by Schwibinger and Normann (1988). This means that although the designer may believe the system will not be excited close to any natural frequency, this may in fact not be the case if coupling effects have been ignored. Although coupling effects complicate the natural frequency analysis, the coupling allows energy due to torsional vibration to be dissipated through vibration in the transverse direction. This may effectively increase the system's torsional damping. Since torsional damping is intentionally minimised to increase mechanical efficiency, the damping in the transverse direction, which is an order of magnitude higher than the damping present in the torsional direction, can be utilized to reduce torsional vibrations. The coupling phenomenon occurs since gears transfer torque by applying an offset load on the gear teeth. This offset load is equivalent to a torque and a transverse load applied at the centre of the rotor. This torque and transverse load set up coupling between the torsional and transverse vibration modes in geared systems. Coupled models represent the most complicated gearbox analyses. More typical are the uncoupled models where transverse and torsional vibrations are considered independently. Uncoupled models implicitly assume that the bearings supporting the gears are infinitely stiff. Since all bearings have finite compliance,

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uncoupled models sacrifice accuracy for simplicity. The assumption of infinitely stiff bearings becomes less valid as bearings are moved further from the gears, a fact of life for most large gear sets. Although the coupling between torsional and transverse vibration in geared systems has been known for some time, only simple systems have been analysed using the lumped mass matrix method or finite element techniques. These methods have limitations: both make unrealistic assumptions about the form of material damping as the damping parameters have to be a linear function of the mass and stiffness matrices defined for the models. These methods also predict system mode shapes but these mode shapes do not act independently; that is, the response at any given frequency is a combination of multiple modes. Receptance theory was used for this investigation. This is a systems approach to vibration modelling and does not have the restrictions associated with both the stiffness matrix method and finite element techniques. The damping parameters are unrestricted and the predicted system response is the deflected shape, including contributions from all the modes. This paper describes the first part of an investigation using receptance theory to model the coupling between torsional and transverse vibration in a gear pair. The second part of the investigation will be reported elsewhere (Sargeant, Drew and Stone (2004)) and describes the modelling of a complete torque regenerative gear box and driving motor. However some initial results of that investigation are included in this paper. 2 PREVIOUS WORK The concept of torsional/transverse coupling effects has been long known, and one of the first methods for analysing these effects was proposed by Lund (1978). This method involved analysing the torsional and transverse vibration modes independently using the stiffness matrix method and then coupling the modes by matching the impedance at the gear mesh. By using equations of motion which incorporated both torsional and transverse motion, Iida et al. (1980) were among the first to investigate both torsional and transverse coupling together. Their study included two shafts, connected via a gear pair and involved many simplifying assumptions. Only one of the shafts was considered flexible and the other one was considered rigid to transverse vibration. The models used lumped parameters (not continuous systems) and the effects of bearing properties were not considered. The motor and load were modelled as lumped inertias. The stiffness matrix method was used and they showed that coupling resulted in both a change in natural frequencies and the associated mode shapes.

Iida, Tamura and Yamamoto (1986) extended the model of Iida et al. (1980) further by introducing the flexibility of the supporting bearings in a 3 shaft gear train by modelling them as springs connected to earth. Only one bearing pair was considered flexible and the flexibilities of the shafts in the transverse direction were not considered. The shaft supported by flexible bearings was also considered rigid for torsional vibration but the other two shafts were flexible for torsional vibration. Damping was not considered and the motor and load were modelled as lumped inertias. Iida, Tamura and Yamamoto (1986) used the stiffness matrix method and found that torsional/transverse coupling can occur if the bearing stifmess is considered even when the shafts themselves are rigid in the transverse direction. The article presented measured system critical speeds and these matched up well compared to predicted critical speeds for a simple geared system.

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Simmons and Smalley (1984) investigated a turbine compressor train gear pair. The study used the stiffness matrix method to predict the system's critical speeds and modal damping levels. A spring and viscous damper modelled the bearing properties in the gearbox. Only the transverse flexibility of the gearbox was considered but the model successfully predicted increases in torsional damping due to torsional/transverse coupling which were replicated in the experimental results. The system's natural frequencies were excited by running the system up from rest and using the inherent excitation sources present in the turbine and compressor to excite the system. An extension of the torsionaVtransverse vibration model was developed by Choi et al. (1999) to investigate the specific vibration problems associated with a 28MW turboset. The turboset consisted of a steam turbine, a double helical gear and a generator. The helical geared system required a model which included axial, transverse and torsional vibration. Choi et al. (1999) used a combination of finite element methods and lumped mass models to explain instabilities in the system that an uncoupled torsional model could not. The flexibilities of the coupling and the bearings were considered and the turbine and generator were modelled as inertias in the model developed. The model considered only the excitation due to mass unbalance and gear tooth error and the system was excited by the inherent vibration sources in the system during normal operation. By replacing the flexible coupling with a more torsionally compliant coupling, and by replacing the bearings with a more stable bearing type, the magnitude of the axial vibration was reduced. Choi et al. (1999) demonstrated the importance of considering the whole system when investigating gearbox dynamics.

3 THEORY Receptances allow a complex system to be broken down into simpler subsystems which are easily modelled. These subsystems can then be added together to represent the complex system. Thus, the strength of receptance theory lies in its ability to replicate a complex system's dynamic response by modelling the simpler subsystems independently. The most general receptance model requires six coupling coordinates. These directions correspond to the six degrees of freedom available to a rigid body, displacements in the x,y,z direction and rotations in the three possible planes. Associated with these six directions are six excitation sources: three force inputs and three moment inputs. For this investigation a coupled model was developed and as a consequence, both a torsional and transverse receptance model were required, which implies three coupling coordinates at each location. 3. I Torsional vibration model A torsional vibration model has one coupling coordinate involving a rotational angle theta (e) and one associated excitation source, torque input (T). For torsion, parallel shafts were modelled as continuous shafts with distributed mass, stiffness and damping. Based on tests with long steel shafts, Deny, Drew and Stone (1996) showed that a hysteretic damping model most accurately predicted the damping in metal shafts. The viscous damping model over predicts the damping present at higher natural frequencies. The torsional continuous shaft model has parameters: length, density, shear modulus, diameter and hysteric damping factor. Since the shafts used were manufactured from mild steel, these values were known from material properties or were measured. The hysteric damping factor was taken from a previously fitted uncoupled torsional model developed for a similar test gearbox (Drew and Stone, 2002).

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3.2 Transverse vibration model For transverse vibration in a plane, two coupling coordinates are needed involving a transverse displacement (x) and tilt angle (cp) and associated with these directions are two associated excitation sources: transverse force (F) and moment input (M). An undamped model was used for modelling the transverse vibration of parallel shafts since the damping contributed by the metal shafts to the complete system is low compared to the damping contributed by the bearings. This is not the case for torsional vibration where the small torsional damping of the continuous shafts is significant since the system's overall torsional damping is small. The transverse vibration model has parameters: length, density, Young's modulus and diameter. Since the shafts were mild steel, these parameters were known from material properties or were measured. 3.3 Coupled Receptance Model The coupled model is a combination of both a transverse and torsional vibration model. Therefore, the coupled model requires three coupling coordinates that involve transverse displacement (x), tilt angle (cp) and rotational angle (e). The associated stimuli are transverse force (F), moment input (M) and torque input (T). Due to the three directions in the coupled model compared to one direction in an uncoupled torsional model, the coupled model is a factor of nine times more complex (3'/12). The coupled model was first used to model a simple geared system and then extended to investigate the torsional vibration response of a torque regenerative (back-to-back) gearbox system (Figure 5 ) . Since receptances are used in a systems based approach to vibration modelling, where a large complex system is broken down into simpler subsystems which are modelled independently, these subsystems must be added together to represent the initial complex system. This is achieved by the addition of subsystems. Receptance addition uses the continuity equations at the connection points to define the receptances for the larger system. Each subsystem is added to the model sequentially until the model replicates the response of the complex system. During addition, the directions in the model define the number of continuity equations. The continuity equations are derived because the response at connecting points must be equal. These mandate that the transverse displacement, rotational angle and tilt angle must be the same at the connection point when the subsystems are added. 3.4 Torsional/Transverse coupling in geared systems using Receptance Theory Torsional/transverse coupling in geared systems can be explained by investigating the simple representation of the gear in Figure la), where 1 is the centre of the gear and 2 is the gear tooth which is in mesh. When the gear tooth meshes, an offset load at point 2 is applied, and is equivalent to the same magnitude force and a torque being applied at location 1, as shown in Figure Ib). By considering this offset load (F2) in terms of the combination of a force (F1) and a torque (TI) at location 1, the receptance a2x,2x can be derived in terms of the receptances at location 1.

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T "J Figure 1: Torsional/transverse coupling in geared systems using receptance theory

3.5 Addition using Receptance Theory The receptances at location 1 are given by equations (1) and (2). U I X =a1,.1,4 (1) since F~R=TI 0 = a,,,,,T,= a,,.,,F2R (2) The transverse displacement at location 2 is the transverse displacement at location 1 plus the relative displacement between locations 2 and 1. This gives rise to equation (3). Due to the small values of 0, sin 8 = 0 which then gives equation (4). U 2 , = U,,+ Rsine (3) u2,= + R e (4) Substituting equations (1) and (2) into equation (4) gives equation (5). Since Fl=F2, the transverse displacement at location (2) can be written in terms of the receptances at location 1 and the offset load F2, equation (6). u2, = a,x.IA+RalI,lx4R (5)

u,,

(6) u2x = alx.lxF2 + alx.lrF2R2 Dividing equation (6) by Fz, receptance a2x,2x is derived, equation (7). This gives the transverse displacement at the gear tooth due to the offset load applied at the gear tooth. "2,

a2x.2x

-

= -- a l x . l x + "l,.l,R2

(7)

F2x This is the fundamental difference between an uncoupled torsional model and a coupled model. In the uncoupled model there is no transverse displacement at the centre of the gear since the model implicitly assumes that the bearings are infinitely stiff and hence, there is no transverse displacement ( alx,lx = 0 ) . It was confirmed that by assigning alx,lx = 0 in equation (7), the coupled model replicates the uncoupled model. In work to be reported [Sargeant, Drew and Stone (2004)l this receptance approach has been applied to a torque regenerative (back-to-back) gearbox system. The complete analysis of such a system is beyond the scope of this paper. However the method was first applied to a simple system coupled via a gear pair and that is described here.

4 SIMPLE GEARED SYSTEM Since coupling occurs due to the way in which gears transfer torque, a simple system of two shafts connected via a gear pair (Figure 2a) illustrates the effects of coupling. This can be modelled as shown in Figure 2b).

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Torsional Simulation

Uncoupled Model Flexible bearings

.

" I

1200

1300

1400

1500

1600

1700

1800

Frequency (&)

Figure 2: a) Simple Geared System b)Modelled System c)Receptance a38,,8 with and without flexible bearings 4.1 Damping Effects When the bearing stiffness approaches infinity, the coupled model replicates the uncoupled torsion model since no transverse vibration occurs across the gears. When the bearing stiffness is reduced, the effect of torsionalhransverse coupling becomes evident. A sample torsional receptance, [rotational angle theta (e) at location 3 due to a torque input (T) at the end of the opposite shaft, location 1 (Figure 2a)], is shown in Figure 2c) for both infinitely stiff and reduced stiffness bearings. Figure 2c) highlights the fact that not all system natural frequencies are affected equally. The figure shows that there is a shift in natural frequencies as well as an increase in damping due to energy dissipation across the supporting bearings. By using a range of stiffnesses, the effect of bearing stiffness on torsional damping can be simulated, as plotted in Figure 3a) which displays the simulated damping of the third natural frequency in Figure 2c). Note that the damping values have been plotted against bearing compliance, where compliance = Vstiffkess and when the bearing stiffness is infinite, the bearing compliance is 0.

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Damping vs Bearing Compliance

Uncoupled model vs Bearing Compliance

Figure 3: Effect of bearing compliance on a) torsional damping and b)torsional natural frequencies Realistic bearing stiffnesses are of the order of the area shaded in Figure 3a. This figure reveals that uncoupled torsional models may significantly underestimate the torsional damping present in geared systems. 4.2 Natural Frequency Effects Figure 3b) also reveals that the torsional natural frequencies are influenced by the effect of torsionalhansverse coupling. The magnitude of this influence is directly related to the amount of coupling in the model. Therefore, since uncoupled torsional models ignore the change in natural frequencies caused by coupling, they become less accurate as coupling effects increase. By assuming that the coupled model accurately predicts the system's natural frequencies, the error associated with uncoupled models can be calculated. By using a range of bearing stiffnesses, the trend of this error for the third natural frequency in Figure 2c), is demonstrated in Figure 3b). Realistic bearing stiffness ranges are presented in the shaded area of Figure 3b) and they demonstrate that (since coupling effects are ignored) the natural frequencies may produce large errors even when the bearings supporting the gears are very close to the gears. 4.3 Coupled transverse natural frequencies Larger changes in torsional damping and natural frequency estimates occur when the bearings are moved away from the supporting gears. This can be modelled as illustrated in Figure 4a). Since the transverse force on the gears is reacted by the bearings, a moment acts on the shafts when the bearings are offset from the gears. This results in coupled transverse natural frequencies (Figure 4b). These coupled natural frequencies occur at frequencies below those predicted by the uncoupled torsional model. Although traditional gearbox vibration analysis also involves analysing the system for transverse vibration, the transverse natural frequencies predicted by these methods do not accurately incorporate the coupled nature of these vibration modes, which leads to errors.

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4.4 Infinitely stiff bearings Several gearbox setups were simulated with infinitely stiff bearings (infinitely stiff in both the transverse and tilt direction). The model shows that even when the 'infinitely s t i f bearings are moved as little as 20-30 mm away from the supporting gears, coupling effects are evident. This is because when the bearings are axially offset from the gears, both the transverse flexibility of the shafts and the bearing flexibility are important. Thus, the stifmess between the gears and the earth is a combination of both shaft flexural stifhess and bearing stiffness. If the bearing stiffness is very high, the flexural stiffness of the shaft dominates and becomes the main influence on the system's coupling effects.

5 TORQUE REGENERATIVE GEAR BOX The coupled model was then extended to investigate the torsional vibration response of a torque regenerative (back-to-back) gearbox system (Figure 5). The model developed (Sargeant et al. (2004)) predicted the response at each measurement location (2-12, in Figure 5) due to the torsional motor/exciter input. The simulated and measured results for measurement location 3 are presented in Figure 6. The model developed predicted the response at each measurement location due to the torsional motor/exciter input. The simulated and measured results for measurement location 3 are presented in Figure 6. The model predicted the first 11 natural frequencies to within 8%. The coupled model more accurately predicted the dynamic response compared to an uncoupled torsional model. The latter predicted only 9 natural frequencies over the relevant frequency range since it was unable to predict coupled transverse natural frequencies.

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Slave

Figure 5: Back-to-Back gearbox schematic, recording devices and measurement locations (after Sargeant et al. (2004))

r

@

O -5

d B k

-15 -20 -25 -30

P

O

g 400

800

1200

Frequency (k)

1600

-10

0

400

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Frequency (k)

Figure 6: Simulated and measured amplitude results for measurement location 3 (after Sargeant et al. (2004))

6 CONCLUSIONS The use of receptance theory enabled a complex coupled system analysis incorporating the complete system dynamics to be modelled, rather than the analysis of the gear pair in isolation. The model of two shafts connected via a gear pair demonstrated that coupled torsional natural frequencies and torsional modal damping levels are directly related to bearing properties. The influence of bearing properties on the torsional system response validate the necessity of including coupling effects into geared system vibration models to accurately model the system’s vibration response.

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7 REFERENCES Choi, S.H., Glienicke, J., Han, D.C. & Urlichs, K. 1999, Dynamic gear loads due to coupled lateral, torsional and axial vibrations in a helical geared system, Trans. ASME, Journal of Vibration Acoustics, vol. 121, pp. 141-148. Derry, S., Drew, S.J. & Stone, B.J. 1996, The torsional vibration of a damped continuous bar, In Proceedings of 8th International Congress of Experimental Mechanics, Nashville, Tennessee, pp. 72-73. Drew, S.J & Stone, B.J. 2002. Torsional damping of a back-to-back gearbox rig: experimental measurements and frequency domain modelling, Proc Instn Mech Engrs Part K: Journal of Multi-body Dynamics, vol. 216, pp. 157-168. Iida, H., Tamura, A., Kikuch, K. & Agata, H. 1980, Coupled torsional-flexural vibration of a shaft in a geared system of rotors (1st Report), Bulletin of the JSME, vol. 186, pp. 21 11-21 17. Iida, H., Tamura, A. & Yamamoto, H. 1986, Dynamic characteristics of a gear train system with softly supported shafts, Bulletin of JSME, vol. 29, pp. 18 1 1 - 18 16. Lund, J.W. 1978, Critical speeds, stability and response of a geared train of rotors. Journal of Mechanical Design, pp. 535-539. Sargeant, M., Drew, S. & Stone, B. 2004, Coupled torsional and transverse vibration of a back-to-back gearbox. Submitted to Journal of Multi-body Dynamics. Schwibinger, P. & Nordmann, R. 1988, The influence of torsional-lateral coupling on the stability behavior of geared rotor systems, Trans. ASME, Journal of Engineering for Gas Turbines and Power, vol. 110, pp. 563-571. Simmons, H.R. and Smalley, A.J. 1984, Lateral gear shaft dynamics control torsional stresses in turbine-driven compressor train, Trans. ASME, Journal of Engineering for Gas Turbines and Power, vol. 106, pp. 946-951.

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Development of a simulation tool for the prediction of dynamic transmission error, the source of transmission whine D PARKIN-MOORE and G DAVIS

Ricardo Driveline and Transmission Systems, Ricardo UK Limited, Leamington Spa, U K D BELL and C H LU

Ricardo Software, Ricardo U K Limited, Shoreham-by-Sea,UK

P BROOKS and A LEAVITT School of Mechanical Engineering, University of

Leeds, UK

ABSTRACT Vehicle sound quality and the reduction of gear whine is an issue of increasing interest to vehicle manufacturers. The primary cause of gear whine is Transmission Error (TE), which acts as the source of excitation for the shaft and bearing system, which in turn excites the transmission casing causing it to radiate noise. This noise is then heard in the passenger compartment as audible gear whine. This paper describes the development and validation of a software tool for the prediction of TE for external spur and helical gear systems whilst subject to dynamic operational conditions.

INTRODUCTION Transmission Error (TE) is the deviation in the position of the driven gear relative to the position it would occupy if both gears were geometrically perfect and un-deformed, and is a result of the combined effects of manufacturing, tooth modifications and mesh deformation. The reduction and control of TE whilst subject to true automotive operating conditions is key to the reduction of gear pair whine in automotive transmissions, and is achieved through the combined optimisation of gear pair macro and micro-geometry. Until recently much work into TE has focussed on gear specimens and operating conditions that are not truly representative of those found in the automotive transmission [ 11. The gears used in these studies have often been of an inappropriate size, design or quality to truly replicate gears found in automotive transmissions. Likewise the operating conditions often do not truly simulate the wide range of applied torque and operating speed in conjunction with the mesh misalignment that can occur during. In response to these findings Ricardo Driveline & Transmission Systems in conjunction with the Automotive Driveline Research Group based at the University of Leeds, have set about to develop an analytical modelling capability and advanced experimental test facility to correct

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this deficiency. This paper discusses the development of an analytical tool used for the simulation of TE. The theory behind the analytical tool is discussed in detail and is followed by an example of its application with validation from experimental results. NOMENCLATURE Subscript used for identifying Gear 1 or 2 Gear pair centre distance, m Base circle diameter, m True involute form (TIF) diameter, m Tip diameter, m Number of gear teeth Contact points compliance matrix, mM Global compliance matrix, m M Global gear co-ordinate system for the nthgear, m Local co-ordinate system for the mthgear, m Equivalent to angle's 6, ,el ,6, for Gear 1 during contact, radian Common tangent line between two gear base circles, radian Equivalent to angle's 6, ,SI ,e2for Gear 2 during contact, radian Interference between two involute curves, m Gear pitch angle, radian Gear rotation angle, radian Transverse contact ratio Helical overlap ratio Gear rotational angles about

X,,, and Y, respectively, radian

Involute start angle on the base circle relative to the tooth central line, radian True involute form ( T F ) angle, radian Tooth tip involute angle, radian Angle from true involute form to the tooth centre line, radian Initial centre line angle, radian Angle of the tooth centre line, radian Search angle for tooth contact, radian Compliance matrix for two contacting gears Contact Force Total contact interference on the contact line ANALYSIS TOOL USED The dynamic computational analysis of the gearbox was carried out using a multi-body dynamics simulation tool that was originally developed to analyse the dynamic motion of

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valvetrain systems in the internal combustion engine. Due to its modular approach and flexibility, the program has been developed over time to simulate the dynamics of many other types of mechanisms. Among it’s features is the ability to model 3-D compliant gear meshes and shafts for all 6 degrees-of-freedom. The software uses a flexible approach to modelling dynamic systems whereby specific parts of a system, represented by individual elements, can be linked together to form the whole system model. This flexibility enables the user to generate models varying from a simple sprung mass to a highly complex complete system model. Each modelling element has been carefklly designed to use the minimum degrees of freedom, whilst retaining sufficient detail to achieve a high level of simulation accuracy. This facilitates rapid model generation and gives short simulation times, making it practical to undertake speed sweeps, which give greater understanding of the model dynamics and faster optimisation. Due to the increasing requirement of transmission manufacturers to move fiom developmentby-testing to development-by-analysis in order to reduce costs and development time, the software has been developed to model gear systems with a quick and simple 2-D model or a more advanced 3-D compliant model. The 2-D model is adequate for predicting gear rattle due to its ability to model gear backlash. Whilst with the more comprehensive 3-D model, the engineer is able to simulate TE ,contact forces, gear stresses and misalignment. All macro geometry of spur and helical gears is included within the model and any form of micro geometry corrections such as lead correction, crowning and tiphoot relief can be considered. To aid in the representation of typical operating conditions the gear may be supported by a compliant 3-D shaft thereby including the effects of gear tipping and centre position deviation. The 3-D gear element allows modelling of gear dynamics with either a simple constant tooth stiffness or a varying tooth stiffness derived fiom a finite element (FE) model. In FE mode, the gear tooth stiffness is defined by a compliance matrix representing the whole tooth face and the contact stiffness is dynamically calculated at every time step depending on the contact position. The compliance matrix is generated and solved automatically within the software. The FE model also allows localised tooth stresses and fatigue analysis to be performed on completion of the dynamic simulation.

3D GEAR ANALYSIS THEORY General The 3D gear element allows users to model the effect of varying tooth stiffness on the dynamic TE. The tooth stiffness is calculated by a compliance matrix that defines the relationship between points on the tooth surface when a unit force is applied to a point. At each time step a new compliance matrix is formed for each contact line determined by the tooth contact analysis. Contact lines on both sides of a tooth (for gears in tight mesh) are accounted for. The tooth surface is assumed to be that of an ideal involute, therefore a matrix is generated to represent the deviations from the involute profile that exist due to manufacturing errors, crowning and tip relief etc.

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Theory There can be contact on both sides of the tooth and more than one tooth may be in contact at the same time i.e. for tight mesh or high contact ratio conditions. For one tooth surface the interference along the contact line may vary due to any misalignment or micro geometrical corrections to the profile. The analysis only takes into account the points where interference is non-zero and a new compliance matrix is generated for the non-zero interference contact points by approximation. When two tooth surfaces are in contact, it is necessary to determine how much interference belongs to each tooth. This is achieved by the new Compliance matrix generated for the contact points. The significant procedures during the calculation procedure can be summarised as: 1. Generate a compliance matrix (imported from FE model) 2. Calculate contact points for the teeth that are in contact, for both sides of the tooth. 3. For each contacting tooth, generate a compliance matrix along contact points. 4. Calculate the interference for each contact point (two tooth surfaces are in contact), which should include the effects of the surface error input by user. 5. Solve the algebraic equation to determine the force at the contact points (concentrated force). 6 . Repeat step 3-5 for all teeth in contact. 7. Add up all forces on all contacting teeth and apply them back to the relevant NODES. The Compliance Matrix The tooth surface is meshed such that the elements are evenly distributed in the horizontal and vertical directions as shown in FiglB. A compliance matrix is defined from an FE analysis lrith respect to the element face centroids. A.

I B.

I

2

3

4

5

6 7 M +

8

9

1011

Fig. 1 A. The local co-ordinate system for the gear pair as viewed along the axial direction and from the side. B. A meshed tooth illustrating the compliance matrix. Contact points The gear centre coordinates are defined as (XI,&, 2,) and in X-Y plane is therefore determined as:

314

(X,, Y,,Z,).

The centre distance

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D,,= J ( X , -XI)' +(Y2 -q)' A local co-ordinate system is established (X,-Y,-Z,) Gear 1 as shown in Fig. 1A.

(1) where the origin point is the centre of

Gear 1 centre in (X,-Y,-Z,) system is (0, 0,O). Gear 2 centre is (0, D12,Z2 - Zl). In the local coordinate system, gear rotation angles about X,, Y, can be calculated as;

e,,,, = tan-' (tan e,,, sin 4 - tane,,, . COS^) e,,,, = tan-' (tan e,,, . cos4 + tan@,,.sin 4) where i = 1,2 corresponds to Gear 1 and Gear 2 respectively. Contact positions and interferences are determined by the gear tooth involute curve positions. We denote three typical involute angles as shown in Fig. 2. 8, is the involute start angle on base circle (relative to tooth central line). 8, is the true involute form angle (TIF) and e, is the tooth tip involute angle. (4)

Fig. 2 The three key tooth angles that are considered in the analytical tool where D, is the true involute form diameter (TIF), D, is the base circle diameter, and D, is the tip diameter. 8, is the angle from TIF to tooth centre line. Because of the symmetry of the

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tooth involute profile about the tooth centre line, the involute angles are only calculated for one side. At initial time the initial state of two gears is as follows, Gear 1 tooth centre line is at the line of centres, and Gear 2 is at the position with the centre line at the middle of two teeth. At a certain time step of simulation, the gear rotation angle can be calculated by,

where $, is the initial centre line angle. i = 1,2 corresponds to Gear 1 and Gear 2 respectively. The central line of the tooth, which is closest to the line of centres, is at angle

p, =E,-INT(E,/A,).A., where A, = 2n / N , is the gear pitch angle, and N , is the number of teeth of the gear. After p, ( i = 1,2) is calculated, searching for the contact between two gears can be started and the search range is in these tooth angles, y , =p, + n . A ,

n=0,1,...4

(9)

In the program, the maximum number of contact teeth is 9. This is sufficient for most automotive problems. The total number of teeth in contact is a function of helical overlap ratio, EP, and transverse contact ratio, E ~ To . simulate one full base pitch of contact the maximum number of teeth in contact would be E=+ q + l . Generally for automotive gears 1<~,<3and O<~p<3. It is possible that both Gear 1 tooth faces are in contact at the same time. Therefore, the program has to search for contact on the left face and the right face. For Gear 1’s left face contact, Gear 1’s three involute angle positions can be expressed as, “0

= PI

+60

a, = a. - 6, a2= a o-6, and Gear 2’s can be expressed as well,

Po = p 2+So -O.SxA, PI =a0 -6, P2 =a0

-62

Note So, 6,and 0, for different gears have different values. There is a common tangent line between the two gear base circles. a is the common tangent line between the two gear base

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circles. The involute contact condition is:

a, < a l a , and p, < a l p , If this condition is satisfied, then the interference can be calculated,

where 6 > 0 means there is a contact between two involute curves and the interference is 6 . For Gear 1 tooth’s right side involute contact, a similar procedure is followed to detect the involute contact. If the involute curves are not in contact, it is still possible that the tooth tip is in contact with the other tooth involute curve, so extra checks are performed to see if the tooth tip is in contact. If one gear tooth tip point is within the other gear tooth involute curve region, contact will occur. The effect of tilt angle about Y, axis can be added into 6 before contact is detected. Compliance matrix on contact lines For each contact line, a new small compliance matrix is generated at every time step. If the contact points Q, (m=1, 2, ...M) are all known, the compliance matrix can be calculated based on the “global” compliance matrix which is based on PNM (n=1,2. ..N and m=1,2. ..M). By applying a unit force on each contact point and measuring the deformation on all contact points, the compliance matrix for the contact line can be determined. Force on contact lines After the compliance matrix for a contact line is calculated, the force on it can be calculated by solving an algebraic equation. Here, the force is assumed along tooth pressure angle direction. For two contacting teeth, [Cl] and [CZ] are the compliance matrices. The total contact interference on the contact line is {X}, the force can be calculated as follows,

EXPERIMENTAL TRANSMISSION ERROR TEST RIG Utilising a transverse 5 speed manual automotive transmission as a basis, Ricardo DTS in conjunction with the School of Mechanical Engineering at the University of Leeds developed a dedicated gear pair test rig for the measurement of TE subject to a comprehensive range of true automotive operating conditions. This was partly in reaction to the findings of Davis. G. et a1 (2001) who identified the need for a greater emphasis on experimental TE research. The overhung nature of the Sth speed gear pair of the chosen automotive transmission rendered itself very suitable for the measurement of TE through the use of optical encoders, whilst simultaneously allowing the measurement of gear pair misalignment.

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In order to develop a dedicated test rig from a production transmission, all unnecessary ancillaries were removed whilst the original primary shaft, main shaft and final drive were maintained. The original transmission casing was replaced with a dedicated housing which maintaining key dimensions as per the original. Drive was transferred through the meshing 5Ih speed gear pair and all remaining gears and synchroniser assemblies were removed. Two 9000 line incremental quadrature optical encoders were fitted to the axial centre of the input and output gears and were secured to a dedicated mounting plinth situated a short distance away from the meshing 51h speed gear pair. The encoders were attached to the gears via a dedicated torsionaly stiff Tufnol shaft, CV coupling and precision bearing arrangement. This ensured that the optical encoders were not subject to any misalignment generated at the 5Ih speed gear pair that can occur during operation. For the measurement of mesh misalignment, four non-contacting dis lacement transducers were equally spaced around the perimeter of the outward face of the 5t R speed wheel and pinion. Each gear was manufactured with a dedicated precision target disc fitted to the front of the main gear body. This allowed the displacement transducers to accurately measure gear body displacement and ensured that a controlled flat surface was measured, thus removing any surface deviation effects that the original gear body will have contained. Utilising a dedicated data acquisition system the gear pair test rig was able to measure TE to sub-micron accuracy whilst subject to static, quasi-static or dynamic conditions with the output from each optical encoder measured simultaneously, synchronously and with zero skew at a max sampling frequency of 8OMHz and up to a maximum input shaft speed of 6777rpm. Simultaneous to the measurement of TE, the data acquisition system was able to measure the gear body misalignment. This therefore enabled a comprehensive understanding of the behaviour of the meshing gear bodies. Utilising the gear pair test rig in conjunction with the extensive test facilities based at Ricardo DTS and a range of dedicated gear specimens, the effects of micro and macro-geometrical parameters on TE whilst subject to true automotive operating conditions were measured. AN EXAMPLE OF THE APPLICATION OF THE SIMULATION TOOL Background Utilising the developed simulation tool a dynamic model that replicates the behaviour of the meshing Sth speed helical gear pair of the dedicated test facility has been developed. To remain faithful to the layout of the test facility and to ensure the accuracy of the predicted TE, the analytical model replicates the primary shaft, main shaft and final drive gear pair in addition to the gear pair of interest and calculates their combined effect on the TE of the 5th speed gear pair. The following discussion includes a brief description of the components of the model and presents the results from an initial series of simulation tests that have been performed for validation purposes, which includes some of the obtained experimental data. Model Description A series of elements within the simulation tool are connected together in a sequential manner to form the basis of the analytical model. The required parameter values for each element are then subsequently applied. Global settings such as the initial excitation speed, range of rotation angle for which to simulate and the integration time step are then applied. An

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annotated schematic of the general layout of the analytical model generated in the simulation tool is shown in Fig. 3, with the elements of the model discussed in the following paragraphs.

Fig. 3 Model layout taken from the simulation tool's Graphical User Interface For the input to the system an angle element provides the initial excitation for the model, setting the model input velocity as a ratio of the global angular velocity. A rotary inertia element coupled to a linear spring element simulates the input shaft to the transmission system and allows the angular velocity of the input shaft to be set to a ratio of the model input speed. The torsional stiffness and damping coefficient of the input shaft are represented within the linear spring element. With respect to the transmission components of the gear pair test rig, the primary and main shafts are modelled using a combination of shaft and mass elements with each shaft divided into sections with the position of the bearings used as the dividing point. A mass element is then situated at this point and connected to the adjacent shaft elements. This allows the threedimensional inertial properties of the shaft elements to be calculated in terms of the bearing positions, which are then entered into the appropriate mass element. Data detailing the 3D geometry, stiffness and damping properties of each shaft section are then supplied to the shaft element. Modelling the shaft sections in this manner enables the simulation tool to determine the deflection of each shaft due to its compliance under load. The macro geometry of the 5th speed and final drive helical gear pairs are represented by 3D gear elements. Combining the shaft deflections with the tooth deflections that are calculated in the contact region by the mesh element allows determination of TE. For the output of the test rig, which is connected to a dynamometer, a linear spring element that is fixed to ground represents the load exerted on the transmission system during operation. A zero magnitude of torsional stiffness is applied to this element and a value of preload is entered only, which applies a constant drag force to the transmission system. Adjusting this value allows the torque under which the model operates to be adjusted. In a similar manner to the input shaft, a combination of a rotary inertia and linear spring element represent

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the output shaft, which connects the transmission system to the absorber. However, in this instance, the value entered in the rotary inertia element scales the output speed in accordance with the two gear ratios represented within the model. A simulation run for a gear pair with a Transverse Contact Ratio (TCR) of 1.733 whilst operating at an input shaft speed of 27501pm and for an applied input shaft torque of 115Nm was completed. The predicted TE in terms of the frequency domain is as shown in Fig. 4. From this data the 1" and 2"dharmonics of the l/tooth component of TE for the 5'h speed gear can be clearly seen at 1787.5 and 3575Hz respectively. In addition the Is' and 2"dharmonics of the mesh frequency component of TE for the final drive can be seen at 975 and 1950Hz respectively. These key frequency components are clearly seen in the experimental data as shown in Fig. 5 . The experimental results for the same gear pair subject to similar operating conditions show the same key frequency characteristics as the simulated results; with the harmonics of the 5* speed gear pair and final drive mesh frequency components clearly noticeable. However, it is noticed that the magnitude of the 2ndharmonic of the 5thspeed and final drive mesh frequency components for the experimental data are significantly smaller in magnitude. Additionally the experimental results contain significant signal content at the low end of the frequency spectrum. These low frequency components are mainly due to the lhev component of TE that has not been included in the simulation. The Ihev component of TE is primarily as a result of cumulative pitch error or incorrect and inconsistent spacing of the gear teeth around the pitch circle diameter.

7RAWJISSIW CRAOR

- G.wn.1

010d. 2750rpn. 115%

Input Torque

VAlDYN 2 . 9 ZB-J0--2004

..............................

...............................

I

...............

._........................

................................ -. ...................... ..........................................................

..........................................................................

...........I................. ...............

.....,..............................

....:.................................

...........................................................

I

I..........

II

1080

g. 4 The predicted dynamic TE response for a gear pair with a TCR equal to 1.7 I and subject to an input shaft torque of 115Nm and an input shaft speed of 2750rpm.

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2.1

I

I

I

1.............................

j............................

;............................

; ...........................

I

It i

1.8 .........................

i .............................

1.5 ......................................................

1000

2000 Frequency (Hz)

3000

4000

Fig. 5 1 ne measured dynamic TE response for a gear pair with a TCR equal to 1.: 0 and subject to an input shaft torque of 115Nm and an input shaft speed of 2750rpm.

Fig. 6 plots the experimental and simulation results of the variation in magnitude of the Utooth component of TE for the 5th speed gear pair as applied torque is increased. It can be seen that as applied torque increases then in general the TE also increases. This is primarily as a result of the increased deformation at the meshing zone. It can be seen that there is some discrepancy between the results at low torques. However as torque increases the trends for both the experimental and analytical data are very comparable. In an attempt to reduce the discrepancy between the simulated and experimental data the representation of the actual gear micro geometries and shaft geometries within the simulation software are being enhanced.

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-1 '

+Simulation

1

f Experiment

" ,

10

30

50

75

100

115

Applied Input Shaft Torque (Nm)

Fig. 6 Variation in the magnitude of the 1" harmonic of the l/tooth component of TE with increasing torque. CONCLUSIONS The analytical tool developed combines the system dynamics with the highly non-linear nature of gear contact. The presented results compare very well with the test results and show that the software is able to identify the key features associated with TE. It is hoped that future development will improve this level of accuracy. Importantly it is also shown that the software can predict the change in response due to changes in operating conditions and macro and micro geometry. Utilisation of this tool will enable a greater level of understanding of the effects of macro and micro geometry gear design combined with operational conditions. FURTHER WORK As with all research tools the analytical tool discussed in this paper is subject to continual development. Future versions of the software may look at including the introduction of pitch errors, both cumulative and adjacent which will allow true replication of the lhev components of TE. Currently the software is used only for the simulation of parallel axis gearing and it is intended that future versions will have the capability to solve application where the gear pair operate at 90" to each other. This will include bevels and hypoids etc. Finally the software could also look to include sliding velocity, temperature and efficiency calculations. REFERENCES 1. Davis, G . Brooks, P. Findlay, M. (2001), "Recent Advances in Automotive Gear Pair Dynamic Behaviour Measurement and Prediction - A Review", MPT2001 JSME International Conference on Motion and Powertransmissions, November 15-17, 200 1, Fukuoka, Japan, pp.90-96, Japanese Society of Mechanical Engineers.

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Low-noise automotive transmissions investigations of rattling and clattering S N DOdAN, J RYBORZ, and B BERTSCHE Institute of Machine Components,University of Stuttgart,Germany

1 Introduction There are numerous sources of noise in motor vehicles, which combine to create a complex acoustic scenario. Apart from the internal combustion engine the transmission is one of the dominant sources of noise in the driveline. The problem of transmission noise in development is becoming increasingly acute with the growing use of light-weight construction of internal combustion engines and other driveline components, greater concentration on energy saving, increased customer expectations and more rigorous legal restrictions on exhaust and noise emissions. Automotive transmission noises can be broken down into several groups according to their causes [1, 21, Figure 1. The most dominant type of transmission noise is rolling contact noise from gear pairs under load known as whining and squealing but also as grinding and singing. This type of noise can be caused by meshing impacts, parametrically excited vibration or rolling contact noise due to variations in pitch spacing.

