Fundamentals of rail vehicle dynamics: guidance and stability

FUNDAMENTALS OF RAIL VEHICLE DYNAMICS

ADVANCES IN ENGINEERING Series Editors: Fai Ma, Department of Mechanical Engineering, University of California, Berkeley, U.S.A. Edwin Kreuzer, Department of Mechanics and Ocean Engineering, Technical University Hamburg-Harburg, Hamburg, Germany

FUNDAMENTALS OF RAIL VEHICLE DYNAMICS GUIDANCE AND STABILITY

A.H. WICKENS Loughborough University, UK

Wickens, A. H. Fundamentals of rail vehicle dynamics : guidance and stability / A.H. Wickens. p. cm. -- (Advances in engineering ; 6) Includes bibliographical references and Index. ISBN 90-265-1946-X 1. Railroads--Cars--Dynamics. I. Title. II. Advances in engineering (Lisse, Netherlands) ;6 TF550.W53 2003 625.2’01’5313--dc21 2003045675

This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: Swets & Zeitlinger Publishers www.swets.nl

ISBN 0-203-97099-3 Master e-book ISBN

ISBN 90 265 1946 X (Print Edition)

Contents Arrangement of Book ..........................................................................................ix Preface ................................................................................................................. xi 1. Basic Concepts.................................................................................................. 1 1.1 Introduction ......................................................................................... 1 1.2 The Railway Wheelset......................................................................... 2 1.3 Creep ................................................................................................... 6 1.4 Stability................................................................................................ 8 1.5 Guidance............................................................................................ 10 1.6 Suspension, Performance and Criteria............................................... 13 1.7 Suspension, Articulation and Curving............................................... 15 References ............................................................................................... 17 2. Equations of Motion ....................................................................................... 19 2.1 Introduction ....................................................................................... 19 2.2 Freedoms and Constraints ................................................................. 19 2.3 Wheel Rail Geometry ........................................................................ 21 2.4 Contact Mechanics ............................................................................ 32 2.4.1 Elasticity and Friction..................................................................... 32 2.4.2 Laws of Friction ............................................................................. 32 2.4.3 Contact Between Wheel and Rail ................................................... 32 2.4.4 Creep .............................................................................................. 33 2.4.4.1 Background ................................................................................. 33 2.4.4.2 Formulation of the Creep Problem .............................................. 34 2.4.4.3 Creep Forces for Small Creepages .............................................. 36 2.4.4.4 Creep Forces for Arbitrary Creepages......................................... 37 2.4.4.5 An Approximate Theory for Arbitrary Creepages....................... 37 2.4.4.6 Heuristic Approximations............................................................ 39 2.4.4.7 Non-Hertzian Effects................................................................... 39 2.4.5 Transient Effects............................................................................. 40 2.5 Creepages .......................................................................................... 40 2.6 Contact Forces ................................................................................... 42 2.7 Kinematics of the Wheelset ............................................................... 44 2.8 Equations of Motion .......................................................................... 46 2.9 Constrained Motion ........................................................................... 48 2.10 Equations of Motion for Small Displacements................................ 52 2.11 Equations of Motion for a Two-Axle Vehicle................................. 61 References ............................................................................................... 66

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CONTENTS

3. Dynamics of the Wheelset .............................................................................. 71 3.1 Introduction ....................................................................................... 71 3.2 The Unrestrained Wheelset ............................................................... 71 3.3 Root Locus for Small Motions of the Restrained Wheelset .............. 76 3.4 Instability and Feedback.................................................................... 81 3.5 Amplitude Dependent Behaviour and Limit Cycles.......................... 84 3.6 Energy Balance.................................................................................. 88 3.7 Dynamic Aspects of Guidance .......................................................... 89 3.8 Alternative Methods of Guidance...................................................... 94 References ............................................................................................. 103 4. Guidance of the Two-Axle Vehicle .............................................................. 107 4.1 Introduction ..................................................................................... 107 4.2 Properties of the Stiffness Matrix .................................................... 109 4.3 Steering on Large Radius Curves .................................................... 111 4.4 Response to Cant Deficiency on Large Radius Curves ................... 116 4.5 The Conflict Between Steering and Stability .................................. 118 4.6 Motion on Sharper Curves............................................................... 121 4.7 Response to Misalignments ............................................................. 127 4.8 Flange Forces and Derailment ......................................................... 130 References ............................................................................................. 131 5. Dynamic Stability of the Two-Axle Vehicle ................................................ 133 5.1 Introduction ..................................................................................... 133 5.2 Equations of Motion ........................................................................ 135 5.3 Stiff and Flexible Vehicles .............................................................. 142 5.4 The Flexible Vehicle with Zero Suspension Damping.................... 147 5.5 Damping and the Long Wheelbase Flexible Vehicle ...................... 151 5.6 Stability of the Flexible Vehicle in General .................................... 153 5.7 The Application of Cross-Bracing and Yaw Relaxation ................. 159 5.8 The Stiff Vehicle or Bogie............................................................... 160 5.9 The Three-Piece Bogie .................................................................... 169 References ............................................................................................. 169 6. The Bogie Vehicle ........................................................................................ 173 1 Introduction ........................................................................................ 173 2 Equations of Motion ........................................................................... 175 3 Dynamics of the Conventional Bogie Vehicle ................................... 177 4 Steering and Stability of Multi-Axle Vehicles in General.................. 182 5 Steering and Stability of a Generic Bogie Vehicle ............................. 185 6 Application to Specific Configurations .............................................. 190 7 Stability of Bogie Vehicles with Steered Wheelsets .......................... 197 8 Simple Bogie Model........................................................................... 199 9 Stability of Simple Bogie Model ........................................................ 201 References ............................................................................................. 206

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vii

7. The Three-Axle Vehicle ............................................................................... 209 7.1 Introduction ..................................................................................... 209 7.2 Steering and Stability of Three-Axle Vehicles ................................ 210 7.3 Steering with Unequal Conicities .................................................... 215 7.4 Stability of Vehicle with Uniform Conicity .................................... 217 7.5 Stability with Unequal Conicities.................................................... 225 7.6 Dynamic Response .......................................................................... 230 References ............................................................................................. 232 8. Articulated Vehicles ..................................................................................... 235 8.1 Introduction ..................................................................................... 235 8.2 Steering and Stability....................................................................... 238 8.3 Application to Specific Configurations ........................................... 244 8.4 Stability of an Articulated Three-Axle Vehicle............................... 246 8.5 Stability and Response of an Articulated Four-Axle Vehicle.......... 252 References ............................................................................................. 260 9. Unsymmetric Vehicles.................................................................................. 261 9.1 Introduction ..................................................................................... 261 9.2 Stability Theorems for Rigid and Semi-Rigid Vehicles .................. 263 9.3 Unsymmetric Rigid Vehicle ............................................................ 265 9.4 Steering of a Vehicle with Unsymmetric Inter-Wheelset Structure 267 9.5 Stability of a Two-Axle Articulated Vehicle................................... 272 9.6 The Influence of Elastic Stiffness on Stability ............................... 276 9.7 Applications of Unsymmetry........................................................... 279 References ............................................................................................. 281 Index ................................................................................................................. 283

Arrangement of the book Sections are numbered serially within each chapter. If reference is made to a section within the chapter containing the section, the section number is cited as a single number. Otherwise, a section is identified by two numbers separated by a decimal point, the first number referring to the chapter in which the section appears, and the second identifying the section within the chapter. Equations are numbered serially within each section. If reference is made to an equation within the section containing the equation, the equation number is cited as a single number. If reference is made elsewhere in the same chapter then the equation number is cited as a two-figure number and if reference is made in another chapter all three numbers-chapter, section and equation are cited. Figures and tables are numbered by chapter.

Preface The fundamental method of guidance of the railway vehicle is the coned and flanged wheelset. Whilst facilitating guidance in curves, coning can give rise to sustained lateral oscillations, termed hunting. This oscillation induces forces which can cause damage to both vehicle and track and there can be, at least, discomfort to the passenger and, at worst, the risk of derailment. Inadequate steering on curves can have similar consequences. This book concentrates on the resulting problem of the conflict between guidance and stability and its resolution by proper design of the suspension connecting the wheels and car body of the railway vehicle. The invention of the wheelset, the progressive development of the bogie and the various schemes of articulation which have been developed over the years in order to resolve the design conflict between stability and steering, all predate the theory of railway vehicle dynamics. Engineering insight brought railway technology a long way but empirical methods were not adequate once the railway renaissance started and train speeds increased. A fundamental change in railway technology took place in which the empirical evolution of railway bogies was replaced by a more scientific and numerate approach. This approach has been very successful; for example, not only has stable operation of steel wheel on steel rail vehicles been demonstrated at speeds of over 500 km/h (more than double the speed of the fastest train fifty years ago) but the analytical and predictive capabilities now available have stimulated a rising tide of innovative designs. The detailed modelling of the dynamics of railway vehicles is made possible by the several excellent computer packages that are available, which provide sufficiently detailed and validated mathematical models that can be used with confidence in engineering design and development. These models permit the simulation of the actual motion on a specified stretch of track so that the performance of a specific design can be analysed, or a particular incident recreated. Thus, by simulation the overall performance of a vehicle can be checked. Realism is, of course, essential in design but equates to complexity, and computer output must be tempered with understanding and scepticism. It is important, therefore, that fundamental principles are well understood. This book is concerned with the fundamental principles of guidance and stability, which are a consequence of the mechanics of wheel-rail interaction as embodied in the equations of motion. For research purposes, where the objective is to achieve an understanding of an innovative system or a particular problem, simple models can be very useful and can provide productive insights. Analytical studies which describe the mechanics of various phenomena by the simplest model possible can be used to explore new suspension and vehicle design concepts.

PREFACE

xii

Attention will be concentrated on the configuration and parametric design of the bogie, in relation to steering, dynamic response and stability. Therefore the treatment of the various configurations of vehicle do not simply concentrate on a current typical set of parameters but attempt to consider the consequences of the complete range of parameters open to the designer. By this approach, it is possible to see why much of current practice, though it pre-dates the availability of theory, is the way it is. Moreover, it becomes clear why many innovations failed in the past. Because an important consequence of a more analytical approach is to separate out the dynamic properties of a system from the detailed design of its components the latter will not be discussed. Moreover, the application of active controls to steering and ride control (including body tilting) will not be covered. Active systems will play a large part in the future and those working in the field will require a sound grounding in passive systems. As the emphasis is on ride quality and guidance, a frequency range of roughly 0  15 Hz is of prime interest. This makes it possible to consider that, in general, wheelsets and track (except in the areas of contact) are rigid and that car bodies are without flexibility. This means that some significant phenomena are not discussed here. Moreover, simple forms of suspension elements are assumed. The more straightforward problems of response in the vertical plane, or in the longitudinal direction are not addressed. The basic concepts are described in Chapter 1. A detailed discussion of the equations of motion follows in Chapter 2 in which a compromise has been made between the mathematical rigour of some investigators and the ad-hoc use of Newton’s Laws of others. In this Chapter, though an engineering approach has been followed, great reliance has been placed on the careful derivations of Professor de Pater. The following Chapters deal with the single wheelset and then with progressively more complex configurations of vehicle. If possible, simple analytical results have been derived as these, if available, provide the best basis for understanding the mechanics of the systems involved. All numerical results have been obtained using standard commercially available software for numerical computation. It is a pleasure to acknowledge the stimulus and help I have received over the years from colleagues, too numerous to mention here, at British Rail Research, Loughborough University and through the International Association of Vehicle System Dynamics. A. H. Wickens Idridgehay, 2002

1 Basic Concepts 1.1 Introduction The railway train running along a track is one of the most complex dynamical systems in engineering. It has many degrees of freedom, the interaction between wheel and rail involves both complex geometry of wheel tread and rail head and nonconservative forces generated by relative motion in the contact area, and there are many non-linearities. The long history of railway engineering provides many practical examples of dynamical problems which have degraded performance and safety. The two essential features of operation, running in a train of vehicles and guidance by the track, cause problems which are unique to railways. Inadequate guidance on curves results in high lateral forces between wheel and rail, rapid wear of wheels and rails and the possibility of derailment. Dynamic and static instabilities, and excessive response to track irregularities and other features of track geometry, can result in poor ride quality and high stresses and can contribute to derailment. Operation in a train involves the control of forces acting between the vehicles in the train as the propulsive and braking forces are varied in response to the train traversing hills and valleys. High frequency interaction between wheel and rail can lead to damage to the contacting surfaces and corrugation of the rails, and excessive noise and vibration. The dynamics of the railway vehicle represents a balance between the forces acting between the wheel and the rail, the inertia forces and the forces exerted by the suspension and articulation. Of these, the basic characteristics of the wheel-rail interface such as friction, geometry, and the elasticity in the contact area are hardly under the control of the designer. But the configuration, suspension and forms of articulation can be varied over a wide range of possibilities, limited mainly by the degree of complexity considered acceptable for each application. The objective of suspension design is, therefore, to control the motion of the railway vehicle so that good ride quality is achieved, at the same time dynamic loads and the tendency to derail are reduced to acceptable levels, whilst running on track with geometry that is economically acceptable. In a complete model of the dynamics of a railway vehicle, the vehicle is considered to be assembled from wheelsets, car bodies and intermediate structures which are flexible, and which are connected by components such as springs and dampers. Similarly, the vehicle is considered to run on a track which has a complex structure

2

RAIL VEHICLE DYNAMICS

with elastic and dissipative properties. Each major component has six rigid body degrees of freedom plus additional degrees of freedom representing the elastic distortion of the component. In the latter case, these additional degrees of freedom might represent a finite element model of the structure or a series of natural modes of vibration. The track can be modelled as a continuous structure with a moving interface at the points of contact, where the interaction between wheel and rail is dependent on the relative motion. This kind of model, with varying assumptions, is provided by various computer software packages which are used in the engineering design and analysis of railway vehicles. So, one objective of the study of the dynamics of railway vehicles is the development of sufficiently detailed and validated mathematical models that permit the simulation of the actual motion, on a specified stretch of line, so that the performance of a specific design can be analysed, or a particular incident recreated. Thus, by simulation, the overall performance of an existing or projected vehicle can be checked and design decisions made. A second objective of the study of railway vehicle dynamics is to develop analytical or numerical models describing the mechanics of various phenomena by the simplest model possible. These can be used to explore new suspension and vehicle concepts and to develop a basis for physical understanding and insight. Ideally, not only analysis but synthesis is required in which various possibilities for design are exposed. Simpler models are typically generated by simplifying assumptions and in this book, concerned with guidance and stability, these are that • the vehicle has a longitudinal plane of symmetry (parallel to the direction of motion on straight track) making it possible, under certain conditions, to separate equations governing those motions which are symmetric with respect to the plane of symmetry from those which govern anti-symmetric motions; • variations in longitudinal motion are not considered so that the vehicle moves at constant forward speed; • the motions of interest are at low frequencies and, in most cases, flexibility of components can be neglected. It is the objective of this chapter to explain the basic concepts of stability and guidance of railway vehicles as a preliminary to more detailed mathematical analysis.

1.2 The Railway Wheelset The basic unit of a railway vehicle is the wheelset, Figure 1.1. The conventional wheelset of today has the following features: it consists of two wheels fixed on a common axle, so that each wheel rotates with a common angular velocity and a constant distance between the two wheels is maintained. Flanges are provided on the inside edge of the treads and the flange-way clearance allows, typically, ± 7−10 mm of lateral displacement to occur before flange contact. Whilst many wheelsets commence life with purely coned treads, typically coned at 1/20 or 1/40, these treads wear rapidly in service, so that the treads come to possess curvature in the transverse direction. Similarly, rails also possess curvature in the transverse direction. All these

BASIC CONCEPTS

3

Figure 1.1 Railway wheelset.

features contribute to the behaviour of the railway vehicle as a dynamic system, and it is important to consider their purpose. The conventional railway wheelset has a long history [1] and seems to have evolved by a process of trial and error. Naturally, in the pioneering days of the early railways most attention was concentrated on reducing rolling resistance so that the useful load that could be hauled by horses could be multiplied. Another major problem was the lack of strength and resistance to wear of the materials then available. Moreover, the level of adhesion between rolling wheel and the track was unknown. As a result, many possibilities were tried. An obvious step was to fit wheels with cylindrical treads. However, if the wheels are fixed on the axle and the treads are intended to be cylindrical very slight errors in parallelism would induce large lateral displacements which would be limited by flange contact. There is no guidance until flange contact and thus a wheelset with cylindrical treads tends to run in continuous flange contact. The position of the flange, either inside or outside the rails, was controversial well into the nineteenth century. Nor was there agreement as to whether the wheels should be rigidly fixed to an axle or free to revolve on the axle, though the usual practice seemed to be that wheels were fixed to the axle. The play allowed between wheel flange and rail was initially minimal. In the early 1830s the flangeway clearance was opened up with the objective of reducing the lateral forces between wheel and rail. A further important point is that the geometry of the wheel and rail as it has evolved is particularly favourable for the method of switching which involves a minimum of moving parts and only small gaps in the running surfaces of the rails. It is not known when coning of the wheel tread was first introduced. It would be natural to provide a smooth curve uniting the flange with the wheel tread, and wear of the tread would contribute to this. Moreover, once wheels were made of cast iron, taper was normal foundry practice. The purpose of coning was partly to reduce the rubbing of the flange on the rail, and partly also to ease the motion of the vehicle in curves. A wheelset with coned wheels in a curve can maintain a pure rolling motion if it moves outward and adopts a radial position. Redtenbacher [2] provided the first theoretical analysis in 1855 which is illustrated in Figure 1.2. From the geometry in this figure it can be seen that there is a simple geometric relationship between the

4

RAIL VEHICLE DYNAMICS

R

2λ

R C D

A

O

B l

y

l

OAB = OCD ( r0 - λ y )/( R - l ) = ( r0 + λ y )/( R + l ) y = r0 l / R λ Figure 1.2 Redtenbacher’s formula for the rolling of a coned wheelset on a curve.

lateral movement of the wheelset on a curve y, the radius of the curve R, the wheel radius r0, the lateral distance between the points of contact of the wheels with the rails 2l and the conicity λ of the wheels in order to sustain pure rolling. In practice a wheelset can only roll round moderate curves without flange contact, and a more realistic consideration of curving requires the analysis of the forces acting between the vehicle and the track. It can be seen, in broad terms, why the wheelset adopted its present form. If the flange is on the inside the conicity is positive and as the flange approaches the rail there will be a strong steering action tending to return the wheelset to the centre of the track. If the flange is on the outside the conicity is negative and the wheelset will simply run into the flange and remain in contact as the wheelset moves along the track. Another factor is the behaviour in sharp curves. If the flange is on the inside then the lateral force applied by the rail to the leading wheelset is applied to the outer wheel and will be combined with an enhanced vertical load. As explained later, this diminishes the risk of derailment. With outside flanges the lateral force applied by the rail applied to the inner wheel which has a reduced vertical load and thus the risk of derailment is increased. These factors can be easily demonstrated with the aid of model wheelsets [3]. Thus, it can be seen that for small displacements from the centre of straight or slightly curved track the primary mode of guidance is conicity and it is on sharper

BASIC CONCEPTS

5

curves and switches and crossings that the flanges become the essential mode of guidance. Though this appears to be a modern view, in 1838 Brunel [4] wrote The flanges are a necessary precaution but they ought never to touch the rail and therefore they cannot be said to keep the wheels on the rails. They ought not to come into action except to meet an accidental, lateral force. A railway with considerable curves might be travelled over with carriages at any velocity and with wheels without flanges. The wheels are made conical, the smaller circumference at the outer edge. The pair of wheels are fixed to the axle and thus if anything throws the wheels in the slightest degree to one side the wheel is immediately rolling on a larger circumference than the other and the tendency to roll back is introduced. The carriage is kept always in the middle of the track. A beautiful arrangement. As a concept, this view led to many significant improvements in the design of railway vehicle suspensions in the 20th century. Coning of the wheel tread was well established by 1821. George Stephenson in his Observations on Edge and Tram Railways [5] stated that It must be understood the form of edge railway wheels are conical that is the outer is rather less than the inner diameter about 3/16 of an inch. Then from a small irregularity of the railway the wheels may be thrown a little to the right or a little to the left, when the former happens the right wheel will expose a larger and the left one a smaller diameter to the bearing surface of the rail which will cause the latter to loose ground of the former but at the same time in moving forward it gradually exposes a greater diameter to the rail while the right one on the contrary is gradually exposing a lesser which will cause it to loose ground of the left one but will regain it on its progress as has been described alternately gaining and loosing ground of each other which will cause the wheels to proceed in an oscillatory but easy motion on the rails.

y = a sin ω t

s = Vt

d y ω y ω r0 l 1 =− = = 2 R ds 2 V2 V Rλ 2

2

2

Figure 1.3 Derivation of Klingel’s formula for the kinematic oscillation of a wheelset from Redtenbacher’s formula in Figure 1.2.