Figure 1: Categorisation of vehicle transmission noises [l, 21

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Rattling and clattering noises are caused by idle transmission components subject to torsional vibration. This noise is known as rattling when the transmission is in neutral, and as clattering when a gear is engaged under power or in overrun (pull/push operation). Rattling and clattering noises are perceived as unpleasant not because of their high airborne sound pressure level, but because of their intrusive characteristic. It therefore constitutes a comfort problem (noise immission) in case of passenger cars, and a problem of both comfort and environmental pollution (noise emission) in case of commercial vehicles. Clattering that arises under power with low load, for example at low road speeds and engine speeds, is called creeping [3]. Gear shifting noise can also arise in automotive transmissions from scraping and grating of the selector teeth as a result of defective synchroniser functioning, and noise can arise from transient load cycle excitation, referred to as clonk or load shift knock [4]. But also bearing noise can occur especially in case of damaged roller bearings, and screeching noise caused by vibration of the idler body within the bearing clearance [SI. 2 Statement of Problem Rattling and clattering noises in automotive transmissions are caused by torsional vibration transmitted from the internal combustion engine to the transmission input shaft. This is due to discontinuous combustion processes, resulting in periodically fluctuating drive torque at the crankshaft. Unbalanced engine masses also have an sustainable effect on the rotary movement of the engine crankshaft. This rotary oscillation is superimposed on the rotary movement of the crankshaft, resulting in the irregular rotational speed of the internal combustion engine, which is in principle sinusoidal. The relevant portion of the torsional vibration in a four-stroke internal combustion engine is the bisector of the number of cylinders. The cyclic irregularity of combustion engines is increasing because of efforts to improve fuel consumption and emissions levels, such as the increasing use of turbo-charging, multi-valve technology and direct fuel injection, combined with reducing idling speeds. The engine speed pattern is affected by the operation of additional consumers such as air conditioning, headlights and rear window demister, as well as ignition defects and clutch slip. They increase the irregularity of rotational speed, resulting in larger angular acceleration amplitudes and changes in their characteristic. The torsional vibration transmitted from the internal combustion engine to the transmission excites idle components such as idler gears, synchroniser rings and sliding sleeves to vibrate within their functional clearances. The fixed components encounter idle components at the clearance limits, resulting in impacts perceived as rattling or clattering noises. The intensity and frequency of the impacts are directly related to the airborne sound pressure emitted from the gearbox housing [6, 71. Rattling and clattering noises arise chiefly in manual transmissions and semi-automatic transmissions. But also automatic transmissions can prone to clattering when the torque converter is locked up. Responsible parameters The parameters responsible for rattle and clatter can be divided into operating parameters and geometrical parameters. The operating parameters include the excitation frequency as the product of rotational speed and engine order, which is responsible for the number of impulses per rotation of the shaft, and the angular acceleration amplitude which is the main factor determining contact between the fixed gear and idler gear flanks. The diameter of the fixed gear and the geometrical parameters of the idler gear such as diameter, moment of inertia, mass, helix angle and the associated reduction ratio are the main parameters that can be influenced at the gear development stage in terms of their rattling and clattering noise behaviour. But there is some goal conflict between designing the gearwheel stages in respect of power trans-

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mission, and their service life and low rattling proneness. But also the torsional backlash and the axial clearance offer scope for reducing this noise. Reducing the torsional backlash and a defined increase or reduction in the axial clearance results in a reduction of the airborne sound pressure level emitted [6,7]. Figure 2 illustrates the example of an idler gear whose circumferential and axial movement behaviour relates to the rotational irregularity and its impact performance. The sinusoidal rotational speed profile of the transmission input shaft (for example that of a four-cylinder

Drive side

thrust collar

Output side thrust collar

Figure 2: Movement of an idler gear as a result of an irregular rotational speed profile in the case of a fixed idler gear pair [6, 81 four-stroke internal combustion engine) is characterized by acceleration and deceleration phases, Figure 2a. When the rotational speed profile is decelerating (falling slope of the curve), the idler gear grips the driven flank of the fixed gear, Figure 2b. As soon as an acceleration phase is entered (rising rotational speed profile), the idler gear comes away from the driven flank of the fixed gear with following flying phase and then impacting against the driving flank of the fixed gear, which is perceived as structure-related noise, Figure 2d. After the torsional flank impact, the helical cut idler gear impacts against the drive side thrust collar, Figure 2c, which can be clearly recognized in the structure-related noise graph, Figure 2d. Each torsional flank impact is followed by an axial impact whose intensity is less than that of the torsional flank impact. Transmission fluid as an engineering design parameter also has a considerable influence on rattling and clattering noises. The important factors include the type of oil, the additives used and the viscosity (which is directly related to temperature), and the level of oil in the transmission which together act on a gear pair as drag torque, resulting in a reduction in rattling and clattering noises, especially at low speeds and when cold [10, 121.

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3 Theoretical and Experimental Investigation - Investigation Strategy Throughout the process of developing automotive transmissions with low levels of rattling and clattering noises, suitable equipment is needed to study noise behaviour right from the initial stages. This requires suitable models to be developed from the experimental investigation and analysis of rattling and clattering, to determine the noise behaviour of automotive transmissions by simulation. The strategy includes investigating automotive transmissions in test stand trials and by mathematical simulation, Figure 3. It is possible to develop guidelines and recommendations for reducing rattling and clattering noises in automotive transmissions with the aid of fundamental investigations and parameter variation.

Test stand trlsls

Parameterstudies

KNOWLEDGE FROM EXPERIENCE Determining the rattling and clattering noise behaviourofautomotive mnsmissions DESIGN CATALOGUE FOR LOW-RATTLING AND -CLATTERING NOISE AUTOMOTIVE TRANSMISSIONS

Figure 3 Investigation strategy for developing low-rattling and -clattering noise automotive transmissions In test stand trials, complete transmissions or their components are investigated in terms of rattling and clattering noise performance. The parameters relevant to rattling and clattering noises are then varied within reasonable limits. The mathematical methods involved can be divided into a simulation and an approximation method. In the case of the simulation method the pattern of movement of the loose parts and the impact intensities can be determined. The approximation method enables the level of noise to be computed. The simulation models are verified on the basis of the test stand trials.

3.1 Experimental Investigations The test stand developed at the Institute of Machine Components at the University of Stuttgart makes it possible to investigate rattling and clattering noises in front-mountedtransverse and standard transmissions, in neutral as well as under power and in overrun, under realistic operating conditions, excluding extraneous influences. A highly dynamic, brushless, permanently excited three-phase synchronous motor provides the drive unit, capable of simulating the rotational speed profiles of internal combustion engines with different numbers of cylinders. The same motor is available as a braking motor for clattering testing under power and in overrun. Both servomotors are rated at 12 kW with a nominal torque of 30 Nm. According to the moment of inertia of the test transmission, angular acceleration amplitudes of up to 4000 rads2 can be achieved at the input shaft in neutral. To simulate various internal combustion engines in different operating statuses, a PC controlled function generator is used to specify the corresponding idealized or realistic set values [6,7,lo].

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3.2 Theoretical Investigations Many years of research into in-gear rattle and clatter in automotive transmissions at the Institute of Machine Components has developed methods of calculation to the point where rattle and clatter can be calculated for complete passenger car and commercial vehicle transmissions [6, 7, 10, 113. There are two methods available for this, numerical simulation and approximation. The simulation method makes it possible to determine how the loose parts move, and their impact intensity, whilst the approximation method makes it possible to estimate the noise level of the transmission. The EKM calculation methods (EKM = “Einfachst-mapper-Modell” - SimplestRattling-Model) are based on simple models for which the linked rotary and translatory movement relations of rotating loose transmission components can be simulated. For an idler gear or loose part mounted on the shaft, three more unlinked degrees of freedom (torsional backlash, axial clearance and radial backlash) can be determined. The small effect of the radial backlash, causing wobbling of a loose part in combination with the other degrees of freedom, is ignored. In the case of torsional backlash, the small rotational oscillation amplitudes are described by an equivalent translatory substitution model, Figure 4a. Here the external frame corresponds to the driving fixed gear with mass mi, exciting the loose part (m2)to vibrate within the torsional backlash s,. The loose part movements are then opposed by the sum of the individual drag torque components as an external force R,. In case of helical cut gear wheels, impacts are induced in the hub area because of the axial load and the necessary axial clearance sa, described by translatory movement, Figure 4b. Here the loose part (m2)impacts against the drive side and output side thrust collar (m3)as a result of the helical gearing within the axial clearance Sa.

Figure 4: Substitution models for a) torsional backlash and b) axial clearance The simplest rattling model (EKM) is derived from the two substitution models to describe the linked rotary and axial vibration of a loose part. The loose part (which can be a no-load idler gear or a cluster gear, e.g. a countershaft or a synchroniser ring) is then modelled as a rigid body, the elastic deformation being negligible. EKMSimulation Method Using these model parameters, movement equations can be created for numerical simulation, describing loose part behaviour with defined excitation. The impacts at the clearance limits extend through shafts and bearings both directly as airborne noise and also as structure-related noise, thus inducing vibration in the gearbox housing, leading to noise emission. The calculation ignores the complex transmission behaviour, but as a first approximation, the pulse transmitted when impacts arise is proportional to the noise level caused by it. The theorem of momentum and angular momentum is then used for the loose part because of the unknown time profile of the impact forces. This results in a connection between the force and move-

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ment profile of the loose part, from which the average impact intensity Zm can be determined, Equation 3.1 [ 6 ] : ,/ m2 ‘ h f‘rbl ’ CIm . (3.1) The average impact intensity is represented by the average total of all individual shock pulses in a simulation period, where rn2 is the mass of the excited loose part, is the angular acceleration amplitude, 5 1 is the pitch circle radius of the exciting part and CI, is the related average impact intensity for tooth flank and axial impacts, Equation 3.2 [6]:

The related impact intensities iiz, iix and iia of the respective coordinate directions are determined from the helix angle p, the idler gear mass m2, the pitch circle radius of the fixed gear q,l, the angular acceleration G1, the excitation frequency L D Aand ~ the two clearances, torsional backlash sv and axial clearance s* The average impact intensity Zm is not an absolute measure of noise and can only serve as a comparative figure between different vibration states and loose parts within a transmission. It is possible to calculate a level of rattling and clattering noises Lp only by correlating between the computed noise value and the measured noise level:

The basic noise level L B can~ either ~ be~estimated, ~ or is known from measurement. The calibration factor k creates the connection between the real average impact intensity Im and the acoustic pressure, which is derived from comparing a measured airborne sound pressure level with the average impact intensity from a simulation. EKM ApproximationMethod The calculation methods can be used to investigate the rattling and clattering noise behaviour of individual gear steps, and also of complete passenger car and commercial vehicle transmissions. An empirical approximation formula seeks to achieve a sufficiently accurate calculation of the impact intensity, and thus of the airborne sound pressure level, independently of a numerical simulation. In the Approximation Method, identical standard parameters are used as in case of the numerical Simulation Method. The related average impact intensity CIm can be determined in the range of small relative axial clearances, using Equation 3.4 [ 6 ] : C,m = Csv

r .

[

1,462-

0,714.Cfa .Csa - 0,Ol 6.Cfa + 0,12.Csv

(3.4)

The related parameters of the clearance in the circumferential direction Csv, of the clearance in the axial direction Csa and the friction force Cfa are determined from the excitation frequency W A ~the , pitch circle radius of the fixed gear 5 1 , the angular acceleration d l , the helix angle p, the idler gear mass m2, the friction force FR and the two clearances, torsional , and axial clearance .,s backlash s

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This makes it possible to expand the average impact intensity 1, for the fixed / loose component pairs under consideration with the related ratio using Equation 3.1 [ 10, 111: I, = rn2 . hl . ib. fbl' C,, . (3.5) This makes it possible to determine the rattle and clatter noise level of a complete automotive transmission by logarithmic addition of the individual loose component noise elements using Equation 3.3: L Pcomp.

Whereas the numerical simulation method can be used without restriction, the Approximation Method has to be used for validity ranges of the parameters in Equation 3.4. The relative circumferential drag torque Cmv must be less than 0.3, the relative axial friction force Cfa must be less than 0.7 and the axial clearance Sa must be greater than or equal to 0.1 mm. The rattling curve is moreover valid from above an angular acceleration amplitude i l of approximately 250 rads2 at the required sinusoidal excitation.

4 Correlation of Measurement and Calculation Results Figure 5 shows the correlation between measurements and calculation results for a single-step gear stage, using the example of the first gear of a front transverse transmission. With a single rattling point in the first gear stage, this shows good correlation between the calculated rattling noise level curve and the actual observed level. The rattling noise level curves determined using the EKM Simulation Method a sufficiently good result. From an angular acceleration of 200 rads2, the characteristics calculated are approximately equal, with the rattling noise level curve derived by the Approximation Method coming closer to the measured rattling noise level curve.

Figure 5 : Correlation between measured and calculated rattling noise level curves with a single-step gear stage There is also a satisfactory correlation between the measurement and the EKM Calculation Method for the standard transmission too. Figure 6 shows the airborne noise pressure levels of a 5-speed manual transmission with an angular acceleration of 800 rad/$. This shows the

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clear influence of the countershaft. The first gear makes a decisive contribution to the overall rattle diagram profile, as well as the fifth gear.

Figure 6: Airborne noise pressure levels of measured and calculated idling rattling noise level curves at an angular acceleration of 800 rads2

5 Noise Reduction Measures Both external and internal measures can be taken to reduce proneness to rattle and clatter in automotive transmissions, Figure 7.

Figure 7: External and internal measures to reduce the rattle- and clatter-proneness of automotive transmissions

5.1 External Measures The purpose of external measures is to tune and optimize the driveline in terms of reducing torsional vibration outside the transmission. A major factor here is decoupling the combustion engine from the driveline. One notable decoupling system is the hydrodynamic torque converter used in automatic transmissions for moving off. But the clutch can also have a long-term effect on driveline

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behaviour. For example a hydrodynamic converter clutch, a torsion damper in the clutch disk, a slip-controlled clutch or electronic clutch management reduces the irregular rotational movement behaviour of the driveline. Main and ancillary clutch dampers and torsional vibration dampers provide further measures to improve driveline performance. The dual-mass flywheel (DMF) is used between the combustion engine and the transmission as a standard measure for reducing engine torsional vibration. This significantly reduces torsional vibration both in car drivelines both with manual and with automatic transmissions, in this case with a locked-up direct-drive clutch. This improves the overall noise and vibration behaviour of the driveline in terms of clatter and hum, as well as load cycle effect. Numerous more recent developments are concerned with improving the characteristics and optimizing the manufacturing costs of the decoupling systems used. The mechanical torsion damper (MTD) represents a cost-optimized refinement of the dual-mass flywheel, which is also based on the principle of spreading the inertia torque of the engine flywheel, but is characterized by simpler design with a smaller number of components. The hydraulic torsion damper (HTD)combines the advantages of spreading the flywheel mass inherent in the DMF with a variable damper system. This damper increases its damping effect in critical system conditions, for example in the event of a natural frequency of the engine transmission system being generated, in order to prevent high oscillation amplitudes at the transmission input. The speed-adaptive damper (SAD) enables resonance-free compensation of the dominant exciting torque at the crankshaft of a combustion engine throughout the speed and load range. This damper principle achieves a significant reduction of cyclic irregularity of the crankshaft and of the vibrations in the belt drive. The integral starter-alternator damper (ISAD) also serves to reduce cyclic irregularity. The primary aim of this system is however to provide electrical energy and manage it in motor vehicles. Fitting the electrical machine directly on the crankshaft can generate motive power when starting and generating power to provide energy. There are also other possible external measures to reduce rattle and clatter that are simple, but involve major disadvantages such as reducing efficiency or high cost; these include encapsulating the transmission and deadening the bodywork. These external measures are however subject to physical and economic limits, and they do not always achieve the desired results. 5.2 Internal Measures Internal measures to reduce the rattle- and clatter-proneness of a transmission involve judicious design of the geometrical parameters of components that are subject to impact load, as well as restricting the freedom of movement of loose parts within the degrees of freedom defined by their function. The latter can reduce or prevent impact events occurring or spreading. The aim of restricting the movement of loose parts mainly relates to judiciously impeding freedom of movement in order to reduce or prevent impact events occurring or spreading. Internal measures include for example idler gear braking, tooth gap bracing, bracing toothed disks, auxiliary transmissions, dampers, magnetism, structure-borne noise barriers, and axial impact reduction. With the idler gear brake, the idler gear is braked by friction. With the aid of tangentially acting elastic elements between the idler gear and a thin toothed disk with an equal number of teeth mounted coaxially to the idler gear, spacewidth bracing is achieved by means of the rotary offset. The bracing toothed disk mounted coaxially to the idler gear is pressed against the idler gear by elastic components. An additional tooth on the bracing toothed disk causes the bracing toothed disk to turn more slowly compared to the idler gear, resulting in friction that acts in the same direction as the drag torque. In the auxiliary transmission, the fixed gear has a auxiliary transmission subject to friction in addition to the main transmission. This gives rise to a realigning effect for the idler gear, opposite to its direction of rotation.

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With the damper, the idler gear vibrations are damped with the aid of a secondary mass connected elastically to the loose part. One means of reducing impact events, developed and patented at the Institute of Machine Components, is based on using permanent magnets in fixed gears and idler gears. This prevents the idler gear detaching from the fixed gear up to a certain angular acceleration, thus reducing rattle- and clatter-proneness. The structure-borne noise barrier serves to prevent structure-borne noise being transmitted between the point of impact and the hub of the gearwheels using various insulating materials. Axial impact intensities are reduced by ductile thrust collars on the idler gear or the shaft shoulder. The use of these internal measures depends not only on the degree of the noise reduction achieved, but also on the production engineering and economic parameters. The wear characteristics and the level of power loss and the intrinsic noise involved must also be taken into account. Each transmission system must be individually derived from the large number of various possible solutions.

6 Engineering Design Catalogue for Low Rattle and Clatter Automotive Transmissions This section systematically sets out the physical principles of operation in the form of an engineering design catalogue to minimize the rattle- and clatter-proneness of a transmissions. This lists possible effects for generating braking forces on idler gears or loose parts, and shows existing solutions (some patented) and new solutions summarized in a convenient form. The physical principles of operation aim to prevent or reduce vibration of loose parts when subjected to excitation. The engineering design catalogue can be used with advantage for designing low rattle and clatter automotive transmissions since it represents a source of knowledge and contributes to rationalizing the engineering design process. It is also a source of new ideas for the design engineer, since the superordinate measures enable many existing partial solutions to be combined to find a new solution. In order to give an overview of measures to reduce rattle and clatter, it is very helphl to summarize the possible principles for generating braking forces at the idler gear or loose part in one solution framework. Possible principles include material locking, frictional locking, elastic power flow, field power flow and momentum change locking, see Table 1. In this context locking or flow after Roth /I51 means the reciprocal dynamic effect between two parts (fixed bodies or fluids) maintained over a particular time. These possible principles for generating braking forces at the idler gear or loose part are described below. Material locking Material locking relates to the connection between two fixed elastic, plastic or fluidic bodies that prevents the bodies separating.No braking force is directly generated, but the tooth flanks are prevented from lifting off by changing the toothing geometry (e.g. tooth flanks with "zero backlash"), Table 2. Frictional locking Frictional locking exists between two touching surfaces where force is transmitted by friction. The friction force is applied to the contact surface by normal force. The contact surfaces have a particular surface texture described by the coefficient of friction ,u, The ratio between friction force and normal force determines the coefficient of friction ,u and gives the force relations resulting from the surface finish, the material mating and the relative speed between the friction bodies if appropriate. Depending on whether there is relative movement between the friction bodies, a distinction is made between the coefficient of sliding friction and the coefficient of adhesion. The level of friction force can be influenced by additional application of springs (compression springs, tension springs, Belleville springs, torsion springs), elastomers and magnets, Table 2.

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Elastic power flow Elastic power flow is created by elastic deformation. Some examples of this are shown in Table 3. Field power flow The line of action is transmitted by a force field between two active areas, which normally do not touch, Table 3. Momentum change locking Momentum change locking is force locking in which the dynamic effect arises from the pulse of one body being changed by a change of velocity. The braking force at the idler gear can be switched on and off in a controlled manner by combining frictional locking with momentum change locking, Table 3. The "single dimension" structure of the engineering design catalogue after Roth /15/ was selected to set out the physical principles of operation for minimizing rattle and clatter as clearly as possible for users. The catalogue contents are divided into Connection, Mechanism, Engagement and Appendix. Connection contains the main types of locking for generating braking forces at the idler gear, which are broken down in to material locking, positive locking (frictional locking), momentum change locking, elastic power flow and field power flow. Mechanism in the engineering design catalogue contains the various alternative solutions in the form of simplified schematic diagrams. The alternative solutions presented include patented solutions and new solutions. Engagement describes limitation of the transmittable braking force for each type of solution, or additional sample applications. Appendix contains remarks and notes on the individual solutions, e.g. manufacturing processes and references to further examples and patents. The advantage and benefits of this engineering design catalogue are the systematic representation, in tabular form, of known noise reduction methods for automotive transmissions, rapid access to information, ease of use for users, uniform and clear representation of the schematic diagrams, expandability of the engineering design catalogue, and the possibility of finding new approaches by combining existing possibilities.

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Table 1 : Overview of measures to reduce rattle and clatter in automotive transmissions

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. .

d

..

31

Table 3: Typical applications for elastic + field power flow and momentum change locking rn

Table 2: Typical applications for material and frictional locking

B

h

a

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d0

335

7 Conclusions Table 4 shows the noise reduction levels achievable for the measures synoptically represented according to the influence of the operating parameters, the main geometrical parameters and some internal measures in a five-speed manual transmission. Optimum design configuration of torsional backlash and axial clearance and the selection of a lubricant with high kinematic viscosity are the primary factors leading to low rattle- and clatter-proneness in automotive transmissions. When designing automotive transmissions from the point of view of minimum rattle and clatter, it is essential to take into account that the airborne noise pressure level output rises as mass and moment of inertia of loose parts increase, and as centre distances increase. Table 4: Noise-reducing effect of parameters and internal measures at the passenger car transmissions

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8 Summary Clatter and rattle are an increasing challenge in developing low-noise automotive transmissions. Efforts to reduce consumption and emissions of combustion engines combined with reduction of idling speed result in ever greater cyclic irregularities in the drivelines of today’s motor vehicles, which in turn lead to increased rattle and clatter in automotive transmissions. Loose parts that are not under load, such as idler gears, are responsible for this noise, as they are excited within their functional backlash by the irregular rotational speed pattern to vibrate, and consequently impact against fixed gears. With the aid of the EKM Calculation Method involving simulation and approximation, it is possible to investigate the rattle- and clatter-proneness of automotive transmissions at the development stage, and of existing transmissions on the basis of existing parameters. Numerous external and internal measures have already been developed to reduce rattle and rattle- and clatter-proneness, but only very few have been used in mass production. The use of these measures depends not only on the degree of noise reduction achieved, but also on the production engineering and economic parameters. The wear characteristics and the level of power loss and the intrinsic noise involved must also be taken into account. Each transmission system and each driveline system must be individually adapted from the large number of various possible solutions. Systematic measurement of rattle and clatter in automotive transmissions using knowledge derived from experience has been compiled in the form of an engineering design catalogue for low rattle and clatter automotive transmissions. This provides both rapid access to information and ease of use due to clear representation with schematic diagrams, and expandability of the principles and measures listed. Nomenclature related average impact intensity related circumferential drag torque related axial fiction force related parameter in axial direction related parameter in circumferential direction fiction force average impact intensity calibration factor force mass rattling and clattering noise noise of one component basic noise level noise level of complete transmission related ratio pitch circle radius axial clearance torsional backlash helix angle time [SI [rads-’] excitation frequency [rads-’] amplitude of angular acceleration [-I coordinate direction [-I coordinate direction

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Indices 1 2

fixed gear idle gear

References Lechner, G.; Naunheimer, H.: Fahrzeuggetriebe, Springer-Verlag,Berlin, Heidelberg, 1994 Lechner, G.; Naunheimer, H.: Automotive Transmissions, Springer-Verlag, Berlin, Heidelberg, New York 1999 Seebacher, R.; Fischer, R.: Triebstrangabstimmungmit Simulationsunterstiitzung, VDI Berichte 1285,1996, S. 395-410 Biermann, J.-W.; Hagerodt, B.; Schrerder, E.; Schuch, A.: Experimentelle Untersuchungen lastwechselbedingterBewegungen und Gerausche von Antriebsstriingen, 5. Aachener Kolloquium Fahrzeug- und Motorentechnik 1995, S. 595-619 Schwarze, B.: Losradkreischen in Zahnradgetrieben,Dissertation, Universitiit Bochum, 1996 Lang, C.-H.: Losteilgerauschevon Fahrzeuggetrieben, Dissertation, Universitiit Stuttgart, 1997 Weidner, G.: Klappern und Rasseln von Fahrzeuggetrieben, Dissertation, Universitat Stuttgart, 1991 Dogan, S. N.; Ryborz, J.; Lechner, G.: Simulation von Losteilschwingungen in Fahrzeuggetrieben,Antriebstechnik 37 (1998) Nr. 7, S. 59-64 Dogan, S. N.; Lechner, G.: Mdnahmen zur Verringerung von Losteilschwingungen in Fahrzeuggetrieben,ATZ Automobiltechnische Zeitschrift 100 (1998) 10, S. 7 10-7 16 Dogan, S. N.: Zur Minimierung der Losteilgerauschevon Fahrzeuggetrieben. Dissertation, Universitiit Stuttgart, 200 1 Ryborz, J.: Klapper- und Rasselgerauschverhaltenvon Pkw- und Nkw-Getrieben. Dissertation, Universitiit Stuttgart, 2003 Dogan, S. N.; Ryborz, J.; Bertsche, B.: Rattling and Clattering Noise in Automotive Transmissions - Simulation of Drag Torque and Noise. Symposium -Transient Processes in Tribology, Lyon, 02.-05.09.2003 Pfeiffer, F.: Rattling in Gears - A Review, VDI Berichte 1230, 1996, S. 719-737 Kiicukay, F.: Berechnung und Optimierung der Rasselschwingungen bei Schaltgetrieben, VDI Berichte 699, 1988, S. 593-630 Roth, K. : Konstruieren mit Konstruktionskatalogen.Springer-Verlag,Berlin, Heidelberg, New York, Band II,2. Auflage, 1994

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NVH-behaviour of side shaft-systems J W BIERMANN

Head of Vehicle Acoustics and Drivetrain Department, lnstitut fur Kraftfahrwesen, RWTH Aachen, Germany

1. Initial situation New vehicle development has to meet increasing demands on the acoustic and vibrational performance ("-behaviour). These demands are arising from the fields presented in Fig. 1: environment, legislation and customer.

Fig. 1:

Vehicle Acoustics in the vicious circle of development criteria

Recently, environmental demands aiming on the reduction of both, fuel consumption and exhaust emissions, force the effort to lightweight design and application of consumption-saving combustion engines. From the viewpoint of vehicle acoustics, there are two counterproductive concomitant factors:

- increased vibrational susceptibility of aggregates and body, - increased vibration excitation and noise radiation by modem high-torque Diesel engines.

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Furthermore, NVH-specialists have to meet the mandatory exterior noise levels required by law, which have been cut continuously in the past. Watching the launches on the recent motor shows, driving fun has become a serious factor in the customers decision to purchase. One of the most requested issues is a fast engine response on actuation of the throttle. With low-damping driveline-designs, the vehicle NVH-specialist faces another aggravation in addition to the increased vibration excitation by modem engines. Besides, even compact-size vehicles are expected to provide a good NVH comfort. This means, that the interior noise level has to be low with - primarily - no annoying NVHphenomena. In his field of activity, the vehicle NVH-engineer has to gain a balanced compromise between partially conflicting aims. 2. NVH-Phenomena caused by the drivetrain

In view of the passengers’ comfort sensibility, the drivetrain is one governing factor, in addition to vibration excitation by chassis and air-flow noise (at high speed). The main components of the drivetrain are combustion engine, clutch, transmission, axle shafts with joints as well as the wheels. This composition is a complex vibrational system with rotational and flexural degrees of freedom. The most important system-parameters are listed in Fig. 2 for the subsystems engine, drivetrain and body.

Fig. 2:

Total system “Drivetrain / Vehicle / Driver“

The vibration characteristics of the system are determined by the physical properties of the single components: moment of inertia, stiffness, damping value and clearance. The excitation of the vibrating system is governed by the torque transfer from the combustion engine. Here, the dominating factors are intake pressure, ignition and fuel injection. The subjective perception of the driver is the operative instrument for vehicle comfort rating. The key points are interior noise, longitudinal vibration of the vehicle and engine response on actuation of the throttle.

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The vibrating system “drivetrain” is the source of a multiplicity of characteristic NVHphenomena, which cause a significant loss of comfort. The best-known phenomena are given in the following Fig. 3.

Fig. 3:

NVH-phenomena excited by the drivetrain

The picture clarifies the correlation between drivetrain components and characteristic phenomena. Side shafts and CV-joints cause noise phenomena as shuffle, boom and shudder. In cooperation with OEMs, the Institut fiir Kraftfahnvesen Aachen (ika) carries out extensive driving and test-bench testing as well as simulation in this subject area. These activities are subject of the following remarks.

3. Vibration excitation by CV-joints All joints applied to nowadays vehicles are assigned either to tripod joints or to ball joints. Both basic designs are presented in Fig. 4. For the wide bending-angle performance, the fixed joint design is used at the wheel side. Plunging joints are used at the transmission side.

Fig. 4:

Tripod Joint Basic designs of CV-joints

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Ball Joint

341

The loss of comfort is primarily traced back to vibrations transferred from the joint via sideshaft and wheel suspension into the body. Two different mechanisms act as root causes: on the one hand, axial forces appear in the joint, on the other hand, transfer behaviour and damping characteristics of the joint are determined by so-called plunging forces. The impact of the periodical axial force is illustrated in Fig. 5 on the left, exemplified for the tripod joint. In a bended joint during rotation, the rollers move back and forth along the tracks of the tulip. With torque applied, a cyclic alternating force is induced in direction of the shaftaxle, caused by the frictional force-transfer of the rollers during rolling-off. The main effect is found corresponding to the number of transferring elements: for the third order of the shaft speed in case of tripod joints, for the sixth order in case of ball joints. The interior NVHphenomenon resulting from axial forces is known as "shudder" with characteristic frequencies up to about 100 Hz.

Fig. 5:

Tripod Joint as root cause for the generation of interior noise

The right side of Fig. 5 depicts the impact of plunging force, which may cause the so-called "idle boom" (20 Hz - 50 Hz). This NVH-phenomenon is of special interest in case of frontwheel-driven passenger cars with automatic transmission. The transfer of both, low-frequency vibrational motion and high-frequency structure borne noise from engine and transmission, follows the path via joint, side shaft and wheel suspension into the body. Keeping the vibrations in the vehicle interior at a low level requires the decoupling of powertrain from wheel suspension or body, respectively. Plunging force denotes the frictional resistance of the joint against plunge. It is a measure of the decoupling performance: the lower the plunging force, the lower the vibration excitation in the body. Axial and plunging force are measured with a special measuring device developed by the Institut fiir Kraftfahrwesen Aachen. This device is fit for both, test bench and vehicle on-board investigations. Measuring device and test bench are shown in Fig. 6 . By means of this device extensive measurements have been carried out within joint development over the last ten years.

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Fig. 6:

Measuring device for CV-joints

Fig. 7 clarifies the significant effect of joint design and grease on axial force. As can be seen in Fig. 8, plunging force is affected to a great extent by joint design and applied torque.

Fig. 7:

Impact of design and grease on axial force

Fig. 8:

Plunging force (n = 0 rpm;excitation frequency 30 Hz)

All in all, it can be stated, that the tripod-joints are advantageous in view of plunging force, whereas ball joints are advantageous in view of axial force. The decision, what kind of joint design has to be applied, has to be made from vehicle type to vehicle type.

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4. Vibrating System "Side Shafts"

In addition to the joints, the side shafts also contribute to the noise behaviour of front-wheeldriven passenger cars. There are two different kinds of assemblies: systems with two shafts (short and long side shaft) and systems with three shafts (two commensurate side shafts, one interim shaft). Fig. 9 gives an overview of a representative side-shaft assembly with three shafts designed for a front-wheel-driven passenger car with transversally installed engine.

Fig. 9:

Assembly of a side-shaft-system with three shafts

The properties of shafts and joints contribute to the vibrational behaviour. In addition, special attention is attracted by the design of force-transmission points in wheel hub, differential and support of intermediate shaft. Both, torsional und translatory vibrations, are excited in the complex shaft system. Passive and active mechanisms act as root causes, similar to the conditions in the joints, shown just before. The power coming from the drivetrain is mechanically transferred to the wheels via the side shaft system. Thus, this connection is also transfer path into the passenger compartment for vibrations excited by engine and transmission (passive way). Usually, engineers loose sight of this path in view of the structure-borne noise transfer via engine mountings. Fig. 10 highlights the contribution of different parts to structure borne noise transfer during run-up. In tests with decoupled components, the partial contribution of the single paths were measured /REI03/. It can be seen that the fraction of structure-borne noise, which is transferred to the seatrail by the side shafts, exceeds the fraction transferred via engine mounts by 10 dB in the speed range between 2000 and 3000 rpm. On the path to the steering wheel, the dominance of the sideshaft-transfer covers the entire speed-range.

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Transfer to Seatrail

Transfer to Steering Wheel

Fig. 10: Structure-borne noise transfer to seatrail and steering wheel via side shafts and engine mounts Besides transfer behaviour, the eigenvibrations of this system are in the focus of investigations (active way). These investigations are carried out at the ika by means of operation deflection shape and experimental modal analyses. Fig. l l shows a photograph of the testing device for experimental modal analysis of side shaft systems. On-road engine torque and bending angles are applied by adjusters. Operation deflection shape analysis gives information about the vibrational behaviour under operational conditions. This advantage is paid for with a high measuring expenditure.

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Fig. 1 1: Testing device for Experimental Modal Analysis of side shaft systems Fig. 12 presents the in-vehicle measuring positions at the side shaft system. The measurements are carried out on a driveline test bench with rotating shaft, in different gears and at various load levels.

Fig. 12: Measuring positions for Operation Deflection Shape Analysis In the scope of a benchmark investigation a total number of 26vehicles were measured /REI03/. Fig. 13 gives an overview of the eigenfrequencies for the first side shaft bending mode. This figure includes both, side shafts from systems with three shafts and short side shafts fiom systems with two shafts.

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Vehicle

Fig. 13: Eigenfrequency of the first side-shaft bending mode; comparable passenger cars The spread of the eigenfrequencies covers the frequency range from 220 Hz up to 720 Hz. The measured shafts - except one - were solid shafts. In order to prevent annoying interior noise, the eigenfrequencies have to be shifted beyond the range of critical harmonics from engine excitation (2nd engine order in case of four-cylinder in-line engine e.g.). As arises from these investigations, the weak point is the long side shaft of two-shaft systems. In order to predict the effect of parameter variations and system modifications on the eigenfrequencies, ika set up a programme with ANSYS simulation-tool. The frequency shift of the first side shaft bending mode caused by several constructive modifications is highlighted by Fig. 14. As arises from this figure, the eigenfrequency is shifted from about 80 Hz up to about 280 Hz by selective constructive measures.

Fig. 14: Frequency shift of the first bending mode caused by side-shaft modifications

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Subsequent on-road testing impressively verified the efficiency of these measures exemplified in Fig. 15 with the interior noise level (2nd engine order) during run-up in the second gear.

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 Speed [rprn]

Fig. 15: Interior noise level (2nd engine order) with modified shaft systems, run-up, fullload in second gear The initial shaft system A causes annoying boom noise in the speed range from 2000 rpm up to 2500 rpm, whereas the constructive changes gain a level decrease by about 5 dI3. Additionally, there is no more annoying perception in the interior noise. 5. Outlook

Precondition for reaching a sufficient NVH-comfort during vehicle-development is the balancing of all contributing factors. Fig. 16 depicts the complexity of this task.

Fig. 16: Target Setting - Cascading Principle There are listed the partial sources contributing to the vehicle interior noise. More detailed information are given on the partial source "drivetrain" with the corresponding single

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components. Following this cascading principle, a limiting value is attributed to each single component by tracing back the interior noise target. This applies for the side shaft system as well! Each development process is a joint venture of several departments of both, OEM and supplier. Precondition to hit the target is a close cooperation of all involved parties, in fact right from the start. Further simulation and testing investigations are required in order to gain a deeper knowledge of the physical characteristics in this complex NVH-systems. Only this knowledge provides a basis for reaching the targets of increasing demands in future vehicle development. Nomenclature f

[Hzl

frequency

FAX

[NI

axial force

FFi

[NI

frictional force per roller

h

IT1

normal force per roller

i

S

[NI

input force

Fo

[NI

output force

M

Wml

torque

MI

[Nml

input torque

Mo

[Nml

output torque

n

[rpml

speed

x, Y,

coordinate system

a

["I

bending angle

0

[s-'l

angular velocity

Literature [BE991

BIERMANN, J.-W., RICHTER, S.: ,,Priifstandsuntersuchungenzu Schwingungsverhalten und Ubertragungsverlusten von Pkw - Achsgelenken", Antriebstechnisches Kolloquium, Aachen, 1999

[REI991

REITZ, A., BIERMANN, J.-W., u.a.: ,,Special Test Benches to Investigate Driveline Related NVH-Phenomena", 8'h Aachen Colloquium ,,Automobile and Engine Technology", Aachen, 1999

[REI031

REITZ, A.: ,,Schwingungsiibertragungvon Seitenwellen bei PKW mit Frontantrieb", Dissertation, k a Schriftenreihe Nr. 6603, RWTH Aachen, 2003

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Friction models of automotive transmissions equipped with tripod joints J-P MARIOT, J-Y K’NEVEZ, and B BARBEDETTE Groupe Composites et Structures MBcaniques, Universite du Maine, le Mans, France

Abstact. The present paper deals with kinetostatic friction models of an automotive transmission consisting of an inboard tripod joint close to the wheel and an outboard ball joint close to the gearbox connected by an intermediate shaft. The ball joint is considered as a constant velocity joint (CVJ) whereas the tripod joint is not. It is known from the literature and from experiments that the unwanted vibrations of fhe transmission originated mainly from the tripod joint with three fold frequency shudder excitations; these excitations are due to friction effects. Under normal working circumstances for vehicles, inertial effects at the tripode joint are found to be negligible compared to static efforts . This justify the restriction to kinetostatic models. The inverse kinetostatic model is composed of a tulip driven by the gearbox at a constant velocity and an intermediate shaft with an external constant torque applied . Due to their smaller mass, rollers between tulip and tripod can be neglected and mechanical contacts between tripod and tulip considered as point contacts. Viscous friction can be introduced analytically and it does not produce any shudder effects. Coulomb friction introduced with numerical simulation gives rise to shudder effects but with no velocity dependence. Others models where friction depends on velocity or/and force are in progress. Keywords: automotive transmission, tripod, multibody dynamics, shudder. NOTATION (nominal distances appear between parenthesis) fl0 absolute frame of reference with origin 0 % tulip frame with origin 0 81 tripod frame with origin at tripod centre I R3R6 3/6- fold frequency vibration (3/6 times that of input velocity) 2R p, p+= u, input angles (gearbox) 3

+-

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Constant input velocity (1OOs-l) output angles (wheel) Output velocity Amplitude of p - 0 Joint angle of the inboard tripod joint Mean and amplitude joint angles roller positions Distance between input and output shafts (+SOmm) tripod centre offset length of the intermediate shaft (L=SOOmm) roller-to-tulip distances tulip radius ( ~ 2 5 m m ) roller-to-tripod centre 1 distances input torque, output torque and complementary torque time revolution for input ( P 0 . 0 6 2 8 ~for 4 = lOOrud / s )

1 INTRODUCTION An automotive transmission usually found as a front drive transmission is made of three parts: i) an inboard tripod joint close to the gearbox with plunging properties for suspension movements; an outboard joint whose main function is the front wheel steering; and iii) an intermediate shaft between the two joints. The ball joint (Fig. 1 ) name is due to the rollers (six or eight) contained in the joint. It is a constant velocity joint (CVJ) whatever the joint angle where as the tripod joint is not in general. The tripod joint (Fig. 2) consists of a tulip with three ramps parallel to the input axis and a tripod with three trunnions at 120' from each other perpendicular to the intermediate shaft. In the past thirty years, such transmissions gradually replaced the traditional Cardan transmissions which are not constant velocity transmissions (CV). Automotive transmission with tripod joints are not CV transmission either but, since their departure from CV is weak, they are termed as pseudo CV . Therefore, their dynamical behaviour can be derived from perturbed CV transmission. Tripode joints invented in 1910 have gradually replaced the traditional Cardan joints for their capability of transmitting motion at almost constant velocity, durability and efficiency at a reasonable cost. The recent evolution in the design of tripod joint has been the reduction of noise-vibration-harshness (NVH) contribution of the driveline [ l ] . The main two drawbacks of tripod transmissions are the shudder due to the third-order (R3) axial force generated by the tripod joint and the complementary ( or deflexion) torque generated (R6) that may lead to unwanted beats with the engine vibrations. To minimise these vibrations nuisances, optimisation procedures were proposed which consist in modifying the trunnionsto-intermediate shaft geometry [2, 31 and improving the quality of lubricants between the tripod and the tulip.

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Fig. 1 Outboard ball joint close to the wheel. Joint angle can reach 40"

Fig. 2 Inboard tripod joint close to the gearbox. Joint angle stays less than IO"

In the literature, one of the first paper is due to Mabie [4].In the seventies, papers start with the work of Durum [SI, Orain [6] and Bellomo [7]. In his dissertation, Orain presented kinematics and dynamics of the tripod joint. Lately, Akbil et. a1 [8] discussed the question of the constant velocity nature of the tripod joint; for the same automotive transmission than the present one, they derived kinematic equations that they solved but with tedious methods. Pandrea [9] underlined that the main property of a CVJ is that its intermediate shaft keeps in a constant direction; this condition is obtained for an infinite shaft length. He also proposed an elegant approach with the introduction of invariant properties which we reconsidered [ 10,

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111 to underline that a real tripod joint is just a perturbed constant velocity joint. We took advantage of these perturbations to derive kinematical approximate analytic expressions that are rather simple and efficient in evaluating the deviation between output and input angle on one hand and joint angle precession on the other hand. Urbinati [12] also, presented complete dynamics for this transmission with numerical Runge Kutta integration , The goal of this communication is to show that vibrations nuisances originated from friction effects.

Roller

Tulip

P

Ball joint

6

Intermediate shaft

Fig. 3 Kinematic model of the transmission with tulip (bold) tripode and rollers. The ball joint is modeled as a spherical pair.

2 KINEMATICS This kinematical model already presented in the litterature 18, 121 is depicted in fig. 3 and from input to output the following pairs are found: revolute at input; sphere grooves for roller ramp movement; cylindrical for roller trunnion movement; spherical at the fixed ball joint and revolute at output. The geometrical data are shown in fig. 4: 0 is the ball joint centre, I is the tripod centre, R is the tulip centre and CI,C2 and C3 are the rollers centres. Kinematics equations [8, 1 11are derived by consideration of kinematical loops KIR, IC2Rand IC& e sin(38) = -dsin(y, - 8) ecos8cos(38) = dcos(y,-Q)-Lsin8

354

where the offset e depends on the joint angle Gaccording to, e = -r( - - 1 1) 2 cos6

Fig. 4 Geometrical data d is the distance between the ball joint and the tulip axis and L the intermediate shaft length. For a CVJ, the difference p - 0 = 0. This occurs for infinite L/r ratio and consequently the joint angle has the constant value arcsin(dL). In practice, L / r = 20 and d = 2 r ; consequently a small deviation to constant velocity associated to a small oscillation of 6 is observed. Due to the 3 B terms of (eq. l), B has a 2~ / 3 period; it is the same for 6 and for o, - 0 . But periodic does not mean harmonic; however, Sand for o, - B can be restricted to the first term in their Fourier expansion,

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6 = 6, + 8,cos(3q) q - B = Bo sin(3q)

(3) (4)

Identification of constants 6, and can be derived by considering minimum and maximum joint angles: S is minimum for q = O and maximum for q = n / 3 . Using second line of (eq.3) > e cos S,,, = d - L sin S,,, -e cos S , = d - L sin S , The extreme values, S , and S , , can be numerically obtained by a Newton procedure and mean angle 6, and amplitude angle 8, are given by,

For identification of Bo , a similar geometrical consideration can be invoked be remarking that for angle So the joint angle 6 is close to its mean value S,,,and the input angle is close to d 6 . Since So is always small,

e - - e= O-d

r(-

1 - 1) COSS, 2d

and for small 6, ,

(9)

These approximate analytic expressions have compared to the exact values derived from equation (eq.3) using a Newton Raphson procedure. They were found to be satisfying for both output angle Band joint angle S with a relative error less than lo”. Closure equations also lead to roller-trunnion and roller-tulip ramp distances,

r, = r + e(l + 2 cos(213)) r, = r + e(l + 2cos(2@)) r, = r + e(l+ 2cos(20+)) and to the roller-tulip ramp distances,

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1, =z+r,6cosy, l2 =z+r26cos(p-)

Z3 = z + r36cos(y,+) where z is given by

Calculations of rI and 11 velocities are needed later for the viscous friction model,

i, = LS s i n s + r; sin6 cosy, + 58 cos6 cosy, - ti+ s i n s siny,

and for a constant input velocity

(13)

4, they can be approximated as,

which shows that amplitude of i, is 1/6 (= 10) times that of 4. Dynamic effects are generated by the tangent components of the intermediate shaft mass centre and depend on acceleration 8 and 6-8 ; with the above numerical values, they are found to be less than 6N for a steel cylindrical shape tripode. Similarly, the dynamical moment of the tripode is found to be less than 2Nm which may be neglected with respect to the input torque. Due to their small mass, dynamical effects of the rollers can also be neglected. Therefore, in the case of a constant input velocity, all dynamic effects can be neglected and the mechanical model of the transmission resumes to a kinetostatic model.

3 KINETOSTATIC MODELS WITH FRICTION When transmitting power to the wheels, all the automotive transmissions generate unwanted vibrations and unwanted noise. These phenomena are related to Noise Vibration Harshness (NVH) [l]. For the tripod joint, there are two main nuisances: shudder vibrations and complementary torque. Shudder vibrations are generated by longitudinal movements of the intermediate shaft . The complementary torque (or deflexion torque) is caused by a non zero joint angle. All the inverse models presented below consider a constant input velocity and a constant output torque.

3.1 Constant velocity joint

For a CVJ without friction, the output velocity equals that of the input; the same applies for the input torque identical to the output one. However, a complementary torque is non zero due to the orthogonal components of the tripod to tulip forces. This complementary torque is found to be,

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6 r, =,
i

07

Figure 5 . Shudder excitation (N) versus time (s) for a Coulomb friction coefficient h=0.1. The revolution time 0.0628s corresponds to an input velocity of 1OOrads.

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0

I

I

I

I

I

I

0.01

0.02

0.03

0.04

0.05

0.m

0.07

time

Figure 6 . Complementary torque (Nm): components x ( solid) and y (dashed) for a Coulomb friction coefficient h=O. 1

No shudder vibrations are generated since the forces (between the tripod and the tulip ) projection are equal and their sum is null. When viscous friction is considered, dominate the roller-to-trunnion movements may be neglected considering their velocity about 10 times lower than the roller-to-tulip velocity. Shudder components now include the viscous fiction effects but their s u m still vanishes. Accordingly, the complementary torque is also affected by friction. For Coulomb fiction, only numerical simulations are available and are shown on figure 5 and 6 for shudder and torque respectively. The shudder excitation linearly depends on the Coulomb coefficient; it is the same for the complementary torque. 3.2 Real joint The transition between a CVJ towards a real joint is simply done by reducing to its usual value the length of the intermediate shaft. (LzOSm). For friction coefficient in the 0.1 vicinity, modifications between CV and real joint are hardly noticeable. This indicates that friction dominates kinematics. 4 CONCLUSION

This model of the tripod joint including friction demonstrates that the effects of friction at the roller-to-tulip pair dominates the effects at the roller-to-trunnion pair both for viscous and Coulomb friction. This has been clearly understood by joints designers who tried to minimise friction using more complex kinematics solutions .

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The viscous model does not generate shudder vibrations while the Coulomb friction model does. However, the Coulomb friction leads to shudder excitation but these excitations depend on input torque only and not on input velocity. Other types of friction models are in progress. References 1. E. Baron, NVH Phenomena in Constant Velocity Joints. A 3 fold approach, I. Mech. E. C3891277, 1992,51-60 2. Bartlett S.C., US PATENT 6034331 1, MAR. 7,2000 3. Santonocito, P., Pennestri, E., A Parametric Study of the Dynamics of the Shudderless Tripode Joint, Proc. of DETC 2OOO/MEC"-14079, Sept. 10-13,2000, Baltimore, USA, 1-6 4. Mabie, H.H., Constant Velocity Joints, Machine Design, May 1948, 101-105 5. Durum, M.M., Kinematic Properties of Tripode Joints, Journal of Engineering for Industry, Trans. ASME., 1975,708-713 6. Orain, M., Etude des joints de transmission tripodes, Thhese de doctorat, Universitk de Paris 6,1976 7. Bellomo, N., Montanari, P., The General Theory of the Mechanics of a Large Class of Multibody Systems for Constant Velocity Transmission between Intersecting Axes, Mechanism and Machine Theory 13, 1978,362-368 8. Akbil, E. and Lee, T.W., On the Motion Characteristicsof Tripode Joints. Part 1, General Case, Part 2, Applications. ASME Journal of Mechanisms, Transmissions and Automation in Design 106, 1984,228-234 and 235-241 9. Pandrea, N., Kinematics of Tripode-Joint Transmissions, Rev. Roum. Sci. Techn., Mkc. Appl., 33,1988,531-537 10. Mariot, J.-P. and. K'nevez, J.-Y Kinematics of Tripod Transmissions. A New Approach, Multibody System Dynamics, 1999,Vol3,85-105 11. K'nevez, J.-Y, Mariot, J.-P., Moreau L. and Diaby M., Kinematics of transmissions consisting of an outboard ball joint and an inboard generalised tripod joint, Journal of MultibodyDynamics, 2001, Vol3, 132-145 12. Urbinati, F. and Pennestri, E., Kinematic and Dynamic Analyses of the Tripod Joint, Multibody System Dynamics, 1999, Vol2,355-367

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Using Taguchi methods to aid understanding of a multi-body clutch pedal noise and vibration phenomenon P KELLY

Powertrain Applications, Ford Werke AG, Cologne, Germany J W BIERMANN

Drivetrain/Acoustics,Institute of Automotive Engineering (IKA), Aachen, Germany

In the automobile world of noise, vibration, and harshness (NVH ) there are numerous sources and areas of possible concern. See figure 1 for a schematic of some noise and vibration terms. This world is normally hidden from the customer due to the extensive development resources applied to the vehicle before production. This paper reveals one of those hidden issues, in this case the phenomena known as "whoop". Automotives with manual transmissions require a clutch system to break the torque transfer to assist in gear changing. During the transient process of disengaging and engaging the clutch pedal, sensitive vehicles, such as diesels, can exhibit an unpleasant noise combined with a tactile foot vibration.

Figure 1: Schematic of various noise and vibration terms. The whoop multi-body NVH concern is exposed late in the development process and leads to palliative solutions added to the clutch actuation system. Masking solutions, such as these, are expensive in terms of resource. Patches highlights the lack of root cause analysis. The insufficient system interaction knowledge, related to the excitation and the complicated system behaviour, provided the stimulus for this investigation. Ford Motor Company led a team comprised of research establishments and clutch suppliers, to investigate this phenomenon. Members from the Institute of automotive engineering (IKA) at Aachen, clutch supplier LuK from Buehl, Germany, together with the Motor Industry Research Association at Nuneaton, and the University of Bradford, UK.

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The full investigation covered phases involving problem-solving study, experimentation and modelling. This paper gives some background introduction and describes the experimentation segment undertaken by IKA (l), into the whoop NVH issue. It gives details of the approach taken, the results and the subsequent analysis. Using Taguchi-Methods, an extensive vehicle test programme showed the effects of several clutch-sub-system components, and provided data to prediction an optimal solution. This paper also shows the validation of the final solution.

Clutch System Introduction The clutch system on manual transmission vehicles is defined as the component parts from the pedal to the flywheel; the system is operated by cable or hydraulics. Figure 2 shows an example of a typical clutch system.

Figure 2: Parts in Clutch System Earlier research by Dixon and Rhodes (2) reported on % order excitation problems of diesel flywheels caused by crankshaft bending. The effect of this bending is an impulse excitation to the clutch system, as shown in figure 3.

Figure 3: Crankshaft bending impulse to clutch system resulting in pedal vibration The experimentation work was an attempt to discover the influence component parts had on the transmission of the subsequent vibration and noise. Undoubtedly the trend to smaller engines with higher power outputs will lead to increasing excitation leading to higher impulses into the clutch system.

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Benchmarking pedal vibration A benchmarking exercise on mid size vehicles built between 1998 and 2000 found that clutch pedal vibrations was a industry-wide issue for diesel built vehicles. Figure 4 shows a tactile level of vibration occurs at some point of the clutch pedal travel. For confidentially reasons, the direct link between vehicle and curve is not shown.

Figure 4: Benchmark vehicles showing pedal vibration All vehicles evaluated had front-wheel drive inline Turbo charged 4 cylinder diesels with five speed manual transmissions. Engines of Ope1 and Volkswagen had direct injections whereas all other engines had indirect injections. Differences regarding rated horsepower and rated torque derive from different turbocharger characteristics. Figure 5 gives an overview of the relevant vehicle data.

(% pedal travel) Clutch disc diameter (mm)

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The engineering trend is towards direct injection with increasing fuel pressures. This produces the higher excitations and there is a corresponding increase in measured pedal vibrations. This highlights the industry-wide need for enhanced whoop understanding.

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Introduction to Experimentation Traditionally, engineering experimentation was based on the ‘one-step-at-a-time’ method. This led to changes with little validation as to their affect on other neighbouring systems. The problem with this approach was that design changes might have a particular effect on the system that depended on how the system was set up when the test was actually made. This was especially true during the development phase of a vehicle, where many components are at different design levels. The subsequent changes to other parts outside the system could then cancel out the effects of the original system changes. Thus the information gained from this approach was, time consuming, usually incomplete, and had a limited useful life. Taguchi proposed an alternative using statistically designed experiments to plan the programme of testing. Statistical Design of Experiments (DOE) meant making many design changes at once, conducting several tests and evaluations before decisions were taken. These changes were so designed to discover the effect of each design change. This way an optimal solution can be established even if the particular test combination was not even evaluated. Many have found that this type of experimentation gives an opportunity for major breakthroughs in Quality.

Objective of the DOECase Study A complete vehicle was investigated in order to evaluate the whole clutch system interaction and so to increase the NVH behavioural understanding. The vehicle chosen was a 1.8 litre Diesel, with a solid flywheel and clutch cable actuation. A screening type design of experiment was selected first because of the many possible factors that involved and the limited resources available. The major objectives were: To identify the important noise (nuisance variables) and control factors (those factors that are easily controlled such as mass of material). To estimate the contribution of the control factors. To be able to predict the whoop performance of a system. Identifying the Response and Factors Before designing the layout of the experiment a thorough knowledge of the clutch system and the whoop NVH aspects was absolutely necessary. With this knowledge the important components that were likely to influence the clutch pedal NVH were identified. To assist with the clutch NVH understanding aspects, a Problem Solving Team investigation was undertaken in parallel (3). At the same time a multi-body dynamic clutch model was developed (4) to improve the understanding of the function of the system and to establish which were the important contributing parameters. Response The measure output in an experiment is called the response. When the response relates to some measure of quality, it is also called a quality characteristic. This should be a measure related to the ideal function, and not an error state like vibration or audible noise. Unfortunately the theoretical ideal clutch response, torque transfer, was difficult to measure; therefore pedal vibration (m/s2) and sound pressure (dB) were the chosen responses.

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Noise Factors The noise factors were determined from known influences available from the problem solving study, Thus the final solution would have to work for different application speeds and engine speeds to be a “robust” fix. The Noise Factors chosen were: 1. Clutch application speeds - Fast and Slow 2. Engine speed - 2000 rpm, 3000rpm and 4000rpm Note the term “Noise Factors” should not be confused with audible noise. Control Factors To run a statistically designed experiment, the control factors have to be identified prior to the experiment. A two level experiment was chosen where two values for each of these factors were selected. For a first experiment, two levels for each factor are normally adequate. After this more levels can be assessed by further experimentation to investigate any unresolved issues. In a brainstorming session a list of factors was identified, and the feasibility of producing these parts was assessed. This resulted in the selection of the following control factors: 1. Flywheel mass 2. Clutch lever masddamper 3. Clutch cable length (damping) 4. Pressure plate mass 5. Clutch disc cushion spring stiffness 6. Clutch cover stiffness 7. Number of clutch release bearing balls To protect the Ford Motor Company confidentially requirements, these factors will only be referred to as letters A to G with no cross reference to which is which. Table 1 shows the Control Factors and the two levels of each part that were tested.

Table 1 : The control factors used in the DOE The concept of control factors could be easily distinguished here. The clutch supplier can easily control the clutch disc cushion spring stiffness to the required quality or production requirements but they cannot control the engine speed or speed of clutch pedal application. Thus the cushion spring stiffness is a control factor and engine speed is a noise factor. The aim therefore is to have a clutch system that is insensitive to noise factors.

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Methodology for the Design of Experiments (DOE) Traditionally experiments were conducted through full factorial designs. A full factorial design, which identified all the possible combinations for a given set of factors, is the most complete way of showing the important factors and their interactions, in an experiment. However, this results in a large number of test runs. As most industrial experiments involve a large number of factors, a full factorial is often difficult or too resource intensive. In the whoop experiment there are 7 factors, each with two levels and two noise factors, the total number of combination was 256. For each of these runs 3 engine speeds and two pedal application speeds are recorded, a total of 1536 specific tests. To reduce the number of experiments to a practical level, only a small set from all the possibilities, was selected. A partial factorial experiment was selected which would then act as a screen to limit the number of experiments, but would still provide the information to identify the effect of the factors. Orthogonal arrays The Taguchi approach to DOE provided a systematic way of conducting the experiment. Orthogonal arrays have been established to describe a large number of experimental situations. The arrays are matrices in which each level of factor is performed in a balanced way. That is, each level of every factor is tested in an equal number of times. The Lg orthogonal array selected for the whoop experiment is shown in table 2. Control Factors

Noise Factors Acceleration Audible Noise

Table 2: Orthogonal array used in whoop experiment Each process configuration is called a run and the Lg array consists of 8 runs. During each run the values of noise at the footwell and clutch pedal acceleration were recorded for each of the engine speeds and pedal application speeds. Six averaged values for each run. Shown here is the run for the fast application speed at 2000 rpm engine speed. This procedure was repeated until all the runs were tested. It can be seen that at each run more than one factor (or design change) was made From the results in table 2, it can be seen that there is significant variation. Variation, however, would be expected even if the design had been unchanged and the purpose of the statistical analysis is to identify the variation in the response related directly to the changes in the factor levels. At first glance this seems confusing, and is often a criticism of this method, however, more can be gained from this approach than is first appreciated. Some runs produced a decrease in the response (as required) but other runs increased them again.

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One complication is that the lowest response for pedal acceleration was not always the lowest for audible noise. When the variation in response is related to the variations in factors an optimised combination can be found. This combination may not have been tested, but it can be calculated from the experimental data. Factors that had an important effect on the response and those that had no effect are evaluated. Firstly the factors that have no effect can be selected for reasons of economics or convenience. Secondly only the important factors can be focused upon and no time wasted on the less important ones.

-

Analysis of Results Effects plots To evaluate the impact of each factor the average value of the response or quality characteristic (acceleration and audible noise) was compared when that factor was at level 1, to the average value when the factor was at level 2. The difference between these two factors is the effect it has on the response. A negative answer means that changing from level 1 to level 2 had a beneficial effect. Finding the effect of a factor is possible because of the balance in the experiment. Looking at the experimental array in table 2 it can be seen that the changes are structured in a systematic way, Each factor is tested exactly four times at level 1 and four times at level 2. Furthermore, the runs where a particular factor is at one level, all other factors are tested twice at level 1 and twice at level 2. This is true for any pair of factors, so the experiment is balanced in regards to the design changes. By referring to table 2, you can see that the effect of factor A can be determined by comparing level 1 from runs 1 to 4, to level 2 shown by runs 5 to 8. This is established by adding the sum of the responses for each factor and level. The difference from the sum of the responses for each level gives the effect. For convenience their effects are displayed in a response table - see tables 3 and 4 for the vibration and audible noise data respectively.

Table 3: Vibration response for the 2000 rpm Fast Application in m / s 2

Table 4: Audible noise response for the 2000 rpm Fast Application in dB(A) The response table can be used to pick out which factor levels are apparently best for the response characteristic, simply by choosing the levels that give the lower average response.

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Vibration plots It is useful to present numbers graphically, and the graphical version of a response table is called an effects plot. Figure 6 shows a sample for the fast application at 3000rpm engine speed. This picture indicates the average response for each factor at each level, so that the vertical distance between points is equal to the size of the effect. It is clear to see that factors B and C contribute most. Combining prior engineering knowledge (3) (4) and the numerical information it is possible to select those factor levels which the data suggest improve the situation but which also make sense.

A

B

C

D

E

F

G

Figure 6: Effects Plot - Pedal vibration, Fast application at 3000 rpm In the case of the cable and a mass lever, prior knowledge in the one step at a time approach showed that these changes were expected to show significant improvements. This input also provided a measuring indication on how effective the other changes could be. Variation is expected in the result as there is variation inherent in any process irrespective of the changes made during the experiment. All the effects calculated are influenced to some extent by variation and it is theoretically possible that the effect of a factor is not real, and that it may not really have any effect at all. Audible Noise plots For convenience a summary of the audible noise plots is not shown. The noise effects are much less distinguishable. The largest audible noise effects were from B, C, D. Daniel plots Statistical methods should be used to determine if the results of experimental could be taken at face value. Conventional statistical practice takes a sceptical view of apparent effects: only those, which clearly stand out from the noise, are treated as real. The only 'significance test' that counts in the end is a demonstration of whether the result can be repeated or not. Two methods that can be used to help decide which effects might be treated as real: normal plots (both full and half normal plots). For normal plots the measured effects can be split into 'real' and 'random' and generally the half normal plots are easier to interpret. Alternatively, a full normal plot may reveal problems with a data set that are hidden in a half normal plot, and with computer software both graphs can be easily produced. The half normal plot is often refer to as a Daniel plot, after the author Cuthbert Daniel, who first suggested using it to analyse experimental data.

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Daniel plot - Vibration Most of the points for vibration in figure 7 cluster around a straight line through the origin of the plot, but three points lie well clear of the line. These three effects C, B and D would be treated as real vibration effects and, in the absence of other evidence, the rest as the results of random variation. A b s o l u t e effects

C

Half N o r m a l S c o r e s

Figure 7: Daniel plot for Vibration at 3000 rpm and Fast application Daniel plot - Audible noise There are 6 noise and 6 vibration sets of data for the effects and Daniel Plots. For simplicity the 3000 rpm and fast application are shown. Figure 8 shows that C, F and G have an audible effect at 3000 rpm. Inherently it is difficult to recognise apparent audible noise differences subjectively heard by the operator by using only one figure from a noise trace. The quality or signature of the noise is lost. In the case of noise the team was unsure if this would work. The experimental results obtained confirmed that one value did give sufficient information to identify meaningful differences. Absolute effects

C

Half Normal Scores

Figure 8: Daniel plot for Audible Noise at 3000rpm and Fast application It is important to know if a factor has no effect or only a small effect and this can be just as important as finding a large effect. When managing a design there are usually good reasons for leaving factors alone unless there is a clear advantage to be gained from changing them. Here, because they are not important contributors, factors E, D and A can be disregarded for future analysis. The selection of the factors is then chosen based on convenience.

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The optimum outcome could even be a selection that was not even tested in the experiment. This optimum configuration could be tested to confirm the analysis. First a prediction of how much improvement is expected can be made. With the knowledge gained in the experiment diagnosis is therefore easier for any future changes in the design process to be made without always re-testing.

Interactions One significant benefit of the Taguchi approach is the evaluation of interaction. Using the one step at a time approach this was not possible. Interactions make the usefulness of experimental results difficult. Interactions can be measured numerically, but at the cost of either reducing the number of factors in the experiment or increasing the number of runs. In most engineering experiments it are just the two-way interactions that are included and this helps reduce the number of runs investigated. The whoop design of experiments followed the pattern of an initial analysis and prediction made on the initial assumption of no interactions. Interaction exists when the effect of one factor is different at different levels of the other factors. Graphically it is obvious to see interactions as they are when the lines are not parallel. So to measure interactions a measure of the parallelism is made. When there is substantial interaction, average effects (known as main effects) alone will not then be adequate, to predict what will happen if a factor is changed.

Figure 9: Interactions for the Fast application & 3000 rpm noise factor For simplicity only one of the vibration interactions is shown. See figure 9 which shows the interaction for the components parts, A to G. The non parallel lines are AxB, DxG and ExF, which show that these are possible interaction effects.

A summary of the interactions for the three engine speeds and two pedal application speeds is shown in table 5.

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The ExF interaction is shown as important for all conditions. The DxF interaction is important only for the slow 2000 rpm condition and would not be a priority to be followed up. To evaluate if the major interactions are real a follow up experiment should include ExF, AxB and GxD.

Table 5: Main interactions for the six noise factor conditions

Predictions Using a simple formula, interactions can be included or ignored. In a fractional factorial experiment only a subset of the possible factor combinations is run. A total of n runs provides some information about (n-1) factors, however, as the number of factors increases, the interpretation becomes more uncertain, because of confounding behveen different effects. The following regression edicts the vibration at the pedal: Equation Coefficient of determination in % (Adjusted for degrees of freedom) 9.01 - 4.8B 78.6% 9.01 - 4.8 B - 2.2 C 98.4% 98.0% 9.01 - 4.8B - 2.2 C -0.105D Using factor B alone a satisfactory determination can be made. Adding factor C improves the accuracy to 98.4%. From figure 6,factors B & C had largest influence on the pedal vibration. When a third factor is added the accuracy is slightly reduced because of the extra degrees of freedom allowed for in the prediction showing that factor D adds no real accuracy. DOE- Results and Discussions The following is a summary of the important points found in the whoop case study: The objective were met: This method resulted in being able to closely predict the important and unimportant whoop factors. The contribution of the control factors was identified. It was possible to predict the whoop performance of a system Lessons learnt from the exercise: A screening exercise is a good idea when there is little known about the system. This should be followed up with a larger test program with fewer single factors to improve the accuracy of the first test.

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Choosing the appropriate output response is one of the most important steps. An Ideal Function analysis should be used to help in that decision. Here the pedal vibration and the footwell noise gave a good impression of the subjective comfort rating. The selection of control and noise factors should be made as a team effort, involving the necessary specialists. The Daniel plots gave good graphical representation of the factors with major influences. Future work should include the interactions found as factors in a larger orthogonal array to establish if their affects are real. With the knowledge gained in the experiment diagnosis is therefore easier for any future changes in the design process to be made without always re-testing. The authors would like to thank the contribution from Ford Motor Company, Institute of automotive engineering (IKA), clutch supplier LuK and the University of Bradford for the excellent teamwork and support to achieve a successful outcome. References: (1) Clutch Pedal Dynamic Noise and Vibration Investigation using Taguchi Methods. P.Kelly, J.-W.Biemann. B.Hagerodt, 6th Aachen Colloquium. Automobile and Vehicle Technology, 22nd October 1997 (2) Dixon, J., & Rhodes, D. M., The generation of half orders by structural deformation, International Conference Vehicle NVH & Refinement, IMechE Conference Paper, C487/032/94, MEP, 1994, pp.9-17 (3) Kelly, P., Rahnejat, H., Clutch pedal dynamic noise and vibration investigation, MultiBody Dynamics: Monitoring and Simulation Techniques, University of Bradford, UK, 25 March 1997 (4) Centea, D., Rahnejat, H., Kelly, P., Non-linear Multi-Body dynamic analysis for the study of In-Cycle vibrations (Whoop) of cable operated clutch systems, Simulation, Diagnosis and Virtual Reality, ISATA, Florence, Italy, 18 June 1997

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Combined multi-body dynamics, structural modal analysis, and boundary element method to predict multi-physics interactions of driveline clonk S THEODOSSIADES, M GNANAKUMARR, and H RAHNEJAT Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, U K

M MENDAY

Ford Engineering Research Centre, Dunton, U K

Abstract: The paper describes a proposed methodology for an integrated approach of vehicular NVH (Noise, Vibration and Harshness) problems. The virtual prototype model creation combines rigid multi-body dynamics analysis and flexible body oscillations, using super-element FEA techniques. Emphasis is made on the inclusion of the system’s local non-linearities (gear lash zones, rolling bearings) and the reduction of the total number of degrees of freedom by applying the component modes synthesis method. The radiated noise is calculated using the boundary element method. The resulting signals of vibration and noise are processed using advanced techniques such as the AutoRegressive Moving Average (ARMA) method. The aforementioned methodology is applied in the case of clonk, which is a high frequency driveline NVH phenomenon. Predictions show good conformance with previously obtained experimental results. Keywords: multi-body dynamics, finite element method, component mode synthesis, boundary element method, autoregressive moving average method, clonk phenomenon. NOTATION b Half of normal gear backlash Wave speed Gear teeth pair contact stifmess rn Number of independent modal co-ordinates n Number of constraints 4 Elastic modal co-ordinates U Linear nodal deformation vector XYYYZ Translational degrees of freedom c, Constraint function D Damping energy dissipation function Generalised applied forces projected on 5 Fc,

f e,

K,

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Ball-to-inner race contact stiffness Ball-to-outer race contact stiffness

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Effective point contact coefficient of proportionality The Lagrangian Pitch radii of pinion and driven gear, respectively Kinetic energy Potential energy Ball-to-race contact force Damping Ratios Matrix Ball-to-inner and outer race effective Hertzian contact deformation Lagrange multipliers Generalised co-ordinates

4 P1,2(1)

W%(P

CD IC

alIN 1

Deformation shape function Pinion and gear rotation angles, respectively Euler angles Physical displacements of the interior degrees of freedom in the constraint modes Physical displacements of the interior degrees of freedom in the fixed boundary normal modes INTRODUCTION

Modern trends in vehicle technology require the use of lighter materials and thin walled components since there is a constant need for an effective combination of lower car weight and increased power. These conditions, in addition to the presence of various lash zones in the vehicular drivelines (transmission and differential gears, spline joints etc.) amplify the appearance and effects of NVH phenomena, resulting in significantly higher vibrations in the powertrain system. A typical driveline NVH spectrum contains the contribution of a significant number of vibration components, ranging from a few Hz to several KHz, depending on the elasticity of the particular components, their natural modal density and the characteristics of the transmitting torque through the system. These contributions have usually strong interaction effects. Therefore, the vibration and noise path make their presence felt and perceived by the driver and passengers with immeasurable effects in terms of warranty issues. One of the most interactive NVH problems is a short duration, audible, high frequency, elastoacoustic phenomenon, which occurs as a load reversal in the presence of lashes in the driveline (300-5,000 Hz) (Biermann and Hagerodt, 1999; Krenz, 1985; Menday et al., 1999; Theodossiades et al., 2004; Vafaei et al., 2001). Causes of this load reversal are related to: rapid applicationhelease of throttle at low gear and speed, rapid engagemenudisengagement of the clutch and sudden loading that causes a low frequency fore and aft motion of the vehicle. The

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result is a short duration impulse that excites a large number of structural modes of the lightly damped driveline system and, thus, produces an acute metallic noise. Particularly, components such as the driveshaft pieces and the transmission bell housing are effective noise radiators mainly because of their low thickness and high modal density. Consequently, in powertrain systems, where a number of components interact with each other, the need for accurate virtual prototypes in the early stages of a vehicle programme is imperative, because of the significant time and costing issues. Advanced virtual prototypes permit the testing of various conditions and allow design modifications to be carried out. 2 METHODOLOGY The proposed methodology is a combination of the following: Elasto-multi-body Dynamics analysis including global and local deformation. 0 Component Mode Synthesis to reduce the total number of structural degrees of freedom. 0 Boundary EIement method to calculate the noise radiation levels. Autoregressive Moving Average in order to identify the main frequencies included in the

response. The governing equations of motion for a rigid and flexible body are derived from Lagrange’s equation for constrained systems:

k=l

where: = {x,y,z,y1,f3,(p}~for the rigidbody degrees of freedom

{{,},=,

{ ~ , ) , = l + 6 + m ={x,y,z,yr,f3,~p,q)~ for the flexible body structural response ( q are the modal coordinates of total number, m)

L = T - V is the Lagrangian: the difference between Kinetic and Potential energies and D = -1q .T Z q . 2

The n constraint functions for the different joints in the driveline model are represented by a combination of holonomic and non-holonomic functions as:

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Therefore, the multi-body model of the mechanical system is a constrained non-linear dynamics model, incorporating the elastic behaviour of the driveshafts by considering small, linear body deformations relative to a local frame of reference, whilst this local frame of reference undergoes large, non-linear motion with respect to a fixed global frame of reference. By following the modal superposition principle (the assumption that the linear deformation of a component’s very large number of nodal DOF, u , in a pre-determined frequency area can be captured as a linear combination of a smaller number of shape vectors, 4 ), then the number of mode shapes, m , is requested:

Thus, the finite element modes can be rewritten in the matrix form as: u=@q

(4)

where, the modes 4, are included in the columns of the modal matrix, @. This matrix is the transformation from the small set of modal coordinates,q, to the larger set of physical coordinates, u . The application of Craig-Brampton (Craig, 1981) Component Modes Synthesis method determines the modal matrix, @ ,using: 0

A set of Boundary degrees of freedom ( uB) which is preserved exactly in the modal basis. A set of Interior degrees of freedom ( u , ). The Constraint Modes ( qc), which are obtained by giving each of the boundary degrees of freedom a unit displacement, while holding all other boundary degrees of freedom fixed. The Fixed Boundary Normal Modes ( q,, ), which are obtained by fixing the boundary degrees of freedom and computing a solution of the eigenvalue problem.

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The relationship between the physical degrees of freedom, the Craig Brampton modes and their modal coordinates is expressed as

where: I , 0 are unity and zero matrices, respectively. Eventually, a mode shape ortho-normalization determines the orthogonal modal matrix @’ and permits the address of the 6 rigid body modes and the association of the rest of the modes to a natural frequency, which helps to classify them physically and simulate non-linear systems with unknown spectral content. The use of finite element models for the representation of flexible components such as the driveshafts permits the computation of initial and boundary conditions (reactions at the driveshafts’ ends) that are required for an acoustic analysis. Using the historical data and applying the boundary element method, the sound pressure fields in the exterior domain can be obtained in a natural way having previously simulated the in-service events of interest. The propagation of small amplitude waves can be represented by the Helmholtz equation in the frequency domain, as: (V’

+ k 2 ) p= 0

d2 d2 d2X d 2 y

where, V’=-+-+-

3‘

(6) w

and k = - - .

8’2

C

The Green’s function is given as: e-Jb

G=4m

(7)

which is a fundamental solution to:

where, S(x, y ) =

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0 when X # y 1 when x = y

377

By applying the indirect method, the pressure and velocity in the boundary surface are given as p = [ G ~ - p g p ) G & and v = r

where, o =

($)+

-

($)-,p p + =

- p - and n is the surface normal vector.

The acoustic pressure at a point of interest r, can be written as:

where, rr is a point at the boundary

(Banerjee and Butterfield, 1981; Becker, 1992)

The short, transient, high frequency response phenomena - such as clonk - can be effectively investigated through the Auto-Regressive Moving Average method (ARMA). If a model can be successfully fitted to a data stream, it can be transformed into the frequency domain instead of the data upon which it is based, producing a continuous and smooth spectrum. This is the basic premise of the spectra produced using ARMA. For a data series x of length N, the model is defined as the reverse prediction for the first p values and forward prediction for the remaining N - p values as the output of a pole-zero filter excited by white noise, as: Uk

k = l ...N

=o P

=xk

4

- ~ a , x , + ,+ c b , u k + ,

k=min(N-p,max(lOO,p+q)) ...l

I=I P

= xk - xa,xk-, -k ,=I

Yk = ' k + ' k

4 xb,Uk-, I=I

(10)

k = p + l ...N

k = l ...N

where, a is the auto-regressive parameter array of order p and b is the moving average parameter array of order q (Kay, 1988; Marple, 1987). Once the parameters of the ARMA model are identified, their spectral density function can be obtained as:

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where, v is the driving white noise variance and At is the sampling interval. The flowchart of the proposed overall methodology and computations is shown in figure 1, taking into account the different methods used and their interactions. 3.