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RAIL VEHICLE DYNAMICS

This is a very clear description of what is now called the kinematic oscillation, as shown in Figure 1.3. Thus, if a wheelset is rolling along the track and is displaced slightly to one side, the wheel on one side is running on a larger radius and the wheel on the other side is running on a smaller radius. Because the wheels are mounted on a common axle one wheel will move forward faster than the other because its instantaneous rolling radius is larger. Hence, if pure rolling is maintained, the wheelset moves back into the centre of the track − a steering action is provided by the coning. However, the wheelset overshoots the centre of the track and the result is the kinematic oscillation. In 1883 Klingel gave the first mathematical analysis of the kinematic oscillation [6] and derived the relationship between the wavelength Λ and the wheelset conicity λ, wheel radius r0 and lateral distance between the contact points between wheels and rails 2l as

Λ = 2π (r0l/λ)1/2 This simple formula follows purely from the geometry of Figure 1.3, and is consistent with Redtenbacher’s formula for the wheelset in a curve. Since distance along the track s = Vt where V is the forward speed and t is time, Klingel’s formula shows that, as the speed is increased, so will the frequency of the kinematic oscillation. Very little else can be deduced about the dynamical behaviour of railway vehicles which must come from a consideration of the forces acting.

1.3 Creep Pure rolling rarely takes place, and wheels and rails are not rigid. The normal load between wheel and rail causes local elastic deformation and an area of contact, the contact patch, is formed. In the case where the surfaces of the wheels and rails are smooth and have constant curvature in the vicinity of the contact patch, Hertz [7] showed that the contact patch was elliptical in shape, and the distribution of normal pressure between wheel and rail over the contact patch is semi-ellipsoidal. If a longitudinal force is applied to the wheel, so that it is braked, a deviation from the pure rolling motion occurs. The deviation in relative velocity divided by the forward speed of the wheel is referred to as the longitudinal creepage. Similarly, lateral creepage is defined as the (incremental) relative lateral velocity divided by the forward speed. In addition, relative angular motion between wheel and rail about the normal to the contact patch is referred to as spin. If the longitudinal creepage is small, it is accommodated by elastic strains in the vicinity of the contact patch. As the wheel rotates, unstrained material enters the contact patch at its leading edge. As the material moves through the contact patch, the relative velocity between the wheel and rail equals the rate of change of strain so that the surfaces are locked together. The magnitude of the resulting longitudinal tangential stress increases linearly with distance from the leading edge. Similarly, lateral creepage gives rise to lateral tangential stresses. Both longitudinal and lateral creepage therefore generate forces which are directly proportional to the corresponding creepage. When there is spin, the pattern of elastic strain is more complicated. In this case, as the material moves

BASIC CONCEPTS

7

(a) slip

locked region σz

σx = µσz

(b)

(c)

slip

locked

Figure 1.4 The contact patch between wheel and rail (a) elevation showing locked region of adhesion at leading edge and region of slip at trailing edge (b) normal pressure σz and tangential traction applied by wheel to rail σx (c) contact patch in plan view.

through the contact region the relative velocity between wheel and rail is directly proportional to the distance from the centre of the contact region and therefore the strain field becomes curved. As a consequence, a lateral force is generated (the couple about the common normal is small and may be safely neglected). As the creepages and spin increase, the tangential stresses increase, and where these stresses exceed the normal pressure multiplied by the coefficient of friction, slipping takes place. The result is that the area of adhesion at the front of the contact patch in which the surfaces are locked together progressively reduces as the creepage increases, Figure 1.4. The relationship between the creep force and creepage is then as shown in Figure 1.5(a). For sufficiently large creepage, slipping takes place over the whole contact patch and the creep force is equal to the normal force multiplied by the coefficient of friction. If both longitudinal and lateral creep occur simultaneously then for small creepages the creep forces can be superposed, but for larger creepages in the area of slipping the tangential stresses act in a direction opposite to the local resultant relative velocity. The result is that then all the creep forces are influenced by both lateral and longitudinal and lateral creepages and spin. Though the lateral force is proportional to spin for small values of the spin, for large values of the spin slipping takes place over a large part of the contact patch and the lateral force reduces to zero. The relationship between lateral force and spin is therefore as shown in Figure 1.5(b). It was Carter's [8] introduction of the creep mechanism into the theory of lateral dynamics that was the crucial step in developing a realistic model of the wheelset.

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RAIL VEHICLE DYNAMICS

15000

(a)

(b)

-T2 (N)

-T1 (N)

0

8000

0

γ1

0.015

0

0

ω3

6

Figure 1.5 (a) Variation of longitudinal creep force T1 with longitudinal creepage γ1 showing the limiting value µT3 where µ = 0.3 is the coefficient of friction and T3 = 39000 is the normal force (b) Lateral creep force T2 as function of spin ω3 (zero longitudinal and lateral creep).

In the light of the creep theory the action of conicity can be considered as follows If a wheelset is rolling along a track and is displaced laterally, the rolling radii are different on the two wheels. Noting that the wheels are fixed on a common axle, The tread velocities are fractionally different and so longitudinal creep is generated. The corresponding creep forces are equivalent to a couple which is proportional to the difference in rolling radii or conicity, and which tends to steer the wheelset back into the centre of the track. This is the basic guidance mechanism of the wheelset. In addition, when the wheelset is yawed, a lateral creep force is generated. In effect, this coupling between the lateral displacement and yaw of the wheelset represents a form of feedback, and this introduces the possibility of dynamic instability.

1.4 Stability An important feature of a railway vehicle is that, in addition to the vertical suspension connecting the wheelsets to the vehicle body, there are lateral and longitudinal springs and dampers. This is illustrated by the plan-view of a two-axle vehicle shown in Figure 1.6. The purpose of the plan-view suspension is to stabilise the tendency of the wheelsets to oscillate and to facilitate the motion of the vehicle in curves. Because of the action of the creep forces the motion of a rolling wheelset incorporated in a vehicle is significantly different from Klingel’s description. At low forward speeds successive overshoots decrease in magnitude as the vehicle moves along the track, eventually to travel in a straight line down the centre of the track. The vehicle is dynamically stable, because following a slight disturbance it returns to its original path. At high forward speeds successive overshoots grow as the vehicle moves along the track, for in this case it is dynamically unstable.

BASIC CONCEPTS

9

The mechanism of this instability may be appreciated by considering the simple vehicle of Figure 1.6 in the special case where the vehicle body has very high inertia and is assumed to move forward at constant speed but does not undergo any lateral motion. At any forward speed the tendency of the wheelset is to oscillate at the frequency of the kinematic oscillation. Since this frequency is proportional to speed, then at low speeds the inertia forces will be small and the main component of the resultant force acting on the wheelset is the restoring force provided by the springs connecting the wheelsets to the vehicle body. In order to balance this force, creep must be developed and this will cause a progressive reduction in lateral displacement as the wheelset pursues its oscillatory path. At high speeds the inertia forces will dominate, as the frequency is correspondingly high. In this case, creep must be developed which will cause a progressive increase in the lateral displacement of the wheelset during its lateral oscillation. It follows that there is a speed at which the successive overshoots neither grow nor decay, the wheelset then, and only then, tracing out a sinusoidal path. Klingel’s solution for pure rolling then emerges as a special case of the dynamics of the wheelset. In general, a railway vehicle is stable at low speeds so that following a disturbance the vehicle, together with its wheelsets, will undergo a decaying oscillation, and the vehicle will return to the centre of the track. As the speed is increased, the decay rate of the oscillation is reduced. In most cases, at a sufficiently high speed the oscillations following a disturbance grow and eventually lead to a limit cycle oscillation where the amplitude is limited by either contact of the flanges with the rails or slipping of the wheels on the rails. The energy required to sustain this oscillation clearly comes from the energy of the forward motion. This fully developed limit cycle oscillation is generally termed “hunting”. The lowest vehicle speed at which sustained oscillations can occur is known as the critical speed. Early measurements of hunting, made in the 1960s, showed that twoaxle freight vehicles of then current design had critical speeds as low as 30 km/h [9]

Figure 1.6 Plan view of two-axle railway vehicle showing lateral and longitudinal suspension springs.

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RAIL VEHICLE DYNAMICS

Hunting Limit Cycle

H Vc

V Figure 1.7 Typical variation of lateral force H acting on wheelset as a function of forward speed V for a two-axle vehicle subject to instability.

and this was a contributory cause of many derailments. Similarly a typical passenger bogie vehicle of the same era had a critical speed of about 110 km/h which though not dangerous was the cause of bad riding [10]. In practice, large lateral forces can be experienced by a hunting vehicle, as indicated in Figure 1.7. The first successful theoretical prediction and experiment on the track is described in [11]. The danger posed by hunting at high speeds was appreciated by Matsudaira who was the first to study the effects of the suspension on stability [12], and was graphically demonstrated by the lateral distortion of the track during high speed trials of a locomotive by French Railways in 1955 [13]. Earlier, the Sevenoaks accident in which a locomotive was derailed at high speed was explained by Carter [14] as a form of static instability in which the wheelbase buckled under the action of the creep forces. Thus the guidance offered by the coning of the wheels is the source of potential instability. This is the fundamental conflict in the design of the running gear of a railway vehicle.

1.5 Guidance Guidance is the ability of a vehicle to follow the geometric layout of the track. Ride is the ability of a vehicle to minimise the dynamic response, in terms of stresses and accelerations, to the layout of the track. The actual track layout will consist of the design layout, largely determined by geographical and operational factors and superimposed irregularities or lack of accuracy of real track. The most important aspect of guidance is the behaviour of vehicles in curves. The application of Redtenbacher’s formula shows that a wheelset will only be able to move outwards to the rolling line if either the radius of curvature or the flangeway

BASIC CONCEPTS

11

Y

C S

S

Figure 1.8 Two-axle vehicle in curve showing the relative magnitude of the forces acting between wheel and rail together with the forces applied to the wheelsets by the suspension S, the centrifugal force C and horizontal component of the normal force between wheel and rail Y.

clearance is sufficiently large. Moreover, as discussed above, for stability the wheelsets in a vehicle must be constrained in some way. Thus in many curves, the wheelsets are not able to take up a radial position, and typically the attitude of a two-axle vehicle in plan view is as shown in Figure 1.8. Because a wheelset is constrained by the longitudinal and lateral stiffnesses connecting it to the rest of the vehicle, it balances a yaw couple applied to it by the suspension by moving further out in a radial direction so as to generate equal and opposite longitudinal creep forces, and it will balance a lateral force by yawing further. In sharp curves the flange will be in contact with the rail. In the case of the leading wheelset of a vehicle, the flange force will be reacted by the creep forces generated by the wheels creeping laterally towards the inside of the curve, Figure 1.8. The first essentially correct description of curving was given by Mackenzie [15] in 1883. His discussion was based on sliding friction, and neglects coning, so that it is most appropriate for sharp curves, where guidance is provided by the flanges. Whilst new wheel profiles are often purely coned on the tread, usually to an angle of 1:20, treads wear rapidly and assume a hollow form, Figure 1.9. For a worn profile the variation of the difference in rolling radius between the right hand and left hand wheels as the wheelset is displaced laterally can be very complicated. A useful concept is the equivalent conicity which is the linearised slope of the difference in

12

RAIL VEHICLE DYNAMICS

Figure 1.9 Geometry for typical worn wheel rolling on worn rail, showing the movement of the contact point as the wheel is displaced laterally.

(a )

(b )

Figure 1.10 Normal and lateral tangential forces acting on wheelset (a) in central position (b) in laterally displaced position, illustrating the gravitational stiffness effect.

BASIC CONCEPTS

13

rolling radius plotted against lateral displacement. In practice, the equivalent conicity of a worn wheelset is much greater than that of a purely coned wheelset. This increased conicity enables a wheelset to roll round a curve of much smaller radius than is possible for a coned wheel. The influence of the hollow wheel tread gives rise to additional, significant forces acting between wheel and rail. When the wheelset is rolling on equal radii in the central position, the contact plane is inclined to the horizontal at a small angle as shown in Figure 1.10(a). When the wheelset is displaced laterally, contact is made at new points and the inclination of the contact planes is changed; on one wheel it is increased and on the other it is reduced, as shown in Figure 1.10(b). As the normal reactions between wheel and rail, which support the weight carried by the wheelset are similarly inclined, it is found that there is a lateral resultant when these normal reactions are resolved horizontally. This lateral restoring force is proportional to lateral displacement and is referred to as the gravitational stiffness. A second effect due to the hollow wheel tread arises from the lateral force generated by spin [16]. As the inclination of the contact plane changes with lateral displacement, the resolved component of the angular velocity of the wheelset due to its rolling motion, taken about an axis normal to the contact plane, gives rise to changes in spin creepage. This generates a lateral force which is proportional to lateral displacement and which is of the opposite sign to that of the gravitational stiffness force. For small displacements, typically the total contact stiffness (gravitational + spin) is reduced to about 20% of the gravitational stiffness, and is consequently rather small. However, as mentioned above, the lateral force due to spin becomes small for the large values of spin achieved in flange contact, and there is a large restoring force due to flange contact. Heumann [17] suggested that profiles approximating to the fully worn should be used rather than the purely coned treads then standard. After re-profiling to a coned tread, tyre profiles tend to wear rapidly so that the running tread normally in contact with the rail head is worn to a uniform profile. This profile then tends to remain stable during further use, and is largely independent of the original profile and of the tyre steel. Similarly, rail head profiles are developed which also tend to remain stable after the initial period of wear is over. These results suggested that vehicles should be designed so as to operate with these naturally worn profiles, as it is only with these profiles that any long-term stability of the wheel-rail geometrical parameters occurs. Moreover, a considerable reduction in the amount of wear is possible by providing new rails and wheels with an approximation to worn profiles at the outset and this has now become common practice in many countries.

1.6 Suspension, Performance and Criteria For satisfactory performance, a railway vehicle must meet certain criteria. The most fundamental of these is concerned with the possibility of derailment. Figure 1.11 shows a yawed wheelset in flange contact with a rail. There are initially two points of contact between wheel and rail, one on the wheel tread and the other on the flange. The latter point of contact lies ahead of the former. The onset of flange-climbing

14

RAIL VEHICLE DYNAMICS

F

Q N

α

Y

F b a

Q

Figure 1.12 Flange contact when wheelset is yawed at angle α with rail, showing first point of contact on tread a and second point of contact with flange b.

derailment occurs when the vertical load Q is carried entirely by the point of contact on the flange, and so the derailment limit is defined by the minimum value of the lateral reaction Y. If the component of the tangential force in the transverse vertical plane is denoted by F, and N is the normal reaction, the balance of forces in Figure 1.12 shows that Y N tan α − F < Q N + F tan α The minimum value of Y/Q occurs when F is a maximum so that by the laws of friction F cannot exceed µN where µ is the coefficient of friction, hence tan α − µ Y = Q 1 + µ tan α This derailment criterion is due to Nadal, but like many succeeding studies, its analysis was based on suspect assumptions. However, the application of the laws of creep by Gilchrist and Brickle [18] has shown that Nadal’s formula is correct for the most pessimistic case when the angle of attack is large and the longitudinal creep on the flange is small. It will be apparent that derailment is most likely when a large lateral force occurs simultaneously with a reduced vertical load on a wheel, typically in a curve with a significant vertical irregularity, or a high degree of track twist. Another mode of failure occurs when the lateral forces imposed by the vehicle are sufficient to shift the track laterally. An empirical criterion for this has been given by Prud’homme [19] and is

BASIC CONCEPTS

15

Y = 10 + W/3 where W is the axle-load, and both Y and W are measured in kN. These are extreme conditions. More generally, the curving ability of a railway vehicle is considered good if the angle of yaw of each wheelset is small, flange contact is avoided on all but sharp curves, the lateral forces between wheel and rail are low and the energy expended in wheel rail contact is small. A measure of curving performance that is often used in practice is the degree of flange wear experienced in service and empirical relationships have been established between rates of wear and the energy expended in the contact area. After the essential stability and curving requirements are met, it is considerations of ride quality which dominate the detailed design of railway vehicle suspension systems. The layout of curves is defined by the maximum cant of the track, and vehicle speed, so that the lateral acceleration applied to passengers is within acceptable limits. Moreover, the length of the transition between straight and curved track, and in some cases the shape, is determined by limits on the time rate of change of cant deficiency. So far, the dynamics of railway vehicles running on perfectly aligned straight or curved track has been considered. A further important problem is that of the dynamic response of vehicles to track irregularities. Track irregularities arise from specific features such as switch and crossing work or from the continuous distribution of roughness which must be controlled by maintenance of the track. In the latter case the irregularities may be considered as randomly distributed. The dynamic response, in terms of the acceleration level on the car body, to this stochastic input will characterise the ride quality of the vehicle. Detailed international standards for passenger comfort criteria exist which define frequency weighting characteristics describing human response to vertical and lateral vibration, which are used in the specification and assessment of vehicles. The excursions of the car body on the suspension must remain within the structural clearance gauge of the route on which the vehicle is to run.

1.7 Suspension, Articulation and Curving It can be seen that a vehicle with a perfect suspension would be stable at all operational speeds, would negotiate curves by minimising the forces acting between wheel and rail and in traversing irregular track would minimise the acceleration levels in the car body and the stresses applied to both vehicle and track structures. These requirements are usually conflicting and compromise, informed by analysis, is required in design. Not only are the parameters that are associated with wheel-rail contact, both geometrical and frictional, not under the control of the designer or operator, but they are not known exactly and can vary over a wide range. It follows that practical designs must be very robust in relation to such parameters. On the other hand, there is enormous scope for the design of the suspension system in terms of the way in which the wheelsets and car bodies in a train are connected.