The multi-body model

The multi-body virtual prototype comprises all the components of the powertrain system, starting from the transmission input shaft up to and including the rear axle halfshafts through the use of Parasolid CAD files in such a way that their physical properties are exactly represented. The total number of degrees of freedom is obtained using the Gruebler-Kutzbach expression as: Number of DOF = Flexible Bodies Modes + 6*(number of rigid parts - I ) - Z(constraints) =(39 + 154) + 6* (27 - 1)- 163 = 186 Hence, the drivetrain model has 186 degrees of freedom. In the simpler case, where flexibility is not included in the system, the same expression yields: Number of DOF = 6*(number of rigid parts - 1) - C(constraints) = 6* (29 - 1) - 163 = 5 Thus, the rigid drivetrain model has 5 degrees of freedom, which are shown in figure 2. Four-noded, shell elements have been used for the finite element models of the driveshaft pieces. The number of elements and their size across the shaft length and in the perimeter has been made sufficiently large so as to capture the higher modal responses with their complicated shapes. A sufficient number of structural modes have been kept during the creation of super-elements in order to obtain accurate results in the specified frequency area, where clonk usually occurs (3005,000 Hz). Since the same mesh is also used in the boundary element models of driveshafts, then particular attention has been paid so that the number of elements per wavelength is adequate for the noise radiation analysis in the exterior domain. The maximum frequency of interest determines the actual wavelength. The location of virtual microphones (data recovery nodes that

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measure the noise levels) with respect to the vibrating structure is calculated in such a way that the minimum frequency of interest can be captured. The calculation of the developing transmission gear forces between the engaged teeth pair@) during the meshing cycle takes two important factors into account: The presence of gear backlash * The variable number of gear teeth pairs which are in contact at a time, causing a variation of the equivalent gear meshing stiffness. The use of piece-wise linear equations of motion with time-dependent coefficients represents the involving backlash and time-dependent mesh stiffness ( k ( f ) ) .The centres of both gears are restricted from any lateral motion. The meshing stiffness depends on the number and position of the gear teeth pairs, which are in contact at any given instance and is a periodic finction of the relative angular position of the gears (Choi and David, 1990, Chen and Tsay, 2002; Zhang et al., 1999). Therefore, the force developed between the pair of gears is given by the product k(t)h(x) where: x ( t ) = R,P, (0- R2P2 ( f ) and: X-b, x t b kl
where, n=3/2 for ball bearings and:

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4

RESULTS - DISCUSSION

Studies of sudden torque application at the input shaft of the drivetrain model are carried out under similar conditions with the experimental rig in Vafaei et al., 2001. The duration of the torsional pulse was set on 70 ms (as required in order to simulate the behaviour of an aggressive driver), inducing a generated lash take-up of the order of 1-2 ms of sufficiently short period to excite the higher frequency clonk modes. The time history of accelerations of the front and rear driveshafts’ centres of mass are shown in figure 3 for the vertical motion z. The regions, where the clonk phenomenon appears - the lash zones are taken up - are clearly distinguishable with higher acceleration amplitudes. Additionally, the rear driveshaft seems to get excited more than the front one, which is understandable because of its higher modal density due to its larger length and middle step. The total duration of the phenomenon seems to last about 35-40 ms (corresponding to a frequency of about 4-4.5 Hz), leading to the conclusion that it is a cycle of shuffle (Biermann and Hagerodt, 1999; Menday et al., 1999), which is the lowest rigid body mode of the drivetrain system and another underlying cause for clonk. The CPU time was approximately 10 hours on a 2.6 GHz Pentium IV Desktop for a simulation of 120 ms, in 60,000 computation time intervals of 2 p . The lash take-up time of the order of 2 ms accounts for the impulsive action at the ends of the driveshaft pieces of the same duration. These are the initial conditions used in the noise radiation analysis, the results of which are shown in figure 4. The FFT spectra of the acoustic pressure reveal the existence of several main frequencies. Again, the spectrum of rear driveshaft tube appears to be more complicated, compared to the one of the front tube. However, an FFT spectrum does not provide information regarding the dominant frequencies that are responsible for noise emission, because of the short duration clonk in addition to the very small time step of analysis. Moreover, a leakage in spectral contributions may have possibly harmed the results (Vafaei et al, 2001). The ARMA technique sheds light on this aspect, as it is shown in figure 5 . It is revealed that there are two main contributing frequencies carrying the highest amount of energy at each driveshaft, namely 3311 Hz for the front tube and 3936 Hz for the rear tube. Vafaei et al, 200 I, have observed similar results experimentally. The linear modal analysis of the drivetrain system exhibits another interesting aspect, as it is shown in figure 6. The two aforementioned dominant frequencies qualitatively correspond to breathing modes, 3368 Hz for the front tube and 3917 Hz and 3923 Hz for the rear tube, respectively. Mode shapes of this type are generally effective noise radiators and this probably

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explains the metallic high frequency clonk noise observed during similar experimental conditions. ACKNOWLEDGMENTS The authors wish to express their gratitude to Ford Motor Company, MSC Software and the Vehicle Foresight Directorate (EPSRC and DTI) for their sponsorship and financial support extended to this research project. REFERENCES Banerjee, P.K. and Butterfield, R. Boundary Element Methods in Engineering Science, 1981 (McGraw-Hill). Becker, A.A. The Boundary Element in Engineering, 1992 (McGraw-Hill). Biermann, J.W. and Hagerodt, B. Investigation into the clonk phenomenon in vehicle transmission - measurement, modelling and simulation. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 1999,213(1), 53-60. Chen, Y.C. and Tsay, C.B. Stress analysis of helical gear set with localized bearing contact. Finite Elements in Analysis and Design, 2002,38,707-723. Choi, M. and David, J. W. Mesh Stiffness and Transmission Error of Spur and Helical Gears. SAE 901764. Craig, R.R. Jr. Structural Dynamics -An Introduction to Computer Methods, 1981 (New York: John Wiley & Sons). Kay, S. Modern Spectral Estimation, 1988, Prentice Hall. Krenz, R. Vehicle response to throttle tip in /tip out. M E 850967. Maple, L. Jr. Digital Spectral Analysis with Applications, 1987, Prentice Hall. Menday, M., Rahnejat, H and Ebrahimi, M. Clonk: an onomatopoeic response in torsional impact of automotive drivelines. Proceedings of the Institution of Mechanical Engineers, Part D: Journal ofAutomobile Engineering, 1999,213,349-357. Rahnejat, H. Influence of Vibration on Oil Film in Concentrated Contacts. Doctoral Thesis, Imperial College of Science and Technology, University of London, 1984. Theodossiades, S., Gnanakumarr, M., H Rahnejat, H. and Menday, M. Mode Identification in Impact Induced High Frequency Vehicular Driveline Vibrations using Elasto-Multibody Dynamics Approach. Accepted for publication at Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2004.

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Vafaei, S., Menday, M.T. and Rahnejat H. Transient high-frequency elasto-acoustic response of a vehicular drivetrain to sudden throttle demand. Proceedings of the Institution of Mechanical Engineers, Part K:Journal of Multi-body Dynamics, 2001,215,35-52. Zhang, J.J., Esat, 1.1. and Shi, Y.H. Load analysis with varying mesh stiffness, Computers and Structures, 1999,70,273-280. Adarndyolver Reference Manual, Version 12.0,2002 (MSC Software) Patranhlastran Reference Manual, Version 2001 (MSC Software)

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................................................................... I

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I Q FFT- ARMAAnalysis on Acoustic Pressure to Identify the Frequency Content

Modify the Design

Continuc with the Design

Figure 1: Methodology - the computational flowchart.

Figure 2: The multi-body model of the drivetrain mechanical system.

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385

50

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0

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Figure 5: ARMA spectra of the acoustic pressure time histories of the front (a) and rear (b) driveshafts corresponding to the case of an aggressive driver.

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Figure 6: Magnified graphical scale of dominant breathing mode shapes corresponding to Figure 5: (a) 3368 Hz, (b) 3917 Hz and (c) 3923 Hz.

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Ve hicle Dy namics, Ride and Handling

Dynamic analysis of semi-active suspension systems using a co-simulation approach R RAMLI, M POWNALL, M LEVESLEY, and D A CROLLA School of Mechanical Engineering, University of Leeds, U K

Abstract The aim of this paper is to demonstrate how the principles of co-simulation can be used to calculate the ride response of a vehicle with semi-active suspension. The ultimate goal is to use the approach developed for the estimation of component service loads, enabling optimisation of components based on fatigue life to be implemented. The co-simulation technique employed allows a dynamic vehicle model to be constructed in a Multi Body System (MBS) environment, while two key components, the tyre and active suspension systems, are concurrently simulated in a mathematical simulation environment. The packages used are MSC Visual Nastran MBS software and Mathworks MATLAB/Simulink software. Since previous research [l, 21 has shown that a quarter vehicle model (QVM) is inappropriate for this type of analysis, a lumped parameter full vehicle model (FVM) has been developed in the MBS environment, though for preliminary validation the QVM is used. Using the cosimulation approach, the effects of the semi-active system and the tyre model on the vehicles response to potentially damaging transient inputs (step and potholes for example) are examined. Results show that while the semi-active suspension system employed in this study significantly affects the response of the sprung mass (vehicle body), it has less effect on the unsprung mass (wheel and hub). The converse is true for the tyre model, where the results show different tyre models greatly affect the loads applied to the hub. The paper concludes that the approach adopted is an effective method of integrating the semi-active device, control strategy, and tyre model into the MBS vehicle model. It has significant benefits over alternative methods and has potential for use in the optimisation of suspension components based on fatigue life. Keywords Co-simulation, multi-body system, semi-active vehicle suspension, vertical tyre modelling, service loads, load histories.

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Nomenclature Chard= high damping coefficient Cs = viscous damping coefficient Csoe= low damping coefficient Fd = damper force Ft = vertical tyre force Ks = passive suspension stiffkess Kt = tyre stiffness Mb = vehicle body mass or sprung mass Mw = wheel mass or unsprung mass x = displacement road profile z = displacement of vehicle components t = velocity of vehicle components z = acceleration of vehicle components kb = sprung mass relative velocity i, = unsprung mass relative velocity Introduction The emerging trend in vehicle durability assessment is to utilise virtual prototyping, as a tool for dynamic design verification and as an alternative to physical durability testing. This is made possible with increased computing speed and improved software performance, allowing significant cost saving compared to physical testing. Research at the University of Leeds is seeking to link results from MBS analysis to FE models with structural optimisation algorithms, to generate a method for automating the design of light weight components with maximum fatigue life [3, 41. Central to this method is the assumption that accurate load histories can be generated from MBS vehicle models when subjected to appropriate road inputs. Further, since the technique is currently being designed to remove the need for physical durability tests, the predicted load histories must be accurate for road inputs specifically intended to apply significant damage to a vehicle's components, such as large amplitude transient inputs generated by a step and potholes for example. Clearly the tyre model is a key component of the model, converting road displacements into forces applied to the suspension system. While some MBS software packages contain purpose made tyre models, the compromise between model complexity and accuracy/solution time as well as the ability to populating these models with appropriate parameters remains an issue. An alternative approach [ l ] is to build a the tyre model from spring and damper elements within the MBS model, though this approach may lack the flexibility to be able to assess quickly various tyre models with different MBS vehicle models. Another key component within the suspension system is the main damper element. While most MBS packages allow complex non-linear passive dampers to be included in the model, few will allow easy integration of active or semi-active elements, which are of increasing interest in the automotive industry. Clearly, different actuators and control algorithms could have a profound affect on service loads applied to the suspension system, which need to be taken into account. The aim of this paper is to show how these two key components can be integrated efficiently within an MBS vehicle model. The approach integrates an MBS based vehicle model representation, with tyre models and semi-active control models developed in a mathematical simulation environment. The intention of evaluating the load histories using this method is to enable the user to combine specific and specialised modelling tasks with different

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complexities, into one simulation code. The paper examines the efficacy of the modelling approach and compares preliminary results from the co-simulation method to a mathematical approach, in terms of tyre force, actuator force, sprung and unsprung mass displacements.

Co-simulation The approach adopted for this study integrates an MBS model of the vehicle, with separate simulation codes for the semi-active controller and the for the tyre model. The main reason behind the approach, is to allow easy integration of subsystems Le. vehicle models, suspension types, and tyre models, to be controlled under one simulation environment as illustrated in Figure 1. This, allows each of the subsystems to be developed, analysed, and validated separately. The inputloutput interface is controlled in MATLAB/Simulink [SI environment. An MBS modelling code, Visual Nastran [6] models the suspension system, whilst MATLAB/Simulink is again used to develop semi-active control laws and tyre models. The output of the system, in the form of acceleration, velocity or displacement, is fed back to the MATL,AB/Simulink code at predefined time steps. This process simultaneously computes the solution for both modelling codes. In Figure 1, the co-simulation plant is divided into two phases. The first phase involves cosimulation of tyre models with a passive suspension system. The initial analysis is centred on vertical tyre forces generated from point contact (PC) and fixed footprint (FFP) models, to allow verification against other research. The co-simulation flow may be understood as follows. The tyre subsystem (PC or FFP models) receives a displacement input from the road profile, x, in the form of transient (steps or bumps) input or random input. The product of this displacement and tyre stiffness generates a force output, Ft , which is then fed to the MBS vehicle subsystem as a force input. The force input excites the unsprung mass producing displacement, velocity and acceleration responses of the bodies within the model. With passive suspension in use, the forces exerted by the spring and passive damper are calculated within the MBS model from the relative displacement and relative velocity, between the sprung and unsprung mass, respectively. The response of the unsprung mass is fed back to the tyre model. The output displays dynamic responses of the body and wheel in the form of displacement, z, velocities, i or accelerations, z . Detailed analyses of the tyre force from the PC and FFP models were performed initially within MATLAB/Simulink for the purpose of validating the models against published data. Once achieved, the same models are linked to the MBS system and used in the co-simulation environment, as illustrated in Figure 1. Sprung and unsprung mass displacement profiles, and the tyre force are compared with the conventional simulation using MATLAB/Simulink. The second phase of the co-simulation introduces the semi-active controller subsystem. The dynamic responses from the MBS vehicle model in the form of displacement, velocity, or acceleration are fed to a semi-active controller subsystem where suitable algorithm are coded in the MATLAB/Simulink environment. Then, the subsystem sends a force output, Fd, back to the MBS vehicle model to represent the actuator force in response to the road excitation. In these preliminary studies, a simple 2 degree of freedom (2DOF) quarter vehicle model (QVM), constructed in Visual Nastran, is used to allow validation against other published data. The MBS model requires a force signal from MATLAEVSimulink to drive the actuator that is built into the hub. The position of the hub is fed back into the MATLAB/Simulink tyre model from the MBS model and simulation parameters are controlled from MATLAB/Simulink. Initial results are given focusing on the effects of the semi-active system on the vehicle and comparing these to a passive system.

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Vehicle Models The vehicle models describe in this section are confined to vertical dynamics. Haiba et. uf. [ 11 concluded that the predicted acceleration power spectral density of a quarter vehicle model (QVM) overestimated the experimental results particularly in vertical acceleration. Since a QVM does not take into account vehicle body roll and pitch, analysis of the suspension arm load histories should consider a full vehicle model (FVM). This was confirmed by Levesley er. ul. [2] with a simplified FVM. Both of the studies were based on vehicle modelling in multi-body system (MBS) alone with passive suspension only. In light of this research, it is planned that the complexity of the vehicle models used will grow in stages, starting with validation of the co-simulation approach (the main focus of this paper) using of a 2DOF QVM. A full vehicle model (FVM) in a simplified format (7DOF) has subsequently been developed (though results are not presented in this paper). Ultimately, an MBS model of realistic complexity, will be required in which each of the main components of the suspension is represented. This will allow investigation into the effects of including a semi active suspension element on the service load histories of the individual suspension components such as the arms, for subsequent stress analysis and optimisation. Through out this study the parameters given in Table 1 have been used which represent a generic Multi-Purpose Vehicle (WV) I Vehicle Parameters Units Sprung Mass, Mb Unsprung Mass, Mw Suspension Stiffness, Ks Damping constant, Cs Tyre Stiffness, Kt

400 kg 47.75 kg 22000 N/m 1500 N.s/m 239000 N/m

Table 1 Quarter vehicle model parameters of a multi-purpose vehicle

Tyre Models The development of the tyre models is based on vertical tyre dynamics, modelled in MATLAB/Simulink. Following an initial study, it was concluded that the fixed footprint tyre model (FFP) [7] offered a substantial improvement in simulating tyre forces over the single point contact (PC) spring model, without unduly effecting model complexity and hence simulation times. Although other models have also been developed, it is the FFP model that has been presented in this co-simulation study. Results are compared to the PC model to due to its simplicity, wide spread usage and hence ease of validation. Both tyre models require a negative tyre force saturation to facilitate the ability of the tyre to leave the ground if a negative step is encountered. This non-linearity is an important element in the tyre model. Semi-active Control Model The principal aim of incorporating semi-active systems is to evaluate its effect on the suspension component service load histories compared to those generated from a passive system, [l]. Unlike the active system, semi-active systems are chosen because of their minimal power consumption, simplicity in developing appropriate control laws, and minimal weight addition. In approximating the ideal skyhook concept [8, 91, a switchable semi-active control strategy is adopted as shown in the equation below.

Fd = ‘hard

.(ib-iw)

.e....

if i .(i - i b

b

w

)>o

if i .(i - i w ) < b

394

b

o

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For this initial study, a 2-state switchable controller is selected for its simple controller development which has been experimentally verified. The controller objective minimises vehicle body response by controlling the relative velocity of the sprung, and unsprung mass,,,2 across the damper. A positive value of the relative velocity and the quantity kb (zb )2 , would switch the damper to a hard setting, Chad. It occurs when the vehicle body velocity, kb is larger than the velocity of the wheel, i,, placing the damper in tension. This large damper force, Fd acts in a direction opposite to the relative motion of the masses. In contrast, if the sprung mass velocity, Ab is smaller than that of the unsprung mass, both quantities, (& e)and z b (zb - )z, shall be negative in which case the damper is compressed. The damper then switches off, creating a low or minimal force, Fd, from the soft damper setting, Csofi. Road Input Representations and Simulation Solver In this preliminary study, the road input is modelled as a simple step with a height of 0.023 m. The model is simulated to run at a constant forward velocity of 47 !an/hr. The input is applied after three seconds, allowing static deflection of the vehicle to reach its equilibrium state. More realistic road inputs, including potholes and a durability pav6 road input [ 11, have been developed but results presented in this paper are confined to the step input, due to ease of validation. The ultimate aim is to generate road inputs that match typical transient inputs known to induce significant damage within a suspension system. All simulation in MATLAB/Simulink uses the variable step (ODE 113 Adams-Bashforth-Moulton PECE multi-step) solver for tight tolerance with relative tolerance of le". In co-simulation the same solver is used in conjunction with the MBS Kutta-Merson integrator having a variable integration step size. Results and Discussions In order to obtain validated results from the co-simulation approach, preliminary analysis of the quarter vehicle model (QVM) focused on comparison with established results. A QVM with point contact (PC) tyre model was simulated both in MATLAB/Simulink and using the MBS software and results compared with those from [2]. Similar sprung and unsprung mass displacements were observed suggesting that the QVM models from both software were valid for further analysis. Co-simulation with Tyre Model - Phase 1 Two basic vertical tyre models were selected for this study i.e. the point contact (PC) and the fixed footprint (FFP). Results for QVM response to the step input are shown in Figure 2. The peak tyre force of the PC model can be seen to exceeded the force of the FFP model by 600700 N, with the latter lagging by a few milliseconds. The corresponding unsprung mass displacement profiles revealed similar characteristics to the tyre force in terms of the delay, however differences in peak amplitude are much smaller. Very minimal difference was observed when comparing the displacement of the sprung mass (not shown). Investigations using QVMs of a variety of other vehicles shows similar trends, i.e. the type of tyre model has a significant influence on the load histories generated but has less effect on the unsprung mass response and very little effect on the sprung mass. To investigate the effects of parameter variation in the tyre model, the tyre stifhess in the PC model was varied between 191,200 and 286,800 N/m, as indicated in Figure 3. As expected and in line with the previous results, varying tyre stifmess can be seen to have a significant impact on the load history but less effect on the unsprung mass response. Less still effect was observed on the sprung mass response (not shown).

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Due to the simplicity of the MBS vehicle model at this stage (2DOF QVM), direct comparison between results from the co-simulation approach and a model simulated in MATLAB/Simulink alone is possible. Results, compared well highlighting some important issues related to efficiency verses accuracy. For ease of validation, this initial study uses a step input with a PC tyre model. As can be seen in Figures 2 and 3, this results in an instantaneous application of force to the MBS model. Clearly this is not a true representation of the actual tyre. In reality there would be a more gradual rise in force due to the geometry and flexibility of the tyre as it rolls over the step. This can be taken into account in a number of ways. The response of the FFP tyre model generates the force in discrete stages depending on the number of springs used. This can be seen in Figure 2. Instantaneous step changes in force still exist, however the more springs used the more representative the model becomes but this occurs at the expense of computational efficiency. An alternative would be to use the PC tyre model but alter the step input to take account of the geometry affects of a rigid wheel rolling over the step. This profile would however need to be applied as a series of discrete steps. Again, more steps would result in a more gradual application of the force and hence a more representative model, but once more at the cost of computational efficiency. Sudden changes in force affect the integration step size required to produce an accurate simulation. This can be seen in Figure 4 where results from the MATLAB/Simulink simulation using a variable integration step size are compared to the co-simulation results with two integration step sizes limits imposed. With a minimum step size of 0.005 seconds, the responses are markedly different. As the step size is reduced to 0.001 seconds the unsprung mass response is seen to agrees closely with the variable step simulation, though significant differences in the force remain. These differences are however as a result of the over simple PC tyre model, which produces a large instantaneous increase in force, hence a simple model with a simple input may not necessarily be computationally efficient if small integration step sizes are require to model it accurately. Co-simulation with Semi-active Controller - Phase 2 In phase 2, efforts were focused on the development and validation of the semi-active suspension system in a form suitable for integration within the co-simulation environment established in phase 1. The control laws and algorithms have been implemented in MATLAB/Simulink. Inputs are in the form of sprung and unsprung mass velocities and output is in the form of an actuator force, based on relative velocity and a switchable damper. Simulation results were obtained and validated against established data [101. The simulation for the MPV is given in Figure 5 . It confirms that the semi-active control strategy improves the ride quality in terms of lower body displacement. Both the unsprung mass and actuator force registered significant differences when compared to the linear passive damper. Clearly this may unduly affect durability of the suspension components at the expense of improving comfort. Conclusions (i) A co-simulation environment has been established and validated for relatively simple vehicle and tyre models. It allows simple development and integration of key subsystems, in this case the tyre model and a semi-active suspension device. (ii) The influence of the tyre model on the sprung mass response was found to be minimal. Though its effect on the unsprung mass displacement was also small, its effect on the load history applied to it was significant. This has important implications if these load histories are to be used for optimisation of suspension components based on fatigue life.

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(iii) A relatively simple point contact tyre model may not be as computationally efficient as expected due to the small integration step sizes required to cope with the large and unrealistic instantaneous changes in force it generates. (iv) Evaluation of the semi-active suspension system revealed that the control strategy improves the vehicle body response, but has a significant influence on both the unsprung mass response and the force applied by the actuator to the suspension components.

References M. Haiba, D. C. Barton, P. C. Brooks, and M. C. Levesley, "Using a Quarter-vehicle Multi-body Model to Estimate the Service Loads of a Suspension Arm for Durability Calculations,"Proc. Instn. Mech. Engrs., Part K: J. Multi-body Dynamics, vol. 217, pp. 121-133,2003. M. C. Levesley, S. A. Kember, D. C. Barton, P. C. Brooks, and 0. M. Querin, "Dynamic Simulation of Vehicle Suspension Systems for Durability Analysis," presented at Proceedings of the 5th International Conference on Modem Practice in Stress and Vibration Analysis, Glasgow, 2003. M. Haiba, D. C. Barton, P. C. Brooks, and M. C. Levesley, "Review of Life Assessment Techniques Applied to Dynamically Loaded Automotive Components," Computers and Structures, vol. 80, pp. 481-494,2002, M. Haiba, D. C. Barton, P. C. Brooks, and M. C. Levesley, "The Development of an Optimisation Algorithm Based on Fatigue Life," Internal Journal of Fatigue, vol. 25, pp. 299-310,2003. Anon., "MATLAB-SIMULINK - User Guides," 6.0 ed. Natick: The Mathworks, Inc., 2002. Anon., "MSC Visual Nastran Desktop - User Guide," The MSC.Software Corporation, 2003.

K. M. Captain, A. B. Boghani, and D. N. Wormley, "Analytical Tire Models for Dynamic Vehicle Simulation," Vehicle System Dynamics, vol. 8, pp. 1-32, 1979. D. Kamopp, M. J. Crosby, and R. A. Harwood, "Vibration Control Using Semi-Active Force Generators," Journal of Engineering for Industry, 1974. D. Moline, S. Floyd, S. Vaduri, and E. H. Law, "Simulation and Evaluation of SemiActive Suspensions,"SAE Technical Paper Series, 1994.

K. Sharma, D. A. Crolla, and D. A. Wilson, "Derivationof a Control Law for a 3 State Switchable Damper Suspension System for Improving Road Vehicle Ride Characteristics," presented at International Symposium on Theory of Machines and Mechanisms, Nagoya, Japan, 1992.

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Figure 1: Co-simulation Plant

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Figure 2: Tyre Force and Unsprung Mass Response of PC and FFP Tyre Models

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Figure 3: The Effects of Varying Tyre Stiffness

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Figure 5: Sprung and Unsprung Mass Displacement and the Corresponding Damper Force, Between Passive and Semi-active Systems

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Smart driver: a research project for closed loop vehicle simulation in MSC.ADAMS R FRUZA and A SACCON Dipartirnentodi lngegneria dell'inforrnazione, UniversitB di Padova, Italy D MlNEN Special Project Development Group, MSCSoftware C ORTMANN Corporate Marketing, Automotive Industry, MSCSoftware

ABSTRACT It is almost universally accepted that predictive control and a hierarchical control architecture are at the basis of any human driver simulator. For human driving simulation, a hierarchical control structure is often subdivided in three levels: a strategic level for task planning; a tactical level for trajectory planning; an operational level for motion control. In the past 15 years, the geometric nonlinear control of mechanical systems and in particular of non-holonomic vehicles has been a very active research area. Concepts, such has differential flatness, which lead to new control paradigms have been introduced. These and other principles have all been applied in the SmartDriver Research Project, that MSC is conducting for defining a new driver model for full vehicle MSC.ADAMS models. The three basic components neeeded to perform an accurate simulation of a control-driven car on a given track are: the Vehicle Model (MSCADAMS), the Road Description and the Car Driver Model (SmartDriver). The paper illustrates some of the principles mentioned and shows a few application examples. 1

INTRODUCTION

Nowadays, the automotive industry relies always more and more on simulation tools for Virtual Product Development and the analysis of the performances of new vehicles. These tools [l]integrate multi-body dynamics together with finite elements and optimization packages to compute performance limits, forces, strains, and fatigue. The simulated vehicles are often called virtual prototypes, and the tools, virtual prototyping tools. Their use in the industry has led to relevant savings in production costs and time to market, but also to a significant increase in safety as the virtual prototypes can be subjected t o 'This work was sponsored by MSCSoftware

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tests that would be impossible to perform in the physical world. A fundamental characteristic of virtual prototyping tools is the possibility of simulating standard maneuvers for the dynamical analysis of vehicles as dictated by international norms. This implies the possibility of driving the virtual vehicle exactly as if it was a real vehicle by acting on throttle, brakes, clutch, gears and steering, just as a real pilot. The task of modeling human driving behaviour is a formidable one. The scientific literature in this field is vast and diverse. A recent survey is [MI. The first papers that model the driver as a controller date back to the 6Oies see, for example [22]. The first contributions towards the understanding of which information is used for feed back by a human driver were given in the 7Oies [7] in which a virtual reality set up was created to analyze the gaze of drivers, an amazing achievement given the computing power available at the time. In the second half of the 80ies, the first striking autonomous road vehicles were demonstrated. Dickmanns and his group [5, 61 showed a Mercedes van capable of driving autonomously on highway and country roads even at high speed. Dickmanns goal was driving the car rather than modeling the driver, even if, it is obvious that his controller may be seen as a driver model. In the 9Oies a number of autonomous vehicles were demonstrated and important research projects claimed objectives in which cars would be driven autonomously on intelligent highways. Today, it is almost universally accepted that optimal predictive control and a hierarchical control architecture are a t the basis of any human driver simulator. An interesting and inspiring work in this sense is [25]. Optimization is often thought to model natural behavior because of the analogy it bears to the optimization presumed to occur as a result of natural learning and selection. The hierarchical architecture has to do with robustness and complexity. If not needed, why use a complex and detailed model of the vehicle? Use it only if the driving task excites complex dynamics neglected in the simpler model. In the past two year, MSC.Software sponsored a research project at the University of Padova in Italy dedicated to the development of a driver model for the simulation of virtual prototypes. The objectives of the project are twofold: on one hand the driver should model human behaviour; on the other hand it should be able to drive the vehicle up to its dynamical limits which are independent of the driver model. From a control perspective, these are two different goals and each requires its own research. In this paper, we address the first objective. In the conclusions we give a few hints to how we are solving the second which will be the topic of a future paper.

FOUNDATIONS OF THE DRIVER MODEL

2

The driver is seen as a controller, i.e. a dynamical system that receives as inputs the sensor measurements and furnishes, as outputs, the vehicle controls: steering, throttle, brake, clutch and gear. The foundations of the proposed driver model are two control theory methodologies: model predictive control [27] and backstepping [16]. In model predictive control one solves repeatedly a finite horizon optimal control problem subject to system dynamics and input and state constraints. The following are the steps that define the control action: 1. based on past and present measurements, the controller predicts the dynamic be-

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haviour of the system over a prediction horizon Tp;

2. according to a specific performance criterion, the optimal open loop input u(t) is computed over a control horizon T,; 3. the input signal u(t) is applied until the next sampling instant t steps are repeated.

+ At when these

MPC is closed loop (the optimal control action depends on the current state) and causal (the control action depends only on past and present outputs). MPC is the second most applied [26] control synthesis technique in industry. The main reason for its success lies in the possibility of considering constraints in the controller design, a fundamental characteristic of a driver model. In driving, lots of constraints must be satisfied, steering is constrained, the maximum engine torque is bouned etc. A major research effort [2] is currently on-going in the control systems scientific community to study the properties of MPC in terms of robustness of stability and performance. In the linear case, its interesting properties are known. In the nonlinear case, besides stability for which many results have been obtained, most of its other properties are still guessed even if, until now, confirmed by applications. Backstepping [16] is a powerful control synthesis technique for systems that are composed by a cascade connection of various components. It lends itself very nicely to the design of controllers for hierarchical systems based on nested models of increasing complexity as we shall see.

3

VEHICLES MODELS FOR CONTROL

Modeling for control design is always subject to a tradeoff between accuracy and complexity. The art of control design is deriving the simplest system model that captures all dynamical effects of interest.

A complex, detailed model, with lots of degrees of freedom, at the worst, renders the control synthesis an unfeasible task or, at the least, leads to complex, non robust controllers as they will be tailored to the specific model. A rough, simplistic model, on the other hand, will lead to poor performance and unpredictable behaviours if unmodeled dynamics are excited. The correct model is in between and finding it is more an art rather than a science. Often, in control synthesis, the correct model is determined by successive attempts checking the performance achieved by the controller at each time. The right model, unfortunately, depends on the control task. For high performance driving, when the vehicle is taken to its limits, it is important to model, for example, the load transfer in roll and pitch to correctly estimate forces at the tires. For a relaxed drive at low speed on a smooth course, it might suffice to model the kinematics and completely neglect lateral dynamics. Clearly, the only way to satisfy these complex modeling requirements is to use a hierarchy of models of increasing complexity. We dedicate a particular effort to define a set of nested models of increasing complexity so that the controller architecture would become a number of nested feedback loops and model selection would then be automatic.

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The simplest higher level model at the basis of the proposed controller is the one-track non-holonomic kinematic model of the car

where (x,y) are the coordinates, in an inertial frame ET, of the mid-point of the rear axle, v is the longitudinal velocity of the same point, $ is the yaw angle of the body frame Cs w.r.t. the inertial frame ET, b is the steering angle, b is the distance among the front and rear wheel oxis and u the control action on the steering rate. Let us augment the model with a simple description of the longitudinal dynamics

where F, and Fb are the longitudinal forces generated by the engine and the brakes, respectively, and the remaining term models the aerodynamics.

Fig. 1 Nonholonomic car model: explaining the relationship o = tan(d)/p between the steering angle 6 and the curvature ~7 Model (1) together with (2) is non-holonomic in the sense that no lateral slip is allowed. It is an interesting model in terms of control design because it is dzflerentially flat [9, 10, 31 which means that there exist an output, the flat output, such that the whole state and the inputs can be determined as algebraic functions of the flat output and its time derivatives. This implies that solving the inverse dynamics problem for the flat output is trivial. For model (1) the flat output is the vector of coordinates (qy). Therefore, assigned a trajectory ( z ( t ) , y ( t ) ) for t E [O,T],the steering action u and the velocity v ( t ) can be computed algebraically. It should be noted that (1) describes the motion of a Frenet frame on the given trajectory and it should not be a surprise that the system is flat. The model at the next level of complexity is still one-track, but it includes yaw and more refined, coupled longitudinal dynamics and allows for lateral slip of tires. The tire forces

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are assumed to be a linear function of slip.

x

= 'UTcos($)

y =

* = s w, = R,.F$ IT,

cj,

-

vI sin($)

+ v~ COS($)

WT sin($)

(3)

+ Tr

+

RfFJ = If,

Tf

where VT and w 1 are the velocity components along the longitudinal and lateral body axis, R,. and R f , IT,and I f t , w, and wf are the radii, the inertia and the angular velocities of the rear and front wheels, F;f and F:f are the longitudinal and lateral forces acting on the tires and rT,fare the torques applied at the wheel by the engine or the brakes, finally

M = [

T=

[

F$

m O 0 o m 0 mb (mb2 mbI,,)

+

1

+ FTf cos(6) - F: sin(6) - FA + FTf sin(6) + Ff cos(6)

FI

.=[

p ( ~ T sin(6) f

+ F:

-m.vlU?

:it

cos(6))

- mbs2

]

1

where m is the mass of the vehicle, FA the aerodynamic forces and I,, the vehicle inertia about the vertical body axis. Models at further levels of complexity are two-tracks and include load transfer in roll and pitch to accurately compute tire forces which are nonlinear functions of slip. 4

CONTROLLER ARCHITECTURE

The fundamental task of the controller is path following. The control task 7 consists of driving the vehicle along a desired path I'd(s) = [ ~ ( s )yd(s) ] at velocity vd(s), both given as functions of the arc-length s

7 = { r d ( s )vd(s) , for s E [o, SI}. Assuming that the given control task 7 is feasible, the path following problem consists Of

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Problem 4.1 (Path following) Given a feasible control task T,determine an admissible control action u = { (6, Tr r ~ f ) } such that the lateral error

stays bounded elat 5 emaZand the longitudinal error elong(t) = JMt) - vd(t))2 goes to zero as t

+ +m

The architecture of the controller is the same no matter which model is used. The key concepts are MPC and backstepping.

Fig. 2 Controller architecture. We first solve a MPC problem assuming we may act on the steering angle 6 directly, which means that we can control the vehicle yaw rate 9.Details on the control law and a proof of stability may be found in [13]. Then, we backstep the computed yaw rate V! for stable path following through the dynamics and track it with an appropriate controller which is, once again, based on MPC. Note, that, if the lateral dynamics are not excited, all that the backstepping controller will do is to compute the steering angle 6 which solves

.

tan6

*=-

V.

P

The higher level MPC that controls the kinematic model solves the following problem:

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Problem 4.2 (Local kinematic trajectory tracking (LKTT)) Let

s ( r )=

LT

vd(s(r))dr + s * ( t )

where S* is defined as illustrated in the figure 3.

Fig. 3 Definition of the s' coordinate.

Determine an admissible control action 6(r) and w(r) in r E [t,t J =

+ T ] such that the cost

lliT

eT(r)Qe(r)d r + eT(t + T)We(t+ T )

is minimized subject to (1).

Apply 6(r) and v ( r ) for a time step A, suficient to determine a new solution based on current measurements in the interval r E [t A,, t A, T ) and repeat recursively the previous steps.

+

+ +

Note that the recursive solution of the LKTT will solve the path following problem because the choice of s ( r ) (5) reparametrizes the desired path at each time step A,. If the control task is not particularly demanding, one may assume that the tires slip angles are neglectable and that model (1) is accurate enough. If the velocity profile v ( t ) is smooth and can be tracked easily, one can also assume that w is a control and be satisfied with the above solution.

As soon as the lateral dynamics become excited, however, this simple controller will not be able, for example, to recover when the vehicle looses stability or controllability. The solution at time t of the LKTT problem provides a desired yaw rate trajectory V ! d ( r ) for r E [t,t + TI. This signal together with the desired velocity vd are given as references to a second faster inner control loop also based on MPC with time step Ai, let us call it LDTT. This loop controls the dynamic model (4)acting on (6, -rr,r,} in order to track the desired yaw rate and velocity vd. The cost is quadratic on the tracking error and on the torque inputs.

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5

SIMULATION OF HUMAN BEHAVIOUR

There are numerous papers on the simulation of human driving behaviour. Recently, this research area has become even more popular because of the intensive, on-going, research effort dedicated to the development of driver assistance systems. In these applications, it is important to understand "normal" driver behaviour in order to recognize deviations from it and intervene to avoid dramatic consequences. In the context of the project presented in this paper, our purpose is different, the goal here is t o model the dynamics of a human driver seen as a controller. This is important to explore the performance limits of the ensemble vehicle plus driver or, for example, the understand the interactions among the driver and driver assistance systems such as the ESP. The literature on the problem of interest to us is vast. The earliest paper in which the driver is modeled as a dynamic controller is by Ornstein [22] in the 6Oies. He models the driver as a second order linear system plus a delay. More recent research include a variety of technical approaches which include: fuzzy logic [8], cascade and adaptive neural network architecture [17], adaptive linear control, optimal finite horizon predictive control [25]. For R recent survey see [18]. It is safe to say that, nowadays, it is almost universally accepted that predictive control and a hierarchical control architecture are at the basis of any human driver simulator. If one looks at some of the hierarchical control structure proposed in the literature on human driving simulation, e.g. [31], often finds a subdivision in three levels: a strategic level for task planning; a tactical and an operational level which correspond respectively to a trajectory planner and to low level motion control. This subdivision matches exactly those proposed in the very much related literature on autonomous vehicles see, for example [30]. The two lower levels of the proposed hierarchical architecture coincide with our outer kinematic and inner dynamic control loops. We, therefore, decided to model natural behaviour by adapting our architecture acting on perception, actuation and modeling. On actuation, we introduced a low pass filter with a bandwith that can be set from 2-12 Hz. In modeling, the user may decide the level of complexity of the model used to control the vehicle. This means to model the experience of the driver.

Fig. 4 Human like driver architecture.

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By perception we mean how a human driver perceives the effects of his actions. This happens through the senses. We modeled, in particular, vision and proprioception, where, by the latter we mean the awareness of its own self which, we simply identify with the capability of measuring body accelerations and angular velocities. Vision is tipically processed in the frequency range of 3 to 6 Hz. Vision measures the profile of the desired path up to a certain visibility distance L in meters. This is the input to the higher kinematic control level. In practice, this means that the time step A, of the LKTT MPC will be somewhere within 0.15 - 0.45 seconds. Proprioception is faster, usually in the range of 6 to 18 Hz. We will, therefore, assume that the time step A, of the MPC of the inner control loop will be somewhere within 0.05 - 0.15 seconds.

6

RESULTS

The following are a set of simulations of increasingly demanding maneuvers in terms of vehicle dynamics (see figure 5). The control of lateral dynamics to compensate loss of stability and controllability of the vehicle is apparent. In figure 7, instead, we show

Fig. 5 D o u b l e lane change maneuver: simulation picture. the simulations of a maneuver performed by drivers characterized by poorer and poorer performances obtained by decreasing the vision and proprioception frequencies.

7

CONCLUSIONS

A driver model for ADAMS has been presented. The driver must be able on one hand to take the vehicle to its dynamical limits and, on the other hand, to model natural human driving behaviour. In this paper, we presented the overall structure of the driver model and a description of how we have simulated the human driving behaviour. A set of simulations has been shown to demonstrate that SmartDriver captures expected human driving behaviours. SmartDriver is currently being delivered by MSCSoftware for the ADAMS CAR environment, but it has been developed so that it could be applied to drive vehicles in any other simulation environment. It could even be a module of an automatic driver for a real vehicle. In the next future we will build on SmartDriver an optimal reference path generator to solve driving tasks such as maximum comfort, minimum fuel

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Fig. 6 Double lane change maneuver: steering angle and lateral velocity for increasing speed.

consumption and so on. Maximum performance driving is also being studied and in a forthcoming paper we will present a quasi steady-stae estimator of maximum performances

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Fig. 7 Human like driver: steering angle and throttle demand for decreasing vision and proprioception frequencies.

which will generate the reference signals for the top performance driver. 8

BIBLIOGRAPHY

References [l]http://www.mscsoftware.com/products/quick_prod.cfm

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[2] Findeisen R., L. Imsland, F. Allgower and B.A. Foss, State and Output Feedback Nonlinear Model predictive Control: An Overview, European Journal of Control, Vol. 9, N. 2-3, pp. 190-206, 2003.

[3] Altafini C., General n-trailer, differential flatness and equivalence, Proc. of IEEE Int. Conf. On Decision and Control, vol. 3, pp. 2144-2149, 1999. [4] Bloch A., S. Drakunov, Tracking in nonholonomic dynamic systems via sliding modes, Proceedings of the 34th IEEE Conference on Decision and Control, vol. 3, pp. 2103-2106, 1995. [5] Dickmanns E.D., V. Graefe, Dynamic monocular ma-chine vision, Machine Vision and Applications, vol. 1, pp. 223-240, 1988. [6] Dickmanns E.D., V. Graefe, Applications of dynamic monocular machine vision, Machine Vision and Applica-tions, vol. 1, pp, 241-261, 1988. [7] Donges E., A Two-Level Model of Driver Steering Be-havior., Human Factors, vol. 20 (6), pp. 691-707, 1978.

[8] El Hajjaji A., M. Ouladsine, Modeling human vehicle driving by fuzzy logic for standardized I S 0 double lane change maneuver, Proc. IEEE Int. Workshop on Robot and Human Interactive Communication, pp. 541-545, 2001. [9] Rouchon P., M. Fliess, J. Lvine, P. Martin, Flatness and motion plannning: the car with n trailers. In Proceed-ings Int. Conf. European Control Conference, ECC’93, pp. 1518-1522, Groeningen, The Netherlands, 1993. [lo] Fliess M., J. Lvine, P. Martin,P. Rouchon, Design of trajectory stabilizing feed-

back for driftless flat systems, Proceedings Int. Conf. European Control Conference, ECC’95, pp. 1882-1887, Rome, Italy, Sep. 1995. [ll]Frezza R., G. Picci, On line path following by recur-sive spline updating, Proc. of the 34th IEEE Con-ference on Decision and Control, vol. 4, pp. 4047-4052, 1995.

[12] Frezza R., S. Soatto and G. Picci, A Lagrangian for-mulation of nonholonomic path following”. In The Con-fluence of Vision and Control, S. Morse et al. (eds), Springer Verlag, 1998. [13] Frezza R., Path Following of Air Vehicles in Coordi-nated Flight, Proc. of the 1999 IEEE-ASME Int. Conf. on Advanced Intelligent Mechatronics, Atlanta, Sep. 1999.

[14] Frezza R., Altafini C., Autonomous landing by computer vision: an application of path following in SE(3), Proc. of IEEE Int. Conf. on Decision and Control, vol. 3, pp. 2527-2532, 2000. [15] Hauser J., R. Hindman, Aggressive Flight Maneuvers, Proc. of 36th IEEE Conference on Decision and Control, pp. 4186-4191, 1997. [16] Khalil, H. K., Nonlinear Systems, Prentice Hall, 1996.

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[17] McAdam C., Z. Bareket, Adaptive Neural Network Characterizations of Driver Longitudinal Control Behavior, Proc. of 4th Int. Symposium on Advanced Vehicle Control (AVEC98), 1998. [18] McAdam C., Understanding and Modeling the Human Driver, Vehicle System Dynamics, Vol. 40, Nos. 13, pp. 101134, 2003. [19] Miyazaki T., T. Kodama, T. hruhashi, H, Ohno, Modeling of human behaviors in real driving situations, Proc. of IEEE Intelligent Dasportation Systems, pp. 643-646, 2001. [20] Morari M., E. Zafiriou, Robust Process Control, Pren-tice Hall, 1997. [21] Murray R., Z. Li and S. Sastry, A Mathematical In-troduction to Robotic Manipulation. CRC Press Inc., 1994. [22] Ornstein, G.N., The Automatic Analog Determination of Human Transfer Function Coefficients. Med. Electron. Bio. Eng. 1 (3),1963. [23] Pentland A., A. Liu, Toward Augmented Control Sys-terns, In Proc. of the IEEE Intelligent Vehicles Sympo-sium, vol. 1, pp. 350-355, 1995. [24] Pomerleau D.A., Reliability Estimation for Neural Network Based Autonomous Driving, Robotics and Autonomous Systems, vol 12, no. 3-4, pp. 113-119, 1994. [25] Prokop G., Modeling Human Vehicle Driving by Model Predictive Online Optimization, Vehicle System Dynamics, vol. 11, No. 1, pp. 1-35, 2001. [26] &in S. J., T. A. Badgwell, A survey of industrial model predictive control technology, Control Engineering Practice, vol. 11 (7), pp. 733-764, 2003. [27] Rawlings, J.B. Tutorial overview of model predictive control Control Systems Magazine, IEEE ,Volume: 20 ,Issue: 3 ,pp. 38 - 52, June 2000. [28] Samson C., M. Le Borgne and B. Espiau, Robot Control the Task Function Approach. Oxford Engineering Science Series. Clarendon Press 1991. [29] Samson C., Control of Chained Systems Application to Path Following and TimeVarying Point-Stabilization of Mobile Robots, IEEE Trans. on Aut. Control, vol. 40, no. 1, pp. 64-77, Jan. 1997. [30] Sinopoli B., M. Micheli, G. Donato, T.J. Koo, Vi-sion based navigation for an unmanned aerial vehicle, Proc. of IEEE Int. Conf. on Robotics and Automation ICRA 8001, V O ~ .2, pp. 1757-1764, 2001. [31] Song, B., D. Delorme, J. VanderWerf, Cognitive and Hybrid Model of Human Driver, Proc. of the IEEE Intel-ligent Vehicles Symposium, vol. 1, pp. 1-6, 2000. [32] Stancliff S., M.C. Nechyba, Learning to fly: modeling human control strategies in an aerial vehicle, Internal report Machine Intelligence Laboratory, University of Florida Gainsville, www.mi.ufl.edu.

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Non-linear response of an all-terrain vehicle on a rough terrain L DAI and J WU

industrial Systems Engineering, University of Regina, Saskatchewan, Canada M DONG

Petroleum Systems Engineering, University of Regina, Saskatchewan, Canada

sYN0PsIs A new computational model and a new elastic-plastic terrain model are established in this research to describe the effects of the nonlinear exertions acting on the vehicle structure and the influences of the terrain to the dynamic behaviour of an all terrain vehicle. The models established are nonlinear for accurately representing the motion characteristics of the vehicle under the actual operation and terrain conditions in the field. Numerical simulation of the vehicle is carried out on the basis of the models developed. Nonlinear behaviours of the vehicle motion and vehicle response to the rough terrain are presented.

NOMENCLATURE

k

I

cs

kd cd 9

ke, Ce M

I el

Y004/03 1/ZOO4

angle displacement (rad); moment of inertia of the beam/axle system (s); profile displacements from the terrain surface (m); vertical displacement of the pin-point (m); vertical displacement of the chassis at CG (m); vertical displacements of the cab and engine (m); interaction force acting on the pin-point (N); distance from pin-point to the shock absorber (m); stiffness and damping coefficient of the front wheels stiffness and damping coefficient of the shock absorber; stiffness and damping coefficient of the cab and seats; stiffness and damping coefficient of the engine assembly; mass of the whole chassis including cargo (Kg), moment of inertia of the chassis assembly; distance between the driver to CG (m);

415

distance from the engine installation place to CG (m); distance from the front and rear pin-joint points to CG (m); mass of engine and hydraulic motor assembly (Kg); mass of the cab including the drivers and seats (Kg); amplitude (m), frequency (Hz) and phase angle (rad); depth of the compressed terrain (m); characteristic constants of the soil; width of the wheel of the ATV (m); diameter of the wheel (m); self-weight of the wheel (Kg).

1. INTRODUCTION Since the development of early ~ O ' S all-terrain , vehicles (ATV) have become widely used offroad vehicles, especially in the areas of military, agriculture and forestry. Along with the technical progresses, the modem all-terrain vehicles are required to provide higher flexibility and suitability for driving in various operational and environmental conditions. These in tum demands a better understanding of the motion and dynamic response of the ATV structures subjected to the loading conditions of high complexity. Linear and nonlinear dynamic analyses are always the noticeable topics in the ATV research and design. Systematic researches on ATV dynamics can be traced from the later 1960's [l] with focus on the aspects of vehicle general motion and the interactions between terrain and vehicles. Investigations of the research in the current literature are found in ATV's performance and handling characteristics, track force distribution, steering ability as well as ride properties [2,3,4], Static analyses are commonly employed in the field, especially in design practice. Linearization and simplification are also the common practice in the dynamic studies on the motion of ATV and dynamic response of ATV structures. Thorough and systematic analysis on the nonlinear motion of ATV and nonlinear dynamics response of the vehicle structures subjected to the loading conditions of the real world is still in need. In this research, analytical and numerical models are established for investigating the motion and the nonlinear dynamic response of an all terrain vehicle to the operation and terrain conditions. The vehicle models established combine all the main vehicle components including chassis, cabin and drivers, engine, tire-track assembly and the solid cargo carried. Interactions between the components under the nonlinear exertions are counted to form an eight-degrees-of-freedom system. The terrain is modelled as an isometric nonlinear system, and the surface shape of the terrain as well as the material proprieties of the terrain are also taken into consideration. With the vehicle model and the terrain model developed, the motion of the vehicle can be conveniently described and analyzed with a given terrain and a specified operation manoeuvre. Numerical analysis with different parameters is performed to give the overall dynamics characteristics of this ATV structure. The models thus developed make the further analyses, such as the strength and fatigue analysis of the ATV become readily available.

2. MODEL SET UP & EQUATION DEVELOPMENT An all terrain vehicle with intermediate cargo loading is represented by the combination of the following spring-mass-damping systems as illustrated in Figure 1. The vehicle is equipped

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with eight wheels and is desired to be operated by two persons. The vehicle is driving on rough terrain with a velocity v. The chassis is considered as an elastic beam as shown in the figure. For the convenience of analysis and computation processes, based on the symmetry of the vehicle considered, the ATV structure is simplified for the analysis to be carried out with the concentration of the overall system moving in the y-z plane as shown in Figure 1. From Figure 1, it can be identified that the ATV structure consists of the following main components: Front Axle-Beam Assembly (FABA), Rear Axle-Beam Assembly (RABA), Chassis Assembly, Driver & Cab, and Engine & Hydraulic-Motor Assembly. For the system of FABA, the nonlinear governing equations can be expressed as the following. (mi + m2) Z3 = -k(Z3 - Z I - z2tgeI)- k(Z3 - z2 + i3tgel)+ f i + k s ( z 4- z3 - atgo1).

+ cs(Z 4 - z3 - a-)dtgel dt

- c(Z 3 -

zI

dtgel dtgel - c ( 2 3 + 13dt dt

12-)

- z2) (')

iIel = k ( z 3- z1- i2tge1)i2 + ks(z4- z3- atgel)a- k ( z 3- z2+ i3tge1)i3 dtgel + c ( 2 3 - z I - I 2 - dtgel )12 + cs(Z. 4 - z.3 - a-)a - c(Z3 + 13- dtgel - z 2 ) z 3 dt dt dt

(2)

Figure 1 Main components of the ATV structure. where, 11 and 12 are the distances from the pin-point of the axle-beam connecting the wheels to the centre of front and rear wheels respectively; m /the mass of front wheel assembly, mz the mass of rear wheel assembly; 81 the angle displacement of the axle-beam; Zl the moment of

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inertia of the b e d a x l e system; Zl and ZZ the profile displacements from the terrain surface; vertical displacement of the pin-point, Z4 the vertical displacement of the point of shock absorber designated by the ks-c, system; Ff the interaction force acting on the pin-point, a the distance from pin-point to the shock absorber; k and c the stiffness and damping coefficient of the front wheels respectively, k, and c, the stiffness and damping coefficient of the shock absorber in between the chassis beam and the wheel axle beam as shown in Figure 1. Z3 the

For the Rear Axle-Beam Assembly, the governing equations can be expressed in the following form. (mi

+ mi>Zs = -k(Zs - 2 7 + i2tge2)- k ( ~ -82 6 - i3tge2)+ Fr + Ks(Z9 - Z S + atge2) dtg& + C s ( z 9 - z 8 + a-)dtgB2 - C ( i 8 -z7 + 12-) dtgB2 - C ( z. S -13----z6) dt

dt

(3)

dt

where 02 represents the angle displace of the rear axle-beam assembly; 12 the inertia moment of the rear axlebeam system; z6 and &the profile displacements from the terrain surface; z8 the vertical displacement of the pin-point; Z9 the vertical displacement of the point of the shock absorber of the rear axle-beam assembly; and F, the force acting on the pin-point. For the chassis assembly of the ATV structure, the corresponding differential equations can be expressed as

I2 B 2

+

= Fd, ks(Z4 - Z3 - atgel)(&- a )

+ c s ( Z r - Z ~- asec2B I ~ I ) ( &- a ) - Frlr - ks(Z9 - Ze+ atgB2)(&- a ) - cS(z9- Za + asec2 e2e2)(& - a ) - ~ ( ~ 1 z1 lo-+ eltge3)el + k e ( Z 1 2 - Z i o + e r t g B ~ ) e 2 - c ~ ( Z 1 1 - Z i o + e i sBe ci 3~ ) e 1

(6)

+ce(Z12-ee2sec2 e 3 e 3 - i l o ) e l where, & and cEare the stiffness and damping coefficient of the shock absorber; kd and cd the stiffness and damping coefficient of the cab and seats; k, and ce the stiffness and damping

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coefficient of the engine assembly; A4 designates the mass of the whole chassis including cargo, I represents the moment of inertia of the chassis assembly; 0 3 is the angle displacement of the chassis; el is the distance between the driver to CG (centre of gravity of the chassis assembly; e2 is the distance from the engine installation place to CG; If and I, the distance from the front and rear pin-joint points to CG respectively; Zlo is the vertical displacement of the chassis at CG; Zlland Z12 the vertical displacements of the cab and engine respectively; Z4 and Zg the vertical displacements of the connect points of the front & rear shock absorbers respectively; a is the distance between the pin-joint point and corresponding shock absorber; Ffand Fr are the forces acting on the chassis’s from and rear pin-points respectively. The Engine & Hydraulic-Motor Assembly is governed by the following equation:

where me is the mass of engine and hydraulic motor assembly; ke and ce are the masses of engine and hydraulic motor respectively; and Z12 is the vertical displacement of the cab and engine. The Driver & Cab system is governed by md i l l

+

= -b(zil - zlo eltge3) - c d ( i l 1 - Z i o

+ eisec’ 0 3 8 3 )

(8)

where md is the mass of the cab including the drivers and seats; and cd are the stiffness and damping coefficient of the cab respectively; 21 I designates the vertical displacement of the cab. In addition to the governing differential equations, the following geometric relationships must be satisfied.

ZI = ZIO - 1, sin el

(9)

ZS= ZIO+ I r sine,

(10)

2 4

=ZIO-(I, -a)sine1

Z9 = ZIO+ ( I , - a)sin e3

(1 1)

(12)

With the governing equations coupled and the geometric relationships established, the motion of the ATV structure can be described completely once the terrain conditions and surface profiles are specified in the domain of time and frequency.

For the convenience of the numeral calculation, a series of equivalent transformations of the above equations can be made with the considerations of the interactions between the system components and the geometric relationships. The transformations lead to the following equations for governing the motion of the ATV structure.

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-I,COS^, e , )+ A

f i = (ml+ m2)(z

F~=(~I+~~)(ZIO+I,COSB~~J)+E M

= -Ff - Fr

+C

= FA, - FJ,

+D

Zlo

l e 3

in which A = (mi+m2)lf sinO38: +k(Z3-Z1-13tanQ,)+k(Z3-22-Z3tan8,)

z3 - a tanel) - c , ( z -~ z3 - asec2 elel) + c(z3 - Z I- I, sec2elel) + c(z3 - z z- sec2 elel) - k,(z4 -

13

B = -(mi + m2)l, sin 0,e: + k(Zs - 2 7 - Zztan e,) + k ( 2 s - 2 6 -Istan&) - k,(Z9 - Z S- a tan&) - c , ( z-~zs- asec’ e2e2) + C ( Z S - 2 7 - I , sec2 e2e2)+ ~ ( z-s - 2ssec2 e2e4

(1 8)

2 6

With the equations above, the dynamic equations governing the displacements of the chassis can be developed as shown below.

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Through the above mathematical developments, the series of governing equations for the ATV structure are thus transformed into such the forms that the P-T [ 5 ] or Runge-Kutta methods can be applied conveniently via the computer programs for solving these nonlinear differential equations describing the motion of the vehicle.

3. NUMERICAL ANALYSIS AND DISCUSSIONS Numerical solutions are obtained on the basis of the differential equations developed in Section 2 by employing the newly developed numerical method, the P-T method [5] which provides a faster convergence rate. The time step length used is 0.01 of the time unit. The corresponding systematic parameters of the ATV are listed below. Front Axle-Beam Assembly: mi = lIlkg,m, = 14O.5kg,l2 = 0.559m,I3= 0.406m,a = 0.15m,Ii = 78.5m4 Rear Axle-Beam System: m,’ = 184kg,m,’ = 145.6kg,12 = 0.559m,13 = 0.406m,a = 0.15m,Zi = 81.5m4 Chassis Assembly: M = 1504kg,a = 0.15m,1, = 1.435m,lr = 0.142m,ei = 1.413m,e2 = -0.156m, J = 355m4 Driver Cab & Cab

Mi = 164kg Engine & Hydraulic-Motor Assembly: M , =294kg Stiffness coefficient and damping coefficient of hydraulic motor & engine assembly: k, = 14.56X1OJ;N/m,C, = 8954N.Slm Stiffness coefficient and damping coefficient of hydraulic shock absorber: k, = IOXlO’N/m,C, = 462N.SIm Stiffness coefficient and damping coefficient of driver cab and seat: k, = 96.61X10’N/m,Cd= 1311N.Slm The angular motion of the chassis assembly, especially the angular acceleration is the mainly concerned aspect for this type of ATV. The angular motion of the ATV considered actually plays an important rule in the analysis for the chassis strength and its fatigue strength. Based on the numerical results obtained, characteristics of the angular motion of the ATV are analysed and a group of figures showing the characteristics in the time domain are plotted for illustrating the dynamic response of the ATV structure excited by the rough terrain surface. Based on different surface profile of the terrain and the terrain material properties, the vehicle’s dynamic characteristics are analysed regarding the following conditions.

3.1 Sinusoidal terrain surface To carry out the analysis, different profiles of the terrain surface are taken into account. For the idealized hard surface, the resistance effects due to the ground deformation along the

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vertical direction can be ignored. Sinusoidal terrain surface represented by the coordinates of Zl,Z2,Z3, and Z4, in the following forms is considered. Z, = Asin( (27, Z2=Asin --

2rr(12 +I,)

I

1

(--

2~(1.576+ 12)

(2f"

2z(1.576 + 212) I

Z, = Asin

Z,=Asin

T)

27,

--

I

1

For the numerical simulation, take the amplitude A = 0.1 m and a running velocity v = 9.72 m/s, which is the normal velocity under the condition of fare road surface. The angular acceleration of the vehicle with respect to time for this case is plotted in Figure 2. As a comparison, for a slower velocity of v = 5 m/s, the angular acceleration changes to the form as shown in Figure 3. As can be seen from the figure, the acceleration varies in such a manner that the smaller amplitudes are added on top of a large amplitude. Also, the amplitude is stabilized faster in comparing with that of the previous case as shown in Figure 2. This matches well with the real situation in practice. Plot of the chassis angular acceleration 10

0

5

10

time axis

Figure 2 Angular acceleration of the ATV with velocity of 9.7 d s .

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Plot of the chassis angular acceleration 4

3

2 I

1

0

.1

-2

-3

4

1

3

2

4

5

6

time axis

Figure 3 Angular acceleration of the ATV with velocity of 5.0 d s .

3.2 Terrain surface of higher nonlinearity To represent the terrain surface of the real world, it is convenient to utilize the Fourier Series in the following form for expressing the profile of the terrain surface. In fact, the following expression can also be employed to represent the function of external excitation acting on the ATV structure. m

y=

C

sin(wnt + p>

n=l

where An, wn and p are the amplitude, frequency and phase angle of the nth term of the Fourier Series. Specifically,let

15

I 315 6w I 515 Vt) + -sinsin(- Vt) + -sin-sin(32 2 T 5’ 2

Ion V t ) -+ ...... T

Take the first three terms of the above equation to consider the third order nonlinearity and take into account the response of the vehicle excited by the terrain profile of the higher nonlinearity.

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Assume v = 9 m/s, as can be seen from Figure 4, the variation of the angular acceleration follows a pattern similar to that of the slower velocity in general. However, the acceleration is added with the perturbations of small scale. Plot of the chassis angular acceleration 50

40 30 20

10 0

-10 -20

-30

-40 0

1

2

3

4

5

6

7

8

time axis

Figure 4 Angular acceleration of the ATV on a highly nonlinear terrain profile with velocity of 9 m/s.

3.3 Random surface When the ATV is subjected to a random loading described by y = 0.1 x [rundn(x)- 0.51, which is due to the uneven surface of a terrain with the heights of slabs normally distributed. Taking the velocity v=9.75 m/s, the angular acceleration is plotted in Figure 5, in which the randomness of the acceleration response can be clearly seen over a very short period of time.

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Plot of the chassis angular acceleration

15

1-

-'"O

0.2

0.4

0.6

0.8 1 time axis

1.2

1.4

]

dd~Q5,I)

1.6

1.8

2

Figure 5 Angular acceleration of the ATV with velocity of 9.75m/s.

3.4 Horizontal resistance The horizontal resistance acting on the vehicle is mainly due to the deformation of the soil on which the ATV is moving. During the operation of the vehicle, partial elastic deformation and plastic deformation will occur on the earth, which will cause the periodic alternative resistance on the horizontal direction when the sinusoidal terrain surface is considered, even in the case that the vehicle is moving along a straight line with a constant velocity. For such a soil, the horizontal resistance R, and the vertical loading IV acting on the wheel of the ATV can be expressed by the following balance equations [ 6 ] .It should be noted that the vertical loading W is dynamically varying in this research, corresponding to the motion of the ATV structure.

According to the traditional terrain theory [ 6 ] , the vertical component of the pressure is

(7 1

expressible as p = -+ Kp Z" , where Z designates the depth of the compressed terrain,

K c , Kp, n are the characteristic constants of the soil, and b is the width of the wheel of the ATV. As such, the horizontal resistance can be expressed as

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2n+2

1

(30)

where D is the diameter of the wheel; and the vertical exertion on the A m ’ s wheels is derived as shown below.

GI

W = -k(z, - z , - 1, sine,) - c(i, - z,- I , cos6,ei) + 2

(31)

where GI is the self-weight of the wheel. Take the velocity of the vehicle to be v = 5 d s , the horizontal resistance can be graphically shown in Figure 6 . As can be seen from the figure, the horizontal resistance varies dramatically at the starting time and then varies with a stable fashion on the sinusoidal terrain surface when the vehicle is operated under a stable condition. Plot of resistanceforce of hunt-wheel-fmnt-axle

600‘ 0

1

I

2

3

4

5

time axis

Figure 6 Horizontal resistance force of the front wheel. 4. CONCLUSIVE REMARKS

This research is to analyse the motion and the dynamic response of an all terrain vehicle moving on a rough terrain surface. New nonlinear analytical and numerical models are established with the considerations of the nonlinear excitations generated by the terrains and

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the interactions of the main vehicle components of the ATV. With the employment of the nonlinear models, the analysis on the motion of the ATV and the interactions of the main components of the ATV becomes available. More significantly, the dynamical interactions between the axle beam assembly and the chassis assembly of the ATV can be quantitatively studied. Numerical analyses on the motion of the ATV are performed with various system parameters and considerations of sinusoidal, highly nonlinear and random terrain profiles. Characteristics of the motion under the operation and terrain conditions are analysed and the corresponding angular accelerations of the ATV under these conditions are graphically presented. Conventionally, the horizontal resistance is considered as a static reaction force acting on the ATV as the vehicle is moving on an ideal horizontal plane without the perturbation of the vertical vibration of the vehicle structure. By the model developed in this research with considerations of the partial elastic and plastic deformations and the nonlinearity of the terrain material, the variation of the horizontal resistance to the ATV can be calculated with given terrain profile and terrain material properties. Moreover, the vertical vibration effects to the horizontal resistance can be taken into consideration. The results of the present research can also be used in the strength and fatigue analysis for the ATV structure under the operation and terrain conditions in practice.

REFERENCES M. G. Bekker. Theory of Land Locomotion. Michigan: The University of Michigan Press, 1956. Z.D. Ma and N.C. Perkins, A Track-Wheel-Terrain Interaction Model for Dynamic Simulation of Tracked Vehicles, Vehicle System Dynamics, Vol. 37, pp. 401-421,2002. M. Demic, Identification of Vibration Parameters for Motor Vehicles, Vehicle System Dynamics, Vol. 27, pp. 65-88, 1997. G. Verros and S. Natsiavas. Ride Dynamics of Non-linear Vehicle Models Using Component Mode Synthesis, Journal of Vibration and Acoustics, Vol. 124, pp. 427434,2002. L. Dai and M.C. Singh, A New Approach with Piecewise Constant Arguments to Approximate and Numerical Solutions of Oscillatory Problems, Journal of Sound and Vibration,Vol. 263, pp. 535448,2003, J.Y. Wong, Theory of Ground Vehicles, John Wiley & Sons, Inc., New York, 2000.

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Multi-objective optimization of quarter car models with passive and semi-active suspensions G VERROS, M KAZANTZIS, and S NATSIAVAS Department of Mechanical Engineering, Aristotle University, Thessaloniki, Greece C PAPADlMlTRlOU Departmentof Mechanical and Industrial Engineering, University of Thessaly, Volos, Greece

ABSTRACT

A methodology is presented for optimising the suspension damping and stiffness parameters of quarter car models, subjected to road excitation. First, models involving passive damping with constant or dual rate characteristics are considered. Then, models where the damping coefficient of the suspension is selected so that the resulting system approximates the performance of an active suspension system with sky-hook damping are also examined. For all these models, appropriate methodologies are first employed for obtaining the second moment characteristics of motions resulting from roads with random profile. This information is next utilized in the definition of a composite vehicle performance index, which is optimised to yield representative numerical results for the most important suspension parameters. Finally, results obtained by applying a suitable multi-objective optimization methodology are also presented in the form of classical Pareto fronts. NOTATION C cg c,

CI, c 2 ,

9

damping matrix of linear model damping coefficients of linear model

f 0) -

damping coefficient in compression of passive bilinear model damping coefficient in extension of passive bilinear model damping coefficients of non linear sky-hook model force vector

Fc

control force in the car suspension system

c2c C2e 52

9

C2I

3

c22

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contact force developed between the wheel and the ground gravity coefficient performance index stiffness matrix of linear model stiffness coefficients of linear model the length of the road segment mass matrix unsprung mass sprung mass number of harmonics kept coefficients according to I S 0 263 1 power spectral density of road profile amplitude of harmonics ground displacement temporal measurement period time constant horizontal velocity of vehicle weighting coefficients displacement vector relative displacement between the sprung mass and the unsprung mass absolute displacement of unsprung mass absolute displacement of sprung mass velocity vector relative velocity between the sprung mass and the unsprung mass absolute velocity of unsprung mass absolute velocity of sprung mass acceleration vector road profile displacement wavelength of a surface profile vector of the parameters to be optimised phases of harmonics of road profile fundamental frequency in radsec spatial frequency in cycle/m characteristic spatial frequency in cycle /m 1 INTRODUCTION

Quarter-car models subjected to road excitation, are commonly employed in many areas of the automotive industry (e.g., [ 1-61). This is mostly due to the simplicity of the quarter car models and the qualitatively correct information they provide, especially for ride studies. Also, information extracted from such simple models provides quite frequently a firm basis for

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more exhaustive, accurate and comprehensive studies with more involved dynarnical car models. The main objective of the present study is to develop and apply a systematic methodology leading to optimum combinations of the suspension damping and stiffness parameters of a ground vehicle subjected to road excitation. Most of the previous studies on the subject have dealt with car models with linear characteristics subjected to deterministic excitation. Moreover, little attention has been paid to application of multi-objective optimisation procedures, Here, the excitation is caused by road irregularities, which may be of random nature. This case is described by frequency spectra, corresponding to profiles which are considered as typical in automotive engineering [ 1, 31. Besides linear models, car systems with passive or semi-active dual rate suspension dampers are also examined [4,6]. For the last case, a control strategy is applied, so that the vehicle approaches a state of sky-hook. The paper is organized as follows. First, the car models examined are presented in the next section. In the third section, a methodology for computing the response of the vehicle models to random road profiles is presented. This information is used in the fourth section to define the vehicle performance index. Then, a methodology is developed for selecting the optimum values of the suspension damping and stiffness parameters based on that performance index. A multi-objective methodology is also briefly described. In the fifth section, some typical numerical results are presented, while the highlights are summarised in the final section.

2 MECHANICAL MODELS The mechanical models examined are shown in Fig. 1. The coordinates x, and x2 represent the vertical displacement of the wheel subsystem and the vehicle body, respectively. For the linear model of Fig. la, the equations of motion can easily be put in the matrix form Mz+C.j+Kx= f(t), where g(t) = (x, x2)' represents the response vector, while the vector f ( t ) includes the forcing terms, arising from the road roughness. In particular, the vehicle is assumed to travel with a constant horizontal velocity v,, over a road with a profile s(z) . The model of Fig. 1b obeys a commonly employed passive control strategy, where the value of the suspension damping coefficient c2 switches between two distinct values [3, 41. This means that the suspension damping coefficient depends on the sign of the relative velocity x = f, -i, between the sprung mass and the unsprung mass, as depicted in Fig. Id. In addition, the restoring force in a typical car suspension exhibits the nonlinear characteristics shown in Fig. le, with x = x2 - x, . Finally, Fig. I C shows an ideal "sky-hook" model, which presents certain advantages [6]. Since it is impossible to materialize this type of suspension, an appropriate control strategy has to be applied, modifying its characteristics. Here, the behaviour of the model is approached by applying the following control force in the car suspension system

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F, =c,X2

+c2(X2 -Xl)-cg(X1

-Xg)

This implies that for an active control of the vehicle a continuous monitoring of the damping coefficient value is required, based on measurement of the quantities XI , i2and ig ,so that

c*x2+ c 2 ( i 2 - i 1 ) - C g ( X 1

-Xg)=C2(X2

-X,).

In practice, the most economical and easily realisable strategy is based on a semi-active control logic, obtained with a dual-switch suspension damper, with damping coefficient

3 RESPONSE CHARACTERISTICS UNDER RANDOM ROAD PROFILES This section deals with the estimation of the second order moment response characteristics of the vehicle models examined, when travelling over road profiles characterized by random fields. These fields are real-valued, zero mean, stationary and Gaussian. Therefore, it is sufficient to specify their second order moment. Here, this requirement is fulfilled by assuming that the irregularities possess a known single-sided power spectral density with the form

sg(n,)(n/n,)-"~, if R I R, sg(n,)(f2/R,)-"~, if n 2 R,

(3)

The spectrum used in the calculations corresponds to good quality roads with R, =1/2n, S,(R,) = 16.10-6 m2/cycle/m, n, = 2 and n, = 1.5, according to I S 0 2631 standards [3]. For vehicles with nonlinear properties, the probabilistic characteristics of the response are evaluated using Monte Carlo simulations [7]. Sample functions of the random road profile are first generated. The response of the vehicle to each sample road profile is then computed by integrating the equations of motion. Finally, the second moment characteristics of the response are estimated using the sample responses. Then, the forcing is simulated by the series

with x,(t) = s(v,t) , where the amplitudes sn are evaluated from the road spectra selected. In addition, w, = 2nv0/L, where L is the length of the road segment considered, while the phases pn are treated as random variables, following a uniform distribution in [0,2n).

432

YO04103 112004

4 OPTIMIZATION PROCESSES

In the present section, systematic methodologies are briefly presented, leading to an optimal selection of the suspension damping and stiffness parameters in ground vehicles subjected to random road excitation. The characteristics of the suspension are selected to optimise the vehicle performance over a range of vehicle velocities and under different road profiles. Typically, evaluation of the vehicle performance is based on examination of the maximum absolute acceleration of the passengers, the suspension travel and the wheel traction, related to passenger comfort, wheel space dimensions and vehicle handling, respectively. When the excitation applied is random, a suitable performance index can be identified by the following normalized expected value

where F, represents the contact force developed between the wheel and the ground. In addition, the constants w,, w2 and w3 denote weighting coefficients, T = L/vo is the temporal measurement period, while the vector @ includes the set of parameters to be optimised. The minimization of the performance index (5) requires repeated computations of the second moments of the response quantities for different values of the parameter set For linear systems and stationary response, the second moments of the responses can be obtained using an analytical formulation for the spectral density function of the response [7]. However, for the nonlinear vehicle models, the estimation is done using Monte Carlo simulations as described in the previous section.

e.

The main disadvantage of the above process is related to the need to select the value of the arbitrary weighting factors appearing in the objective function (5). For cases with multiple and conflicting objectives, like the one considered here, it is best to develop and apply multiobjective optimisation procedures. In such cases, there is no longer a single optimal solution but rather an infinite set of acceptable solutions, known as Pareto solutions [8]. These solutions are optimal in the sense that they cannot be improved in any objective, without degrading at least one other objective function. The multi-objective optimisation methodology applied in the present work is based on an appropriate modification of a methodology employed recently in structural model identification context [9].In brief, the basis of this method is the “strength Pareto evolutionary algorithm”, which can capture the optimal solutions and distribute them uniformly along the so called Pareto front. These optimal solutions constitute an acceptable compromise resulting from a trading-off between the satisfaction level of the various criteria applied.

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5 NUMERICAL RESULTS

In the present section, typical numerical results are presented for example vehicle models. First, linear models are examined, with nominal parameter values: m, = 60kg, m2 = 375kg, k, = 200kN/m, k, = 15kN/rn, c, = 7 Ns/m and c2 = 1425 Ns/m . Figure 2a shows the optimum values of the suspension damping coefficient c2, while Fig. 2b presents the optimum values of the stiffness coefficient k, , for w,= w 2 = w, = 113 . The thick curves were

e

determined by applying the optimisation process to the parameter set = (c, k , ) T ,while the thin lines were determined by considering the corresponding parameter alone. Clearly, the optimum values of both parameter depend strongly on the vehicle velocity. Next, results are presented for car models with a passive bilinear suspension damper, with cle = 475 Ns/m and c , ~= 1425Ns/m . The weighting coefficients are w,= w3 = 0.5 and w, = 0. The thick lines in Fig. 3 were determined by optimising with respect to the set

e = (c,~

c,,)~, while the thin lines were obtained by considering the corresponding damping

e

parameter alone. Moreover, the broken lines represent data obtained for = (k, c , ~ c , ~ ) ~ . The optimum value of the suspension damping coefficient in compression presents qualitatively similar trends with those observed for the linear models. However, the suspension damping coefficient in extension follows an opposite trend. As a result, the ratio of these coefficients starts from a value of less than unity for low velocities but it climbs up to 3.7 at the higher velocity values. This is in agreement with an empirical rule of thumb appearing in the literature, where it is claimed that the optimum damping effects are realized when the damping coefficient in rebound is three times larger than the damping coefficient in jounce [3]. Next, results referring to sky-hook models are presented. The optimum values of the parameters = (c, c2 cg)’ were first determined as: c, = SOkNs/m, c2 = 3.4kNslm and cg = 450 Ns/m . Then, the optimum values for the two different suspension damping

coefficients of the equivalent nonlinear model were located. Figure 4 presents results obtained for w ,= w3= 0.5 and w 2= 0. The optimum suspension damping coefficient shown in Fig. 4a exhibits a different trend in different velocity intervals, while the damping coefficient depicted in Fig. 4b presents a monotonically decreasing tendency. In closing and taking into account the existence of multiple terms in the objective function (5), the multi-objective programming procedure mentioned at the end of the previous section was also applied. Here, the road is assumed to have a harmonic profile, with ten? Wavelengths. Also, the vehicle speed was assumed so that the resulting excitation frequency is lOHz More specifically, Fig. 5 shows results for a model with linear suspension characteristics, while Fig. 6 presents results for a model with bilinear suspension damping characteristics.

.

6 SUMMARY In the first part of this study a systematic methodology was presented, yielding optimal values for the suspension damping and stiffness parameters of quarter-car models moving on roads with randomly varying geometrical profiles. Car models with passive linear and dual rate suspension dampers as well as models with semi-active sky-hook damping were examined. The control strategies applied led to the appearance of strong nonlinearities in the equations of

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motion. Then, numerical results were presented, referring mostly to optimum selection of the suspension parameters for typical quarter car models and different combinations of the weighting factors in a suitably defined performance index. In the second part, numerical results were presented, obtained by applying an effective multi-objective optimisation methodology.

REFERENCES 1. Dodds, C.J. and Robson, J.D., 1973, “The description of road surface roughness,” Journal ofSoundand Vibration,31, 175-183. 2. Karnopp, D., Crosby, M.J. and Harwood, R.A., 1974, “Vibration control using semi-active generators,” Transactions of ASME, Journal of Engineeringfor Industry, 96,619-626. 3. Gillespie, T.D., 1992, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA. 4. Surace, C., Worden, K. and Tomlinson, G.R., 1992, “An improved nonlinear model for an automotive shock absorber,” Nonlinear Dynamics, 3,413-429. 5. Hrovat, D., 1993, “Application of optimal control to advanced automotive suspension design,” Transactions of ASME, Journal of Dynamic Systems, Measurement and Control, 115,328-342. 6. Verros, G., Natsiavas, S . and Stepan, G., 2000, “Control and dynamics of quarter-car models with dual-rate damping,” Journal of Vibration and Control, 6, 1045-1063. 7. Roberts, J.B. and Spanos, P.D., 1990, Random Vibration and Statistical Linearization, J. Wiley and Sons, New York, NY. 8. Matusov, J., 1995, Multicriteria Optimization and Engineering, Chapman & Hall, New York, NY. 9. Haralampidis, Y., Papadimitriou, C. and Pavlidou, M., 2004, “Multi-objective framework for structural model identification,” Earthquake Engineering and Structural Dynamics, (in press).