16

RAIL VEHICLE DYNAMICS

It has long been the objective of vehicle design to incorporate wheel and steering arrangements that permit a vehicle to follow a chosen path or a track by a motion which involves pure rolling of the wheels, apart from the necessary transmission of traction forces and the reaction of centrifugal force. In the first place pure rolling might be achieved by a choice of configuration of the wheels and the way in which they are articulated. Such rolling motions may or may not be statically or dynamically stable. Additionally the introduction of creep, in response to inertia and suspension forces may stabilise or destabilise the system. It follows that there are many possible mechanisms of guidance and stability which can be considered. There have been many attempts to provide an alternative to the railway wheelset and reference will be made to some of these in Chapter 3. However, in using the conventional railway wheelset, there are many ways to improve performance in curves by making a vehicle more flexible in plan view, thus encouraging the axles to take up a moreor-less radial position in curves. It will be shown later that a two-axle vehicle that is capable of radial steering on a uniform curve will be dynamically unstable at all speeds so that the design of a two-axle vehicle requires a compromise between stability and curving. This is the subject of Chapters 4 and 5. As discussed in Chapters 6 and 7 for a vehicle with three or more axles it is possible to arrange the suspension so that radial steering and dynamic stability are both achieved. One approach is to provide elastic or rigid linkages directly between wheelsets in a vehicle. This can be referred to as self-steering as the vehicle body is not involved. Alternatively, a linkage system can be provided which allows the wheelsets to take up a radial position but provides stabilising elastic restraint from the vehicle body. This is so-called forced steering as it can be considered that the vehicle body imposes a radial position on the wheelsets. There are many designs in which there is articulation of the vehicle bodies of a vehicle or train. Articulation, in the present context, describes an arrangement in which the relative motion between the vehicle bodies is used to influence the stability and guidance of the vehicle. In many cases the interaction between the vehicles in a train is minimised by the form of coupling between the vehicles, so that longitudinal forces can be transmitted between car bodies, but the coupler is capable of transmitting little or no lateral force or yaw couple. In this case it is a good approximation to treat each vehicle as if it were isolated and the lateral dynamics of each vehicle can be considered to be largely independent of that of the rest of the train. In an articulated vehicle the connections between vehicles form an essential part of the running gear. A common design feature is to link the relative angle between vehicle bodies to yaw of the wheelsets. As is discussed in Chapter 8, such designs improve curving performance and other aspects of vehicle design but can exhibit a wide spectrum of various hunting instabilities. All the configurations discussed so far have been symmetric fore-and-aft. Unsymmetric configurations make it possible, in principle, to achieve a better compromise between curving and dynamic stability, at least in one direction of motion. But additional forms of instability can occur as discussed in Chapter 9. The equations of motion are fundamental for all configurations of vehicle, and the derivation of these for a wheelset and a simple two-axle vehicle is discussed in Chapter 2.

BASIC CONCEPTS

17

References 1. Wickens, A.H.: The dynamics of railway vehicles-from Stephenson To Carter. Proc. I. Mech. E. 212, Part F (1998), pp. 209-217. Gilchrist, A.O.: The long road to solution of the railway hunting and curving problems. Proc. I. Mech. E. 212, Part F (1998), pp. 219-226. 2. Redtenbacher, F.J.: Die Gesetze des Locomotiv-Baues. Verlag von Friedrich Bassermann, Mannheim, 1855, p. 22. 3. Wickens, A.H.: Dynamics and the advanced passenger train. Speaking of Science 1977, Proceedings of The Royal Institution of Great Britain, 50 (1978), pp. 33-65. 4. Vaughan, A.: Isambard Kingdom Brunel − Engineering Knight Errant. John Murray, London, 1992, p. 102. 5. Dendy Marshall, C.F.A.: History of British Railways Down to the Year 1830. Oxford University Press, Oxford, 1938, p. 165. 6. Klingel.: Uber den Lauf der Eisenbahnwagen auf Gerarder Bahn. Organ Fortsch. Eisenb-wes. 38 (1883), pp. 113-123. 7. Timoshenko, S.P.: A History of the Strength of Materials, Mcgraw-Hill, New York, 1953, p. 348. 8. Carter, F.W.: The electric locomotive. Proc. Inst. Civ. Engs. 221, 1916, pp. 221252. 9. Pooley, R.A.: Assessment of the critical speeds of various types of four-wheeled vehicles. British Railways Research Department Report E557, 1965. 10. King, B.L.: The measurement of the mode of hunting of a coach fitted with standard double-bolster bogies. British Railways Research Department Report E439, 1963. 11. Gilchrist, A. O., Hobbs, A.E.W., King, B.L. and Washby, V.: The riding of two particular designs of four wheeled vehicle. Proc. I. Mech. E. 180 (1965), pp. 99113. 12. Matsudaira, T.: Hunting problem of high-speed railway vehicles with special reference to bogie design for the New Tokaido Line. Proc. I. Mech. E. 180 (1965), pp. 58-66. 13. Knothe, K. and Bohm, F.: History of stability of railway and road vehicles. Vehicle System Dynamics, 31 (1999), pp. 283-323.

18

RAIL VEHICLE DYNAMICS

14. Carter, F.W.: The running of locomotives, with reference to their tendency to derail. Inst. Civil Engs, Selected Engineering Paper, No. 81, 1930. 15. Mackenzie, J.: Resistance on railway curves as an element of danger. Proc. Inst. Civ. Engs. 74 (1883), pp. 1-57. 16. Johnson, K.L.: Effect of spin upon the rolling motion of an elastic sphere on a plane. Trans. A. S. M.E. Ser. E, 80 (1958), pp. 332-338. 17. Heumann, H.: Zur Frage des Radreifen-Umrisses. Organ Fortschr. Eisenb.-wes. 89 (1934), pp. 336-342. 18. Gilchrist, A.O. and Brickle, B.V.: A re-examination of the proneness to derailment of a railway wheelset. J. Mech. Eng. Sci. 18 (1976), pp. 131-141. 19. Birmann, F.: Theoretical and experimental solutions of track problems for high speeds. Monthly Bulletin of the International Railway Congress Association 45 (1968), pp. 391-460.

2 Equations of Motion 2.1 Introduction The basic physical phenomena involved in the dynamics of railway vehicles have been described in Chapter 1. Equations of motion governing the stability and dynamic response of vehicles will now be derived which encompass the essential features of the wheel-rail geometry, the frictional forces acting between wheel and rail and the elastic and damping forces generated by the suspension. As attention will be confined to the dynamics at low frequencies, the wheelset and track are assumed to be rigid apart from local elasticity in the contact patch between wheel and rail, and the contributions of the local deflections near the contact patch to the overall motion of the wheelset are neglected. The wheelset, which is assumed to be axisymmetric about the axle centreline, is considered to be constrained to run along the track at constant speed. The track is arbitrarily curved in plan view and may be canted. The kinematics of the wheelset is considered first, and this is followed by a discussion of wheel rail geometry. An evaluation of the creep forces acting between wheel and rail makes it possible to formulate equations of motion of a freely running wheelset. This is followed by the derivation of the equations of motion of a complete two-axle vehicle in which the action of the suspension is taken into account.

2.2 Freedoms and Constraints The track possesses curvature in a horizontal plane with radius R0, cant or crosslevel φ0, and can be displaced locally through a lateral displacement y0, Figure 2.1. R0, φ0, and y0 vary with the distance s along the track. The wheelset reference frame Oxyz is attached to the centreline of the undistorted track, Figure 2.1, and moves along the track at the speed of the vehicle V. Thus the irregularities y0 are measured from this centreline. The origin of this set of axes is located at the centre of mass of the wheelset when the wheelset is central on the track. Ox lies along the tangent to the track centreline, Oy lies along the wheelset axle centreline when the wheelset is central and lying in the radial direction on the curve, and Oz is mutually perpendicular. The coordinates X, Y and azimuth Ψ of the origin of the frame Oxyz, Figure 2.1,

20

RAIL VEHICLE DYNAMICS

X

Y

O*

O

Ψ x

x*

y*

ψ

y

O G

φ0

y

φ y*

z

z*

Figure 2.1 Wheelset axis systems and coordinates.

with reference to an axis system fixed in the earth are given by dX/ds = cosΨ

(1)

dY/ds = sinΨ

(2)

dΨ/ds = 1/R0

(3)

A second set of axes O*x*y*z* has origin at the centre of mass of the wheelset. O*y* coincides with the axle centreline, O*x* is perpendicular to O*y* and O*z* is mutually perpendicular. To specify the orientation of the frame O*x*y*z* it will be convenient to select successive rotations, yaw ψ, and roll φ, about the carried axes Oz*, Ox*. Thus the rotations are taken about the position the axes have taken following the previous rotation. The rotation of the wheelset about the carried axis Oy* is denoted by θ. The displacements of the wheelset centre of mass O* relative to Oxyz are denoted

EQUATIONS OF MOTION

21

by the vector x0 with components ux, uy and uz, but as O* is in the plane yOz and both O and O* move forward at constant speed V x0 = [ 0 uy

uz ]

(4)

and the longitudinal position along the track s = Vt. Thus the position and orientation of a wheelset can be defined in terms of the six variables, s, the lateral and vertical displacements uy and uz and three rotations, yaw ψ, roll φ, and θ. As the area of contact between wheel and rail is small compared with the dimensions of the track contact between the wheelset and the rails can be considered to take place ordinarily at two points. As will be discussed later, two point contact gives rise to two constraint equations which makes it possible to eliminate two of the above coordinates. It will be convenient to eliminate the vertical displacement and roll angle of the wheelset as independent coordinates, so that they become simply dependent functions of lateral displacement and yaw. As the vehicle speed is constrained to be constant, the system has three degrees of freedom.

2.3 Wheel Rail Geometry As discussed in Chapter 1 the most important geometrical characteristics of the wheel rail geometry are (a) the variation of rolling radius with lateral displacement as this governs the conicity effect and (b) the variation of the slope at the contact point with lateral displacement as this governs the gravitational stiffness effect. For a typical wheel and rail combination, Figure 2.2, both profiles have curvature which varies continuously across the rail head and wheel tread and are defined by

ζw = f (ηw)

ζr = g (ηr)

(1)

ζr

ηr Figure 2.2 Typical wheel and rail profiles relative to the rail coordinates ζr , ηr (mm).

22

RAIL VEHICLE DYNAMICS

δl

ηl

δr

ηr

ζl

ζr uy-y0-r0φ ηwr

ζ wr

B

-uz-lφ A

ζ rr

δ0

C central

ηrr

displaced

Figure 2.3 Wheel rail geometry (right-hand). A is the origin of the rail axes ζr, ηr, B is the origin of the wheel axes ζw, ηw, so that A and B are coincident when the wheelset is central. When the wheelset is displaced, contact takes place at C.

where ηw, ζw are the wheel coordinates and ηr, ζr are the rail coordinates, the profiles being the same for the right-hand and left-hand wheels, Figure 2.3. In order to derive equations of motion, the position of the contact points, and the slopes and curvatures at these contact points, as functions of the wheelset lateral displacement and yaw are required. It will be assumed that the cross-sectional geometry of the wheel-rail system does not vary with distance along the track. It follows that the cross-sectional geometry is independent of s and θ. The wheels and rails will be assumed to be rigid in so far as their mutual geometry is concerned. When the wheelset is in the central position on the track, and is not yawed, the angle made by the contact plane with the horizontal is δ0 and the tread circles of the wheels have the same radius r0. When the wheelset is displaced laterally, the angles made between the contact planes and the axle centreline at the new points of contact are δwr and δwl. Similarly, the radii of the tread circles become rr and rl. The position of the contact points is determined by noting that the wheel and rail contact points must occupy the same position in space, the angles made by the contact planes at the points of contact must be the same for wheel and rail, and the contacting bodies cannot penetrate each other. As the angle of yaw of a wheelset is small in most ordinary circumstances, the two-dimensional case where the influence of yaw is neglected will be considered. Consider the right hand wheels and rails, Figure 2.3. When the

EQUATIONS OF MOTION

23

wheelset is in the central position the contact point is A. The centreline of the track is displaced laterally by y0 from the reference axis from which uy is measured. When the wheelset centre of mass is displaced laterally through a distance uy from the reference axis the wheelset rotates about a longitudinal axis through a small angle φ and so the lateral displacement at the contact point is uy - y0 - r0φ, and contact is made at a new point C. If the lateral movement of the contact point on the right hand wheel is ηwr and that on the rail is ηrr then uy - y0 - r0φ - ηwr + ηrr = 0

(2)

Similarly, considering the vertical movement of the wheelset ζwr at the right hand rail uz + lφ + ζwr - ζrr = 0

(3)

Also, if if δrr denotes the angle between the rail axes and the contact plane

φ - δwr + δrr = 0

(4)

The three corresponding equations for the left hand wheel are uy - y0 - r0φ + ηwl - ηrl = 0

(5)

uz - lφ + ζwl - ζrl = 0

(6)

φ + δwl - δrl = 0

(7)

In addition, the slopes at the contact points are given by tan δwr = dζwr /dηwr

(8)

tan δwl = dζwl /dηwl

(9)

tan δrr = dζrr /dηrr

(10)

tan δrl = dζrl /dηrl

(11)

For profiles specified by (1), equations (1) to (11) can be solved to yield the contact positions and slopes, the vertical displacement and roll angle as functions of uy. Measuring equipment has been developed to measure wheel and rail profiles with the necessary accuracy, [1, 2, 3]. The solution has been implemented in computer programs and typically uses a Newton-Raphson iterative procedure to solve the nonlinear algebraic equations. The first step is to determine the points of contact when the wheelset is central. This then defines a new origin for the wheel rail geometric data. Then equations (1) to (11) are solved for the contact points. Once the contact points have been established the various geometrical characteristics can be

24

RAIL VEHICLE DYNAMICS

0

1 φ (mr)

uz (mm)

0

-0.2

-1 -10

0 r

10 ηr (mm) 0 -10 -10

-0.4 -10

10

10

20 ηw (mm) 0 10

0

0

-20 -10

r l 0

10

2 ζr (mm) 1

r

ζw (mm) 2

r

0

l

0

l

-10

10 0 uy (mm)

-10

0 10 uy (mm)

Figure 2.4 Roll angle φ, vertical displacement uz and location of contact points as a function of lateral displacement for the wheel rail combination of Figure 2.2.

determined. Detailed discussions of the analytical aspects of wheel rail geometry have been given by de Pater [4] and Yang [5]. As an example, for the wheelset and rail combination shown in Figure 2.2 the variation of φ, uz and the location of the contact points is shown in Figure 2.4. Figure 2.5 shows the variation of rolling radius, contact angle and the transverse curvatures of the rail and wheel with lateral displacement. The examples shown refer to worn wheel and rail profiles and in this case yield quite smooth characteristics. However, in practice, many rail and wheel profile combinations yield characteristics with major discontinuities. Though, as the wheelset is displaced laterally, the rotation φ and vertical displacement uz are small it will be seen later that their derivatives with respect to uy play an important part in the equations of motion. Expressions for these derivatives will now be derived.

EQUATIONS OF MOTION

25

δ

r

r (mm)

r

l

l

l

l Rw (mm)

Rr (mm)

r

r

Figure 2.5 Variation of rolling radius, contact slope, and radii of curvature with lateral displacement for the wheel rail combination of Figure 2.2.

I l - rrtanδwr

l - rltanδwl

δuy

δz δφ

rl

rltanδwl

δu y

rr

rrtanδwr

Figure 2.6 Tilting and vertical displacement of wheelset due to lateral displacement. Wheelset rotates about instantaneous centre I.

26

RAIL VEHICLE DYNAMICS

Differentiating (3) with respect to uy dζ dζ du z dφ +l = rr − wr du y du y du y du y =

dζrr dηrr dζ wr dηwr − dηrr du y dηwr du y

and using (8) and (9) = tan δrr

dηrr dηwr − tan δ wr du y du y

and as φ is small ⎛ dη dη ⎞ = tan δrr ⎜⎜ rr − wr ⎟⎟ du y ⎠ ⎝ du y and differentiating (2) and substituting ⎛ dφ ⎞ ⎟ = − tan δrr ⎜⎜ 1 − r0 du y ⎟⎠ ⎝

(12)

Similarly ⎛ du z dφ dφ ⎞ ⎟ −l = tan δrl ⎜⎜ 1 − r0 du y du y du y ⎟⎠ ⎝

(13)

Hence from (12) and (13)

φy =dφ/duy = - ( tanδrl + tanδrr)/(2l - r0 tanδrr - r0 tanδlrl)

(14)

zy = duz/duy = l ( tanδrl - tanδrr)/(2l - r0 tanδrr - r0 tanδrl ) (15) which also follow from the geometry of Figure 2.6. For small displacements from the central position

φy = - σ/l ( 1 - r0 δ0 / l )

zy = -εuy/l ( 1 - r0 δ0 / l )

(16)

where

σ = δ0

ε = (δrr - δrl)l/2uy

(17)

ε is the parameter that determines the change of inclination of the normal force between wheel and rail as the wheelset is displaced laterally and therefore influences the gravitational stiffness. σ is the roll parameter. Figure 2.7 shows the variation of the geometrical derivatives φy and zy. Numerical values of the parameters for the wheel rail combination of Figure 2.2 are

EQUATIONS OF MOTION

27

r0 = 0.4500 m, δ0 = 0.0493 and l = 0.7452 m. If uy* is the lateral displacement of the wheelset in the plane of the original points of contact, then from Figure 2.6

u *y = 2luy/(2l - r0 tanδwr - r0 tanδwl)

(18)

because the wheelset is rotating about the point I. For small displacements this becomes u *y = u y/( 1 - r0 δ0 / l ) (19) From (19) the difference between the lateral displacement of the wheelset at the axle and at the contact points for the profiles of Figure 2.2 is only 3%. 0.4

0

φy (1/m)

0.2

zy

-0.2

0 -0.4

-0.2 -0.4 -10

0

10

-0.6 -10

uy (mm)

0

10

uy (mm)

Figure 2.7 Variation of derivatives zy and φy with lateral displacement for the wheel rail combination of Figure 2.2.