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(4 (e) Fig. 1 (a) Linear car model. (b) Piecewise linear model. (c) Sky-hook model. (d) Force characteristics of suspension damper. (e) Restoring force of suspension spring.

2200

4000

40

75

ll0

I45 vo [kink]

I80

40

7s

110

I45 v. F m h l

I80

Fig. 2 Optimum values for: (a) the suspension damping coefficient c2 and (b) the suspension stiffness coefficient k , , for a car model with linear suspension damper.

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3500

4000

I c,,

WmI 3500

3000

2500

2000 40

75

110

I45 vo [kmhl

I80

40

75

110

145 v.

I80

NnhI

Fig. 3 Optimum values for the suspension damping coefficient in: (a) compression and (b) extension, for a quarter car model with bilinear suspension damper. 4200

4000

c,, [ N h ]

c,,

Wml 3600

4000

3200

3800 2800

/

I

3600

d

1 40

2400

75

II O

145 V"

Fnhl

I80

40

75

110

145

I80

vg F m h l

Fig. 4 Optimum values for the suspension damping coefficients: (a) c2, and (b) c22 of a nonlinear sky-hook car model.

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0.7

6

0.6 J, [x

IO']

k, [ 10'N/mj

4

0.4

2

0.2

0

0

IO

5

0

J, [x IO']

15

0

0.4

0.2

0.6

0.7

c, [ 5 10"s/m]

Fig. 5 Pareto oDtimal solutions for car model with linear suspension characteristics: (a) objective space (b) arameter s ace. 4

0.9 0.8

:,.[5 IO'Ns/m]

I, [x IO']

3 0.6

2 0.4

1

0.2

0

0 0.2

0.6

I

I .4 J, [x IOb]

I .8

0

0.6

I .2

1.8

c, [SIO'Ns/m]

Fig. 6 Pareto optimal solutions for car model with bilinear suspension characteristics: (a) objective space (b) parameter space.

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Torque steer influences on McPherson front axles J DORNHEGE Global Vehicle Dynamics -NVDT, Ford Werke AG, Germany

Jens Dornhege Ford Werke AG Global Vehicle Dynamics-NVDT

1 Abstract For front wheel driven vehicles the influence of the engine torque on the steering is called Torque Steer. Especially during full acceleration the steering may pull strongly to one side, which is very disturbing to the driver. As the Torque Steer Effect is directly related to the engine torque capabilities this problem becomes more and more evident with the upcoming high power diesel and petrol engines. Engine torque transferred through the driveshaft as well as traction force can generate a torque about the kingpin axis. Ideally the other side of the vehicle counterbalances this torque about the kingpin axis, but when different angles between driveshaWwheelkingpin axis or different drive torques left to right occur, the residual torque influences the steering.

Figure 1: Secondary Driveshaft Torque

Root causes for Torque Steer are: Nonsymmetric driveshaft angles, e.g. due to o Nonsymmetric design of the vehicle, e.g. different driveshaft length o Transient movement of the engine

Different driveshaft torques left to right Suspension geometry tolerances Unequal traction forces due to road surface in combination with Kingpin Offset

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In order to minimize Torque Steer for upcoming carlines a study on the main influencing parameters was conducted. The simulation model, an ADAMSPRE full vehicle model including a powertrain for front wheel drive, was enhanced to cover all influences observed in vehicle dynamics testing. Friction forces and torques had to be added at several significant joints. Wide-open throttle acceleration in first and second gear was simulated while straight line driving. This model was validated based on several measurements and used to set up a “Design of Experiments” in ADAMWINSIGHT. Based on the resulting response-surface the sensitivities of the observed factors as well as their interactions were assessed. With these sensitivities reliable predictions on the influence of design changes can be made. The enhanced model can now be used to optimize the vehicle design in an early phase of the development. The experience gained during setup and validation of the model will help to improve pre-program simulations to cover Torque Steer disturbances before first physical prototypes are built.

2 Objective Vehicle dynamics testing observed severe influences of unbalanced driveshafts in terms of torsional rigidity left to right. Furthermore moving the engine within the engine mount tolerances changed the initial angular configuration of the driveshafts, what also had a big impact on Torque Steer behaviour. There are a couple of other known influencing factors, Kingpin Offset (mainly package driven) Toe, Caster and Camber setting (general vehicle dynamics behaviour, tire wear) Engine Torque Roll Axis (NVH decoupling of vibration modes) Asymmetric mass distribution (loading of the vehicle) that cannot be changed with regard to Torque Steer during the vehicle development phase. Therefore the driveshaft arrangement might be a design option to avoid or compensate Torque Steer. This research was started to get a better understanding on how the driveshafts contribute to

this.

3

Approach

A simulation model was set up for a first evaluation of effects. This model was refined until the effects noticed during vehicle dynamics testing could be qualitatively reproduced. Simultaneously a test vehicle was equipped for measurement. The vehicle tests were used to validate the simulation model on a system and a vehicle level. The validated model was used for parameter variations to establish well-founded knowledge of the driveshaft influence on Torque Steer.

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4

Simulation Model

The simulation model, an ADAMSiPre full vehicle model including a powertrain for front wheel drive, was enhanced to cover these influences. To reproduce the influence of torsional unbalanced driveshafts differential gear friction had to be added. To achieve stability during “hands free acceleration” steering rack friction was introduced.

4.1

Differential Friction

As researched by the Powertrain Testing Department differential friction occurs as soon as there is differential speed between the left and the right driveshaft. The value of the frictional torque only depends on the differential input torque, not on the speed difference. This can be written as:

To increase numerical stability the SIGN function is replaced with a STEP function in the ADAMS/Solver Dataset. Please refer to MSC/ADAMS documentation for an exact definition of this function.

In the validation process ,u and 6, had to be adjusted to the measurement drives. p Was predicted well from gearbox rig tests, 6, should be as small as possible. Results converge asymptotically when lowering6, but making it too small causes smaller simulation steps when the speed difference changes its sign or even numerical problems and simulation halts.

4.2

Steering Rack Friction

For Steering Rack Friction the correspondent switch in ADAMSPre was activated resulting in a FRICTION statement in the ADAMS/Solver dataset. This statement caused strong numerical problems during the simulation, which could not be resolved. A simple friction approach was chosen to improve stability. A viscous damper paired with an in-line combination of a spring and a friction element. This was realised as a VFORCE user-subfunction embedded in the I ADAMS/Solver library. All three elements are set to be linear, therefore spring stiffness, Figure 2: Steering Rack Friction Approach maximum friction force and damping coefficient must be supplied for the subroutine. This simple friction model proved to work without any instability.

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5 Manoeuvre This project up to now concentrated on the influence of the driveshafts on Torque Steer during acceleration. To evaluate these influences a manoeuvre had to be defined that can be easily transferred between CAE and vehicle testing.

5.1

Steering Control

During the first simulations a closed loop steering controller was used to keep the car driving on a straight line, which was skipped due to bad repeatability on the test track. As an open loop manoeuvre two possible solutions are to accelerate “hands free” and evaluate the steering wheel angle or to drive with a fixed steering wheel and assessing the steering wheel torque. Both methods were driven on the proving ground.

5.2

Throttle Control

The longitudinal definition of the manoeuvre is to drive in 1’‘ or 2”d gear with a constant engine speed of 2000rpm, what reflects maximum torque. When the vehicle is driving quasistatic at this speed the throttle is suddenly pressed down for maximum acceleration up to 4500rpm. At this engine speed the throttle is released suddenly to apply maximum engine brake, which reverses the Torque Steer effect. This manoeuvre was driven in first and second gear and proved good repeatability on the track.

5.3

Future research

Some important aspects of Torque Steer remain to be investigated: Acceleration during cornering Acceleration over single wheel bump impacts Acceleration over double wheel bump impacts Acceleration on p-Split Quasi static Torque Steer at high speed or during hill climbing

6 Measurement The measurements were carried out at Ford’s Lommel Proving Ground, Belgium. The test vehicle was equipped with: Acceleration and yaw rate sensor String pots on all wheels Correvit sensor for vehicle speed measurement Measurement rims to acquire wheel forces in all six degrees of freedom

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One aim of the measurement was to evaluate the influence of driveshaft angles on Torque Steer. To reduce transient, uncontrollable effects rigid engine mounts replaced the rubber bushings. These engine mounts were crafted to be adjustable flOmm in the vehicle driving direction as well as the vertical direction. With these adjustable mounts the relative angle between left and right driveshaft was adjusted by moving the engine into different positions. In each position several reference points of the engine block were measured with a 3D-system to document the engine position for the measurement. Several engine position configurations were tested with torsional unbalanced as well as balanced driveshaft combinations.

7 7.1

Validation of Simulation Model System Level

Figure 3:

Front Axle

Elastokinematics

On system level the elastokinematic behaviour of the model was adjusted to comply with test vehicle measured on the Lommel K&C Rig.

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Some hardpoints of the simulation model had to be moved in their tolerance range to reflect the kinematical behaviour of the test vehicle. Further some bushing stiffness had to be scaled for a good elastokinematic representation. 7.2

Vehicle Level

To represent the full vehicle behaviour during the Torque Steer test, the introduced parameters for differential friction and steering gear friction had to be re-adjusted. For the validation runs the simulation models were prepared to follow the longitudinal acceleration of the corresponding test run.

Figure 4: Configuration 2,lst gear acceleration, Engine centre position

In Figure 4 the measurement data are plotted together with simulation data. All graphs show a good correlation except for the differential friction torque during throttle off. This friction torque directly influences the steering wheel torque and therefore we see a similar deviation in that plot too. The root cause of this deviation between test and simulation is not finally clarified, but as torque steer is mainly evident during acceleration and the deviation occurs during deceleration these models are credible.

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Further, Figure 5 and Figure 6 show the correlation of differential friction torque and steering wheel torque for different engine mounting positions.

Figure 5: Configuration 4 , l s t gear acceleration, Engine tilted to right

Figure 6: Configuration 5, 1st gear acceleration, Engine tilted to right, balaneed driveshaft stiffness

8 Next Steps The validated model is used to set up a “Design of Experiments” in ADAMS/INSIGHT. On the basis of the resulting response surface the sensitivities of the observed factors as well as their interactions are assessed. With these sensitivities reliable predictions on the influence of changes in the car can be made. Therefore in a vehicle concept phase targets can be derived for systems and components. Additionally, a design guideline is created for mutual compensation of different influences, if tradeoffs must be made to comply with other attributes. The enhanced model can now be used to optimise the vehicle design in an early phase of the development. The experience gained during set-up and validation of the model will help to improve pre-program simulations to cover Torque Steer disturbances before first physical prototypes are built.

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9 Appendix - Nomenclature Tfiicrion

Differential Gear Friction Torque

Tlnpur

Differential Gear Input Torque

P

Friction Torque scaling coefficient

~(leff,righr)

Rotational driveshaft speed at differential gear 1

for x > O

-1

for x < O

-imt t

STEP, 8,

446

For an exact definition of the STEP function please refer to MSCIADAMS Documentation

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Computer-based development of control strategies for ground vehicles M A NAYA and J CUADRADO Escuela Politecnica Superior, Universidadde La Corufla, Ferrol, Spain

SYNOPSIS During the last years, our group has worked on real-time formulations for the dynamics of multi-body systems. Now, in order to find out if such methods are suitable to address real industrial problems, we intend to develop control algorithms for a car on its computer model (virtual prototyping), and evaluate the performance of such controllers when implemented on the corresponding physical prototype. This paper addresses the first part of the work. Two maneuvers are to be considered: straight line and obstacle avoidance. The computer model of the car has been written in Fortran language. Fuzzy logic has been chosen to design the control algorithms, which have been implemented on Matlab environment. Several alternatives to connect Fortran- and Matlab-based functions have been studied, concluding that the most appropriate election depends on the purpose being pursued: controller tuning or onboard use of an already tuned controller. Simulator capabilities have been given to the program by means of a realistic graphical output and game-type driving peripherals (steering wheel and pedals), so that comparison may be established between human and designed automatic control. 1 INTRODUCTION

During the last years, our group has worked on real-time formulations for the dynamics of multi-body systemsI.2. As a result, a robust and efficient method has been developed? an index-3 augmented Lagrangian formulation with projections of velocities and accelerations, which features natural coordinates for the modelling, the trapezoidal rule as numerical integrator and sparse matrix technology for calculations. The method has shown to be robust and accurate, successfully facing singular configurations, changing topologies and stiff systems, as well as efficient, achieving real-time performance on a conventional PC when

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simulating the full model of a car vehicle undergoing rather violent manoeuvres, like stairs descent. Now, in order to find out if such formulation is suitable to address real industrial problems, we intend to develop control algorithms for a car on its computer model (virtual prototyping), and evaluate the performance of such controllers when implemented on the corresponding physical prototype. Figure 1 shows the general diagram describing the mentioned objective. The actual prototype has been built, and its virtual counterpart has been implemented on a computer through the mentioned dynamic formulation. The instrumentation of the former and the programming of the latter have been carried out in such a way that the inputs (actuators) and outputs (sensors) of the model and the physical prototype are exactly the same. Then, control algorithms can be designed and tested on the computer model of the car, until satisfying behaviour of the controller is achieved. If the simulator is accurate enough, the resulting control algorithms should also work properly when implemented on the actual car. Two manoeuvres are to be considered: straight line and obstacle avoidance.

Fig. 1. General context of the work.

Many references can be found in the literature regarding the automatic control of car vehicles. Two good reviews of the state-of-the-art concerning the trajectory tracking problem were presented by Antos and Ambrosio3, and Gordon et a14. Some other works have been focused on more specific aspects of the problem, like the so-called intezligent cruise control (ICC)’s6, or the measurement of some kinematic magnitudes during the motion’. Conversely, the present work aims to address the problem from a more holistic and general point of view.

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The paper is organized as follows: Section 2 focuses on the computational model of the car; Section 3 justifies the use of fuzzy logic in the present work; Section 4 explains the different alternatives available to connect the Fortran code containing the dynamics of the car with the Matlab functions implementing the fuzzy logic control algorithms, and points out their preferred contexts of application; Section 5 shows the comparison, for the two manoeuvres considered, between human and automatic control at simulation level; finally, Section 6 outlines the conclusions of the work. 2 COMPUTATIONAL MODEL OF THE CAR

The mathematical model of the car, illustrated in Figure 2 along with the physical prototype, has been carried out in natural coordinates'? 44 points, 7 unit vectors, 5 distances and 1 angle have been used as problem variables, leading to a total problem size of 159. Some few out of the mentioned variables have not been included for strict mechanical reasons, but rather for graphical representation requirements.

Fig. 2. a) The prototype; b) its model in natural coordinates.

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The equations of motion have been derived by means of an index-3 augmented Lagrangian formulation with projections of velocities and accelerationsz. The steering wheel is kinematically guided. Forces which deserve to be described are the following: - Suspension forces: they have been considered through linear models of springs and dampers. - Tire forces: lateral force and self-aligning torque have been introduced through the magic formula' with coefficients provided by the tire maker. The longitudinal effort has been neglected. - Power transmission forces: from the engine relationships torque-speed and the gear ratios, both provided by the engine maker, the automatic gearing has been modelled. Then, for a certain value of the car velocity, the engine speed and, consequently, the engine torque, can be easily derived. The torque is applied on the rear wheels. - Brake forces: the braking torque has been estimated from disk geometry", and applied on the four wheels. A code that calculates the dynamics of the described model has been implemented in Fortran

language, due to its high efficiency. 3 REASONS FOR THE USE OF FUZZY LOGIC

Designing a conventional controller, such as a proportional, integral and derivative (PID) controller, normally follows a standard procedure of modelling the plant, constructing a controller and evaluating the performance". A complete ground vehicle is naturally a highly nonlinear system. Developing a model which preserves the nonlinear characteristics of the system, and is simple enough to represent the plant of the complete system, is not easy at all. In fact, it's common to resort to a simplified model, as the bicycle model, in order to design a controller for a whole vehicle. Fuzzy control is knowled e-based control technology that can mimic human strategies to control complex systems". Due to its capability of handling systems nonlinearity, this technique seems a good choice to control a ground vehicle, and has been used recently to control different parts of actual vehiclesI3. Reviewing the accomplishments reported on applications of fuzzy control to vehicle systems, a common feature was that natural language models were used to describe the control process14, being similar for plants having similar implementation mechanisms. This characteristic of fuzzy control makes it possible to design a generic fuzzy controller for similar plants. Moreover, another valuable reason is that a reliable toolbox for fuzzy logic programming is provided by Matlab environment. This fact highly simplifies both the programming and tuning of the controller. 4 FORTRAN-MATLAB CONNECTION

In order to introduce the control on the prototype model, the Fortran code containing the dynamics of the car and the Matlab functions implementing the fuzzy logic control algorithms must be combined. Matlab always employs double-precision variables, so that the different types of Fortran variables must be converted to double-precision for Matlab compatibility. To connect Fortran and Matlab, two alternatives have been investigated: Matlab Engine and

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MEX files. For both of them, compatibility between Matlab and the Fortran compiler is required. Fortran-Matlab communication through Matlab Engine requires opening a communication channel from Fortran to Matlab. Matlab features several compilation functions which enable data transference between both languages, as well as executing functions in Matlab”. Then, when the Fortran program is run, a Matlab session is started, which implies some delay. Likewise, data transmission through the communication channel and the execution of Matlab instructions both slow down the program. In fact, to execute a Matlab function it must be written and executed on Matlab command window through the communication channel. Despite this fact, the simulation CPU-times obtained are kept moderate and, therefore, Matlab Engine can be considered as a suitable tool to design new control algorithms. Matlab allows the user to write new functions by means of the so-called MEX files. Through this method, the user can write the whole program in Matlab language with the exception of bottleneck functions, which could be written in more efficient languages, like Fortran or C, and executed directly from Matlab as its own functions. A command library is available in order to communicate Matlab with the other programming language. The compilation is carried out in Matlab. Consequently, this option is just opposite to the previous one. In the present work, the whole Fortran program containing the dynamics of the car has been converted to a MEX file, which can be executed in Matlab, thus enabling the access to the functions of the fuzzy logic toolbox. The efficiency has shown to be a little lower than that obtained with the first option. 5 WMAN VS AUTOMATIC CONTROL IN SIMULATION

The computational model of the car has been used to create a driving simulator, shown in Figure 3a, by combining the already mentioned vehicle dynamics code, along with a realistic graphical output and game-type driving peripherals (steering wheel and pedals), so that comparison may be established between human and designed automatic control.

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Fig. 3. Driving simulator: a) general view; b) first manoeuvre; c) second manoeuvre. As said in the Introduction, two manoeuvres have been performed. In the first one, the car starts from rest, covers a distance of 20 m following a straight line, and stops. The maximum speed allowed is 5 d s , but the time spent in the manoeuvre has not been limited. First, a human driver has carried out the manoeuvre. Figure 3b shows the corresponding simulator environment. The control scheme for automatic driving has been represented in Figure 4. It receives position, velocity and acceleration of the car, and acts upon gas and brake. The values sent to both actuators range from 0 to 1, Corresponding to null and maximum displacement of the respective devices. Simultaneous operation of gas and brake has been avoided. No error function has been used, but the position of the car. The total travelling distance has been divided into four intervals -start, taxiing, brake and stop-, and control rules have been specified for each interval. A fifth interval has been defined for the rest period after the arrival, in order to avoid an undesired behaviour in that phase.

Fig. 4. Control diagram. Figure 5 shows the comparison of the position history, gas actuation and brake actuation carried out by both the human driver and the controller. As it can be observed, the manoeuvre performed by the controller is cleaner. The controller does not need to make several approximations, as the human driver does. Actuation on both the gas and brake is more efficient when the car is automatically controlled, and less time is needed to complete the manoeuvre.

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n.

.

.

.

-

I.

I

Fig. 5. Comparison between human driving (dashed) and automatic control for the first manoeuvre: a) position history; b) gas actuation; c) brake actuation. The second manoeuvre consists in obstacle avoidance: starting from rest, the car covers an initial straight path of 20 m,then follows a full period (from peak to peak) of a sinoidal path of amplitude 1.75 m and, finally, must return to the straight line. The speed is kept under 8 m/s. First, a human driver has carried out the manoeuvre. Figure 3c shows the corresponding simulation environment.

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For this second manoeuvre, the controls of both the steering and the gas-brake couple have been addressed separately, as illustrated in Figure 6. The pursued objective is that the controller slows down the car when the error in path tracking increases. For steering control, two error functions have been defined. The first one is the position error of the vehicle at the current time. The second one aims to anticipate the behaviour of the car in the next instants. For this purpose, the tangent to the trajectory at the current point is compared with the tangent to the desired trajectory at a more advanced point. The anticipated distance depends on the car speed, being greater for larger speeds. For gas-brake control both the velocity and acceleration of the car are taken into account, as well as the path tracking error. In this case, the error is calculated as the mean of the two errors used for the steering control.

Fig. 6. Control diagrams: a) steering wheel control; b) gas-brake control. Figure 7 shows the comparison of the velocity and the trajectory carried out by both the human driver and the automatic controller. It can be seen that the human driver tends to smooth the trajectory at the turns. His maximum error is 0.5 m, and occurs at the time of returning to the straight path. It is surprising the fact that the driver does not anticipate the last turn, likely due to an excessive speed. The controller shows a more moderate trend to smooth the trajectory at the turns. During the curved part of the trajectory, the error is negligible, and the speed reduction is less acute than in the human-driven case. Again, the most difficult point arrives at the time of returning to the straight path, with an error of 0.22 m. For lower speeds, such error is almost zero. However,

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as observed when looking at the trajectory performed by the human driver, the manoeuvre has been managed at a quasi-critical speed from a stability point of view.

0

Fig. 7. Comparison between human driving (dashed) and automatic control for the

second manoeuvre: a) velocity; b) trajectory. The CPU-times obtained for the two manoeuvres reported when using the Matlab Engine option on a Pentium IV @ 2 GHz are listed in Table 1. CPU-times' include the time spent in opening a Matlab session, while CPU-time;' are obtained when the running program is attached to an already opened Matlab session. It must be said that, when no control is considered, the Fortran program comfortably reaches real-time performance when solving for the dynamics of the vehicle. It can be seen that the second manoeuvre needs a greater computational effort, due to the evaluation of two controllers, as described above. Evaluation of a fuzzy controller implies carrying out aJirzziJication (Le. determining the membership degree in the input fuzzy sets), applying the inference rules and, finally, performing a defirzzificution(Le. mapping an output value to its appropriate membership value in the output fuzzy sets). The CPU-times needed when using the MEX file option are slightly higher.

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Table 1 Efficiency.

The Matlab Engine option seems to be preferable, since the computational effort required is lower, and since less code must be added to the original program. Indeed, only commands for the opening and closure of the communication channel along with those relative to data transference must be added (when the MEX file alternative is chosen, a new heading file must be added too, in order to enable the program to be called from Matlab). Therefore, Matlab Engine represents a good solution when, as in the present case, only some parts of the program need to be executed on Matlab. Furthermore, it is adequate for the stage of controller tuning, since real-time is not required. However, for the use of an already tuned controller on a real application, real-time performance must be achieved. At the view of the CPU-times shown in Table 1, it is obvious that another alternative must be searched. In order to find a solution, it must be taken into account that the fuzzy logic generates, by means of rules of membership and actuation, a hyper-surface which relates the input (error, velocity, etc.) and output variables (gas, brake, steering). Then, it is possible to evaluate the controller by sweeping a mesh of the different input values, and storing the resulting output values on a matrix. The mesh will be more refined at those points in which the controller behaviour is more nonlinear. Matlab provides function evurjs to this end. For those elements which are not in the matrix, fast interpolation can be done, thus assuring real-time performance. 6 CONCLUSIONS Based on the results previously described, the conclusions can be drawn as follows: a) By applying an authors’ method for the dynamics of multibody systems, the computer model of an actual prototype car has been generated, and a Fortran program to determine its motion has been implemented. b) Algorithms for the automatic control of the car during two manoeuvres have been developed by using fuzzy logic functions -which mimic human strategies-, provided by the corresponding Matlab toolbox. c) Different alternatives to connect the Fortran and Matlab programs have been studied. Matlab Engine seems to be the best option for controller tuning, while encapsulation of the controller hyper-surface in a matrix appear to be more convenient for real-time applications. d) Simulator capabilities have been given to the program by means of a realistic graphical output and game-type driving peripherals (steering wheel and pedals), so that comparison may be established between human and designed automatic control.

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e) The two manoeuvres -straight line and obstacle avoidance- have been performed by both a human driver and the automatic controllers developed, and the obtained results have been compared, showing an excellent behaviour of the controllers. In a future work, the authors intend to address the experimental validation of their formulation for the dynamics of multibody systems, by implementing the developed control algorithms onboard the actual prototype car and verifying whether a good behaviour is still achieved. ACKNOWLEDGMENTS This research has been sponsored by the Spanish CICYT (Grant No. DPI2000-0379) and the Galician SGID (Grant No. PGIDTOlPXI16601PN). REFERENCES [l] J. Cuadrado, J. Cardenal and E. Bayo, “Modeling and Solution Methods for Efficient Real-Time Simulation of Multibody Dynamics”, Multibody System Dynamics, 1, 259-280 (1997). [2] J. Cuadrado, J. Cardenal, P. Morer and E. Bayo, “Intelligent Simulation of Multibody Dynamics: Space-State and Descriptor Methods in Sequential and Parallel Computing Environments ”, Multibody System Dynamics, 4, 55-73 (2000). [3] P. Antos and J.A.C. Ambrosio, “A Control Strategy of Vehicle Multibody Model for Trajectory Tracking”, Multibody Dynamics 2003, CD Proceedings, Lisbon, Portugal (2003). [4] T.J. Gordon, M.C. Best and P.J. Dixon, “An Automated Driver Based on Convergent Vector Fields”, Proc. Inst. Mech. Engrs. Part D: J. Automobile Engineering, 216, 329347 (2002). [5] K. Yi, S. Lee and Y.D. Kwon, “An Investigation of Intelligent Cruise Control Laws for Passenger Vehicles”, Proc. Instn. Mech. Engrs. Part D: J. Automobile Engineering, 215, 159-169 (2001). [6] N.D. Matthews, P.E. An, J.M. Roberts and C.J. Harris, “A Neurofuzzy Approach to Future Intelligent Driver Support Systems”, Proc. Instn. Mech. Engrs. Part D: J. Automobile Engineering, 212,43-58 (1998). [7] D.M. Bevly, J.C. Gerdes and C. Wilson, “The Use of GPS Based Velocity Measurements for Measurement of Sideslip and Wheel Slip”, Vehicle System Dynamics, 38 , 127-147 (2002). [SI J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems -The Real-Time Challenge-, Springer-Verlag, New York (1994). [9] E. Bakker, and H.B. Pacejka, “The Magyc Formula Tyre Model”, I” International Colloquium on Tyre Models for Vehicle Dynamics Analysis, Proceedings 1-18, Delft, Netherlands (199 1). [IO] J.E. Shigley and C.R. Mischke, Mechanical and Engineering Design, McGraw-Hill, 6Ih edition, Singapore (2001). [l I] G. Ellis, Control System Design Guide, Academic Press, San Diego (1991). [121A. Hopgood, Intelligent Systems for Engineers and Scientists, CRC Press, New York (2001).

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[ 131Q. Zhang, “A Generic Fuzzy Electrohydraulic Steering Controller for Off-Road Vehicles”, Proc. Instn. Mech. Engrs, Part D: J Automobile Engineering, 217, 791-799

(2003). [ 141L A . Zadeh, “Fuzzy Logic=Computing with Words”, IEEE Transactions on Fuzzy Systems, 4, 103- 1 1 1 ( 1996). [151 Matlab 6.1 Documentation, User’s Guide Version 2. I (2000). [161M.A. Naya and J. Cuadrado, “Real-Time Determination of the Position of a Car with Tri-

axial Accelerometers”, Multibody Dynamics 2003, CD Proceedings, Lisbon, Portugal (2003).

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Machines and Mechanisms

Modelling of a smart spindle unit P HYNEK, M JACKSON, R PARKIN, and N BROWN Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, UK

SYNOPSIS Modelling of a small-scale smart spindle unit is described in this paper. The smart spindle unit uses piezoelectric actuators to control the movement of the spindle nose. The front bearing of the spindle is supported by four piezoelectric actuators, which control the movement of the spindle nose in the plane perpendicular to the spindle axis. The spindle itself is modelled with the use of Finite element method (FEM). A state space model of the electrical part of the smart spindle unit, which includes the piezoelectric actuators and their driving electronics, is combined with the finite element model in order to obtain an overall model of the smart spindle unit.

NOMENCLATURE piezoelectric actuator cross-sectional area piezoelectric actuator capacitance piezoelectric actuator layer thickness electrical displacement piezoelectric charge constant identity matrix mechanical strain electric field vector of external forces piezoelectric actuator load

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V/m

N

461

force generated by piezoelectric actuator electrical current piezoelectric actuator stiffness relative dielectric constant at constant mechanical stress actuator length global finite element model matrices number of actuator layers charge stored in piezoelectric actuator global displacement vector displacement vector of nth node driving amplifier impedance mechanical compliance at constant electric field input matrix (selects input DOFs) output matrix (selects output DOFs) nodes displacements amplifier input voltage amplifier output voltage piezoelectric actuator displacement vacuum permittivity nodes angular displacements actuator displacement constant mechanical stress spindle speed 1

N A N/m m

C

R Pa*'

m V

V m F/m rad mN

Pa rad/sec

INTRODUCTION

In rotary machining processes, the relative movement between spindle nose and machine tool bed is a common problem affecting machining performance, and in particular the surface finish. This problem can be reduced by active control of the spindle movement. Significant efforts are being made to apply active vibration control to rotating machinery. Active magnetic bearings (AMB) have been used as actuators in the majority of the active vibration control research mentioned in the literature. However, despite unquestionable advantages of active magnetic bearings, the technology possesses some inherent disadvantages. AMB occupy relatively large space in relation to the magnitude of the generated magnetic forces (typically the force per bearing surface area reaches about 40 N/cm2 [l]) and for certain applications they can be prohibitively expensive. These disadvantages have motivated several researchers to investigate active vibration control utilising other types of actuators. Perhaps the most promising alternative to AMB for active vibration control of rotating machinery is the use of active materials, such as piezoelectric material. Palazzolo et al. [2] explored the use of piezoelectric actuator for vibration

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suppression of rotating machinery. The actuation principle proposed by the authors is a hybrid system where rotor is supported by piezoelectric actuators via bearings. They demonstrated the principle on a small-scale test rig. Dohner et al. [3] reported on design of a smart spindle unit for active chatter suppression of a horizontal octahedral hexapod milling machine. A model of an actively controlled rotor system with piezoelectric actuators is important at the design stage of the system. Moreover, the model can be used to design a control algorithm for the system and evaluate its performance through simulation. The rotor system typically comprises of electrical subsystem .encompassing the piezoelectric actuators and driving amplifiers and mechanical subsystem encompassing spindle, bearings and support structure. The two subsystems influence each other and therefore both have to be included in the rotor system model. A common approach for modelling such electromechanical system is to create a model of each subsystem separately and combine them into an overall model. Lin et al. [4] used a rotor model with a number discrete masses connected via flexible elements and a simplified piezoelectric actuator model consisting of an infinitely stiff pusher and spring connected in series. The authors reported that the simplified actuator model significantly simplifies the analysis since it can be represented merely in terms of scale factor in the model. The drawback to this simplified model is its inability to predict the feedback gains at which the electromechanical system will become unstable. Dohner et al. [ 5 ] described modelling of a horizontal octahedral hexapod milling machine with a smart spindle unit. The actuators, embedded into the spindle unit, were modelled as a bar with voltage dependant forces acting at each end. Thus, the actuator model could be incorporated into a finite element (FE) model of the milling machine containing over 16000 degrees of freedom (DOF). The FE model was reduced to down to 69 DOF, transformed to state space model and further reduced. The final model contained 28 states and was used to design the control algorithm for the spindle unit. This paper describes a modelling approach for rotor systems with embedded piezoelectric actuators that has been developed during development of a smart spindle unit. The smart spindle unit, described in section 2, has been developed as a part of a woodworking research project aimed at surface quality improvement by real time displacement of the tool trajectory. However, the spindle unit can potentially be used also for active vibration control. The mechanical subsystem of the spindle unit is modelled with a 1D version of FEM. The electrical subsystem includes piezoelectric actuator model and a simple model of the driving amplifier. 2

SPINDLE UNIT

The spindle is supported by two precision angular contact ball bearings arranged in a face-toface configuration. The back bearing is fitted in the spindle unit housing. The front bearing is fitted in the front ring, which is retained in the axial direction by four spacers as shown in Fig. 1. The bearings are preloaded by a pair of disc springs to eliminate any axial play of the spindle. The spacers and the front ring provide a flexible support for the front bearing. The stiffness of the support, without actuators, is low (approx 2 N/pm) in the radial direction, which is desirable, because the actuator then needs to exert less force to move the spindle. This is also the reason for choosing the face-to-face bearing configuration as this arrangement has generally lower stiffness than back-to-back configuration. The four piezoelectric actuators provide the main support for the front bearing. Applying appropriate voltage levels to the

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piezoelectric actuators controls the movement of the spindle in the plane perpendicular to the rotation axis of the spindle. Backbearing

Spindle housing

Actuator housing

\

Spindle’

(a)

(b)

Fig. 1 Spindle unit

The piezoelectric actuators are able to act only in one direction, because they are very susceptible to tensile stress. If there is a possibility of tensile forces acting on the actuator, special care must be taken to protect the piezoelectric stack itself from tensile stress in order to avoid actuator damage. Therefore, two opposing actuators for each axis have been chosen in order to achieve a “push-pull” operation. This approach was also adopted by other researchers [2], [3].

3

SPINDLE UNIT MODEL

3.1 Spindle model The mechanical part of the spindle unit is essentially a rotor system and modelling techniques used for modelling rotor systems can be applied to modelling the mechanical part of the spindle unit. Rotor systems could be modelled as a lumped-parameter structure (e.g. a massless rotor and a disc with mass). However, this simplified modelling approach is not satisfactory if the mass of the rotor cannot be neglected relative to the mass of the disc. The FEM has been chosen because it can model the rotor with continuously distributed mass and thus it provides a versatile modelling approach. The basis of the FEM (Le. element matrixes) was adopted from [6] and used to implement a simple FE pre-processor in MATLAB. A brief description of the FEM follows. The rotor system is divided into elements with the use of nodes. In order to achieve maximum flexibility, 6 DOF is considered in each node. While FE model with 4 DOF per node is necessary to model rotor transverse vibration, FE model with 6 DOF per node also allows for modelling torsional and longitudinal vibration. Equation (1) shows the vector of the node displacements that correspond to 6 DOF

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where u, v, w are displacements in direction of x, y , z axes respectively and 4, y, 9 are angular displacements around axes x, y , z respectively. The FE model of the rotor system is created by assigning elements to the nodes. For example, rotor-representing element is assigned to two nodes and disc-representing element is assigned to just one node. The main element type is the element for modelling the rotating part of the rotor system (ROT) and is depicted in Fig. 2. The element has two nodes A (for x = 0 ) and B (for x = I). The lateral deformation (i.e. v, w , v, s) are approximated by a third order polynomial along the element length in order to ensure continuity and smoothness of the deformed rotor centre line. The longitudinal and torsional deformations (Le. x, 4 are approximated by a linear polynomial along the length of the element. The model of the rotor system obtain using FEM has a form of a matrix equation of motion which can be expressed as follows

Mq + (B + wG)q + Kq = f

(2)

where M is the global mass matrix, B is the global damping matrix, G is the global gyroscopic matrix, K is the global stifmess matrix, f is the vector of external forces, q is the global displacement vector and w is rotor speed. The list of available elements is stated in Table 1

Fig. 2 Rotor element (ROT) The main parts of the smart spindle unit depicted in Fig. 1 are spindle unit housing, spindle itself and spindle support structure consisting of the front ring, spacers and bearings. The spindle unit housing, which is a ridged barrel, is considered foundation reference for the FE model. The FE model of the smart spindle unit is depicted in Fig. 3. The model has 5 nodes. The nodes 3 and 4 are placed in the middle of the bearing supports. The node 2 was placed at point where the diameter of the spindle changes. The spindle itself is modelled by four ROT elements (El - E4). The front bearing and the front ring are represented by a DISK element (E5) connected to the node 3. The stiffness of the front ring support structure (Le. the four spacers, actuators) is represented by the SUPPORT element (E6) connected into the node 3. The back bearing is directly connected to the spindle unit housing. Thus, it is modelled by introducing boundary conditions into the node 4 for displacement in directions y and z (coordinate system depicted in Fig. 2) so that the movement in those directions is prevented

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(Le.

v4 =

0,

w4

= 0). However, the angular stiffness of the back bearing is modelled by the

SUPPORT element (E7),because the angular displacements in the back bearing (i.e. node 4 4~494) are not tied

1

to the spindle unit housing.

El

Cutterhead

241

E2

E3

E4

54

AE6

Front bearing and actuator

Fig. 3 Finite element spindle model

A cutterhead can be modelled, if required, by attaching the DISK element (E8) to the node 2. The DISK element (E8) can be displaced from the node 2 along the spindle length, as depicted in Fig. 3, in order to accurately describe the position of the cutterhead. The element parameters for elements E l - E5 and E8 have been obtained from the geometry of the corresponding parts. The parameters for the elements E6 and E7 have been identified from the dynamic response of the test rig as described in section 3.5. The purpose of the model is to study displacement in the directions of axis y and z. Therefore, the DOF corresponding to longitudinal displacement u and torsional deformation 4 are omitted from the model by introducing zero boundary conditions for all DOFs (Le. u = 0 4 = 0) in order to lower the number of DOF of the final model. The FE model has initially 30 DOF. The final FE model has 18 DOF after introducing boundary conditions for omitting longitudinal and torsional displacement (-10 DOF) and boundary conditions for modelling of the back bearing (-2 DOF).

3.2 Actuator model The mechanical part of the spindle unit is modelled with the use of FEM as described in section 3.1. The inputs of the resultant FE model are forces acting in the nodes. The actuator model is required to provide force as an output so that it can be connected to the spindle FE model. The piezoelectric stack type actuator, depicted in Fig. 4(b) is constructed of n layers of piezoelectric elements, which are connected mechanically in series but electrically in parallel. Each layer has a thickness d and the overall length of the actuator is L = nd. It can be assumed that the only stress component in piezoelectric stack type actuator is in the direction of positive polarization (Le. direction 3). The constitutive equations of the actuator in direction 3 are expressed as follows [7] e, = S i c 3+ d,,E,

D,= d , p , + KYE,,E,

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(3)

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where e3 is mechanical strain, cq is mechanical stress, E3 is the electric field and D3 is electrical displacement. The piezoelectric material constants are relative dielectric constant at constant mechanical stress T3,mechanical compliance f 3 3 at constant electric field and d33 is the strain constant relating the mechanical strain to the applied electric field.

9 Piezoelectric actuator

t' 1' t' 4'

d

JCP ,+:

v,

L

-

*

vo

co+

I' 1'

The stress in the piezoelectric actuator can be considered uniformly distributed, if the actuator operates below its natural frequency. Moreover, the thickness of a single layer is very small (e.g. 0.1 - 0.3 mm) in comparison with the cross-sectional dimensions. Thus, the electric field inside the layer can be considered homogenous. This allows simplifying the constitutive equations (3) into following form 1 y = -F, + nd,,V, kP q = nd,, F, + C,Vo

(4)

where y is the actuator displacement, q is the charge stored in the actuator, C, is the capacitance of the piezoelectric actuator and k,, is the actuator stack stiffness which are expressed as follows

AK,"&,, d A k =-

C,=n-

SiL

The actuator is driven by an amplifier with output impedance R,. The circuit diagram of the electrical part of the actuator model is shown in Fig. 4(a). The circuit is described by the following equation

V, = Roi + V,

(7)

where Vi is the amplifier input voltage, V, is the amplifier output voltage (Le. voltage applied to the piezoelectric actuator) and i is the current flowing in the circuit. The behaviour of the piezoelectric actuator is governed by its constitutive equations. The current flowing through

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the actuator can be calculated as a rate of change of the charge q stored in the actuator. Therefore, the current i can be expressed from the constitutive equations (4)as follows. dY dV, dV, i = k,nd,, -- k,n2df, -+ C dt dt dt The product nd33 (Le. number of actuator layers n and strain constant d33), which appears in equation (S), describes the relation between the actuator displacement y and the applied voltage V, for an unloaded actuator. The number of layers and strain constant are not always available for commercially available actuators and therefore it is more convenient to establish the following constant, which can be estimated from actuator’s specification. 8, = 4

(9)

3

Inserting equation (8) into equation (7) yields first order differential equation for the amplifier output voltage V,.

The force the actuator exerts onto the attached structure Fpo is expressed according to the constitutive equations (4)as follows.

It should be pointed out that the force Fpo is taken as negative, because the force Fp that appears in equation (4) is force acting on the actuator while Fpo is the force acting on the attached structure (e.g. the spindle front bearing). The equations (IO), (7) and (11) can be assembled into a state space model as follows

ry

The resultant state space model has the following inputs: structure displacement y and velocity dy/dt at point where actuator is connected to the structure and driving amplifier input

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voltage vi. The outputs of the model are: actuator generated force F,,, output voltage V, and current flowing in the circuit i.

driving amplifier

3.3 Overall model The overall spindle unit model consists of the FE spindle model described in section 3.1 and the actuator model described in 3.2. The FE spindle model is in the form of a matrix equation of motion (2) while the actuator model is in fonn of a state space model (12). The two models need to be converted into a compatible format so that they can be integrated. The state space model representation has been chosen as a format for the overall spindle unit model, because it is easier to convert FE model into a state space representation. The conversion of an FE model into a state space representation can be expressed as follows

The output of the state space model described by (13) is displacement and velocity in selected DOF determined by the output matrix To,,. The input of the model is a vector of external forces fin.The external forces act in selected DOF that are determined by the input matrix Ti,. The input and output matrixes are assembled by the FE pre-processor from information supplied in its input file. In the case of the smart spindle unit FE model depicted in Fig. 3, the inputs of its state space representation are actuator forces acting in node 3. The outputs are displacement and velocity in node 3, which are needed as inputs for the actuator model. Moreover, the displacement in node 2 (Le. place where cutterhead is attached) is also selected for output. The overall smart spindle unit model consists of the FE spindle model and two actuator models (Le. one for each direction) as shown in Fig. 5. In reality, two pairs of opposing actuators support the spindle in order to achieve “push-pull” operation. However, two actuators, one from each pair, are considered “passive” and are modelled as springs. The other two actuators are considered capable of a “push-pull” operation and are modelled by the model described in section 3.2. The interconnection between the parts of the overall model is shown in Fig. 5. The resultant model of the spindle unit is a state space model with 38 states (Le. 36 states from the FE spindle model and 2 states from the two actuator models).

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Actuator model far y direction

L-----==~~~ Actuator model for x direction

Input vollage

~

dxJdt

Fig. 5 Overall model Note that the coordinate used to describe the spindle displacement on the test rig is different from the coordinate system of the FE spindle model as can be seen in Fig. 5. While, this might appear as rather confusing and establishing one coordinate system seams more appropriate, the coordinate systems used servers each their purpose. The coordinate system established for the overall model is more intuitive (i.e. x-axis in horizontal direction and yaxis in vertical direction). The coordinate system for the FE model as depicted in Fig. 2 is used because by convention, x-axis of coordinate system describing deformation of a rotor or a beam like structure is along the length of the rotor.

3.4 Implementation The commercially available FEM software is usually aimed at building complex finite element models (e.g. comprising of thousands of elements) and implementing custom models in those software packages is sometimes difficult. The complexity of a FE model needed to describe a rotor system is usually low (e.g. tens of elements) as opposed to much more complex FE models (e.g. thousands of elements), used for stress analysis. Moreover, the rotor system can be modelled with sufficient accuracy by one-dimensional elements. Therefore, FEM has been implemented in MATLAB environment in order to simplify the integration of the two models. The MATLAB Control toolbox can be conveniently used to manipulate and analyse the state space models The FEM implementation in MATLAB uses a simple pre-processor to assemble the model system matrixes according to the model configuration. The information about the model is

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supplied to the pre-processor in an input file. The input file is a plain text file that contains model information, which are divided into several sections. Each section is marked with HTML-like tags. The FE model can be parameterised by specified element parameters as MATLAB variables. A snippet of the pre-processor input file is shown in Fig. 6. A number of element types, listed in Table 1 have been implemented to model typical parts of a rotor system. The state space actuator model is combined with state space representation of the FE spindle unit model with the use of MATLAB Control toolbox. Element name

Table 1 Elements for modelling rotor system Numbe ofnode

6 Spindlr unit finite element model 6 El.mentm definition CldSSP> EN N-0

6 E E E E

1 3

rot rot rot

4

rot

2

1 0.010 0.037 0.087

d 0.016 0.027 0.0115

do dona

4

Par P P P

I

P

0.054

0.008

o

Nod.. N 1 N 2 N 3 N 4

2 3

6 EN N-0 Nod* Par m E 5 dimk N 3 P 0.15 6 EN N-0 E 6 .upport

Nod. N 3 E 7 mupport N 4

IO 0.0540-3

0 0

0

7800 7800 7800 7800 I 0.028.-3

Par

kx ky

ku

P

0 0

0 0

P

kn 1406 14.6 0 0

0 3 . 7

E

nu

2.1011 1.1.11 2.1811 2.1.11

0.3

I

e

0

0

0.3 0.3

0.3

k 1 1 1 1

fi ni k 0 0 1

kz= bx by b i bxx byy bZz I 0 0 20 20 0 0 . 1 0.4 0 0 0 0 0 0

3 . 7

Boundary rrondition deriniton N 1 ~ 0 1 1 0 0 0 ~ 0 0
6 Output DOR d.rinitiOn

N 3 N 2

Fig. 6 FE pre-processor input file snippet

3.5 Model tuning The overall suindle unit model has a number of inuut uarameters such us suindle limensions, piezoelectric actuator capacitance, driving amplifier output impedance, bearing damping etc. The parameters define the dynamic properties of the model and must be determined in order to obtain an accurate spindle unit model. Some of these parameters can be easily measured. For example, spindle dimensions can be measured; piezoelectric actuator capacitance can be measured or obtained from actuators datasheet. However, there are parameters of which measurement is difficult. For example, it is difficult to experimentally determine damping in rolling bearings supporting the spindle [8]. Therefore, parameters of the spindle unit model that are unknown such as bearing damping and angular bearing stiffness have been selected as tuning parameters. The tuning parameters have been adjusted so that the frequency response of the model matches the frequency response of the actual spindle unit that was determined experimentally by measuring the unit response to a linear swept-frequency signal. Fig. 7 1

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shows comparison between the measured frequency and the frequency responses of the tuned model and the initial model. The initial model has all the tuning parameters set to zero. The tuning procedure allows obtaining model that describes the spindle unit more accurately. The comparison between measured spindle response and simulated response of the tuned model to an input quadratic pulse (width 555 ps, magnitude IOOV) shows a good agreement.

Fig. 7 Frequency response comparison

Fig. 8 Spindle response to pulse

4

DISCUSION AND CONCLUSION

This paper summarises a modelling approach for creating an electromechanical model of a rotor system with embedded piezoelectric actuators. The modelling approach has been successfully implemented in programming system MATLAB. The choice of MATLAB for the implementation proved to be advantageous, because a number of tasks needed for the model implementation, such as numerical algorithms, state space model manipulation and analysis, can be offloaded to standard MATLAB functions.

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The implementation of the modelling approach described in this paper provides a flexible modelling tool. The model can be easily adopted for different design scenarios by changing the pre-processor input file. Moreover, the model generation can be parameterised and fully automated, which is useful for design optimisation, sensitivity analysis and model tuning. The unknown model parameters, typically damping in the system, can be obtained via tuning procedure outlined in section 3.5. However, based on experience or known parameters from similar designs, those unknown parameters can be estimated and a reasonably accurate model can be obtained in the design stage of the spindle unit. For example, it can be seen from Fig. 7 that there is a good qualitative agreement between the actual response and the initial model, which is effectively based on design parameters of the spindle unit. Clearly, the magnitude of the resonance peaks of the initial model is much larger than the actual spindle unit response, because the damping coefficients were unknown and therefore set to zero. The modelling approach has two limitations. Firstly, the mechanical subsystem is modelled with the use a 1D version of FEM, which limits the use only to modelling of beam-like structures (e.g. spindle). Secondly, the derived model is essentially a linear model. Thus, all non-linear properties of the subsystems are neglected. However, the advantage of having a linear model is that the theory of linear dynamic systems can be used for analysis. 5

REFERENCES

[I] Bleuler, H. Survey of magnetic levitation and magnetic bearing types. JSME International Journal, Vibration, Control Engineering, Engineering for Industry 3 [3], 335342. 1992. [2] Palazzolo, A.B., Lin, R.R., Alexander, R.M., Kascak, A.F., & Montague, J. (1991) Test and theory for piezoelectric actuator-active vibration control of rotating machinery. Journal of Vibration, Acoustics, Stress, and Reliability in Design, 113, 167-175. [3] Dohner, J.L., Lauffer, J.P., Hinnerichs, T.D., Shankar, N., Regelbrugge, M., Kwan, C. M., Xu, R., Winterbauer, B., & Bridger, K. Mitigation of chatter instabilities in milling by active structural control. Journal of Sound and Vibration 269[1-21, 197-21 1.2004. [4] Lin, R.R., Palazzolo, A.B., Kascak, A.F., & Montague, G.T. (1993) Electromechanical simulation and testing of actively controlled rotordynamic systems with piezoelectric actuators. Journal of Engineering for Gas Turbines and Power, Transactions of the ASME, 115,324-335. [SI Dohner, J.L., Hinnerichs, T.D., Lauffer, J.P., Kwan, C.M., Regelbrugge, M.E., & Shankar, N. (1997) Active chatter control in a milling machine. Proceedings of SPIE - The International Society for Optical Engineering, Smart Structures and Materials 1997: Industrial and Commercial Applications of Smart Structures Technologies, 3044,28 1-294. [6] Slavik, J., Stejskal, V., & Zeman, V. (1997) Zaklady dynamiky stroju (Fundamentals of machine dynamics). Prag, Vydavatelstvi CVUT. [7] Clark, R.L., Saunders, W.R., & Gibbs, P.G. (1998) Adaptive Structures (dynamic and control). John & Wiley. [8] Zeillinger, R., Springer, H., & Kottritsch, H. (1994) Experimental determination of damping in rolling bearing joints. Proceedings of the International Gas Turbine and Aeroengine Congress and Exposition, 1-5.

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A dynamic modelling and simulation of cutting process in turning W S E MOUGHITH, A A ABDUL-AMEER, and A KHANIPOUR

School of Engineering, Design, and Technology, University of Bradford, UK

Abstract A dynamic model of a lathe machine and cutting process is presented for condition monitoring purpose. The model presented permits a better understanding of the cutting process and its interaction with the machine tool, and this can be used as a part of a model based fault diagnose system. Keywords: Turning machine, modelling, simulation, dynamic, spindle drive

1. INTRODUCTION

The role of mathematical modelling in design, optimisation and product development is crucial. This knowledge also can be utilised in condition monitoring and diagnosis resulting in improved fault finding procedures and techniques during cutting [l]. The computer aided analysis generating the cutting forces has been developed by many researchers in the past but these models are either based on mechanistic approaches, which are highly complicated and labour intensive or on statistical methods. There is clearly a need for a simplistic approach, which simulates the dynamic response of the cutting process. The output of the model then can be used as inputs to other CAD packages such as finite element or multi-body dynamic for the structural analysis of the machine. The overall model then can result in a virtual prototyping environment for design optimisation. The analysis of the cutting forces can reveal valuable information about the machine tool structure. This is because the cutting process is the one relationship of the whole machine tool dynamic which closes the loop between the axis feed subsystem and the spindle subsystem [2]. Also it is the cutting process that affects the control of the servo-mechanism, and generates the disturbance torque on the feed and the spindle motors. The turning process has been investigated by many researchers. Mackinnon [3] developed a method of wear estimation for carbide tools using multi-component force measurements for the purpose of tool condition monitoring during adaptively controlled metal cutting on a turret lathe. Dan and Mathew [4] presented a review for cutting force measurements for on-line tool wear and

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breakage sensing. El Baradie [SI discussed a closed loop system with two main elements representing the machine tool structure and the cutting process to develop a generalised statistical theory of chatter to predict a threshold of stability in terms of mean values. Balkrishna, et al [6] presented a model of the dynamic cutting force process for the three-dimensional.

Figure 1 Condition monitoring Scheme

They linked the mechanistic force model to a tool-work piece-vibration model to predict the dynamic cutting forces. Stein [7] proposed a cutting force monitoring approach based on the spindle motor current and speed as well as a model relating these measurements to the cutting force. The non-linear nature of cutting processes makes it impossible to be represented as a simple mathematical function in these CAD packages. Therefore, in this paper a dynamic model of a lathe machine and cutting process is presented, where a block diagram model of the cutting process in the turning operation has been developed through an analytical approach in order to predict the cutting forces. This gives a fundamental idea about cutting forces acting on the axis drive and spindle drive systems which can be used in machine tool virtual prototyping. The aim of this paper is to analyse the process in physical terms and hence to develop a cutting process model that relates the feed rate and spindle speed to the cutting forces generated. The model presented here permits a better understanding of the cutting process and its interaction with the machine tool, and can be used as a part of a model based fault diagnoses system and as a part of an integrated computer aided engineering design tool. Model parameters were measured experimentally so that the model can be simulated. The results of the simulation were compared with data obtained during cutting operations. 2. DRIVE SYSTEM

The main components of the turning machine are illustrated in figure 2. The spindle axis drive generates a rotary motion of the work piece. The X-axis drive system, which is not presented in this paper, is mainly used to develop a longitudinal motion of the table over the slide-ways.

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I

- x Axis Driw System

1

I

Figure 2 Schematic of the turning machine The spindle drive system comprises a DC motor connected to a gearbox, which transfers the motion to the spindle shaft using a belt and pulley mechanism. The input to the DC motor is a controlled voltage VSM,which is supplied from the velocity loop controller. The input to the velocity loop is the command signal Vc. Figure 3 shows a schematic diagram of the velocity loop, which is a pulse width modulated system (PWM).The speed regulator is a proportional plus integral controller. The power control loop has a current feed back IS through a sensor with a gain, KN.Figure 3 illustrates the block diagram model of the velocity loop [2]. GPWM

=-

1

K,s+l

where, Kp is the time constant of GPWM, and s is the Laplace operator.

P a r t Program

Position Processing

CNC Controller

Drive

Control

Control

System Velocity Regulator k Drive Motor

1 Position Feed Back

I

I

Figure 3 Drive system control loop Figure 4 shows the free body diagram of the spindle axis drive system.

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Figure 4 Spindle drive system Where, BSMis the lumped coefficient of the viscous friction in the motor, gearbox, gearbox pulley and bearings, Bs is the lumped coefficient of the viscous friction in the spindle bearings and Mthe lumped moment of inertia of the spindle motor, gearbox, and pulley and pulley. Also, J ~ is Js is the lumped moment of inertia of the spindle, pulley, and work piece. TSMFis the coulomb friction of the spindle motor and gearbox. TSF is the coulomb friction of the spindle shaft bearings. II and j2 represent the spindle motor pulley and spindle axis pulley angular speeds, respectively. PIand P2 are the forces on the tight and slack toothed belt, respectively. TSGis the gross torque generated by the spindle motor armature. TC is the cutting torque generated by the cutting process. Bs 6, and BS 4, are the resistance torques due to viscous friction in the spindle motor and the spindle shaft, respectively. TSDis the spindle drive torque exerted at the spindle pulley to overcome the cutting torque and friction torque. The coulomb friction in the spindle motor and gearbox TSMFis ignored in the beginning to simplify the equations of motion for the motor pulley, which is given below.

Taking Laplace transformation,

where, rsl is the motor pulley radius. Assuming there is no longitudinal strain through the drive belt, then:

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4 2 = r42 (6) where, r n is the spindle pulley radius. r is the pulley ratio. The angular position of the spindle 4s is less than the angular position of the spindle at the pulley end h due to the elongation in the toothed belt as shown below.

where, 84 is the angular deflection due to the elongation in the toothed belt assuming that the angular deflection through the spindle shaft is neglected. The elongation in the tight side of the toothed belt is:

Where, PIis the force in the tight side, PO is the pre-load force, and KB is the linear stiffness of the toothed belt. The reduction in the slack side is equal to the elongation of the tight side of the belt, then:

where, Pz is the force in the slack side of the toothed belt, Assuming that the spindle drive torque T ~ causes D the toothed belt elongation.

The elongation in the toothed belt transfers to a position change in the spindle, 64 64 =d,/r,

where, Kc is the equivalent angular stiffness in the toothed belt. The equation of motion of the spindle drive shaft is

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-Tc +(PI -Pz)rsz -ESds = J ,

5,

-Tc+TsD-Bs~4z= J S s z ~ ,

where, TCis the cutting torque acting on the spindle shaft and is related to the tangential force, F,.

where, rwp is the work piece radius. Figure 7 represents the block diagram model of the spindle drive system.

3. TURNING CUTTING PROCESS The cutting force, F, acting on the tool is generated by the engaged part of the cutting edge including the main cutting edge, nose radius, and part of the secondary cutting edge. This force comprises three components as shown in figure 8 [7]. Where, FJis the feed force component, F, is the radial force, and Ff is the tangential force, considered for spindle drive modelling. There are many variables affecting on the cutting force in turning, some of theses variables are tool geometry, depth of cut, d, feed,J cutting speed, Vc which is function of spindle speed, N, work piece hardness, state of lubrication, and tool wear. These variables are summarised in the standard equation for the tangential force component as follows [8].

Figure 7 Block diagram model of the spindle drive system

F; =Ksfd' p (24) Where, Ksi is the specific cutting force. b, and a, are constants. The effect of cutting speed Vc on the cutting force can be ignored [3]. This model when logarithmically transformed becomes: In& =lnK,, +b, Indi-a, In f

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(25)

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Then, equation (24) can be rewritten as: 56 =bo ++I +9x2 (26) where, j F ,is the estimated tangential force component and bo, bl, and b2 are the model parameters. X I and x2 are the natural logarithmic value of depth of cut and feed, respectively. Multiple regression analysis estimates these three parameters for the first-order model in equation (25). If there was statistical evidence of lack of fit a second-order or third-order model can be developed. The third-order mathematical model developed will be of the following form.

A second or third-order mathematical model is essential when the true response (tangential force) function is non-linear or unknown. 32 tests were conducted to identify these cutting model parameters. A statistical package MINITAB software was used to identify the cutting model parameters, which are bo, bl, bZ , b3, bq, bs, bg, and b7. A further regression is performed after eliminating x12 and x13 from the model. The analysis of variance of the tangential force model is given in [IO], which shows that the regression model is significant since P-value, which is the smallest level of significance that would lead to rejection of the null hypothesis, is 0.00. The value of F0.99,16,10 is found to be 4.53. Since F h c k o f ~ r< F0.99,16,10, a lack of fit is not indicated. The mathematical model of the tangential force is shown below.

j4= 8.2748+ 1.3844 X, +3.5819 x2 + 2.2287 xZz+OS329 X: +0.2944 X , x2

(28)

Figure 8 Cutting forces in turning process

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4. OVERALL BLOCK DIAGARM

The overall block diagram model of the spindle drive system and turning cutting action is derived by connecting the output of equation 27 after taking the exponential value of jF,of the spindle drive model, as shown in figure 9.

Depth o f cut

F,

Spindle drive - system model

Cutting force model

Spindle speed feed

Velocity c o m m a n d signal

( U s i n g multiple regression analysis)

-

Tangential force

model

Figure 9 Block diagram model of the machine

5. SIMULATION the The model has been simulated using MATLAB software. The input to the simulation is the the spindle velocity command, feed rate, and depth of cut. The outputs are the motor current, the spindle speed, as shown in figures 10.

0

-101

482

.....,.......................................................................... I

I

I

I

.,. ..................

I

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6. CONCLUSION

Engineers of machine tool design need mathematical models in order to understand the system and its response. The spindle drive model in this paper provides an understanding in showing the linkage between the machine elements and the cutting process. This model can be used in a condition monitoring system for turning processes. In this work the model was produced in a theoretical form and simulated to confirm viability thereafter. 7. Nomenclature

at and bt Constants bo, bl, Parameters used to estimate PO,Pland P 2 respectively Lumped viscous friction coefficient in the motor, gear box, gear box pulley and bearings Lumped viscous friction coefficient of the spindle bearings and pulley Cutting force (N) Feed force component (N) Radial force component (N) Tangential force component (N) Transfer function of the pulse width modulator Feed back current (A) Lumped moment of inertia of the spindle pulley and workpiece Lumped moment of inertia of the spindle motor, gear box and pulley Linear stiffness of the toothed belt Equivalent angular stiffness in the toothed belt Sensor gain Time constant Specific cutting force (N) Spindle speed PI and P2 Tight and slack toothed belt forces respectively Pre-load force (N) Spindle pulley radius workpiece radius Cutting torque Spindle drive torque Coulomb friction of the spindle shaft bearings Coulomb friction of spindle motor and gear box Spindle motor armature gross generated torque Cutting speed (mm/min) Command signal Controlled voltage (v) Natural logarithmic of cut depth and feed depth Pulley ratio Angular position of the spindle Angular deflection Angular speed of the spindle motor pulley and spindle axis pulley respectively

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REFERENCES 1. Ebrahimi M., Victory, J., Web-based

machine tool condition monitoring, SPIE International Symposium Boston, MA, USA, 2000. 2. Martin,K.F., Ebrahimi, M., Modelling and simulation of the milling action. Proc. IMechE, 213, part B, pp. 539-554, 1999. 3. Mackinnon R., Wilson, G.E. and Wilkinson, A.J. Tool condition monitoring using multicomponent force systems. Matador conference, 317-324, 1986. 4. Dan and Mathew, Tool Wear and Failure Monitoring Techniques for Turning - A Review, International Journal of Machine Tools and Manufacture, Vol. 30, no. 4, pp. 579-598, 1990 5. El Baradie (1991) discussed a closed loop system with two main elements representing the machine tool structure and the cutting process to develop a generalised statistical 6. Balkrishna C. Rao, and Yung C. Shin, A Comprehensive dynamic cutting force model for chatter prediction in turning, International Journal of Machine Tools & Manufacture, Vol. 39, pp. 1631-1654, 1999. 7. Stein, J. L., Monitoring cutting forces in turning: a model-based approach. Transactions of the ASME, 124,26-3 1,2002. 8. Anon, Adaptive Control of Machining Conditions, MTIRA Research report No. 35, pp. 89, July 1970. 9. Tlusty, G , ManufacturingProcesses and Equipment, Prentice hall, Inc., New Jersey, 2000. IO. Ebrahimi M., Mougthith W and Victory, J. L., Block Diagram Model of Lathe Machine, Advances in Simulation, Systems Theory and Systems Engineering, WSEAS Press, 2002, pp74-79.

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Modelling and simulation of a precision pneumatic actuation system R C RICHARDSON Department of Computer Science, University of Manchester, UK M BROWN Lockheed Martin UK Integrated Systems, Hampshire UK B BHAKTA School of Medicine, University of Leeds, UK M LEVESLEY School of Mechanical Engineering, University of Leeds, UK

SYNOPSIS: Low friction pneumatic cylinders are now being considered in applications for which only electric motors or hydraulics were previously considered suitable. One potential application of low friction pneumatics is robotic physiotherapy where the high power to weight ratio and low cost could be exploited. As part of an ongoing project to develop a pneumatic physiotherapy robot, this paper presents an analysis of pneumatic cylinder characteristics that simplifies controller design. Using mathematical modelling and simulation, non-linearities of modern pneumatic systems have been investigated. The derived models give an excellent representation of the system, despite the inclusion of a simplified friction model. Nomenclature a = Orifice area (m') $,i = Orifice area pipe saturation (m2) A, = Piston area of bottom chamber = 1.98e-4 m2 Ab = Piston area of top chamber = 1.78e-4 m2 cp= Constant pressure specific heat for air = 1003.5 J kg" K ' cq= Orifice discharge coefficient = 0.9 (poppet valve) c, = Mass flow coefficient cmPl = Mass flow coefficient for poppet 1 cmPl = Mass flow coefficient for poppet 2 c, = Constantvolume specific heat for air = 718.6 J kg.' K ' C = Poppetdamping= 13 N/ms" D = Cylinder damping = 15 N/ms"

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E = Area of Spool = 4.5e-4 mz FI = Force from bottom of cylinder (N) F2 = Force from top of cylinder (N) FpI = Force on poppet 1 (N) Fp2 = Force on poppet 2 (N) k. = Poppet circumference = 3e-2 m K, = Poppet spring constant = 1500 N/m m, = Mass in chamber A (kg) mb = M a s in chamber B (kg) M = Mass of piston and load = 0.5 kg P, = Pressure in chamber A (Pa) Pb = Pressure in chamber B (Pa) Pd = Pressure from pilot valve (Pa) P, = Exhaust pressure = l*lOs(Pa) (abs.) P, = Supply pressure = 6.5’10’ (Pa) (abs.) R = Gas constant for air = 287 J kg.’ K 1 T,= Supply temperature = 300 K T, = Chamber A temperature = 300K ult= Valve 1 control signal (V) uzl = Valve 2 control signal (V) Vi =Voltage input = 0 - 5V V. = Volume of chamber a (m’) Vb = Volume of chamber b (m’) Vda= Chamber A ‘dead volume’ = 5.9e-6 m’ Vdb = Chamber B ‘dead volume’= 5.34e-6 m’ x = Displacement (m) xI=Cylinder stroke = O.lm ‘T = Link torque (Nm) p = Density of air (kg m-’) y = Specific heat ratio= 1.4

1. INTRODUCTION In 1990 the annual incidence of stroke in the general population was around 2 per 1000 [l]. Around 50% of surviving stroke victims incur chronic motor disability, i.e difficulty moving an arm or leg. These patients require physiotherapy to increase muscle strength and to relearn the ability to use the limb. To a limited extent, prototype robots are now able to apply physiotherapy exercises, and do so with greater consistency than their human counterparts. However current prototypes tend to be complex and expensive [2,3,4]. The high power-toweight ratio, low cost and direct drive capabilities of pneumatic actuators, however, mean that the potential exists to make such devices simpler and more affordable. Traditionally, pneumatic systems were only used for end stop positioning. For this method of control, only a limited understanding of the underlying dynamics was needed to obtain required performance. Recent developments in low friction pneumatic cylinders [5] and electropneumatic proportional valves, however, allow pneumatics to be considered for precision tasks requiring a greater understanding of the underlying dynamics. One of the first people to analyse moving pneumatic systems was Shearer [6], who developed a detailed model of a double-ended cylinder by considering both isothermal and adiabatic heat transfer processes. Also, Liu and Bobrow [7] controlled a simple one-degree-of-freedom pneumatic robot, experimentally determining linearised spool valve coefficients and

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developing a control algorithm around the model. Mc Donne11 [8] developed a model of a pneumatic cylinder and spool valve using established techniques, forming experimental quadratic models of mass flow. Research has been performed into the behaviour of asymmetric actuators. Backe and Ohligschlager [9] performed a detailed analysis of heat transfer in asymmetric pneumatic cylinders to enable their thermodynamic properties to be understood further. Wong and Moore [lo] analysed the behaviour of double-ended asymmetric cylinders when driven from a single end. They found significant differences in maximum response time, velocity, acceleration and force for extension and retraction. Kawanaka and Hanada [ 1 11 controlled a double-ended pneumatic cylinder with two pressure proportional valves. The top chamber was maintained at constant pressure while the bottom was varied, limiting the potential system performance. Several active robotic physiotherapy devices have been developed [2,3,4,12,13], but most of these devices use electric motors as joint actuators. White et al. [12], however, used a pneumatic cylinder to perform elbow flexiodextension physiotherapy. They used pressures either side of the cylinder to obtain information on the forces applied to the limb. Austin [131, who investigated elbow flexiodextension robotic physiotherapy using a single-degree-offreedom robot, compared control strategies using experimental results from a cross-section of stroke patients. As part of a project to develop a 3 degree of freedom (d.0.f) pneumatic physiotherapy robot [14], this paper presents an analysis of the pneumatic joint actuators. Firstly, a model of a single valve is developed. This is then augmented to model a pneumatic cylinder controlled by two valves. Section 3 develops a model of the valve response, which is then simplified to omit less significant elements of the transient response. The full and simplified models are simulated and compared to results obtained experimentally. In section 4, a model of two valves driving a low friction cylinder presented then simulated and compared to results obtained experimentally. 2 EXPERIMENTAL APPARATUS

Experiments were performed on one degree of a three-degree-of-freedom robot, custom designed for robotic physiotherapy [ 141 (Fig 1).

Figure 1 Three degree-of-freedom physiotherapy robot

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Joint angles are measured using a potentiometer. Air was supplied to the pneumatic cylinder by two electro-pneumatic pressure proportional valves, which supply each cylinder chamber independently. A single demand signal was used to operate both valves around a pre-set equilibrium pressure (the pressure either side of cylinder for a zero demand signal). For a positive control signal, one valve increases the lower chamber pressure, while the upper chamber pressure is reduced by the same amount, via the other valve [17]. The equilibrium pressure was set to ensure the maximum range of sonic flow when supplying and exhausting air.

3 VALVE RESPONSE MODEL For applications such as robotic physiotherapy it is important to analyse actuator performance to ensure accurate control. In order to understand the valve transient characteristics, the cylinder displacement was fixed at its mid-point while a single valve was used to supply pressure to the bottom chamber. The valves are designed so that pressure output is proportional to voltage input. Examining the structure of the valve (Fig. 2 ) a controlled pilot stage supplies pressure proportional to voltage into a small volume. The pressure differential between pilot pressure (Pd) and actual pressure (pa)moves the spool, which in turn operates the poppet valves. A mathematical model was developed and compared to experimental responses to explain the operation of the valve.

Figure 2 Valve cross section and poppet model

3.1. Model A simplified model of the spool and poppet system (Fig. 2 ) was constructed, neglecting the dynamics of the spool, but including stiction and spring pre-load (Le. all force generated on the spool, above the stiction threshold, is applied to the poppets). This model allows independent movement of both poppets. When supplying air to chamber A the mass flow rate, m, ,through an orifice can be written as [9]:

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Where cq is based upon the valve type, in this case cq = 0.9 (Poppet valve), T, is supply temperature, P, is supply pressure, a is orifice area and cmplis a non-linear term based upon pressures either side of an orifice. cmplis given by:

where R is the gas constant for air, P, is downstream pressure (pressure in chamber A) and y is the specific heat ratio. The orifice area, a, is given as: a = kcxpl (3) where is poppet circumference and xpl is poppet displacement.

Also, the force generated on the spool is given as (5) (Pd - P,)E = F, where P d is the pressure supplied by pilot valve, E is spool piston area and F, is the total force applied to the spool. Due to stiction and spring preload, small forces are not sufficient to cause poppet motion: If abs(F,) <1 then F = 0, otherwise F,

=F

(6)

As the spool is never in contact with both poppets, a simple set of rules was formulated. i f F < 0 thenF,I = 0 andFp2 = F (7) if F 1 0 then F,l = F and F,2 = 0 (8) where F,1 and Fp2 are the forces applied to poppets 1 and 2 respectively. So from figure 4 the motion of, for example, poppet 1 is given by:

F

ifF?O

x p I =-

if F < 0

x p l ( C s+ K , ) = 0

Cs+K,

where C is poppet damping and K, is poppet stiffness. For an ideal gas (adiabatic heat transfer)

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PLl =

P a -

YRT a

where pais the density of air and T, is the temperature of air in chamber A. (Note that y = 1 if the heat transfer is isothermal). Actual heat transfer varies between adiabatic and isothermal. In this case, adiabatic heat transfer is assumed) Multiplying equation (1 0) by chamber A volume (V,) gives: P

TI

Where mais the mass in chamber A. Differentiating (1 l), assuming T, is constant, gives:

__

Remember that the actuator is fixed at this modelling stage, so V, is constant. The total mass flow rate into the chamber is the sum of the mass flow rates from both poppets (mass leaving chamber is considered as negative mass flow rate). From equation (4) we then have:

Where cmp2is the mass flow rate coefficient poppet 2. Hence, by combining equations (13) & (14) and rearranging (assuming T, = TJ we have a model for p, as:

Also, Pd (absolute) responds to voltage input (due to the control circuitry) as: Pd = (V, - 1) * 2.2e5 + le5 (16) Where Vi is the voltage input. Equations (1) to (15) were simulated in Matlab and the results of this simulation are compared to experimental results taken from the actual test rig.

3.2 Model Validation 3.2.1, Maximum rate of change ofpressure For maximum rate of change of pressure to occur the spool displacement (fig 2) needs to be at its maximum (x = 1.5e-3m) with sonic flow (c, = 0.0404). Examining the experimental step response, the maximum rate of change of pressure of the valve differed considerably from the theoretical. This was due to the area of the supplying pipes being less than the maximum valve opening area, limiting the performance of the valves. Therefore, this saturation effect is included in the simulated system as:

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where apip, is the maximum valve opening due to pipe area restrictions. The effect could be overcome by increasing the diameter of the pipes, but this is not feasible due to design constraints.

3.2.2. Step response The step response obtained from the valve was compared with the simulated response (Fig. 3). The response time of the simulation closely matches that of the experimental. The general shape of the overshoot agrees with the experimental results. It is apparent from inspection that the pressure overshoot for supply is greater than exhaust. This is due to the difference in supply pressure and chamber pressure (P, and P,). For example, with P, = 4.5 bar (abs.) and saturation of orifice area due to pipes, the maximum mass flow rate for supplying air is 1.5 times larger than the mass flow rate for exhausting air even though air entering the chamber is slightly unchoked. As both poppets have the same delay in closing, a larger pressure overshoot is created when supplying pressure.

Simulated and experimental pressure response

x10

1

1.2

1.4

1.6

1.8

2

2.2

Time, t (s)

Figure 3 Simulated and experimental pressure response Examining the poppet displacements (Fig. 4) at approximately 2s, for a pressure increase, illustrates the valves operation: 1. Poppet 1 opens due to desired pressure being greater than actual pressure 2. Due to overshoot in pressure, poppet 2 opens 3. Poppet 1 closes (lag due to damping) 4. Then poppet 2 closes (again with lag)

(Note that the delay in poppet closing allows both poppets to be open simultaneously).

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IO-^ I '

I

1.51

-2

0.8

I

n

1

1.2

1.4 1.6 Time, t (s)

1.8

2

2.2

Figure 4 Simulated poppet displacement at transient point of step response

3.3 Simplified valve model The small overshoot caused by the damping in the poppets would not have any significant effect when controlling the cylinder, due to its bandwidth being much larger than that of the cylinder. This assumption enables the valve model to be simplified by ignoring the poppet damping. For poppet 1, equations (8) and (9) then become: if F > O x p , = E K, if F < 0 x,,, = 0 (19)

(18)

And for poppet 2 if F < o x p l = E(20)

K,

if F > 0 x P z = 0 (21) The simplified model was simulated using Matlab. The steady-state and response times of the simplified model (Fig. 3) are similar to the experimental but the transient peaks have not been modelled.

3.4 CYLINDER MODEL

A mathematical model was constructed for two valves supplying air to both ends of the pneumatic cylinder based on the work of Shearer [4] and Lui et a1.[5]. Due to the cylinder's asymmetric nature the exposed piston surface area was greater in the bottom chamber. To compensate for this effect, the equilibrium pressure of the top chamber was increased in proportion to the area ratio. The dead volumes of the cylinder result from volume that is not

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effected by cylinder movement (this also includes the volume of air in any connecting pipes). Due to the asymmetric nature of the cylinder, different mathematical models need to be derived for chambers A and B. Equations 22 & 23 describe the cylinder response. A detailed deviation can be found here [181. The model for pressure change in chamber A.

The analysis for chamber A can be repeated for chamber B to give: c p RTS = P, (23) m b +X

Where all b subscripts refer to chamber B variables.

3.5 Cylinder dynamics Finally, the dynamic behaviour of the piston and link can be described as [ 161: M x + DX = T (24) where M is link moment of inertia, D is an approximation for various friction effects, which in reality are nonlinear, and is the torque applied to the link. It was necessary to resolve cylinder force (Pa& - P d b ) into link torque. The low friction nature of these cylinders reduces friction effects, but small amounts of friction and stiction are still present [17].

4 MODEL VALIDATION Using the simplified model of the valves, and the cylinder model, a simulation of the valves and cylinder was constructed. A simple stiction model which caused internal cylinder forces between *l N to be negated due to cylinder stiction was included in the model. The system was experimentally excited using proportional control. Proportional control is a simple control method utilising one forward path (k,) and unity negative feedback . This control method was chosen as its simplicity masks very few of the underlying system dynamics, the purpose being to verify the model rather than achieve optimum control. The experimental results were obtained with k, =3e-2. These were then compared to the simulation. The simulation position response (Fig. 5) is similar to the experimental although approximating the non-linear damping @) as linear and using a simplified stiction model has caused differences in the response. (Note the position results are not symmetrical due to a non-linear relationship between cylinder position and angle). 5 CONCLUSIONS

The overall behaviour of the simulation closely resembles that of the actual system, although the model could be improved by introducing a more accurate non-linear representation of friction and stiction within the cylinder. Even though the pressure proportional valves and low friction cylinders make accurate positioning of pneumatic cylinders possible, it is important understand the dynamic characteristics of such systems. Knowledge of non-linear

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elements can lead to successful controller designs. The model derived in this paper will form the basis of an impedance control method that will be implemented for robotic physiotherapy.