For small displacements equations (1)-(11) have a simple solution. In the vicinity of the point of contact Rw and Rr are the radii of curvature of the wheel tread and rail head respectively. When the wheelset is displaced laterally through a distance uy it can be seen from Figure 2.3 that the lateral displacement of the contact point on the rail is, approximately,

ηrr = Rr ( δrr - δ0 )

(20)

to the first order in δ0 and δrr The lateral displacement of the contact point on the wheel is

ηwr = Rw ( δwr - δ0 )

(21)

Substituting in (2) from (20), (21) and (4) and noting from (16) that φ = φyuy, yields for the right hand side

28

RAIL VEHICLE DYNAMICS

δrr , δrl = δ0 ± ε0uy/l

(22)

where, since to first order (1+ r0 δ0 / l)-1 = (1 - r0 δ0 / l)

ε0 = l ( 1 + Rwδ0 /l)/ (Rw - Rr)(1 - r0 δ0 / l)

(23)

and

δwr , δwl = δ0 ± ε 0* uy/l

(24)

where

ε 0* = l ( 1 + Rrδ0 /l)/ (Rw - Rr)(1 - r0 δ0 / l)

(25)

For example, for the wheel rail combination of Figure 2.2, ε0 = 6.423 and ε0∗ = 6.372. Note that ε0 - ε0∗ = σ. For conical wheels, Rw → ∞ , ε0∗ = 0 and

ε0 = δ0 /( 1 - r0 δ0 / l )

(26)

For profiled or worn wheels Rw δ0 / l << 1 and Rr δ0 / l << 1. For example, for the wheel rail combination of Figure 2.2, Rw δ0 / l = 0.01795 and Rr δ0 / l = 0.00989. Hence, in this case

ε 0 = ε 0* = l / ( Rw - Rr )( 1 - r0 δ0 / l )

(27)

Reference to Figure 2.3 shows that

ζwr = - Rw (cosδwr - cosδ0 ) = - Rw (δ02 - δwr2 )/2

(28)

approximately, so that substituting from equations (22) and (24), to first order rr, rl = r0 ± λ0uy

(29)

λ0 = δ0 Rw( 1 + Rrδ0 /l)/ ( Rw - Rr )( 1 - r0 δ0 / l )

(30)

where λ0 is the effective conicity

For example, for the wheel rail combination of Figure 2.2, λ0 = 0.1144. For conical wheels λ0 = δ0( 1 + Rrδ0 /l) /( 1 - r0 δ0 / l ) (31)

EQUATIONS OF MOTION

29

δr - δl

4

rr - rl (mm)

0.2

2

0.1

0

0 -0.1

-2 -4 -10

-0.2 0

10

-10

uy (mm)

0

10

uy (mm)

Figure 2.8 Variation of rolling radius difference and contact slope difference with lateral displacement for the wheel rail combination of Figure 2.2.

and for profiled or worn wheels where Rw δ0 / l << 1 and Rr δ0 / l <<1,

λ0 = δ0 Rw/ (Rw - Rr)(1 - r0 δ0 / l)

(32)

as first derived by Heumann [6]. For purely coned wheels the vertical displacement of the wheelset is negligible. Considering therefore the case of profiled or worn wheels so that ε 0 = ε 0*

ζrr = - Rr (cosδrr - cosδ0 ) =- Rr ( δ02 - δrr2 )/2

(33)

approximately, so that substituting (28) and (33) into (22) and (24) and then adding (3) and (6) the vertical displacement of the centre of mass is found to be uz = - ( Rw - Rr ) ε 02 uy2/2l2

(34)

consistent with (16). For completeness, to this could be added the small vertical displacement which occurs when the wheelset yaws through a small angle ψ, which is δ0lψ2, so that the total vertical displacement is uz = - ( Rwr - Rrr ) ε2y2/2l2 + δ0lψ2

(35)

The above analysis was first given in [7] and was subsequently elaborated by Blader [8] and Joly [9]. A very precise consideration of the linearisation of the wheel rail geometry has been given by de Pater [4]. For real wheel and rail profiles linearisation has severe limitations and whilst the above approximate approach is adequate to illustrate the nature of the problem, in practice a more precise numerical analysis is necessary. This is illustrated by the plots of the variation of the difference in rolling

30

RAIL VEHICLE DYNAMICS

radii and in contact angles with lateral displacement of the wheelset shown in Figure 2.8 for the wheel rail combination of Figure 2.2. Thus the wheel rail geometry is essentially non-linear. An equivalent conicity can be defined in terms of the difference of rolling radii on opposite wheels when the wheelset is displaced laterally [10]. It is dependent on the amplitude of the lateral displacement in a wheelset oscillation and is approximately equal to the mean slope out to the amplitude in question. Similarly, equivalent contact slope and roll parameters can be defined so that rr -rl = 2λ(a)uy

(36)

δr - δl = 2ε(a)uy /l

(37)

φ = - σ(a)uy /l

(38)

A number of more refined approaches to quasi-linearisation have been developed. The describing function method assumes that the motion is approximately sinusoidal. Hence a sinusoidal variation of the lateral displacement of the wheelset is assumed uy = asinωt

(39)

Then the resulting rolling radius difference can be expanded as a Fourier series, in which it will ordinarily be sufficient to take only the first term. Thus, the following amplitude dependent describing functions can be defined 2π

λ(a) = (1/πa)

∫ (r

r

− rl ) sin sds

(40)

0

2π

σ(a) = (1/πa)

∫ φ sin sds

(42)

0

0.25 0.2

λ

0.15

10

0.1

5

σ

0.05 0

ε

0

2

4

a (mm)

6

0

0

2

4

6

a (mm)

Figure 2.9 Describing functions for the wheel rail combination of Figure 2.2.

EQUATIONS OF MOTION

31 2π

∫

ε(a) = (1/πa) (δr − δl ) sin sds

(41)

0

In practice, the values of λ can vary over a wide range of values, 0.05 < λ < 0.5 and, as will be seen later, it is this variation that imposes difficulties on suspension design rather than the detailed shape of the rolling radius difference graph. Various practical aspects of equivalent conicity have been discussed by Pearce [11]. The above discussion generally neglects the effect of wheelset yaw on the wheelrail geometry, which is a realistic assumption except in the case of flange contact at large angles of wheelset yaw, such as occurs during flange climbing derailment. Three-dimensional geometry analyses have been given by Cooperrider and Law [3], Hauschild [12], Duffek [13], Yang [5], de Pater [14] and Muller [15]. The major effect of yaw at large angles of attack is to provide an additional torque on the wheelset, see Section 5. Many wheel-rail combinations experience contact at two points on one wheel for certain values of the lateral wheelset displacement. This commonly occurs, for example, when contact is made between the throat of the flange and the gauge corner of the rail, Figure 2.10. If the wheels and rails are considered to be rigid, as was done in the case of single-point contact, discontinuities occur in the geometric characteristics such as the rolling radius difference and slope difference graphs. This is revealed by singular solutions of equations (2)-(7). The mathematical aspects of two-point contact in this case have been considered by Yang [16]. However, in this case the distribution of forces between the points of contact depends on the elasticity in the contact areas [17,18].

Figure 2.10 Two point contact between wheel and rail.

32

RAIL VEHICLE DYNAMICS

2.4 Contact Mechanics 2.4.1 Elasticity and friction So far, the size of the contact patch has been assumed to be small compared with the dimensions of the wheelset. However, the forces acting between wheel and rail depend on their elastic interaction in the vicinity of the contact patch, as well as the effects of friction. A comprehensive treatment of contact mechanics is given in [19]. 2.4.2 Laws of friction Laws of friction were postulated by Leonardo da Vinci, Amontons and Coulomb. The nature of frictional phenomena is complex and simple laws cannot be supposed to be rigorously true under all conditions. Nevertheless, rail and wheel surfaces are sufficiently smooth that it can be usually assumed that the laws of friction are obeyed at each point in the contact patch, so the tangential friction traction σt acting at a point (i) opposes the direction of the relative motion that would occur if the friction did not exist, or opposes the relative motion if it occurs; (ii) is independent of the position in the area of contact; (iii) has magnitude always just sufficient to prevent relative motion at the point under consideration, provided that σt < µ σz where µ is the coefficient of limiting friction; (iv) is proportional to the normal pressure σz between the bodies when relative motion takes place so that σt = µ σz. It is assumed that µ is independent of the relative velocity. 2.4.3 Contact between wheel and rail When an elastic body, such as a wheel, is pressed against another elastic body, such as a rail, so that a normal load is transmitted, a contact area is formed. As the elastic deformation in the vicinity of the contact area is small its effect on the geometrical analysis of Section 3 can be neglected. Then, assuming that the curvatures of wheel and rail are constant in the vicinity of the contact patch, that the contact patch is small compared with the radii of curvature and the dimensions of the wheel and rails, the contacting bodies can be represented by elastic half-spaces and their shape can be approximated by quadratic surfaces. Usually it can be assumed that the material properties of wheel and rail are the same and in this case it can be shown that the tangential tractions do not affect the normal pressures acting between the bodies. Then with these assumptions, for the case where the wheels and rails are smooth, the dimensions of the contact area can be obtained from the theory of Hertz [20], which is described by Love [21] and Timoshenko and Goodier [22]. In this case the contact area is elliptical and the normal pressure distribution is semiellipsoidal. The shape and orientation of the contact ellipse depends only on the transverse radius of curvature of the wheel tread at the point of contact Rw (as calculated geometrically, measured positive if the wheel profile is hollow), the radius of curvature of the rail head at the point of contact Rr, and the wheel radius r. Then if N is the normal force acting in the contact area the major and minor semi-axes of the contact area, a and b are given by

EQUATIONS OF MOTION

33

(a/m)3 = (b/n)3 = 3N(1 - ν2)/E(1/r + 1/Rr - 1/Rw)

(1)

where E is Youngs modulus, ν is Poissons ratio, and m and n are functions tabulated in [22] as a function of β where cos β = (1/r + 1/Rr - 1/Rw)/(1/r - 1/Rr + 1/Rw) Here if cos β is positive then a lies across the rail, and if cos β is negative then a lies along the rail. Table 2 gives some example results for the wheel-rail geometry of Figure 2.2 for the cases where the wheelset is central or displaced laterally by ± 6 mm. In this Table and henceforth a and b are the semi-axes of the contact area in the forward and lateral directions. The variation of the shape of the contact area with wheelset displacement is noteworthy and, as illustrated in Table 2.1, in some wheel Table 2.1 Contact ellipses for geometry of Figure 2.2 N = 39240 N.

y r Rr Rw

mm mm mm mm m n β deg. a mm b mm

-6 449.59 336.50 784.93 1.099 0.9174 82.28 4.67 5.59

0 450 149.55 271.28 1.110 0.9082 81.42 5.14 4.20

6 452.42 55.935 86.43 1.461 0.7275 61.34 5.75 2.86

and rail combinations contact can take place over a large area in the flange throat and corner of the rail. The assumptions of the Hertzian theory are then invalid and the calculation of normal pressures and tangential tractions must be considered simultaneously. Methods of analysis have been developed for this case by Nayak and Paul [23] and by Kalker [24]. 2.4.4 Creep 2.4.4.1 Background As discussed in Chapter 1, Carter introduced the fundamental concept of creep in relation to the rolling wheel. The relationship between the creepage and the creep forces has been investigated for a variety of practical situations. Carter [25] gave a solution to the creep problem for the two-dimensional case of two long cylinders, pressed together by a normal force, and transmitting a tangential force across the contact strip. Similar results were obtained by Poritsky [26], and in the discussion of [26], Cain [27] pointed out that the region of adhesion must lie at the leading edge of the contact area. A three-dimensional case was solved approximately by Johnson [28] who considered an elastic sphere rolling on an elastic plane. This solution was based on the assumption that the area of adhesion is circular and tangential to the

34

RAIL VEHICLE DYNAMICS

area of contact (which is also circular) at the leading edge. Good agreement with experiment was obtained. The influence of spin about an axis normal to the contact area was first studied by Johnson [29]. The general case where the contact area is elliptical was considered by Haines and Ollerton [30] who confined their attention to creep in the direction of motion and assume that Carter's two-dimensional stress distribution holds in strips parallel to the direction of motion. A general theory for the elliptical contact area, based on similar assumptions to those made in [28], was developed by Vermeulen and Johnson [31], yielding the relationship between creepage and tangential forces for arbitrary values of the semi-axes of the contact area. De Pater [32] initiated the complete solution of the problem by considering the case where the contact area is circular, and derived solutions for both small and large creepages, without making assumptions about the shape of the area of adhesion. However, this analysis was confined to the case where Poisson's ratio was zero; Kalker [33] gave a complete analytical treatment for the case in which Poisson's ratio is not zero. The agreement between these theoretical results and the experimental results of Johnson [28] is very good. Kalker gave a full solution of the general three-dimensional case in [34] covering the case of arbitrary creepage and spin, and subsequently gave simpler approximate solution methods [35]. Kalker’s theory is described in [36]. 2.4.4.2 Formulation of the creep problem A local set of axes is defined with origin at the centre of the contact patch, with the axis Oz normal to the contact area, and fixed in the contact patch so that the material forming the contacting surfaces moves backward in the direction -Ox at speed V. If tangential tractions are applied by the wheel to the rail in the contact patch elastic strains result which cause a departure from the pure rolling motion, which is measured in terms of the longitudinal creepage

γ1 = (Vxw − Vxr )/ V

(4)

γ2 = (Vyw - Vyr )/ V

(5)

ω3 =( Ωzw - Ωzr )/V

(6)

the lateral creepage

and the spin

where Vxw and Vyr are the rigid body velocities of the wheel in the Ox, Oy directions, Vxr , Vyr are the rigid body velocities of the contact point of the rail, the mean velocity of the wheel along the rail is V = (Vxw + Vxr )/ 2, and Ωzw and Ωzr are the angular velocities of the wheel and rail about the Oz axis. In the plane of the contact area, the relative velocities vary and are vx = vxw - vxr

(7)

EQUATIONS OF MOTION

35

vy = vyw - vyr

(8)

Similarly, the relative displacements in the plane of the contact area will be denoted by ux and uy in this Section and are given by u x = u xw - u xr

(9)

u y = u yw - u yr

(10)

A particle will experience a change in velocity through moving to a position where the displacement has a different value, so that at time t + δt the particle which was originally at x is at x + Vδt. Therefore the change in displacement in time δt is ux(x + Vδt, t + δt) - ux(x, t) and hence the rate of change of the relative displacement is V∂ux/∂x - ∂ux/∂t. To this must be added the contribution of creepage from (4), (5) and (6) so that vx = V(γ1 - ω3y) + V∂ux/∂x - ∂ux/∂t

(11)

vy = V(γ2 + ω3x) + V∂uy/∂x - ∂uy/∂t

(12)

The normal pressure acting at any point in the contact area is σz, given by the theory of Hertz, and there are tangential tractions σx and σy, with resultant σt, applied by the wheel to the rail

σt = ( σx 2 + σy2 )1/2

(13)

The contact area consists of a region in which there is adhesion and a region in which there is slipping. In the former vx = vy = 0 and if µ is the coefficient of limiting friction | σt | ≤ µσz

(14)

and in the region in which there is slip |σt| = µσz

(15)

and the direction of the resultant traction opposes the slip velocity

σx / vx = σy / vy =- σt/ ( vx 2 + vy2 )1/2

(16)

A particle entering the contact area is initially unstrained, so that at the leading edge the tangential tractions must be zero

σx ( xl, y) = 0

(17)

36

RAIL VEHICLE DYNAMICS

σy ( xl, y) = 0

(18)

where xl = a{1 - (y/b)2}1/2

(19)

defines the leading edge. 2.4.4.3 Creep forces for small creepages For small values of the creepages and spin, there is adhesion over the complete contact area and there is a linear relationship between the creep forces and the creepages. This is of the form T1 = - f11γ1

(20)

T2 = - f22γ2 - f23ω3

(21)

M3 = f23γ2 - f33ω3

(22)

The moment M3 can ordinarily be neglected. The general linear case of a three dimensional wheel rolling on a three dimensional rail has been analysed by Kalker [37]. In this case the fij are given by f11 = Gc2C11

f22 = Gc2C22

f23 = Gc3C23

(23)

where c2 = ab, G is the elastic modulus of rigidity and the coefficients Cij are tabulated in [37]. Table 2.2 gives some example results for the wheel-rail geometry of Figure 2.2 for the cases where the wheelset is central or displaced laterally by ± 6 mm. The longitudinal and lateral creep coefficients f11 and f22 are similar in size and assuming their equality is sometimes a useful approximation.

Table 2.2 Example linear creep coefficients N = 39240 N. y mm a/b C11 C22 C23 f11 MN f22 MN f23 kNm

-6 0.835 3.95 3.46 1.29 8.20 7.19 13.7

0 1.222 4.33 3.95 1.72 7.44 6.79 13.7

6 2.014 5.10 4.90 2.64 6.67 6.40 14.0

EQUATIONS OF MOTION

37

2.4.4.4 Creep forces for arbitrary creepages Kalker [34] has solved equations (7)-(18) in the case of steady motion using the basic equations of the theory of elasticity to relate the displacement differences ux, uy to the tractions σx, σy. The creep forces are functions of the creepages, the ellipticity a/b of the contact patch, the normal force N and µ the coefficient of friction. Kalker has developed computer programs and tables of results which are in nondimensional form [34]. A set of representative results of these calculations is shown in Figure 2.11. In the absence of spin, and constant lateral creep, as the longitudinal creep increases the longitudinal creep force eventually achieves the limiting value of µN, whilst the lateral creep force is reduced. An analogous variation occurs for lateral creep. In the presence of spin, the lateral force is increased at the expense of a reduction in the longitudinal force. The significant contribution of spin to the lateral force is to be noted. For small values of the spin the lateral force is proportional to the spin, particularly at low creepages, but as the spin increases the lateral force reaches a maximum and then reduces in value at large values of spin. As might be expected from considerations of symmetry, the lateral creep force is affected equally by positive or negative values of longitudinal creepage and the longitudinal creep force is zero if the longitudinal creepage is zero. At very large values of the creepages and spin, there is pure sliding and the forces correspond to the laws of friction discussed above. Experimental measurement of creep forces carried out by Hobbs [38], Illingworth [39] and Brickle [40] suggest that Kalker’s results are verified when the contacting surfaces are clean. If the surfaces are contaminated the situation is less clear, the most relevant experiments being those of Pearce and Rose [41]. It seems likely that the coefficient of friction is affected by surface contamination but the creep coefficients are not. Measurements of the resultant forces acting on wheelsets exerting large tractive efforts are not consistent with the theory as it stands. Knothe et al [42] point out three main deviations: (a) measurements show large dispersion in the tractive force-creepage characteristic (b) the maximum value of the tractive force occurs at a higher value of the creepage than theory predicts (c) beyond this value of the creepage the measured tractive force decreases. This latter effect can be explained by assuming that the coefficient of friction depends on the surface temperature and hence on the sliding velocity. An extension of Carter’s theory to cater for this effect shows that the traction forces at large creepages not only depend on the creepages but also on vehicle speed [43].

2.4.4.5 An approximate theory for arbitrary creepages An alternative to the complete numerical solution of the problem is the approximate and faster approach given by Kalker’s simplified theory [35] which is implemented in the computer program FASTSIM. If the flexibility in the contact area was isotropic then ux = fσx uy = f σy (24)

38

RAIL VEHICLE DYNAMICS

γ2= 0.001

20

10

ω3=0.6

ω3=0 -T1 (kN)

-T2 (kN)

ω3=0.6 2

γ1

0

ω3=0

0.01

0

γ1= 0.001 20

-T1 (kN)

0.01

γ2

0.01

20

-T2 (kN) ω3=0.6

ω3=0

ω3=0

ω3=0.6 0

γ1

γ2

0.01

0

γ1= 0.002 20

10

γ2=0.001 -T1 (kN)

γ2=0

-T2 (kN)

γ2=0

γ2=0.001 0

ω3

1

0

ω3

1

Figure 2.11 Typical relationships between creep forces and creepages and spin. N =39240 N, µ = 0.3, r0 = 0.45 m, Rr = 149 mm, Rw = 271 mm.

then substitution in (11) and (12) would give for the area of adhesion ∂σ x γ 1 ω 3 y = − ∂x f f

(25)

EQUATIONS OF MOTION

39

∂σ y ∂x

=

γ 2 ω3 x + f f

(26)

These equations are approximated by ∂σ x γ 1 ω3 y = − ∂x f1 f3

∂σ y ∂x

=

(27)

γ 2 ω3 x + f2 f3

(28)

where f1, f2 and f3 are determined so as to obtain results which agree with linear theory for small creepages. This leads to f1 = 8a/3C11G

f2 = 8a/3C22G

f 3 = πa a / b / 4C23G

(29)

Equations (27) and (28) can then be integrated, subject to the boundary conditions (13-19). 2.4.4.6 Heuristic approximations The analysis by Vermeulen and Johnson [31] is valid for arbitrary creepages but not for spin, and represented a simple way of accounting for creep saturation. It was, therefore, useful for dynamics analysis. Shen et al [44] proposed an extension to the method of Vermeulen and Johnson which includes the effects of spin. If the resultant of the creep forces calculated from (20) and (21) is TL then the effect of creep saturation is represented by if T ≤ 3µΝ

⎧⎛ T L ⎞ 1 ⎛ T L ⎞ 2 1 ⎛ T L ⎞ 3 ⎫ ⎪ ⎪ ⎟− ⎜ ⎟ + ⎜ ⎟ ⎬ T = µN ⎨⎜ 3 27 µ µ µ N N N ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪⎩⎝ ⎭

if T > 3µΝ

T = µN

(30)

(31)

and then the longitudinal and lateral forces from (20) and (21) are reduced by the ratio T/TL. Another approximation has been given by Polach [45]. 2.4.4.7 Non-Hertzian effects Kalker [24] has extended the simplified theory of rolling contact to cover this case which can occur particularly in the case of worn wheels and rails. The case of nonelliptical contact areas has also been considered by Knothe and Le-The Hung [46]. In the case of two-point contact, the complicated non-Hertzian contact patches can be approximated by one or more elliptic patches [18]. A further assumption of the Hertz theory is that the contacting surfaces are perfectly smooth. Nayak and Paul

40

RAIL VEHICLE DYNAMICS

[23, 47] have given an alternative theory which assumes that the friction in the contact area arises from the deformation of friction junctions and thus exploits the rough surface theory of Greenwood [48]. An analysis by Bucher [49] has shown that the longitudinal creep coefficient is reduced when the wheel and rail surfaces are very rough. 2.4.5 Transient effects Finally, the above analysis considers that steady-state conditions have been established and transient effects have been neglected. Newland [50] and Knothe and Gross-Thebing [51] have considered the calculation of the complex transfer functions for the case of periodically varying creepages. Their results show that the creep forces deviate significantly from the steady state case only when the wavelength of the motion approaches the diameter of the contact patch, a situation unusual in vehicle dynamics. A crude representation of the transient behaviour for small creepages is given by d }T1 = - f11γ1 dt d {1 + (2a/V) }T2 = - f22γ2 - f23ω3 dt

{1 + (2a/V)

(32) (33)

which will suffice later for a preliminary examination of the effect.

2.5 Creepages The longitudinal velocity components of the wheel relative to the origin of a coordinate system O123 fixed in the contact patch are V1rw= Ωrr - l ψ

(1)

01

ψ

.

Ω r r - lψ

uy

.

u y * + Ω r rψ

02

Figure 2.12 Velocity components of wheelset at contact point on right hand wheel.

EQUATIONS OF MOTION

41 l

l V/(R 0 - rsin φ 0 )

φ0 R 0 - lcos φ 0 R0

Figure 2.13 Velocity components in curve.

V1lw = Ωrl + l ψ

(2)

where Ω = θ (= -V/r0 when the wheelset is in the central position), so that Ωrr is the forward velocity at the tread in the plane of the wheel. The velocity in the Oy direction of the wheelset axes is Ωrrψ. The velocity of the wheel normal to the contact patches must be zero, and so the velocity components of the wheel in the plane of the wheel at the contact points are, Figure 2.12 V2rw = ( u *y + Ω rrψ ) sec δr

(3)

V2lw = ( u *y + Ω rlψ ) sec δl

(4)

The angular velocities about the normal to the contact patch are

Ω3rw = Ω sinδr + ψ cosδr

(5)

Ω3lw = −Ω sinδl + ψ cosδl

(6)

The forward velocity of the wheelset is measured at the centre of mass and Figure 2.13 shows that the angular velocity of rotation due to the curve is V/(R0 - r0sinφ0). Hence the longitudinal velocity of the right hand and left hand rails relative to the contact patch will be V(R0  lcosφ0)/(R0 - r0sinφ0) or since φ0 is small V1rr = - V (1 - l/R0)

(7)

V1lr = - V (1 + l/R0)

(8)

42

RAIL VEHICLE DYNAMICS

The other velocity components of the rail relative to the contact area axis system are V2rr = 0

(9)

V2lr = 0

(10)

Ω3rr = −-Vcosδr /R0

(11)

Ω3lr = − Vcosδl /R0

(12)

Substituting into equations (4.4), (4.5) and (4.6) yields the following expressions for the creepages

γ1r = Ω rr /V + 1 - lψ / V - l/R0

(13)

γ1l = Ω rl /V + 1 + lψ / V + l/R0

(14)

γ2r = ( u *y + Ω rrψ ) sec δr/V

(15)

γ2l = ( u *y + Ω rlψ ) sec δl/V

(16)

ω3r = (Ω/V)sinδr + ( ψ / V )cosδr + cosδr /R0

(17)

ω3l =− (Ω/V)sinδl + ( ψ / V )cosδl + cosδl /R0

(18)

2.6 Contact Forces The forces acting in each contact area, in the contact axis system, are obtained by evaluation of the creep forces as described in Section 4 using equations (5.13) to (5.18) to evaluate the creepages. From Figure 2.14 these forces referred to the wheelset axis system are Txr = T1r

(1)

Tyr = T2rcosδr + T3rsinδr

(2)

Tzr = T3rcosδr - T2rsinδr

(3)

Txl = T1l

(4)

Tyl = T2lcosδl - T3lsinδl

(5)

Tzl = T3lcosδl + T2lsinδl

(6)

EQUATIONS OF MOTION

43

01

01 Tx

T 1l

T 1r Ty

Mz

Mx Ty 02 T 2r T 3l T 2l

Tz

T 3r

02

03 03

Figure 2.14 Definition of axes and creep forces.

The resultant creep forces acting on the wheelset in wheelset axes are derived by assuming that the forces act on the wheelset at the positions of the contact areas when the wheelset is in the central position (with one exception discussed below). They are then given by Tx = Txr + Txl

(7)

Ty = Tyr + Tyl

(8)

Tz = Tzr+ Tzl

(9)

Mx = Tzrl - Tzll - Tyrrr - Tylrl

(10)

My = ( Txr + Tyrψ − Tzrψ tanδr )rr + ( Txl + Tylψ+ Tzlψ tanδl )rl Mz = Txll - Txrl

(11) (12)

44

RAIL VEHICLE DYNAMICS

T xl rltanδ l

ψ

Txl

rl

T yl

rltanδ lψ Tzl Figure 2.15 Effect of longitudinal shift of contact point.

The terms Tzrψ tanδr and Tzlψ tanδl in equation (11) arise from the longitudinal shift of the contact point from vertically below the wheelset axis when the wheelset is yawed, Figure 2.15. The importance of this term was pointed out by Gilchrist and Brickle [53] who refer to Matsui [54] and Bodecker [55].

2.7 Kinematics of the Wheelset Referring to the axis systems defined in Section 2.2, the vector velocity of the wheelset with respect to fixed axes instantaneously aligned with Oxyz is v where v = av + bv

(1)

the velocity of O is a

v= [V 0

0 ]T

(2)

and the velocity of O* relative to O is b

v = [ u x u y u z ]T

The angular velocity vector of the wheelset ω with respect to Oxyz is

(3)

EQUATIONS OF MOTION

45

ω = aω + bω + cω

(4)

where aω is the angular velocity vector of the reference frame Oxyz with respect to Oxyz (due to track geometry and vehicle forward speed), bω is the angular velocity vector of O*x*y*z* with respect to Oxyz, and cω is the angular velocity vector of the wheelset due to wheelset rotation about its axis of revolution. Therefore

ω = [ φ0 Vsinφ0/R0

a

ω = [ φ

b

0

Vcosφ0/R0]T

(5)

ψ ]T

(6)

and if the angular velocity vector of the wheelset about its axis is dω where

ω=[0

d

θ

0 ]T

(7)

then

ω = G dω

(8)

c

where G is the rotation matrix ( G is orthogonal, GTG = I33 the 3 × 3 unit matrix), the elements of which depend on the relative angular orientation of the two frames. By carrying out successive rotations of the axes it is found that ⎡cosψ G = ⎢⎢ sin ψ ⎢⎣ 0

− cos φ sin ψ cos φ cos ψ sin φ

sin φ sin ψ ⎤ − sin φ cos ψ ⎥⎥ ⎥⎦ cos φ

(9)

If the rotations ψ and φ are small the order in which the rotations take place is immaterial and (9) reduces to ⎡ 1 G = ⎢⎢−ψ ⎢⎣ 0

ψ 0⎤ ⎥ 1 φ⎥ − φ 1 ⎥⎦

(10)

As the variation in the rotational speed of the wheelset will usually be small

θ = - V/r0 + χ where χ is small. Then to first order

ω = [ φ0 + φ + Vψ/r0, Vsinφ0/R0 - V/r0 + χ, Vcosφ0/R0 + ψ - Vφ/r0]T

(11)

RAIL VEHICLE DYNAMICS

46

= [ ωx

ω y ω z ]T

(12)

2.8 Equations of Motion With the axis systems defined as above, the wheelset reference axes are not fixed in the wheelset and are themselves moving with respect to inertial space. The formulation of the equations of motion therefore require special care and their logical derivation has been considered by de Pater [4] and Schiehlen [56]. The derivation given here follows that of de Pater whose paper should be consulted for rigorous detail, and applies the principles of linear and angular momentum referring the forces and moments, and the linear and angular displacements to the wheelset reference axis system Oxyz. Assuming that the displacements ui and their derivatives are small, the vector equations of motion if the axes are moving are p + aϖ p = X

(1)

h + aϖ h = L

(2)

where

⎡ 0 ⎢ a ϖ = ⎢ aω 3 ⎢ − aω 2 ⎣

ω2 ⎤ ⎥ − aω1 ⎥ 0 ⎥⎦

− aω 3

a

0 ω1

a

(3)

(the tilde will be used to indicate the skew symmetric matrix corresponding to a vector quantity with the elements arranged as above). In (1) and (2)

[

p = m u x + V

[

h = ℑ ωx

u y

ωy

u z

ωz

]

]

T

T

(4) (5)

where ℑ is defined with respect to Oxyz and therefore varies with φ and ψ. If I* is defined with respect to O*x*y*z* then ⎡I 0 I * = ⎢⎢0 I y ⎢⎣0 0 so that, to first order,

0⎤ 0⎥⎥ I ⎥⎦

(6)

EQUATIONS OF MOTION

47

⎡ I ⎢ ℑ = GI G = ⎢− ( I − I y )ψ ⎢ 0 ⎣ *

T

⎤ ( I − I y )ψ 0 ⎥ ( I − I y )φ ⎥ Iy ⎥ I − ( I − I y )φ ⎦

(7)

Evaluation of (1) and (2), neglecting second order terms, then leads to the equations of motion of the unconstrained wheelset written in the form Ar + F (r ) = 0

(8)

where r = [ ux uy uz φ

χ

ψ ]Τ

(9)

and A = diag [ m m m I

Iy

I]

(10)

The forces F(r) are given by F(r) = cF + gF + wF + fF + sF + nF

(11)

where cF are the centrifugal forces c

F = [ 0 mV2cosφ0/R0 - mV2sinφ0/R0

φ0 g

IyV (d/dt)(sinφ0/R0)

IV(d/dt)(cosφ0/R0) ]T

(12)

F are the gyroscopic forces g

F=[0 0

0 IyV ( ψ + Vcosφ0/R0)/r0 0

-Iy V ( φ + φ0 )/r0 ]T

(13)

In (12) and (13) it is assumed that V/R0 is a first order quantity. wF are the gravitational forces w

F = [ 0 - mgsinφ0 - mgcosφ0 0

0

0 ]T

(14)

f

F are the creep forces dependent on the terms in equations (6.1)-(6.6) in T1r, T2r, T1l, T2l only and nF are the normal forces dependent on the terms in equations (6.1)- (6.6) in T3r and T3l only, so that f

s

F + nF = [ - T x

- Ty

- Tz

- Mx - My - Mz ] Τ

(15)

F are the forces applied by the rest of the vehicle and which are necessary to sus-

RAIL VEHICLE DYNAMICS

48

tain the motion of the wheelset along the track Lψ ]Τ

(16)

mux - Tx + Px = 0

(17)

2 mu y - Ty + mV cosφ0/R0 - mgsinφ0 + Py = 0

(18)

muz - Tz - mV2sinφ0/R0 - mgcosφ0 + Pz = 0

(19)

Ix φ + Ix φ0 - Mx - Iy θ ( ψ + Vcosφ0/R0) + Lφ = 0

(20)

Iy θ - My + IzV(d/dt)(sinφ0/R0) + Lθ = 0

(21)

Iz ψ + Iy θ ( φ + φ0 ) - Mz + IzV(d/dt)(cosφ0/R0) + Lψ = 0

(22)

s

F = [ Px Py Pz

Lφ Lθ

Written out in full, these equations are

2.9 Constrained Motion The actual motion of the wheelset is constrained. The motion of the wheelset is assumed to be constrained to be at constant speed V so that u x = V and relative to the wheelset frame of reference u x = 0. Furthermore, the wheelset is constrained to remain in contact with the track at the two points of contact. As nF are the normal forces dependent on the terms in equations (6.1)-(6.6) in T3r and T3l only, from equations (6.7)-(6.12) it is possible to write n

F = HT

(1)

where T = [T3r T3l]T. The normal forces T3r and T3l are given by the theory of Hertz, and may be expressed as an equivalent linear stiffness kh so that if the normal deflections at the contact points on the right and left hand sides respectively are e = [ e3r e3l ]T. T=ke

(2)

e = HΤr

(3)

where k = diag[ khr khl ] and

Substituting (3) and (2) into (1) and (8.8) results in the following form of the equations of motion

EQUATIONS OF MOTION

49

A r + F0(r) + HkHTr = 0

(4)

where F0 = cF + gF + wF + fF + sF. One approach to the solution of the equations of motion is to solve these equations directly. (This may be particularly appropriate in the case of two-point contact. For methods of computation in this case see [57], [58] and [59], for example). However, as the Hertz stiffness is large, the normal deflections will be small in comparison with the displacements associated with the gross motion of the wheelset. Therefore, a new set of generalised coordinates qθ, qy and qψ is defined which represent motions in which there is no normal displacement of the contact area, so that r = Jq

(5)

or ⎡ux ⎤ ⎡0 ⎢u ⎥ ⎢0 ⎢ y⎥ ⎢ ⎢uz ⎥ ⎢0 ⎢ ⎥=⎢ ⎢ φ ⎥ ⎢0 ⎢ θ ⎥ ⎢1 ⎢ ⎥ ⎢ ⎢⎣ψ ⎥⎦ ⎢⎣0

0 0⎤ 1 0⎥⎥ zy 0⎥ ⎥ φy 0⎥ 0 0⎥ ⎥ 0 1⎥⎦

⎡ qθ ⎤ ⎢ ⎥ ⎢q y ⎥ ⎢qψ ⎥ ⎣ ⎦

where zy and φy are dependent on qy through the specified wheel-rail geometry as expressed by equations (3.14) and (3.15). Since u = 0 for arbitrary values of q, on substitution from (5) into (3), to first order HT J = 023

(6)

where 023 is the 2 × 3 null matrix. Therefore premultiplying (8.8) by JT the motion of the system is governed by the set of three equations of motion

 + JTF = 0 JTAJ q

(7)

Hence the equations of motion become Iy qθ - Rθ + IzV(d/dt)(sinφ0/R0) + Lθ = 0

(8)

{m( 1 + zy2) + Ixφy2} q y + Ixφy φ0 + mV2cosφ0/R0 - mgsinφ0 - zymV2sinφ0/R0 - zymgcosφ0- φyIy qθ ( qψ + Vcosφ0/R0) - Ry + Py + Pzzy + Lφφy = 0

(9)

Iz qψ + Iy qθ (φy q y + φ0 ) + IzV(d/dt)(cosφ0/R0) - Rψ + Lψ = 0

(10)

where

RAIL VEHICLE DYNAMICS

50

Ry = Ty + Tzzy + Mxφy

(11)

Rθ = My

(12)

Rψ = Mz

(13)

are the generalised creep forces. Substituting into equation (11) from equations (6.8) to (6.10) and (3.14) and (3.15) and neglecting terms of third order in contact angles Ry = T2rcosδr + T2lcosδl - zyT2rsinδr + zyT2lsinδl - φylT2rsinδr

−- φylT2lsinδl - φyrrT2rcosδr - φyrlT2lcosδl

(14)

which is independent of T3r and T3l to first order, as expected, as the normal reactions do no work, the normal relative velocities being zero at the contact points. Similarly, substituting in equations (6.11) from equations (6.1) to (6.6) My = (T1r + T2rψ/cosδr)rr + (T1l + T2lψ/cosδl)rl

(15)

T3r and T3l can be determined by substituting uz = z y uy and φ = φ y u y (where zy and φy are given by (3.14-15)) into (8.19) and (8.20) respectively, and eliminating uy using (8.18). Py, Pz, Lx and Lz are forces and moments transmitted by the suspension and are a function of the relative motion between the wheelset and the rest of the vehicle. The evaluation of these forces requires consideration of a complete vehicle which is considered later. If the applied braking or tractive torque Lθ is zero then the solution of the coupled equations (8)-(10) will yield the wheelset rotational speed θ , and (8.17) will give the value of the longitudinal force necessary to sustain the motion at constant speed V. Otherwise both θ and V can be specified and (8) will give the torque, and (8.17) will give the longitudinal force, necessary to sustain the steady values of V and θ . The various steps in the solution of the equations of motion are shown in Figure 2.16. Consideration of practical values shows that a number of small terms can be neglected in the equations of motion (8)-(10). In (9) as zy and φy are small {m( 1 + zy2) + Ixφy2} can be replaced by m. φyqy is small compared with unity and can be neglected though the gyroscopic terms will be retained for future reference. It will be adequately accurate to put cosφ0 = 1 and sinφ0 = φ0. The term zymgcosφ0 = zymg represents a restoring force generated by the gravitational stiffness and arises from the raising of the centre of mass as the wheelset is displaced laterally. The rate of working of the various forces acting on an unrestrained wheelset when running on straight level track is obtained by multiplying equations (8-10) respectively by qθ , q y , and qψ , and multiplying (8.17) by V and adding, giving

EQUATIONS OF MOTION

• •

51

•

uy ψ θ y ψ

GEOMETRY Equs (3.1-11)

δ, r

φ0 R0 y0 r Rr Rw CONTACT AREAS

CREEPAGES Equs (5.13-18)

Equs (4.1)

ab COEFFICIENTS

Equs (4.23) T3

fij

CREEP FORCES Equs (4.20-22, 30-31)

NORMAL FORCES T1 T2

Equs (8.19-20)

T1 T2 FORCES Equs (6.7-12) Qy My Mz EQUATIONS OF MOTION

Equs (9.8-10) •

u y ψ θ y• ψ• PATH Equs (2.1-3) XYΨ Figure 2.16 Steps in the solution of the equations of motion (heuristic creep model).

52

RAIL VEHICLE DYNAMICS

d (T + U G ) − Rθ q θ − R y q y − Rψ qψ + Lθ q θ + PxV = 0 dt

(17)

where the kinetic energy T of the system is T=

1 1 1 1 {m(1 + z y 2 ) + I x φ y 2 }q y 2 + mV 2 + Iqψ 2 + I y q θ2 2 2 2 2

(18)

the potential energy UG is UG = −mgzy qy

(19)

Substituting for Ry, Rθ, and Rψ from equations (12)-(14), (6.1)-(6.12) and (5.13)(5.18) d (T + U G ) − VT1r γ 1r − VT1l γ 1l − VT2 r γ 2 r − VT2 l γ 2 l + Lθ q θ + PxV = 0 dt

(20)

This follows from the definition of the creepages, equations (5.13−18).

2.10 Equations of Motion for Small Displacements Much of importance can be discussed using a linearised form of the equations of motion. These may be derived either by assuming that the motions are of very small amplitude, as in this section, or, more realistically, by adopting describing functions as discussed in Section 3. To the first order in qy (1) Ω = θ = −V / r0 so that the equations of motion for the linearised system will have two generalised coordinates qy and qψ. For simplicity, qy and qψ will be represented by y and ψ henceforth. The corresponding creepages are given by equations (5.13) - (5.18), if φ0, δr and δl are small and r0 << R0, as

γ1r = - rr /r0 + 1 - lψ / V - l/R0

(2)

γ1l = - rl /r0 + 1 + lψ / V + l/R0

(3)

γ2r = y /V - ψ

(4)

γ2l = y /V - ψ

(5)

EQUATIONS OF MOTION

53

ω3r = − δr/r0 + ψ / V + 1/R0

(6)

ω3l = δl/r0 + ψ / V + 1/R0

(7)

The above equations show that when the wheelset is central, on straight track,

γ1r = γ1l =γ2r = γ2l = 0

(8)

ω3r = - ω3l = - δ0 / r0

(9)

but

Substituting from equations (4. 20) - (4.21) there are corresponding forces T = T = 0 1r

(10)

1l

T = -T = -f ω = f δ /r 2r

2l

23

3r

23 0

0

(11)

In addition, T = T = -N 3r

3l

(12)

0

so that N0 is the normal force when the wheelset is central. As pointed out by de Pater [55], because of this steady-state spin there will be a first-order variation of the lateral creep force due to the change in the normal force as the wheelset is displaced, Figure 2.17. Note that by symmetry ∆Nr = - ∆Nl = ∆N. De Pater [4] and Yang [60] have carried out the analysis for arbitrary values of δ0. Equations (4.20)-(4.22) must be extended and to first order T1r = - f11γ1r

∆N

∆N

N0

N0

(13)

δ 0f23/r0 ∆Nδ 0f23/r0N0 ∆Nδ 0f23/r0N0

δ 0f23/r0

Figure 2.17 Variation of the lateral creep force due to the change in the normal force as the wheelset is displaced.