Figure 5 Experimental and simulated position response

REFERENCES [I] Bamford JM,Sandercock P, Dennis M, Burn J, Warlow C. A prospective study of acute cerebrovascular disease in the community: the Oxfordshire Community Stroke Project 1981-1986. Incidence, case fatality rates and overall outcome at one year of cerebral infarction, intracerebral haemorrhage and subarachnoid haemorrhage. Journal of Neurol, Neurosurgery and Psychiatry, 1990; 54:16-22. [2] Krebs H, Hogan N, Aisen M, Volpe, B. Robot-Aided Neurorehabilitation. IEEE transactions on rehabilitation engineering, 1998; 6( 1):75-87. [3] Buckley MA, Yardley A, Platts RGS, Marchese SS. MULOS system prototype. Web page. http://www.ncl.ac.uk/crest/Prototype.htm#specification; [4] Nagai K, Nakanishi I, Hanafusa H. Development of an 8 DOF Robotic orthosis for assisting human upper limb motion. Proceedings of the IEEE international conference on robotics and automation, Belgium 1998; 3486:3491. [5] Gabermann M. Near frictionless air cylinders provide precision pneumatic motion control system. Power conversion and intelligent motion, 1995; 21(11):48-51. [6] Shearer JL. Study of pneumatic processes in the continuous control of motion with compressed air 1,II. Transactions of ASME,1956; Feb:233-249.

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[7] Lui S, Bobrow JE. An analysis of a pneumatic servo system and its application to a computer-controlled robot. Joumal of dynamic systems, measurement, and control, transactions of the ASME, 1988; 110:228-235. [SI McDonnell B. Modelling, Identification, and control of a pneumatically actuated robotic manipulator. University of California, Irvine, Department of Mechanical and Aerospace EngineeringJ996; Ph.D. thesis. [9] Backe W, Ohligschlager 0. A model of heat transfer in pneumatic chambers. Journal of Fluid Control, 1989; 20(1):61-78. [lo] Wong PJ, Moore PR. Acceleration characteristics of a servo controlled pneumatic cylinder. ASME fluid power systems technology, 1996; 3:119-130. [111 Kawanaka H, Hanada K. Many points positioning of pneumatic cylinder for a vertical axis actuator using two degrees of freedom PI control method. Proceedings of the 4th international workshop on advanced motion control, 1996; Japan 18-21 March 1:71-74. [12] White CJ, Schneifer AM, Brogan W. Robotic orthosis for stroke patient rehabilitaition. Proceedings IEEE 15th annual international conference on engineering in medicine and biology society, 1993 San Diego, 28-31 Oct; 15(3):1272-1273. [ 131 Austin ME, Cozens JA, Seedhom BB. Motion control for robotic physiotherapy. Movic 98', Switzerland 1998; 3:1203-1206. [14] Richardson R, Austin ME, Plummer AR. Development of a physiotherapy robot. Proceedings of the international biomechatronics workshop, 1999; Enshede 19-21 April 116-120. [15] Astrom, K., Wittenmark, B. Computer-Controlled Systems, Theory and Design., Third edition, 1997, ISBN 0-13-314899-8 [16] Gross, D.C., Rathan, K.S. A feedforward MNN controller for pneumatic cylinder trajectory tracking control. Proceedings international conference on neural networks, 1997,2:794-799. [17] Richardson R, Plummer AR, Brown MD, Self-tuning control of a low friction pneumatic actuator under the influence of gravity. IEEE Transactions on , Volume: 9 Issue: 2 ,March 2001. [ 181 Richardson R. Actuation and control for robotic physiotherapy. Phd thesis. School of Mechanical Eng. , University of Leeds, UK, 2001.

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Transient dynamic behaviour of deep-groove ball bearings W W SUM, E J WILLIAMS, and S McWlLLlAM School of Mechanical, Materials, Manufacturing Engineering, and Management, University of

Nottingham, U K D R ASHMORE

Rolls-Royce plc, Derby, U K

ABSTRACT The dynamic behaviour of a ball bearing is modelled and solved using a time-stepping solution. The dynamic models are used to simulate critical transient conditions within the ball bearing. The results show the limitations of the conventional quasi static methods. The gyroscopic effect on the bearing cage motion is also simulated and is shown to be a potentially significant factor in the cage dynamics. NOTATION mass cylindrical coordinates locating cage and ball centre forces resolved in axial, circumferential and radial directions moments resolved in azimuth frame moments resolved in body fixed frame mass moment of inertia of ball mass moment of inertia of cage in body fixed frame angular velocity of ball in azimuth frame angular velocity of cage in body fixed frame angular acceleration of cage in body fixed frame

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z

h U

P FD rb

V cd

P Subscripts b C

shear stress at elastohydrodynamic contact lubricant film thickness lubricant velocity inclination angle of rotational axis of ball drag force radius of ball velocity of ball drag coefficient pressure distribution at cage-inner race hydrodynamic interface ball cage

1. INTRODUCTION

Computer modelling of the dynamics of rolling element bearings has advanced in sophistication since Jones and Harris [ 1, 21 introduced the quasi-static model. Essentially, the quasi-static approach is a force-balance, equilibrium-type analysis. While the quasi-static model is useful for determining steady-state values for a given loading on the bearing, in practice it has been established that most bearing seizures are caused by transient instabilities such as ball skidding [3], cage lap [6],and ball - cage-pocket hammer. There are two major limitations in the quasi-static model. Firstly, this approach does not include the cage. Hence many of the critical transient instabilities involving cage-ball or cage-raceway interaction cannot be analysed. Secondly, it has to employ a raceway control condition to artificially balance the gyroscopic moment acting on each ball [4]. It has been shown that while outer raceway control is approximated in lightly-loaded high speed bearings, it is flawed in many other cases especially in gas turbine engine applications where the applied thrust loads are significantly higher. This paper presents two dynamic models: (1) a ball-raceway dynamic model, and (2) a cageraceway hydrodynamic model, which enable most of the aforementioned instabilities to be analysed. The ball-raceway interaction is established by releasing the positional constraint on the balls and tracking the ball motion around the bearing. The equations of motion are solved using the Runge-Kutta-Merson time-stepping solution. Only four degrees of freedom for each ball is considered. Equilibrium constraints are imposed on the other two degrees of freedom by utilizing values obtained from the quasi-static analysis. With this model, the influence of the applied axial load on ball skidding is investigated. The ball axis rotational axis angle is also compared against the outer-raceway control method. The results presented indicate that improved accuracy is achieved using a fully dynamic analysis. The cage-raceway hydrodynamic support for an inner-piloted cage is modelled by solving the generalised Reynolds equation in conjunction with the continuity equation to calculate the pressure distribution and extent of the oil film. The inclusion of the continuity equation provides the model with the ability to simulate all conditions between fully flooded and highly starved. The resultant forces are incorporated in the general equations of motion for the cage and solved using the Runge-Kutta-Merson method. In particular the developed model is used to investigate the influence of the gyroscopic term and comparisons are made to a model which neglects the gyroscopic term [ 6 ] .

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2.

BALL-RACEWAY DYNAMIC MODEL

Figure 1: Releasing positional constraint of balls

Figure 1 shows the ball motion described in the azimuth and inertial reference frames. Ball motion is considered in four degrees of freedom where the translational motion of the masscentre is governed by:

This equation governs the orbital motion of the ball around the bearing in the inertial frame. Equilibrium constraints are imposed on the axial and radial motions of the mass-centre and these values are obtained from the quasi static analysis. This simplification is justified as the high-frequency axial and radial motions are relatively insignificant compared to the orbital motion. The rotational motion of the ball is considered in an azimuth frame and the governing equations of motion are given by:

I b h , - I , w ,=~G,

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(3)

499

The traction model used to establish the friction and lubricant characteristics at the elastohydrodynamic (EHD) contact is that developed by Kannel and Walowit [7] such that:

where

A,

=

p p ' p

uyu,-- 24,)' 8K

; u is the lubricant velocity; p is the lubricant

viscosity; h is the lubricant film thickness; and K is the thermal conductivity. In addition, the equation used to model the thickness of the EHD film at the contact is that developed by Hamrock and Dowson [8]:

Q

U

where W = -, Q is the applied load, and k = - is the ellipticity ratio, where u and b E'R b are the semimajor and minor axes of the contact ellipse. U and G are the nondimensionalised speed and elasticity parameters respectively. The slip velocities between the ball and raceways, and the normal loads from the Quasi Static analysis are fed into the traction model. The resultant forces and moments, from the integration of the traction forces/moments over the contact ellipse, are then included in the equations of motion and solved in a time-stepping routine using the Runge-Kutta-Merson method.

3.

RESULTS FROM THE BALL-RACEWAY MODEL

3.1

Inclination angle of rotational axis of ball

It has been mentioned that, due to the raceway control assumption, considerable doubt has been raised about the validity of some of the results from the quasi static approach. One of the parameters which can be used to illustrate this is the ball-rotational-axis inclination angle p as defined in Figure 1. A comparison (see Figure 2) of the ball-rotational-axis inclination angle from the Ball Dynamics Program against values obtained from the conventional Quasi Static Analysis indicates that, while at lower thrust loads the values tend to approximate that of outer raceway control (ORC), there is considerable deviation at higher thrust loads, highlighting the limitations of the conventional approach.

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I

Comparison of Ball-Rotational-Axis Inclination Angle 35 30 25 20 15 10 5 0

--b Dynamic

+ORC

0

50000

100000

150000

200000

Thrust Load (N)

Figure 2: Comparison of inclination angle of ball rotational axis

3.2

Analysis of Ball Skidding

With the developed ball-raceway dynamic model, investigations were carried out on the effects of the applied axial load on baN skidding, which is defined as the gross sliding motion of the ball over the raceway. This condition is often associated with premature bearing failure as there is significant heat generation and metallic interaction at the contacts. The most significant parameter in the investigation of ball skidding is the drag force acting on the ball. In this model, the ‘ball in an oily wind tunnel’ representation of the drag forces is used, such that: 1 F - - C , p , V 2A (7) D-2 1 A = - 4 2 r b)’

4

In the drag force expression above, the effective density of the oil-air mixture, p, is difficult to estimate. According to Rumbarger [9], “high speed bearings never are completely flooded with lubricant, and seldom are more than 15 to 20 percent full of oil within the bearing for very high speed operation”. Here these suggested values are used as an initial estimate. From a series of runs from the ball-raceway program, the cage to shaft speed ratio is plotted against different thrust loads. Intuitively, the cage to shaft speed ratio would taper off below a certain minimum axial load. Below this minimum load, excessive sliding (skidding) occurs. Figure 3 shows a comparison of the results obtained using the dynamic program against experimental results from Poplawski [SI. It can be seen that the results from the dynamic program show reasonably good correlation with the published experimental results from the literature. In this particular case, the ball dynamics program predicted the onset of skidding more accurately than the methods employed by Harris [3]. Harris predicted the onset of

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skidding to be approximately 240 Ib, whereas the dynamics program gives a value of approximately 180 Ib. Comparison of ball skidding trends against experimental results from Mauriello [IO] also shows good agreement. These results emphasize the usefulness of the developed model as an efficient design tool for minimizing ball skidding.

CagelShaft Speed VS Thrust 27600 rpm, pe=SO, cd=0.46

0.44

fn

0

0.4

tExperimen

0.34

0

100

200

300

400

Thrust (LE)

Figure 3: Comparison of ball skidding results with experimental data

4.

CAGE HYDRODYNAMIC MODEL

4.1

Basis of Hydrodynamic Model

Most of the hydrodynamic cage-support interactions used previously have been simplified by considering fully-flooded conditions. The simplifications made were justified on the basis that the fluid film forces tend to be much less than the other interaction forces acting on the bearing, and hence a detailed model of the oil film is not necessary. However, in recent years, the phenomenon of cage lap has become a major cause of concern in the bearings industry and forms one of the motivations for the current study in which the hydrodynamic support for an inner-piloted cage incorporating starved-film conditions is considered. The Cage Hydrodynamic Program presented here is based on the model proposed by Ashmore et al. [6]. The model neglects the rolling elements and considers a cage with an imposed (constant) angular velocity, rotating around an inner race with oil films supporting the cage at the two lands of the inner race. The oil film is fed into the system through the inner race to each land through oil feeds at circumferentially equidistant positions forming separate films of varying extents that support the cage. The model is developed by assuming that the bearing is equivalent to two rigidly connected short journal bearings. By judiciously applying certain simplifying assumptions to the Reynolds equation, an analytical solution for the pressure distribution in each oil film can be established:

P = -[QcEsin(O 6P h’

502

- a)- mh

(9)

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The pressure distribution for each oil film can be integrated to give the forces experienced by the cage. The generalised Reynolds equation is then solved for each film in conjunction with the continuity equation, and enables starvation effects to be investigated.

4.2

Inclusion of gyroscopic effects

One of the simplifications made in the hydrodynamic model [6] is the omission of the gyroscopic term in the analysis. The Cage Hydrodynamics Program presented here improves upon the original model by including the effects of the gyroscopic moment on the cage dynamics. This is achieved by considering the cage motion in two reference frames as follows:

mc(r,B,

+ 2 i c B , ) = Fee

(1 1)

m c ( f c - r,B,‘ 2 ) = F ,

I,a, = GI + ( I ,

-

13)w2w3

I,a, = G, + ( I , - I , ) o , w ,

(13) (14)

where equations (10) to (12) use an inertial frame to define the translational motion of the mass-centre of the cage and equations (13) to (15) use a body fixed frame to define the rotational motion of the cage.

5.

RESULTS FROM THE PROPOSED CAGE HYDRODYNAMIC MODEL

Figure 4 (a) shows a plot of the position of the cage mass-centre neglecting gyroscopic effects at a cage speed of 550 rads, inner race speed of 1100 rads and a fully flooded condition. Figure 4(b) shows the angle of misalignment, b and g of the cage in the two different planes for the same situation.

Y 004/009/2004

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-

X-Y eccentricity of maan centre of cage capsped

Angle of Misalignment

5 5 3 m d s . i m r m e = llO'Jm@Jy~

Cage Speed = 550 rads, Inner race = 1100 rads

0.01

,

1

I

4.01 I

I '

I

1

Time fmsl

Figure 4: Results from Ashmore's model (omitting gyroscopic effects)

Figure 4(a) shows the mass-centre of the cage whirling out to a large eccentricity initially, but quickly settling into a much smaller eccentricity which is close to zero. Figure 4(b) indicates that both edges of the cage are in constant contact with the inner race once the steady-state condition is reached. This type of cage motion is known as 'coning'. These plots show that, even for a fully-flooded condition, the hydrodynamic film is incapable of supporting the cage. This same case has been considered with gyroscopic effects included and the results are plotted in Figure 5. The plots indicate that the gyroscopic effect appears to have a stabilizing effect on the cage motion. As a result of this, the mass-centre of the cage whirls at a much higher eccentricity (e = 0.66), but the angle of misalignment is reduced by about 25%. Thus, rather than constantly being in contact with the inner race, the edges of the cage exhibit periodic contact with the inner race. These results indicate the importance of including the gyroscopic effects in the cage dynamics.

Angle of mlrnllgnmsnt Cage Speed = 550 rad/%Inner nee = 1100 nus 0 WE03

; 6WE03 4WEW

-

.o"

2WE43

m-

.=~oooE+w E C 5 2WE.03 2 4WE03 cp

2

6WE03

4WE03 lime (ms)

(b) Figure 5: Results from the Cage Hydrodynamic Program (with gyroscopic effects)

504

Y004/009/2004

6.

CONCLUSIONS

Two detailed dynamic models that simulate the dynamic behaviour of a deep-groove ball bearing have been presented: 1. The Ball Dynamics Program provides a solution which is superior in accuracy to previous methods where the Raceway Control assumption had to be employed. The program enables ball skidding to be analysed and the minimum axial load to prevent skidding to be established. 2. The Cage Hydrodynamic Program, which is an improved version of a previous model [6], incorporates the gyroscopic term into the dynamic analysis. The gyroscopic moment appears to have a stabilizing effect on the cage motion. Rather than going into an extreme coning motion where both edges of the cage are in constant contact with the inner race (indicating no support from the hydrodynamic film), the cage still exhibits coning motion, but with a smaller misalignment and a higher eccentricity. The new results indicate that the gyroscopic effect is not negligible, but has the potential to alter the cage motion significantly. The aim of ongoing work is to integrate the two programs to enable transient conditions such as ball-pocket hammer to be evaluated. ACKNOWLEDGEMENTS The authors wish to thank Rolls-Royce plc, Aerospace Group, for their financial support of the research, which is being carried out at the University Technology Centre in Gas Turbine Transmissions Systems at the University of Nottingham. The views expressed within this paper are those of the authors and not necessarily those of Rolls Royce plc, Aerospace Group. REFERENCES 1. Jones, A.B., “Ball Motion and Sliding Friction in Ball Bearings”, Journal of Basic Engineering, Transactions ASME, D, Vol. 81, No.1, Mar. 1959 2. Harris, T.A., “Ball Motion in Thrust Loaded, Angular Contact Bearings with Coulomb Friction”, Journal of Lubrication Technology, Jan 1971 3. Harris, T.A., “An Analytical Method to Predict Skidding in Thrust-Loaded, AngularContact Ball Bearings”, Journal of Lubrication Technology, Jan 1971, pp 17-24 4. Harris, T.A., “Rolling Bearing Analysis”, 2”dEd., John Wiley 1984 5. Poplawski, J., and Mauriello, J., “Skidding in Lightly Loaded, High Speed Ball Thrust Bearings,” ASME Paper 69-LubS-20 6. Ashmore, D., Williams, E.J., McWilliam, S., “Hydrodynamic support and dynamic response for an inner-piloted bearing cage”, Proc. Instn Mech. Engrs Vol. 217 Part G: Journal of Aerospace Engineering, 2003 7. Kannel, J.W. and Walowit, J.A., “A Simplified Analysis for Traction Between RollingSliding Elastohydrodynamic Contacts”, Journal of Lubrication Technology, 93: 39-46, 1971

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8. Hamrock,B.J., Dowson,D., “Ball Bearing Lubrication: The Elastohydrodynamics of Elliptical Contacts”, John Wiley & Sons, Inc., 1981 9. Rumbarger,J.H.,Filetti,E.G.,andGubemick,D., “Gas Turbine Engine Main Shaft Roller Bearing System Analysis”, Journal of Lubrication Technology, ASME Trans., 95: 401416,1973 10. Mauriello,J.A., Lagasse,N., Jones,A.B., and Murray,W., “Rolling Element Bearing Retainer Analysis”, USAAMRDL Tech Report No. 72-45, Nov 1973

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Determination of the effect of contact kinematics of squeeze caving phenomenon through general covariance M KUSHWAHA and H RAHNEJAT Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, UK

Abstract The paper describes impulsive action causes by an imposed stop-start type motion in gaps of the order of tenths of a micrometer. This impulsive action often causes the emergence of localized deformation in the form of a dimple, referred to as a squeeze cave. This phenomenon is more pronounced and better understood under impacting conditions, caused by a falling ellipsoidal solid body upon a flat surface. Its emergence under relative motion of contiguous solids such as a loaded roller along a flat race, however, has been attributed to a host of phenomena, like thermal wedging effect or prevailing kinematic conditions. It is noted that stop-start motion is equivalent to inertial dynamics of a falling object, which in turn is the same as gravitational action by the virtue of the equivalence principle. Thus, with constrained vertical inertial dynamics of the loaded roller against the flat race, the emergence of the squeeze cave must be equivalent to the introduction of a micro-gravity effect. The proof of this hypothesis gives further credence to the importance of kinematic conditions, being responsible for the mechanism of dimple formation. Using hyperbolic geometry, proper time parameterization of the cave roof motion, and the principle of general covariance, the paper describes the complex motion of this point. These findings conform to the observations of motion and kinematic behaviour arrived at by an independent detailed analysis of the lubricated contact under transient conditions. Nomenclature

h h,

: Contact half-length : Proper acceleration of cave roof : Hertzian half-contact width : Reduced modulus of elasticity : Film thickness : Geometrical profile of the roller

ho

: Initial guess for central film thickness

a A 6 E'

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507

[JI

: Dimensionless Jacobian matrix

m,ny

: Number of grid points in x and y directions respectively : Pressure : Maximum Hertzian pressure : Reduced radius of non-conforming contact : Arc length : Time : Speed of entraining motion : Velocity of cave roof in the inertial frame of reference : Proper velocity of cave roof in the accelerated frame of reference : Load on roller : Cartesian co-ordinates

P p* R S

t Y” V

V W x, Y

Greek symbols 6, 60

: Elastic deflection

Y

: Scalar acceleration : Absolute and atmospheric viscosity respectively : World-curve : Density : Proper time : Spatial co-ordinates : Accelerated frame of reference

7 l J O

K

P z

4’ 6

: Central elastic deflection

Superscripts J : Jacobian R : Residual value Subscripts XY Y

: Define directions along or about the respective axes.

1- Introduction

Many machine elements are often subjected to acceleration motions. Away from any significant gravitation field, these deviations from uniform motion often account for the main source of divergence from idealized functions, such as micro-gravity effect in satellite thruster mechanisms in the stratosphere. The same conditions also occur in curvilinear motions at high speeds such as in motor sport, where centrihgal accelerations of the order of a few g are not unexpected. Other induced gravitational action include impact dynamic phenomenon at high speeds such as in the collision in modern motor vehicles: typically several to few tens of g. Whilst in few instances the introduction of such induced gravitational fields are desired, as in the lift-off of the space shuttle, in most other cases, their effects are largely unwanted, such as whiplash in rear collisions. For most machinery, load bearing and transmitting members are used to attenuate the effect of such impulsive actions, such as through the use of bearings with suitably lubricated contacts.

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Acceleration-deceleration or stop-start motions in lubricated contacts correspond to the introduction of impulsive actions by inertial dynamics, normal to the plane of contact. This is equivalent to impacting conditions. When the contiguous bodies are constrained not to separate by rigid body motion, then the increased lubricant reaction under deceleration (a diminishing gap due to reduced lubricant entrainment) causes localized squeeze caving of elastic contiguous surfaces by the prevailing conditions. This is analogous to gravitational action other than that which would have resulted fiom their unconstrained vertical inertial dynamics by the virtue of the equivalence principle. These kinematic conditions have been investigated, but not fundamentally understood. This paper attempts to bridge the prevailing gap in the knowledge in this respect.

2- Theoretical Formulation A simultaneous solution of Reynolds’ equation for transient conditions, the elastic film shape, lubricant rheology, and the speed distribution in the rolling bearing of the nominated roller is required.

2.1- Reynolds’ Equation The Reynolds’ equation in its dimensionless form for transient condition is given as:

where the non-dimensional terms are given as:

The ultimate term in the equation can be simplified as:

2.2- Contact condition The contact of a roller against the race may be approximated by an equivalent roller of reduced radius R near a semi-infinite elastic half-space. Except for a number of analyses, most others have considered the counterformal contact to be of an infinite line configuration. The reason behind this assumption is that under lubricated conditions, the lubricant side flow (Le. the side leakage) is ignored, thus making the problem one-dimensional. This, however, does not conform to practical situations, as no element in contact has an infinite dimension. Therefore, the contact width, transverse to the flow direction, should be made finite, particularly that due to the abrupt changes in the vicinity of roller extremities large pressure

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spikes are generated as shown in [ l , 21, inhibiting the flow of lubricant into these regions, yielding the absolute minimum film thickness, as shown later on. The analysis, with finite equivalent roller length is termed finite line contact. Since an equivalent system is used, where the roller is considered to be in contact with an elastic half-space, only the deformation of the elastic surface is considered.

2.2.1- The Elastic Film Shape Assuming the undeformed profile of the roller to be parabolic, the lubricant film thickness at any location within the contact domain can be expressed as (see figure 1):

By the principle ofsuperposition, the total elastic deflection at node (k, l ) can be formulated as [3]:

where: m = l k - i + l ) and n = l l - j + l l and the influence coefficient matrix Dm.n is given in reference [4].

2.3- Lubricant Rheology Roelands [5] derived an expression for the lubricant viscosity variation with pressure for mineral oils under isothermal conditions as:

where, 3 = 5 . 1 ~ 1 0m2/N. ~ ~ It must be noted that in the Roelands’ equation, the lubricant viscosity is defined, using three different parameters (Le. the atmospheric viscosity ( qo), the and the pressure-viscosity index (% =0.67)). asymptotic iso-viscous pressures The variation of lubricant with pressure is given by Dowson and Higginson [6] as: 9FPh jT=l+--(7) 1+ pFPh where 9 and pare constants, dependent upon the properties of the fluid. The values used in ~ 1.68xlO-’ respectively, which are based on mineral the current analysis are 5 . 8 3 ~ 1 0 - ’and oils.

51 0

Y004/040/2004

3- Method of Solution The solution is obtained by Effective Influence Newton-Raphson (EM) method. Reynolds’ equation is represented in the following form:

The

The right-hand side of the above equation is the Reynolds’ equation itself, referred to as the residual function. The Jacobian matrix in equation (8) contains the partial derivatives of the flow induced terms in the Reynolds’ equation in terms of pressure as:

[4,,k1)] =A ” . The “k

,I

elements of Jacobian matrix and the residual function are based upon point-wise discretisation in a constructed mesh. This form of discretisation uses 5 Jacobian terms and their typical form can be found in reference [4]. The system state equations are then solved using Gauss-Seidel iteration and EIN method. The density and viscosity of the fluid also alter at each computational node. Their variations with pressure in Jacobian form can also be found in reference [4].

4- Boundary conditions The boundary conditions employed are: 1)

At the boundaries of the rectangular computational - zone-the pressures are zero. Thus, for a mesh of 129 by 109; = &,) = = qI,,)= 0

2)

To avoid the generation of negative pressures in the fluid film, at the cavitation

6,))

boundary, the Reynolds’ Numerically,

condition:

-

P

=

aF aF ax =

dY = 0 whenever the pressure is negative.

=0

is employed.

4.1- Computation domain Once the dimensions of the elastostatic footprint for a particular load and geometry are calculated, a domain; which is generally larger than the contact area is identified. Assuming 1 the entraining velocity vector; u,, = -(u, + u 2 ) to be in the X direction, a regular mesh with 2 finite nodal points is constructed over the rectangular computational zone. As shown in the figure, the size of this zone is such that it is about 4.5 to 5.0 times the elastostatic footprint half-width in the inlet region of the lubricated conjunction, and about 1.5 to 1.6 times that at the outlet. Along the axial length of the roller (Le. in the Y direction), the zone is extended at both sides to about 4% to 8% of the elastostatic footprint half-length. The percentage extension of the boundary can be varied to obtain the best starting position of the pressure curve in the axial direction.

Y004/040/2004

51 1

4.2- Convergence criteria The solution to equation (8) can be obtained, using an error tolerance with the NewtonRaphson method. If a solution obtained is within the limits of the required tolerance, then the numerical procedure is deemed to have converged. The two convergence criteria required are: 1. The convergence of generated pressures to evaluate the carried load given as:

If convergence is not achieved, the pressures are updated as:

where, A is the under-relaxation factor.

2. The convergence of the integrated pressure distribution providing the calculated load with the contact/applied load. The integrated pressure in the dimensionless form is given as:

The load convergence criterion, which is regarded as the constraint function in this transient analysis is:

If the load has not converged, then the central lubricant film thickness is adjusted according to the evaluated unbalanced load, using the following relationship:

where, q is a damping coefficient used to reduce the sharp variations in the iterative values of the central oil film thickness.

5- Results and Discussion A simulation study is carried out for a Steel roller of modulus of elasticity 211 GPa and Poisson's ratio 0.3 against a rigid flat race. The roller length is 0.0127 m, with an applied constant bearing load of 1500N. The roller is blended at its ends by dub-off radii of 0.0762 m

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in order to reduce the generated contact edge pressure spikes caused by geometric discontinuities [3].The length of the flat land in the axial profile of the roller is 0.0076 m. The roller is subjected to deceleration-acceleration motion as shown in figure 2, which is analogous to equivalent gravitational action. Thus, the lubricant film is subjected to an equivalent impact. During the decelerated motion, the speed of lubricant entrainment decreases and the two contiguous surfaces approach each other. The lubricant reaction ordinarily increases with diminishing gap and the inertial balance should alter in the vertical direction. The opposite effect should be observed during the acceleration phase. However, the load balance condition acts as a constraint equation here, and as the result the inertial dynamics along the contact normal is prohibited. This causes the contiguous surfaces to deform in order to account for the increased lubricant pressure during approach of the bodies. The phenomenon observed is the formation of a dimple in the film shape, which is entirely due to prevailing kinematic conditions, and has aroused considerable interest. The lubricant used was a mineral oil of dynamic viscosity 0.041 1 Pas at atmospheric pressure and a piezo-viscosity index, a =13.3 GPa-'. The spatial mesh density for the contact domain was set at 129 by 109 with the former in the direction of entraining motion. The time step of transient analysis was 0.5 ms,making 36 time steps of simulation. Figure 2 also shows the entraining velocity of the lubricant into the conjunction. Note that under decelerated motion the entraining velocity decreases and the bodies will mutually converge, and, the lubricant pressure will increase accordingly. In such an elastohydrodynamic condition the film thickness is almost insensitive to load, and the action of this load is merely a constraining effect against vertical inertial dynamics. Consequently, as the acceleration phase commences, separation of bodies in a rigid sense is inhibited. The sudden change of state of motion and lubricant entrainment causes a localised micro-gravity effect (analogous to impulsive action), causing squeeze caving effect. This is evident in a series of elastic film shape plots in figure 3 under transient conditions. These plots correspond to the indicated points on the entraining velocity graph in figure 2. The squeeze cave, also referred to as the dimple, is evident almost from the outset of the analysis due to the high applied load of 1500N (see figure 3). Its depth variation with time is shown in figure 4. Its position alters in time, reaching a maximum depth approximately at the vicinity of motion reversal from a deceleration mode to that of acceleration (Le. where the surface of the roller undergoes an impulsive action). Here, the sudden change in the depth of cave is evident with significant change in kinematic conditions. In the finite line contact analysis, the position of the dimple also changes in the contact footprint. This is shown in figure 5, where the ordinate is the spatial location of the roof of the dimple. Local spatial incremental change as observed from the usual inertial frame of reference is given as: As =.-/, This represents the arc length of motion as an increment along the curve in Euclidean geometry. However, the actual arc length of motion in the accelerated frame indicated by the co-ordinate system (T,<),< = { ' , i = 1+ 3 (spatial co-ordinates, here aligned with, s) is given by A T , which is the proper time of movement of the cave roof through the contact according to the general theory To of relativity and in non-Euclidean geometry. In hyperbolic geometry: dt = JG'.

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obtain the contact kinematic conditions, one should determine the scalar acceleration of the cave roof, y(r)which corresponds to the curvature of deformation as a function of s in Euclidean geometry

[2).

The world-curve of the cave roof as a material point (Le. its

locus in time) in the frame of reference (r,<)can be considered as: ~ ( z ) .Thus, its scalar acceleration is obtained as: y (z) = llA(r)11,where: A ( r )= V' ( r )= K' ( r ),where:

):,:(

V(r)=Ic'(r)= - - =( d t -d s d t ) =-(Idt dz'dt dr dr The term

'4

(14)

dt is the local time dilation factor, which is expected to be very near unity for the

dr problem at hand. Since, the magnitude of V ( r )is bounded by unity (here representing the speed of light) according to special relativity, then it follows that: dr = -.Note that the dr relationship here is given in geometric unit (i.e. v = l for speed of light in vacuo). Thus:

47

A ( r ) =-d V ( r ) =--d dr dr dtdr

4

='(")'{l,v} dt dz

i

2v

l+v2 l d v

Therefore:

514

Y004/040/2004

Where:

dv d 2 s = - is the acceleration of the cave roof within the inertial frame of reference

dt dt2 ( t , s ),s = x, y, h , or in other words, the local micro-gravity, s‘ . Thus: y ( r )= s’ . Note that micro-gravity is described by the local curvature of the cave roof at any instant of time. Clearly, the motion of the cave roof in all the three spatial dimensions indicates the local deformation of the elastic solid as being equivalent to a small gravitational field. If for simplicity of demonstration one assumes the micro-gravitational field to be one dimensional in the direction 4’ of the fixed co-ordinate frame of reference (r,<) in which the world-curve K(r)resides, then by the principle of general co-variance, one can obtain the relativistic equation of motion for the cave roof (i.e. localized deformation of the elastic solid) by assuming the frame of reference (t,s)to fall freely in space with respect to(r,<), and map the results back to the latter. Thus: d2r d r d4’ -2y(r)-dt dt dt

-=

dr Note that for this case: - 1 , thus, the above equations simplify to: dt d2r dt

-= -2y(r)-

e’ dt

In Newtonian space, the first of the above equations yields a zero value and the second d24’ equation yields: -= -y ( r ). This confirms the hypothesis that the mechanism of dimple dt formation is a kinematic phenomenon. The relationship (16) shows that for small values of v, which is the case (compared to speed of light), the scalar acceleration of the cave roof is very small compared to the imposed ah accelerated motion. This can be seen in figure 6 , which shows the squeeze film velocity, at

and its derivative,

a2h a24’

-oc -(for the assumption made here) at the transient location of the

at2 atz squeeze cave. Note that these values are 2-3 orders of magnitude smaller than the imposed inertial motion, shown in figure 2.

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6- References

[ 11 Kushwaha, M., Rahnejat, H. and Gohar, R., “Aligned and misaligned contacts of rollers to races in elastohydrodynamic finite line conjunctions”, Proc. Inst. Mech. Engrs., Part C: J. Mech. Engng. Sci., Vol. 216,2002, pp. 1051-1070. [2] Mostofi, A. and Gohar, R., “Elastohydrodynamic lubrication of finite line contacts”, Trans. ASME, J. Lub. Tech., Vol. 105, 1983, pp. 598-604. [3] Johnson, K. L., Contact Mechanics, Cambridge University Press, 1985. [4] Kushwaha, M. and Rahnejat, H., “Transient elastohydrodynamic lubrication of finite line conjunction of cam to follower concentrated contact”, Journal of Physics D: Applied Physics, 35,22nd October 2002, pp 2872-2890. [SI Roelands, C. J. A., Correlation Aspects of Viscosity-Temperature-Pressure Relationship of Lubricating Oils, Ph.D. Thesis, Delft University of Technology, The Netherlands, 1966. [6] Dowson, D. and Higginson, G. R., “A numerical solution to a elastohydrodynamic problem”, J. Mech. Engng. Sci., Vol. 1, 1959, pp. 6-15. Deformed

, Rigid

Figure 1: Determining the elastic film shape

516

Y004/040/2004

2.58+03

2-

2.OE+03

1.n -

1.5EI03

1.n 1.4.

j z

!

10E*03 5.OEt02

1

1.2 -

O.OE+W

1

1-5.OE102

0,s -1.OE+03

06. 0.4

-

0.2

-

1

.1.5E+03 -2 OE+03 LlnvI

0 O.OE+OO

2.OEM

4.OEM

EOE-M

S.OEM

1.OE-03 12E-03 Tlm (swondl)

1.4E-03

1 BE-03

i.EE-03

2.

.2 5E.03 13

Figure 2: Imposed motion 1

08

on

i

O7

5 on

-E Y

1

O5

Io4

!

03 02

oi 0 80

7.0

no

9.0

10.0

11 0

12 0

13.0

N o n . D l m ~ l o XQlnct(0n ~l

Figure 3: Transient elastic film shape in the entraining direction (Dashed lines indicate decelerative mode

Y004/040/2004

517

4.5E-07

6 OE-07

3 5E-07

-

E

3 OE-07

I

25E47

20E-07

15E47

1 OE-07

5.0E48

O.OE+W 0.0

~

0

2 OEM

4 OEM

0 OEM

8 OEM

1 OE43

12143

1OEm

43

n miutom)

Figure 4: Transient depth of squeeze cave 4.5143

30E.03

i a

e Z~E-O~

'6 '6

[

20E-03

f

15E43

l.OE-03

5 OE-W

O.OE+W 0 OE+W

L

O.OE+W 0 OEtW

2 OEM

4 OE-04

0 OE-M

8 OEM

Tlm. t iwconds)

Figure 5 : Space-time location of the cave roof

518

Y 004/040/2004

20

2 OE43

15E43

r

10

1 OE43

0

-

-

1

0.c

K)

B OE-M

1 OE-03

8 OE

1

03

5 OE44 -10

OOEIW

.i jz2

.20

I

-5 OE4d

/

-30

1

-1 OE-03

/ 40

.1 5E-03

-50

-2 OE43

Tlm (second.)

Figure 6: Local squeeze velocity and surface film acceleration at the cave roof

Y004/040/2004

519

Authors’ Index

Index Terms

Links

A Abdul-Ameer, A A

475

Antonya, Cs

111

Ashmore, D R

497

137

B Barbedette, B

351

Basu, B

147

Bell, D

311

Bertsche, B

323

Bhakta, B

485

Biermann, J W

339

361

Broderick, B M

147

155

Brooks, P

311

Brown, M

485

Brown, N

461

155

C Coltrona, M L

287

Crolla, D A

391

Cuadrado, J

99

447

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

D Dai, L

123

Davis, G

311

Dietz, S

73

Doğan, S N

323

Dong, M

415

Dornhege, J

439

Drew, S J

243

415

E Eichberger, A

73

F Foellinger, H

207

Frezza, R

401

Fritzson, D

57

83

91

G Germain, G M

51

Gnanakumarr, M

373

Goudas, I

231

Gutierrez, R Guzzomi, A L

99 243

H Hamidzadeh, H R

175

Han, Q

123

Hynek, P

461

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

J Jackson, M

461

K K’nevez, J-Y

351

Karlsson, U

255

Kazantzis, M

429

Kelly, P

361

Kerkkänen, K

191

Khanipour, A

475

Kushwaha, M

507

L Leavitt, A

311

Levesley, M

391

Liu, A

123

Loibnegger, B

255

Lu, C H

311

LUO. A C J

163

485

M Ma, M T

255

Mariot, J-P

351

McWilliam, S

497

Menday, M

373

Metallidis, P

231

Mikkola, A

191

Minen, D

401

Moughith, W S E

475

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

Moxey, L C

175

Murtagh, P J

147

155

Natsiavas, S

231

429

Naya, M A

447

N

O Offner, G

255

Okuma, M

51

Orlandea, N V

31

Ortmann, C

401

P Papadimitriou, C

429

Parkin, R

461

Parkin-Moore, D

311

Pownall, M

391

Priebsch, H H

255

R Rahnejat, H

273

Rainer, G

221

Ramli, R

391

Richardson, R C

485

Rothberg, S J

273

Ryborz, J

323

373

507

This page has been reformatted by Knovel to provide easier navigation.

Index Terms

Links

S Saccon, A

401

Sargeant, M

299

Schiehlen, W

3

Shabana, A A

15

Siemers, A

57

Sopanen, J

191

Stavrakis, I

231

Stone, B J

243

Sum, W W

497

83

91

287

299

T Talaba, D

111

Teodorescu, M

273

Theodossiades, S

373

137

V Verros, G

429

W Wikström, A

255

Williams, E J

497

Wu, J

415

This page has been reformatted by Knovel to provide easier navigation.