54

RAIL VEHICLE DYNAMICS

T1l = - f11γ1l T =-f γ -f ω +δ 2r

22 2r

23

3r

(14)

0

T =-f γ -f ω +δ 2l 22 2l 23 3l 0

∂f 23 ∆N/r 0 ∂N

(15)

∂f 23 ∆N/r 0 ∂N

(16)

where the fij are evaluated with the wheelset central. From equation (4.23) it can be seen that f23 is proportional to c3 and hence from (4.1)

∂f 23 = f /N 23 0 ∂N

(17)

The variation of normal force with lateral displacement of the wheelset is found from the equation of rolling moments, (8.20). The rolling moment of the creep forces is given by equation (6.10). The creep forces are given by equations (13)-(16) and the creepages and spins by (2)-(7). Using the geometric parameters from (3.24) and (3.30) then M = 2f22 r0 ( y /V - ψ ) + 2f23 r0 (- ε 0* y/ r l + ψ /V + 1/R0) 0

x

+ 2N0 (λ0δ0y + ε 0* r0 y/l) - 2∆Nl(1 + f23δ0/N0l)

(18)

where it is assumed that δ0l << r0 and δ0r0 << l and second order terms in the displacements are neglected. Substituting from (18) into (8.20) and using (1) yields

∆N = {-I φ - I φ 0 - I V( ψ + V/R0) /r0 + 2f22 r0 ( y /V - ψ ) + x x y 2N0(λ0δ0y + ε 0* r0 y/l) - Lφ + 2f23r0 (- ε 0* y/ r0 l + ψ /V + 1/R0)}/ 2l(1 + 2f23/N0) (19) The generalised creep forces are given by (9.11-13) and to first order are R = - 2f22( y /V - ψ ) - 2f23(- ε 0* y/ r0 l + ψ /V + 1/R0) + 2∆Nδ0f23/N0 r0 y

Rψ = - 2f11(λ0yl/ r0 + l2 ψ /V + l2/R0)

(20) (21)

Substituting from (20) and (21) into (9.9) and (9.10) the equations of motion take the form

EQUATIONS OF MOTION

55

[ As2 + (B/V + GV)s + C + Eg + E] q

= Q

(22)

where q = [ y ψ ]T, s is the operator d/dt and Q represents the externally applied forces. A is the inertia matrix, B and C are the creep damping and stiffness matrices, G is the gyroscopic matrix, Eg is the gravitational stiffness matrix and E is the elastic suspension stiffness matrix. A, B, Eg and E are symmetric matrices but C is not symmetric indicating that the system is non-conservative. The contribution of the variation of the lateral creep force due to change in the normal force is, for typical values of the parameters, negligible except in the case of G where G12 is multiplied by a factor κ

κ = δ ( 1 - f /N r ) 0

23

0 0

(23)

If, moreover, the small term M3 from equation (4.22) is retained, then the system matrices adopt the simple form ⎡m 0 ⎤ A=⎢ ⎥ ⎣0 I⎦

⎡ 2 f 22 B=⎢ ⎣− 2 f 23

2 f 23 ⎤ 2 f 11l 2 ⎥⎦

(24)

(25)

⎡− 2 f 23 ε 0* / r0 l − 2 f 22 ⎤ C=⎢ ⎥ 2 f 23 ⎥⎦ ⎢⎣ 2 f 11 λ0 l / r0

(26)

− I yκ / r0 l ⎤ ⎡ 0 G=⎢ ⎥ 0 ⎣ I y σ / r0 l ⎦

(27)

⎡Wε * / l 0 ⎤ Eg = ⎢ 0 ⎥ − Wlδ0 ⎥⎦ ⎢⎣ 0

(28)

and representing the action of a lateral spring of stiffness ky and a yaw spring of stiffness kψ 0⎤ ⎡k y E=⎢ (29) ⎥ ⎣ 0 kψ ⎦

56

RAIL VEHICLE DYNAMICS

The generalised applied forces are given by ⎡− mV 2 / R0 + mgϕ 0 + σI yV 2 / R0 r0 + 2 f 23 / R0 ⎤ ⎥ Q=⎢ I yVϕ0 / r0 − I zVsR0 − 2 f 11 l 2 / R0 ⎢⎣ ⎥⎦

(30)

Written out in full, the equations of motion are my + 2 f 22 y / V + K y − 2 f 22ψ + (2 f 23 / V − I yκV / r0l )ψ = Qy

(31)

2 f11λ0ly / r0 − (2 f 23 / V − I yδ 0V / r0l ) y + I zψ + 2 f11l 2ψ / V + Kψψ = Qψ

(32)

where Ky = ky + (2N ε 0* /l)( 1 - f /N r ) 0

23

0 0

Kψ = kψ + 2N l (- δ + f /N l) 0

0

23

0

(33) (34)

In the absence of any external forces, premultiplying (22) by q T yields the equation of energy balance  d (T + U + U G + U F ) + q T Bq / V + q T Cq = 0 dt

(35)

where

The skew symmetric matrix

T=

1 T q Aq 2

(36)

U=

1 T q Eq 2

(37)

UG =

1 T q Egq 2

(38)

UF =

1 T q (C + C T )q 4

(39)

EQUATIONS OF MOTION

57

 1 C = (C − C T ) 2

(40)

clearly plays an important role in the stability of the system. The kinetic and potential energy may be increased or decreased depending on whether or not the dissipation represented by the damping coefficients is greater than the energy added by the skew-symmetric stiffnesses. It should be noted, however, that this equation of energy balance (35) does not take into consideration the forces that are necessary to sustain the forward motion of the wheelset and is not the complete energy balance. It does not indicate the source of energy which distinguishes the non-conservative system from a conservative system. In order to maintain a constant forward speed V work must be done on the system by a corresponding longitudinal force Px, or tractive torque Lθ. Moreover, − q T Bq + q T Cq is not the total rate of working of the creep forces. This is given by 2F = V(f γ

2

22 2r

+f γ

2

22 2l

+f γ

2

11 1r

+ f γ

2

11 1l

)

(41)

so that F is a dissipation function for the creep forces, analogous to Rayleigh’s dissipation function. Note that f23 does not appear in this equation. Substituting into (9.20) from (13)-(16) then yields the corresponding version of the energy balance equation d (T + U + U G ) + 2 F − Lθ V / r0 − PxV = 0 dt

(42)

Reference to Figure 2.18 which shows the forces acting on the wheelset including the forces necessary to sustain the motion shows that to the second order in the generalised coordinates Px = −2 f 22ψ ( y / V − ψ ) Lθ = 2 f 11λy (

λy ψ + ) r0 V

(43) (44)

Substitution of (43) and (44) into (42) shows that (42) and (35) are equivalent and thus the source of energy which promotes the active behaviour of the system is the tractive effort applied either as a longitudinal force or as a driving or braking torque. The discussion of energy balance demonstrates the active and nonconservative nature of the system which is indicated by the assymmetry of the terms coupling lateral displacement of the wheelset with yaw, C12 and C21, which arise from the creep forces acting in conjunction with the conicity of the wheels. Stiffnesses also arise from forces acting in the contact area and comprise gravitational stiffnesses in the

58

RAIL VEHICLE DYNAMICS

Lθ

rr

Px T 2rψ

rl

T 1l

T 2l

T 2lψ

T 1r

ψ T 2r

Figure 2.18 Energy balance for wheelset.

matrix Eg (Eg11 is generated by the tilting of the normal reaction as the wheelset is displaced) and terms arising from the lateral creep force generated by spin. The gyroscopic couplings G12 and G21 are due to the tilting of the angular momentum vector as the wheelset is displaced laterally. G12 is modified because of the first-order variation of the lateral creep force due to the change in the normal force as the wheelset is displaced, as discussed above. For a wide range of practical parameters f23/N0r0 = 4/5 approximately, so that κ = δ0/5, and G12 = G21/5 approximately. The total contact stiffness C11 + Eg11 = (2N0 ε 0* /l)( 1 - f23/N0r0)

(45)

so that four-fifths of the gravitational stiffness effect is counteracted by the lateral force due to spin. Similarly, C22 + Eg22 = 2N0( - δ0 + f23/N0l)

(46)

so the stiffness is positive though very small. These considerations can be illustrated by a representative set of parameters for a restrained wheelset shown in Table 2.3. Parameters for the wheel rail geometry are taken from Figure 2.5 and the creep coefficients are from Table 2.2 and are consistent with Figure 2.2 and the axle-load. A significant feature of these parameters is that they give system creep coefficients B11 = C12 = 13.6 MN, B22 = 8.26 MNm and C21 = 2.82 MNm, which are large

EQUATIONS OF MOTION

59

Table 2.3 Parameters for an elastically restrained wheelset. Iz = 700 kgm2

m = 1250 kg

Iy = 250 kgm2

ky = 0.23 MN/m r0 = 0.45 m

W = 78.48 kN

kψ = 2.5 MNm/rad

λ0 = 0.1174

l = 0.7452 m

f11 = 7.44 MN

c y = cψ = 0

ε0 = 6.423

δ0 = 0.0493

f22 = 6.79 MN

σ = 0.0508

f23 = 13.7 kN

relative to the other elements in the system matrices. It was assumed in the above derivation that m >> δ02Ix/l2 and δ0 r0 /l << 1. For the above parameters δ02Iz/l2 = 3.064 and δ0 r0 /l = 0.0298 thus confirming the validity of this assumption. The contact stiffnesses C11 + Eg11 = 0.151 MN/m and C22 + Eg22 = 0.0245 MNm which, in this case, are small compared with kψ . In view of the relative magnitudes of the system coefficients it is a useful approximation to neglect f , E and G giving the simplified equations of motion 23

Y

Σ

+

g

y 1/(ms2 + 2f22s/V + ky)

A

2f11λl/r0

-2f22

1/(Is2 + 2f11l2s/V + kψ)

ψ

Figure 2.19 Feedback representation of wheelset.

Σ

+ Ψ

60

RAIL VEHICLE DYNAMICS

m  y + 2f ( y /V - ψ ) + k y = 0 22

y

(47)

2

2f λly/r + I ψ + 2f l ψ /V + k ψ = 2f11l/R0 11

0

z

ψ

11

(48)

the similarity to a closed loop feedback control system being indicated in Figure 2.19. It is this form of the equations of motion that were originally derived by Carter as discussed in Chapter 1. For certain calculations, it is useful to replace s/V by D. D therefore represents the operation of differentiation with respect to distance travelled along the track. At low speeds, so that the terms in V2 may be neglected, the equations are path dependent rather than time dependent and the equations of motion become [BD + C + E + E] q g

= Q

(49)

In many applications it is advantageous to cast the equations of motion into the equivalent first-order or state-space form. For future reference these are x T = [q

q ]

(50)

x = Ax + Y

(51) where the system matrix A is O22 I 22 ⎤ ⎥ −1 ⎣− A (C + E + Eg ) − A ( B / V + GV )⎦ ⎡

A =⎢

−1

(52)

where 022 and I22 are the 2 x 2 null and unit matrices and YT = [

012

A-1Q ]

(53)

where 012 is the 1 × 2 null matrix. In many wheelset motions the creepages are often small and it is then reasonable to express the creep forces in terms of the linear relationship between creep forces and creepages. This leaves only the geometry as the significant nonlinearity and the rolling radius difference, the slope difference and the roll angle may be expressed as linear functions of the lateral displacement y in terms of the describing functions λ, ε and σ as discussed in Section 3. The equations may then be described as quasilinear as the coefficients will be dependent on a the amplitude of the lateral displacement of the wheelset. It is obvious that the equations so derived will be of the same form as (47) and (48).

EQUATIONS OF MOTION

61

Figure 2.20 Generalised coordinates for two-axle vehicle.

2.11 Equations of Motion for a Two-Axle Vehicle Consider an idealised two-axle vehicle with a body and two wheelsets, Figure 2.20. The wheelsets and car body are assumed to be rigid and connected by mass-less elastic structures and mass-less damper elements. The vehicle is symmetric about a longitudinal plane of symmetry. The car body has six degrees of freedom, longitudinal xb, lateral yb and vertical translation zb, roll φb , pitch θb and yaw ψb and, as defined above, each wheelset has three degrees of freedom, lateral translation, rotation about the axle centreline and yaw. Consequently, the motion of the vehicle is defined by the following set of twelve generalised coordinates q =[ y1 ψ1 θ1 xb yb zb ϕb θb ψb y2 ψ 2 θ2 ]T

(1)

For the wheelsets the generalised coordinates yi and ψi are measured with respect to the centreline of the track and the normal to the centreline of the track in the plane containing the axle when the wheelset is central. For the leading wheelset, the origin of these coordinates is defined by X1, Y1 and azimuth Ψ1 of the wheelset, Figure 2.21. With reference to fixed axes these are given by s1

X1 = ∫ cosΨ 1 ds

(2)

0

s1

Y1 = ∫ sinΨ1 ds

(3)

0

s1

Ψ1 = ∫ (1/R1) ds

(4)

0

Similarly, X2, Y2 and Ψ2 for the trailing wheelset are given by integrals similar to (2), (3) and (4) with a limit of integration s2 = Vt - 2h. For the vehicle body, the origin of

62

RAIL VEHICLE DYNAMICS

X2

O

X1

f(X, Y)

Ψ2

Y2

O2

Ψ1

Ob h

Y1

O1

Ψb Figure 2.21 Path coordinates for two-axle vehicle.

the axis system is defined by the intersection of O1O2 and the normal to the track centreline, Figure 2.21. The arrangement of the suspension for this simple two-axle vehicle is shown in Figure 2.22. For ease of exposition the dampers will be assumed to be located similarly to the springs. Adaptation for the more general case is obvious. In terms of the chosen generalised coordinates the compressive strains ε in the springs are given by the equation of compatibility

ε =aq

(5)

where

ε = [δx1 δy1 δz1 δφ1 δψ1 δx2 δy2 δz2 δφ2 δψ2 ]Τ ⎡0 ⎢ ⎢1 ⎢0 ⎢ ⎢0 ⎢0 a=⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢0 ⎢0 ⎣

0 −1 0 0 0 0 0 −1 0 d 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 d 0 0 0 0 −1 0 0 0 0 0 0 −1

0 0 0 0 1 0 0

0 0

0

0

0

0

0 0 0 0 0 −h 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 h 1 0 0 0 0 0 0 0 0 0 0 −1 0 1

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ 0⎥ ⎥ 0⎥ 0⎥⎦

(6)

(7)

The forces exerted by the suspension on the car body are given by P = kaq + ca q

(8)

EQUATIONS OF MOTION

63

k y /2

k y /2

kψ

kψ k y /2

d

k y /2 h

ky

kφ

h

Figure 2.22 Arrangement of suspension for simple two-axle vehicle.

where P = [Px1 Py1 Pz1 Lφ1 Lψ1 Px2 Py1 Pz1 Lφ2 Lψ2]

(9)

and k is a matrix of component stiffnesses k = diag [kx ky kz kφ kψ kx ky kz kφ kψ ]

(10)

c is a matrix of component dampings c = diag [cx cy cz cφ cψ cx cy cz cφ cψ ]

(11)

The suspension is strained when the generalised coordinates are zero in the chosen axis system. Firstly, it is necessary to account for the fact that the vertical suspension is loaded when the vehicle is running in the central undisturbed position on the track, and this initially deflected vertical displacement will define the vertical origin of the axis system for the car body. The vertical suspension forces Pz10 and Pz20 when zb = 0 are given by Pz10 = Pz20 = - (mbV2sinφ0/R0 + mbgcosφ0)/2

(12)

For simplicity, if the vertical stiffness is assumed to be linear then the initial vertical deflection will be given by z0 = { mbV2sinφ0/R0 + mbgcosφ0 }/2kz

(13)

Secondly, the yaw suspension is strained when the vehicle is on curved track and the generalised coordinates are zero due to the choice of axes. Yaw moments are exerted

64

RAIL VEHICLE DYNAMICS

on the car body as follows Lψ10 = kψ( Ψ1- Ψb )

(14)

Lψ20 = kψ( Ψ2- Ψb )

(15)

and these must be added to the forces defined by equation (8). It will be noted that the origin of the axis system for the car body is laterally displaced by h2/2R0 from the centreline of the track. The equations of motion will then consist of two sets of equations of the form of (9.8) to (9.10) for the wheelsets, together with the six equations for the car body. mb xb − Px1 − Px 2 = 0

(16)

mb x + mbV 2 cos φ0 / R0 − mb g sin φ0 − Py1 − Py 2 = 0

(17)

mb  z + mbV 2 sin φ0 / R0 − mb g cos φ0 − Pz1 − Pz 2 = 0

(18)

I xb ϕb - Lφ1 - Lφ2 + dPy1 + dPy2 = 0

(19)

I yb θb + hPz1 − hPz2 − dPx1 − dPx2 = 0

(20)

I zbψb + IzbV(d/dt)(1/R0b) - Lψ1 - Lψ2 − hPy1 + hPy2 = 0

(21)

The equation (9.9) for the wheelsets will have the terms - zymV2sinφ0/R0 - zymgcosφ0+ Pzzy replaced by - zy(m + mb/2){V2sinφ0 /R0 + gcosφ0} by virtue of (12), so that the gravitational stiffness is proportional to axle-load. The generalised forces arising from the suspension corresponding to the coordinates of equation (1) are given by the application of the principle of virtual work and are Qs = - aTP

(22)

Τhe equations of motion will be of the form

 + F( q , q, V) + D( q - q 0 ) + E( q - q0) = Q Aq

(23)

EQUATIONS OF MOTION

65

where q0 represents the straining of the suspension due to the choice of axes and A is the inertia matrix A = diag [m I Iy mb mb mb Ixb Iyb Izb m I Iy ]

(24)

and E and D are the suspension elastic stiffness and damping matrices obtained by substitution of (8) into (22) so that E = aTka

D = aTca

(25)

In simple discussions of stability and guidance it is frequently possible, and is very convenient, to make use of the fact that a vehicle possesses a longitudinal plane of symmetry. Small disturbances of the car body which are symmetric about the plane of symmetry, namely xb, zb and θb, will not introduce variations in the lateral forces and yaw moments acting on the wheelsets. The symmetric degrees of freedom therefore do not couple with the antisymmetric degrees of freedom. It is the latter motions which describe the stability and guidance of the vehicle. Then, the motion of the vehicle is defined by the following set of nine generalised coordinates q =[ y1 ψ1 θ1 yb ϕb ψb y2 ψ 2 θ2 ]T

(26)

As far as the wheelset is concerned, for small displacements the angular rotation θ = −V / r0 , and then the motion of the wheelset is determined by the two generalised coordinates y and ψ which describe motions which are antisymmetric about the plane of symmetry. Then q =[ y1 ψ1 yb ϕb ψb y2 ψ 2 ]T

(27)

Of course, these approximations are not necessary when numerical solutions of the full nonlinear equations of motion are carried out but simplification of the equations is a valuable aid to an appreciation of system behaviour. Consistent with (27), E is then given by ⎡ ky 0 ⎢ kψ ⎢ 0 ⎢−k 0 y ⎢ E = ⎢ k y d − kψ ⎢ ⎢− k y h − kψ ⎢ 0 0 ⎢ 0 ⎢⎣ 0

− ky

kyd

− kyh

0

0

0

− kψ

0

2k y

− 2k y d

0

− ky

− 2k y d

2k y d 2 + 2kφ

0 2kψ + 2 k y h

kyd 2

kyh

0

0

− ky

kyd

k yh

ky

0

0

− kψ

0

0⎤ ⎥ 0⎥ 0⎥ ⎥ 0⎥ ⎥ kψ ⎥ 0⎥ ⎥ k ψ ⎥⎦

(28)

66

RAIL VEHICLE DYNAMICS

and D is of similar form to E. For a real vehicle with a complex suspension system with possible flexible structural components E and D (or the nonlinear equivalents) would obviously be much more complex and depend on design detail.

References 1. Gostling, R.J.: The measurement of real wheel and track profiles and their use in finding contact conditions, equivalent conicity and equilibrium rolling line. British Rail Research Technical Note TN DA 22, 1971. 2. Anon.: Geometry of contact between wheelset and track, part 1, methods of measurement and analysis. ORE Report C116 RP3, 1973. 3. Cooperrider, N.K., Hedrick, J.K., Law, E.H. Kadala,, P.S. and Tuten, J.M.: Analytical and experimental determination of nonlinear wheel/rail geometric constraints. Report FRA-O&RD 76-244, US Dept. of Transportation, Washington, 1975. 4. De Pater, A.D.: The motion of a single wheelset along a curved track. Delft University of Technology, Laboratory for Engineering Mechanics Report 1072, 1995. 5. Yang, Guang.: Dynamic Analysis of Railway Wheelsets and Complete vehicle systems. Delft University of Technology, Faculty for Mechanical Engineering and Marine Technology, Doctoral Thesis, 1993. pp. 42-50. 6. Heumann, H.: Zur Frage des Radreifen-Umrisses. Organ Fortschr. Eisenb.-wes. 89 (1934), pp. 336-342. Heumann, H.: Gründzuge der Führung der Schienenfahrzuge, Oldenbourg, Munchen, 1953, p. 133. 7. Wickens, A.H.: The dynamic stability of railway vehicle wheelsets and bogies having profiled wheels. Int. J. Solids and Structs. 1 (1965), pp. 319-341. 8. Blader, F.B.: Free lateral Oscillations in long freight Trains. Doctoral Thesis, Queens University, Kingston, Canada, 1972. 9. Joly, R.: Study of the transverse stability of a railway vehicle running at high speed. Rail International, 3, No. 2 (1972), pp. 83-118. 10. Gilchrist, A.O., Hobbs A.E.W., King, B.L. and Washby, V.: The riding of two particular designs of four-wheeled railway vehicle, Proc. Instn. Mech Engrs. 180, Part3F (1965-66), pp. 99-113. 11. Pearce, T.G.: Wheelset guidance-conicity, wheel wear and safety. Proc. Instn. Mech Engrs. J. Rail and Rapid Transit. 210, Part F (1996), pp. 1-10.

EQUATIONS OF MOTION

67

12. Hauschild, W.: Die Kinematik des Rad-Scheine Systems. Institut für Mechanik, Technische Universität Berlin, 1977. 13. Duffek, W.: Contact geometry in wheel rail mechanics. Proc. Symp. Contact Mechanics and Wear of Rail/Wheel Systems, (Ed.) J. Kalousek et al., University of Waterloo Press, 1982, pp. 161-179. 14. de Pater, A.D.: The geometric contact between wheel and rail. Vehicle System Dynamics, 17, No. 3 (1988), pp. 127-140. 15. Müller, C.Th.: Kinematik, Spurführungsgeometrie und Führungsvermogen der Eisenbahnrädsatz. Glasers Annalen, 77 (1953), pp. 264-281. 16. Yang (ibid), pp. 16-42. 17. Netter, H. Schupp, G. Rulka, W. and Schroeder, K.: New aspects of contact modelling and validation within multibody system simulation of railway vehicles. In L. Palkovics (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 15th IAVSD Symposium, Budapest, August 1997, pp. 246-269. Swets and Zeitlinger Publishers, Lisse, 1992. 18. Pascal, J.P.: About multi-Hertzian contact hypothesis and equivalent conicity in the case of S1002 and UIC60 analytical wheel/rail profiles. Vehicle System Dynamics 22 (1993), pp. 263-275. 19. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge, 1985. 20. Hertz, J.: Maths. (Crelle‘s J.) 1881. 21. Love, A.E.H.: Theory of Elasticity. 4th Ed., Cambridge University Press, Cambridge, 1927, pp. 193-198. 22. Timoshenko, S. and Goodier, J.N.: Theory of Elasticity. 2nd Ed. McGraw-Hill, New York, 1951, pp. 377-379. 23. Nayak, P.R. and Paul, I.L.: A new theory of rolling contact. MIT Report DSR76109-7, 1968. 24. Kalker, J.J.: A simplified theory for non-Hertzian contact. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983, pp. 295-302. Swets and Zeitlinger Publishers, Lisse, 1984. 25. Carter, F.W.: On the action of a locomotive driving wheel. Proc. Royal Soc. Ser. A 112 (1926), pp. 151-157.

68

RAIL VEHICLE DYNAMICS

26. Poritsky, H.: Stresses and deflections of cylindrical bodies in contact with applic-ation to contact of gears and of locomotive wheels. J. Applied Mech. Trans. ASME 72 (1950), pp. 191-201. 27. Cain, B.S.: Discussion of reference 25. J. Applied Mech. Trans. ASME 72 (1950), pp. 465-466. 28. Johnson, K.L.: The effect of tangential contact force upon the rolling motion of an elastic sphere upon a plane. J. Appl. Mech Trans. ASME 80 (1958), pp. 339-346. 29. Johnson, K.L.: The effect of spin upon the rolling motion of an elastic sphere upon a plane. J. Appl. Mech Trans. ASME 80 (1958), pp. 332-338. 30. Haines, D.J. and Ollerton, E.: Contact stress distributions on elliptical contact surfaces subjected to radial and tangential forces. Proc. Inst. Mech. Engrs. 177 (1963), pp. 95-114. 31. Vermeulen, P.J. and Johnson, K.L.: Contact of non-spherical elastic bodies transmitting tangential forces. J. Appl. Mech Trans. ASME 86 (1964), pp. 338-340. 32. de Pater, A.D.: On the reciprocal pressure between two elastic bodies. Proc. Symposium on Rolling Contact Phenomena, pp. 29-74. Elsevier, Amsterdam, 1962. 33. Kalker, J.J.: The transmission of force and couple between two elastically similar rolling spheres. Proc. Kon. Ned. Akad. Wet. Amsterdam, B70 (1964), pp. 135177. 34. Kalker, J.J.: On the rolling of two elastic bodies in the presence of dry friction. Doctoral Thesis, Delft University of Technology, 1967. 35. Kalker, J.J.: Simplified theory of rolling contact. Delft Progress Report No 1, 1973, pp. 1-10. See also: Kalker, J.J.: A fast algorithm for the simplified theory of rolling contact. Vehicle System Dynamics 11 (1982), pp. 1-13. 36. Kalker, J.J.: Three-dimensional elastic Bodies in Rolling Contact. Kluwer Academic Publishers, Dordrecht, 1990. 37. Kalker, J.J.: Survey of wheel-rail rolling contact theory. Vehicle System Dynamics 8 (1979), pp. 317-358. 38. Hobbs, A.E.W.: A survey of creep. British Railways Technical Note DYN 52, 1967.

EQUATIONS OF MOTION

69

39. Illingworth, R.: The mechanism of railway vehicle excitation by track irregularities. Doctoral Thesis, University of Oxford, 1973. 40. Brickle, B.V.: The steady state forces and moments on a railway wheelset including flange contact conditions. Doctoral Thesis, Loughborough University of Technology, 1973. 41. Pearce, T.G. and Rose, K.A.: Tangential force-creepage relationships in theory and practice. Proc. Int. Symp. on Contact Mechanics and Wear of Rail/Wheel Systems, Vancouver, Canada, 1982. Pearce, T.G. and Rose, K.A.: Measured force-creepage relationships and their use in vehicle response calculations. In: O. Nordstsrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 427-440. 42. Knothe, K., Wille, R. and Zastrau, B.W.: Advanced contact mechanics-road and rail. Vehicle System Dynamics 35 (2001), pp. 361-407. 43. Ertz, M. and Knothe, K.: Die Temperaturentwickling im Rad-Scheine-Kontakt und ihre Auswirkungen auf Kraftschluss und Materialverhalten. Tagungsband zur VDEI Fachtagung Bahnbau 2000, Berlin, September 12-15, 2000, pp. 232-237. 44. Shen, Z.Y., Hedrick, J.K. and Elkins, J.A.: A comparison of alternative creepforce models for rail vehicle dynamics analysis. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 591-605. 45. Polach, O.: A fast wheel-rail forces calculation computer code. In: R. Frohling (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 16th IAVSD Symposium, Pretoria, August 1999. Swets and Zeitlinger Publishers, Lisse, 2000, pp. 728-739. 46. Knothe, K. and Le-The Hung.: Determination of the tangential stresses and the wear for the wheel-rail rolling contact problem. In: O. Nordstsrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 264-277. 47. Nayak, P.R.: Surface roughness effects in rolling contact. J. Applied Mech. Trans. ASME 39, Ser. E, No. 2 (1972), pp. 456-460. 48. Greenwood, J.A. and Tripp, J.H.: The elastic contact of rough surfaces. J. Applied Mech. Trans. ASME 34, Ser. E (1967), pp. 153-167. 49. Bucher, F.: Normal and tangential contact problem of surfaces with measured roughness. Paper presented at the 5th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems, Tokyo, Japan, 25-28th July 2000. To be published in Wear, 2001/2002.

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50. Newland, D.E.: On the time-dependent spin creep of a railway vehicle. J. Mech. Eng. Sci. 24, No. 2 (1982), pp. 55-64. 51. Knothe, K., and Gross-Thebing, A.: Derivation of frequency dependent creep coefficients based on an elastic half- space model. Vehicle System Dynamics. 15, No 3 (1986), pp. 133-153. 53. Gilchrist, A.O. and Brickle, B.V.: A Re-examination of the proneness to derailment of a railway wheelset. J. Mech. Eng. Sci 18, No. 3 (1976), pp. 131-141. 54. Matsui, N.: On the derailment quotient Q/P. Railway Technical Research Institute, Japanese National Railways, 1966. 55. Boedecker, Chr.: Die Wirkungen zwischen Rad und Schiene und ihre Einflüsse auf den Lauf und den Bewegungswiderstand der Fahrzeuge in den Eisenbahnzügen. Hahn’sche Buchhandlung, Hannover, 1887. 56. Schiehlen, W.: Modelling of complex vehicle systems. In J.K.Hedrick, (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 548-563. 57. Kik, W., Knothe, K. and Steinborn, H.: Theory and numerical results of a general quasi-static curving algorithm. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 414-426. 58. Piotrowski, J.: A theory of wheelset forces for two point contact between wheel and rail, Vehicle System Dynamics, 11 (1982), pp. 69-87. 59. Elkins, J.A. et al.: The effect of a restraining rail on the curving behaviour of a transit vehicle. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 132-147. 60 Yang (ibid), pp. 90-95.

3 Dynamics of the Wheelset 3.1 Introduction As the wheelset is the basic guidance element of the railway vehicle it is appropriate to consider the dynamics of the wheelset as a preliminary to a discussion of the behaviour of a complete vehicle. In the simple two-axle vehicle of Figure 2.22, attention is now concentrated on a single wheelset which is connected by means of a mass-less suspension to the vehicle body which moves forward at constant velocity with negligible lateral and vertical motion. The vehicle body is assumed to apply a constant vertical force through the vertical suspension so that the total axle load is W. As discussed in Chapter 2, the system has three degrees of freedom, with the corresponding generalised coordinates lateral translation y of the centre of mass, angle of yaw ψ about a vertical axis and the rate of rotation of the wheelset Ω. The importance of this rather simple system lies in the fact that for a wide range of practical values of the parameters, the body of the vehicle does not strongly participate in the lateral motions of its wheelsets at high speeds, as will be discussed later, so that the elastically restrained wheelset exhibits behaviour which is typical of complete vehicles. Moreover, it is a system which lends itself to experimentation using a roller rig [1-4]. In addition, alternatives to wheelset guidance such as the peg-in-slot system, exhibit some contrasting aspects of the behaviour of guided wheeled systems, and will be briefly discussed.

3.2 The Unrestrained Wheelset The kinematic oscillation of a wheelset has already been discussed in Chapter 1. Before considering the stability and response of an elastically restrained wheelset it is useful to discuss the behaviour of an isolated wheelset in motion at low speed. If the solution of the full set of equations of motion, equations (2.9.8)-(2.9.10), is considered for the unrestrained wheelset in which the suspension stiffnesses ky and kψ are zero, then the behaviour conforms broadly to the simple model considered in Chapter 1. Figure 3.1 shows the response of a wheelset with the parameters of Table 2.3 and the wheel-rail geometry of Figure 2.2 to a step change in alignment of the track of 6 mm at low speed. After an initial transient, there is a lightly damped os-

72

RAIL VEHICLE DYNAMICS

(a)

(b) Tyr (N)

y (m)

ψ

Tyl (N) (c)

t (s)

(d) Tyr + Tyl (N)

ω3l

s-1

ω3r

zwW (N)

(e)

(f) Ω + V/r0

N - W/2

Figure 3.1 Oscillation of unrestrained wheelset in response to a step change in alignment y0 = 6 mm at V = 10 m/s.

cillation which is similar to the kinematic oscillation in that the angle of yaw is approximately 90° out of phase with lateral displacement and it is approximately sinusoidal, Figure 3.1(a). The longitudinal creep is small, and as shown in Figure 3.1(b) the lateral creep forces are mainly generated by spin (which is mainly proportional to the contact slope through equation (2.5.17) and (2.5.18)) and are, therefore, in phase with lateral displacement. Figure 3.1(c) shows the spin on each wheel and, as discussed in Section (2.10), even when the wheelset is in the central position equal and outward lateral creep forces are generated by spin. Figure 3.1(d) shows that the resultant lateral force is almost completely reacted by the gravitational stiffness force. Though the amplitude of the oscillation is relatively large at 6 mm, the peak creep forces, at 6000 N. are far from saturation (µ = 0.3, µN0 = 11772 N.). Figure 3.1(e) and 3.1(f) shows that the variation of the normal forces and rotational speed is small and occurs at double the frequency. Accordingly, consider the extent to which purely kinematic motions of a wheelset are possible. On straight track equations (2.5.13-18) reduce to, if r0δ0/l<< 1,

γ1r = Ωrr /V + 1 - l ψ /V

(1)

γ1l = Ωrl /V + 1 + l ψ /V

(2)

DYNAMICS OF THE WHEELSET

73

γ2r = ( y + Ωrrψ ) sec δr/V

(3)

γ2l = ( y + Ωrlψ ) sec δl/V

(4)

ω3r = (Ω/V)sinδr + ( ψ /V)cosδr

(5)

ω3l =− (Ω/V)sinδl + ( ψ /V)cosδl

(6)

In view of the near equivalence of the lateral creep force due to spin and the gravitational restoring force their effect will be neglected and equations (5) and (6) discarded. Equating (1) and (2) to zero and adding, the variable rotational velocity of the wheelset is found to be

Ω=−

2V (rr + rl )

(7)

Equating (3) and (4) to zero, adding and substituting from (7) y −ψ = 0 V

(8)

Equating (1) and (2) to zero, subtracting and substituting from (8)  y+

V 2 (rr − rl ) =0 (rr + rl )l

(9)

describing a periodic motion with wavelength independent of speed. The numerical solution of (7), (8) and (9) for the response to a step change in alignment of the track of 6 mm at low speed for the wheel-rail geometry of Figure 2.2 yields a time-history for y, ψ and Ω closely similar to that in Figure 3.1(a) and 3.1(f). Consider motion at small amplitudes, and replace s/V by D so that D represents the operation of differentiation with respect to distance travelled along the track. Then substitution of the equivalent conicity from (2.2.36) into (9) yields 2

( D + λ/r l ) y = 0 0

(10)

indicating that the motion consists of a single undamped oscillatory constituent of constant wavelength, with circular frequency

ω = V ( λ/r l ) 0

1/2

(11)

the frequency given by Klingel’s formula discussed in Chapter 1. The mode shape of this oscillation is, if the amplitude is a,

74

RAIL VEHICLE DYNAMICS 1/2

y = a sinωt

ψ = a ( λ/r l ) cosωt 0

(12)

Exactly these latter results can, of course, be recovered from the simplified linearised equations of motion (2.10.47-48). As the wheelset is unrestrained, the suspension stiffnesses ky and kψ are zero, and in terms of the operator D, these equations reduce to 2 2 (13) (mV D + 2f D)y - 2f ψ = 0 22

22

2 2

2

2f λly/r + (I V D + 2f l D)ψ = 0 11

0

z

11

(14)

2

It can be seen that at low speeds, since the terms in V can be neglected, equations (13) and (14) become 2f Dy - 2f ψ = 0 22

22

(15)

2

2f λly/r + 2f l Dψ = 0 11

0

11

(16)

Elimination of y and ψ then gives (10). This recovers the solution first given by Carter [5]. As discussed in Chapter 1, closely related to the kinematic oscillation is the behaviour of a wheelset on curved track, with zero cant deficiency. For steady motion on curved track equations (2.5.13-18) reduce to

γ1r = Ωrr /V + 1 - l/R0

(17)

γ1l = Ωrl /V + 1 + l/R0

(18)

γ2r = Ωrrψ secδr/V

(19)

γ2l = Ωrlψ secδl/V

(20)

ω3r = (Ω/V)sinδr + (1/R0)cosδr

(21)

ω3l =− (Ω/V)sinδl + (1/R0)cosδl

(22)

As before, neglecting the effects of spin, rolling round the curve can occur if the wheelset moves outward so that the longitudinal creep is zero and adopts a radial position ψ = 0 so that the lateral creep is zero. Adding (17) and (18) the rotational velocity of the wheelset is given by (7) and subtracting (17) and (18) yields (rr − rl ) l =− R0 (rr + rl )

(23)

DYNAMICS OF THE WHEELSET

75

which determines the lateral displacement of the wheelset for zero longitudinal and lateral creepage. Thus, in response to track curvature the unconstrained wheelset attempts to roll, without creep, round the curve adopting a radial position which causes the two rolling circles to form cross-sections of a cone having its apex at the centre of curvature of the track. On substitution from (2.2.36), (23) reduces to Redtenbacher’s formula discussed in Chapter 1. Redtenbacher’s formula can, of course, be recovered from the simplified linearised equations of motion (2.10.47-48). The suspension stiffnesses ky and kψ are zero so that for steady motion in a uniform curve with zero cant deficiency these equations reduce to - 2f ψ = 0 (24) 22

2f λly/r = - 2f l/R0 11

0

(25)

11

giving the expected results.

V/r0 + Ω (s-1)

0 -2

0.2

-4

0.15

-6

0.1

-8

0.05

y (mm)

-10

0

1000

2000

R0 (m)

0

0

1000

R0 (m) 2000

Figure 3.2 Lateral displacement and change in rotational velocity for wheelset with geometry of Figure 2.2 on curve with zero lateral and longitudinal creep, equations (7) and (23).

For the wheel and rail profiles of Figure 2.2, the application of equations (7) and (23) yield the results shown in Figure 3.2. On a very large radius curve, say R0 > 1000m., there is agreement with Redtenbacher’s formula as might be expected. It is noteworthy that rolling can occur within the flangeway clearance down to relatively small radius curves. On sharper curves, even if the wheelset is able to take up a perfectly radial position, the lateral displacement is limited by the available flangeway clearance. However, the worn wheel and rail combination is capable of supporting motion without lateral and longitudinal creep down to relatively low radii; for example, on a radius of 150 m the lateral displacement of the wheelset is 7.28 mm (just within the flangeway clearance). The reduction in rotational velocity is negligible. It can also be seen from the equations of motion that if a lateral force, such as

76

RAIL VEHICLE DYNAMICS

that arising from centrifugal force or cant deficiency, is applied to a wheelset then the wheelset yaws in order to generate a reacting lateral force. Thus, as discussed in Chapter 1, it can be seen that for small displacements from the centre of the track the primary mode of guidance is provided by conicity. It is on sharp curves and switches and crossings that the flanges become the essential mode of guidance. As already discussed, as a wheelset is displaced laterally the normal reaction forces change direction and give a lateral resultant force.

3.3 Root Locus for Small Motions of the Restrained Wheelset Consider the wheelset elastically restrained by a lateral spring ky and a yaw spring kψ, both connected to ground. For motions of small amplitude, the linear equations of motion in first order form, equations (2.10.51), are valid and, in the absence of forcing, the motions can be represented by the trial solution st

x(t) = Xe

(1)

where X is a column matrix with elements independent of t. Substitution of (1) into (2.10.51) gives sX = A X

(2)

The condition that equation (2) will have non-zero solutions is det (A - s) = 0

(3)

so that for a system with N degrees of freedom there will be 2N eigenvalues sp which satisfy (3). For asymptotic stability, in which the displacements of the system eventually decay to zero irrespective of the initial conditions, it is necessary that the eigenvalues, if real be negative, and if complex should have negative real parts. A real positive eigenvalue indicates divergence or static instability. In this case an initial displacement given to the system can result in growing displacements. A complex eigenvalue possessing a positive real part indicates oscillatory instability. In this case an initial displacement given to the system can result in growing oscillations. The requirement that the eigenvalues sp, if real must be negative, and if complex must have negative real parts is a necessary condition for stability, but in general it is not sufficient. For if there are equal imaginary eigenvalues, there may be a general solution of the form X = (a1 + a2 t )est

(4)

in which case the solution grows indefinitely with time and is unstable. However, if the system is conservative a2 is zero, and the system is not unstable [6, p.50]. Exact

DYNAMICS OF THE WHEELSET

77

repetition of the eigenvalues is unlikely to be encountered in the case of the railway wheelset and will not be considered here. Corresponding to each eigenvalue sp there will be a solution to (2) so that the eigenvector Xp satisfies spXp = A Xp

(5)

and the p solutions can be consolidated into the single equation UΛ = A U

(6)

where U is formed from the eigenvectors Xp and Λ is a diagonal matrix with elements consisting of the eigenvalues, U and Λ being ordered consistently. By virtue of equation (2.10.50) the structure of U has the form in which the pth column is ⎡ up ⎤ Up = ⎢ ⎥ ⎣u p s p ⎦

(7)

where up is, of course, an eigenvector of the original second order equations, (2.10.22). For further details of the eigenvalue problem reference may be made to [6] and [7]. In the case of the wheelset (3) can be expanded to obtain the characteristic polynomial 4

3

2

p4s + p3s + p2s + p1s + p0 = 0

(8)

where p = mI 4

2

p = 2( mf l + If )/V 3

11

22

2

2

p = m K + I K + 4f f l /V + ( 2f /V - I σV/r l )( 2f /V - I κV/r l ) ψ

2

11 22

y

23

2

p = 2f K /V + 2f l K /V - 4f ( f 1

ψ

22

11

p = K K + 4f f λl/r 0

y

ψ

22 11

y

23

22

0

y

23

y

+ f λl/r )/V + 2 I V( σf 11

0

y

22

0

+ κf λl/r )/r l 11

0

0

0

Of prime interest is the way in which the eigenvalues vary with speed V. At low speeds there is one complex or oscillatory root A and two large roots B and C which are real. Figure 3.3(a) shows a typical root locus as V is varied, and Figure 3.3 (b)

78

RAIL VEHICLE DYNAMICS

(a) 100

100

80

80

60

60

ω (1/s)

ω

20

A

0

µ (1/s)

B,C

0 0

10

D

40 20

A

0 -10

(b)

100

200

V (m/s)

Figure 3.3 Variation of eigenvalues for elastically restrained wheelset with parameters of Table 2.3. The root locus does not show branches B and C as their real parts are too large. In this, and other root locus diagrams, only the positive imaginary part of the complex pair of roots are shown.

shows the variation of the imaginary parts with V. The imaginary part of the oscillatory root, branch A, is closely equal to the frequency of the kinematic oscillation, and the real part is negative − the damping is positive. However, as the speed increases, the damping reaches a maximum and then decreases until it becomes negative beyond a speed VB. The system is unstable at all speeds V > VB. Thus, small motions will not remain small, and the behaviour of the system will then have to be examined using the nonlinear equations of motion, equations (2.9.8)-(2.9.10). This is done below. At higher speeds, the two large real roots at B and C coalesce to form a complex pair shown as D in Figure 3.3, which remains over-critically damped. It is necessary to form some appreciation of the way in which the various parameters of the system influence the eigenvalues, and in particular, the stability of the system. Noting that the branch of the root locus which becomes unstable is that associated with the kinematic oscillation, a perturbation procedure will be adopted [8]. It has already been noted that the creep coefficients f11 and f22 in the equations of motion are much larger than the other coefficients, and equations (2.15-16) show that these same coefficients define the kinematic oscillation. Thus let f now represent the typical creep coefficient f11 or f22 and relying on the fact that f is large con2 sider the characteristic equation if all terms of order less than f are neglected. Then 2

p =p =p =0 4

3

1

2

p = 4f f l /V 2

11 22

p = 4f f λl/r 0

22 11

0

(9)

and the characteristic equation reduces to 2

ps + p =0 2

0

(10)

DYNAMICS OF THE WHEELSET

79

The solution of (10) is, of course, s = ± iω1 =± iV( λ/r0l)1/2

(11)

2

The condition that all terms of order less than f be neglected may now be relaxed 2 by regarding terms of order f as small perturbations δpn of the terms pn of order f . From (8) the corresponding perturbation in the eigenvalue si will be given by n

n-1

δs = ∑ δp s / ∑ n p s i

n

(12)

n

provided that there are no repeated roots which is the case for the system under consideration. The initial values are given by equation (9) and the perturbed value of the polynomial coefficients are 2

δp = 2( mf l + If )/V 3

11

22

2

δp = 2f K /V + 2f l K /V - 4f ( f 1

ψ

22

11

y

23

22

+ f λl/r )/V + 2 I V( σf 11

0

y

22

+ κf λl/r )/r l 11

0

0

Using (12) two roots of the characteristic equation are found to be s = ± iω + µ 1

1

(13)

1

where 2

2

2

µ = - V [ f K + f l K - λV ( f l m + f I )/r l 1

22

ψ

11

y

11

- 2f ( f 23

22

22

0

+ f λl/r ) + I V2( σf 11

0

y

22

2

+ κf λl/r )/r l]/4f f l 11

0

0

22 11

(14)

The frequency of the oscillation corresponding to s1 is the same as that of the kinematic oscillation. Insertion of typical values of the parameters from Table 2.3 shows that the terms in f23 (due to spin creep) and Iy (due to gyroscopic effects) are of secondary importance and the damping at low speeds (where the V2 terms can be neglected) depends on the stiffnesses Ky and Kψ. The inertia forces depend on V2 and are destabilising, becoming dominant at high speeds. The damping becomes negative beyond a speed VA given by

VA2 =

{Kψ f 22 + K y f 11 l 2 − 2 f 23 ( f 22 + f 11 λl / r0 }r0 l {λ ( mf 11 l 2 + If 22 ) − I y (σf 22 + κf 11 λl / r0 )}

(15)

The explicit terms in f23 in (15), which arise from the coupling terms in the creep damping matrix, reduce VA slightly (for the example parameters, of less than 1%)

80

RAIL VEHICLE DYNAMICS

and the terms in Iy increase VA slightly (for the example parameters, by 4%). These trends can be confirmed by comparison of the eigenvalues obtained numerically. It follows that a good approximation to VA is given by

VA = 2

( Kψ f 22 + K y f 11 l 2 ) r0 l

(16)

λ ( mf 11 l 2 + If 22 )

so that then VA is inversely proportional to the square root of the conicity λ, elastic restraint is stabilising and inertia destabilising. Moreover, f11 and f22 possess similar values and appear in both numerator and denominator of (16) with the effect that VA is insensitive to small variations of creep coefficient. To the same accuracy as (13), and neglecting f23 and Iy, the corresponding eigenvector is 1 + i0 ⎡ ⎤ u1 = ⎢ 2 2 2 1/ 2 ⎥ (17) λ l )} / 4 i ( λ / lr ) K f − K f l − V lr m f − I f + { / / ( / )( / / 11 0 22 11 0 ψ ⎣ y 22 ⎦ 2

2

Using (13) to form the quadratic factor (s + 2µ s + ω ) the characteristic equation 1 1 can be completely factorised in the form 2

2

(s + 2µ s + ω ) (s + µ ) (s + µ ) = 0 1

1

2

3

(18)

where approximately

µ = - 2f /mV 3

22

(19)

2

µ = - 2f l /IV 4

11

(20)

These two real eigenvalues correspond to heavily damped subsidences. Substitution of the eigenvalues into the equations of motion yields the corresponding eigenvectors, which to the same degree of approximation, indicate that µ3 is associated with a motion in which only lateral translation of the wheelset takes place, yawing displacement being negligible, u 3 = [1 0]

T

(21)

Similarly, µ4 is associated with a motion in which yawing of the wheelset is predominant, lateral translation being negligible, u 4 = [0 1]

T

(22)

DYNAMICS OF THE WHEELSET

81

The time constants associated with these large real roots are such that at low, or even moderate, speeds they are comparable with the time taken for a material point to move through the contact area as discussed in Section 2.4.5. In this case, transient creep effects will be significant, and the equations of motion must be modified as discussed in Section 2.4.5. It will then be found that the large roots are changed significantly but the eigenvalues corresponding to the kinematic oscillation are virtually unchanged. Thus little error is incurred in the discussion of stability if creep transient effects are neglected.

3.4 Instability and Feedback The approximate solution derived above is useful in summarising the behaviour of the system at various speeds. However, it is based on the supposition that the creep coefficients are large compared with the other parameters. If attention is restricted to the motion at the onset of instability, the origin of the instability in the feedback that exists between the lateral displacement and yaw modes of the wheelset becomes apparent. As discussed in Chapter 2 the equations of motion can be represented by the block diagram shown in Figure 2.19 which indicates that the coupling terms which arise from creep and conicity in the equations can be interpreted as providing feedback. It is therefore appropriate to apply the Nyquist criterion for stability of linear feedback systems [9]. Referring to Figure 2.19, the open loop transfer function of the system is defined by breaking the loop at some point such as A, and measuring the output at A as a function of frequency ω for unit input at A; the open loop transfer function of the system, F(iω), is therefore F (iω ) =

{2 f11λl / r0 − (2 f 23 / V − σVI y / r0l )iω }{2 f 22 − (2 f 23 / V − κVI y / r0l )iω } ( −mω 2 + 2 f 22iω / V + K y )( − Iω 2 + 2 f11l 2iω / V + Kψ )

Now, for sustained oscillations to occur, Nyquist’s criterion states that F(iω) = -1

(2)

the open loop gain of the system must be unity with a phase lag of 180°. If F(iω) < -1 when the phase lag is 180° then the system will be unstable as the output will increase with time. If F(iω) is plotted in the complex plane, then for stability the point (-1 + i0) must lie to the left of the F(iω) locus as ω is varied from zero to infinity. Another stability criterion, exactly equivalent to the Nyquist criterion, is that of Routh [9] which is expressed in terms of the coefficients of the characteristic polynomial. For the quartic (3.8), Routh’s criterion for stability is that T3 = p1 p 2 p 3 − p 0 p 32 − p 4 p12 > 0

(3)

82

RAIL VEHICLE DYNAMICS

1 0.5

ω=0

ω =∞

0

(-1,0)

-0.5 -1

Im(F) -2 -2.5 -3 -3

-2

-1

0

Re(F)

2

3

Figure 3.4 Open loop transfer function for elastically restrained wheelset with parameters of Table 2.3, V = 115 m/s, creep coefficients one third of nominal giving VB = 110 m/s.

Figure 3.4 shows the open loop transfer function plotted in the complex plane for the elastically restrained wheelset, for a speed slightly above the onset of instability. At ordinary speeds, terms in f23 and Iy may be neglected as discussed above. At the speed VB separating instability from stability, the motion is sinusoidal and on separating real and imaginary parts (2) expressions for VB and the corresponding frequency ωB can be found. The speed VB in terms of speed VA found in the previous Section and given by equation (3.16) is, V A2

V B2 = 1−

r0 l ⎛ mKψ − I z K y ⎞ ⎜ ⎟ λ ⎜⎝ 2( f 22 I z + f 11ml 2 ) ⎟⎠

(4) 2

and the corresponding frequency ωB is given by ω B2 =

f 22 Kψ + f 11l 2 K y f 22 I z + f 11l 2 m

(5)

so that ωB = ωA. It is noteworthy that this frequency is independent of the speed at which instability occurs. In addition, ωB always lies between the two natural frequencies possessed by the uncoupled degrees of freedom. This fact also follows from a consideration of phase since for sustained oscillations the open loop transfer function must cross the real negative axis in the complex plane. As the numerator is positive, this can only occur when the denominator is negative, and this can only occur when the transfer functions of the uncoupled systems are of opposite signs, i.e. at a frequency between the two natural frequencies.

DYNAMICS OF THE WHEELSET

83

The motion of the system, in the mode in which the damping is zero at the speed VB is y = a1 sin(ωt + ε)

ψ = a2 sinωt

(6)

where 1/2

a2/a1 = ( λ/r l ) 0

2

2

(7)

tanε = 2f ω/V( K - mω ) = -2f l ω/V( K - Iω ) 22

y

2

11

ψ

The amplitude ratio is therefore identical to that found in the kinematic mode of a slowly rolling wheelset. As the critical frequency always lies between the two natural frequencies possessed by the uncoupled degrees of freedom, if Kψ /I < Ky /m tanφ is positive, but if Kψ/I > Ky/m , tan φ is negative. Normally, the creep coefficient f22 is much larger than m and Ky, and then φ will be slightly less or slightly greater than -90°. Equation (4) shows that VA is a close approximation to VB provided that either the creep coefficients are very large in relation to the elastic stiffnesses or that the uncoupled modal frequencies are sufficiently close. In fact, if the modal frequencies are equal reductions in a factor on the creep coefficients have no influence on VB. As mentioned above, in ordinary cases, f11 and f22 are large and nearly equal in value so VB is relatively insensitive to their value, Figure 3.5. Calculations show that, consistent with equation (3.15), the gyroscopic terms are then slightly stabilising and the f23 terms slightly destabilising. Figure 3.5 also shows the stabilising effect of large reductions in creep coefficient in accordance with equation (4). For the parameters of the example wheelset, Table 2.3, reductions of up to 50% are only slightly stabilising, but if the creep coefficients were significantly smaller then VB increases and the instability associated with the kinematic oscillation disappears. However, another form of instability appears and this is associated with gyroscopic effects, for if Iy is zero there is no insta-

Figure 3.5 Effect of large reductions in magnitude of the creep coefficients on VB. Also shown is the solution for the case in which Iy = 0.

84

RAIL VEHICLE DYNAMICS

bility at low values of the creep coefficients. Examination of equation (2) for small f shows that if Ky/m>Kψ/I then there is instability above a speed given by V2 =

f 22 ( K y I / m − Kψ ) + 2 f 23 ( f 22 + f11λl / r0 ) I y ( f 22σ + f11κλl / r0 ) / r0l

(8)

and the corresponding frequency is ω = (Κy/m)1/2. Otherwise, if Ky/m< Kψ/I then instability occurs above a speed given by V2 =

f11l 2 ( Kψ m / I − K y ) + 2 f 23 ( f 22 + f11λl / r0 ) I y ( f 22σ + f11κλl / r0 ) / r0l

(9)

and ω = (Κψ/I)1/2. Using the parameters of Table 2.3 it can be seen that for ordinary configurations this instability seems of no practical importance.

3.5 Amplitude Dependent Behaviour and Limit Cycles In deriving the equations of motion, two major nonlinearities were identified as being those due to the wheel-rail geometry and those due to creep saturation. Unless the displacements of the wheelset are small the behaviour of the system will depend on the amplitude of the motion. In the linear solutions discussed so far, the solutions are, of course, independent of amplitude. Figure 3.6(a) shows the lateral displacement amplitude a as a function of speed for the linear case. There are two possibilities, for the wheelset, for steady motions − the steady motion along the track centreline A which has been shown above to be asymptotically stable for V < VB , and unstable for V > VB . For V = VB, B is a point of bifurcation and steady oscillation is possible for any amplitude a as shown by the stability boundary D, separating stability from instability. If the wheelset motions cannot be considered small, nonlinearities must be taken into account. The most direct way of investigating the stability of the system in this case is to examine the dynamic response to a disturbance using a numerical solution of the equations of motion, equations (2.9.8-10). Figure 3.6(b) shows the lateral displacement amplitude a as a function of speed for the nonlinear case. For small amplitudes the motion is consistent with the linear model and stability exists for V < VB , and instability for V > VB . As the amplitude increases, the motion associated with the kinematic oscillation is approximately sinusoidal and the creepages remain small. However, the rolling radius difference increases and therefore the stability boundary D of Figure 3.6(a) now moves to lower speeds as the amplitude increases. For larger amplitudes, creep saturation increasingly occurs with two results: creep saturation is itself stabilising (compare the stabilising effect of reducing creep coefficients in the linear case discussed above), and as the creep lateral force due to spin is reduced, the stabilising effect of the flange force (or gravitational stiffness) comes fully into play. Consequently, the

DYNAMICS OF THE WHEELSET

85

(b)

(a)

E D

a

a

C D

B

A

A

V

VB

B VC

VB

V

Figure 3.6 Limit cycle diagram for (a) linear system and nonlinear system (b).

stability boundary E moves to higher speeds. Separating the boundary E from boundary D is point C which occurs at speed VC . It is this speed that is the critical speed of the vehicle in engineering terms, for if V < VC the vehicle is asymptotically stable and any disturbance from the motion along the track centreline will decay exponentially. When VC < V < VB there are three possibilities for steady motion. The motion along the track centreline A is asymptotically stable. The steady oscillation D is unstable, for if the amplitude is slightly greater than that corresponding to the boundary D, oscillations will grow until the boundary E is reached. If the amplitude is slightly less than that corresponding to the boundary D, oscillations will decay until A is reached. The steady oscillation E is stable for if the amplitude is slightly less than or slightly greater than that corresponding to the stability boundary, oscillations will converge on to the boundary. The stable or sustained oscillation corresponding to boundary E is termed a limit cycle, and it is this limit cycle that is properly referred to as “hunting”. For V > VB , the motion along the track centreline is unstable and any disturbance µ = 0.15

~ 6

µ = 0.10 ~

a (mm)

Linear creep

Linear geometry

~

3

µ = 0.05

~ 0 50

80

V (m/s)

110

Figure 3.7 Effect of nonlinear geometry and creep saturation on limit cycles. The points marked by ~ correspond to nonlinear solutions with µ = 0.3. Parameters for vehicle of Table 2.3 and wheel rail geometry of Figure 2.2.

86

RAIL VEHICLE DYNAMICS

will result in the oscillation growing until the limit cycle E is reached. Similarly, if the amplitude is greater than that corresponding to boundary E the oscillation will also converge to the limit cycle E. Figure 3.7 shows the limit cycle diagram for the example wheelset of Table 2.3 and wheel-rail geometry of Figure 2.2 for two cases. The first case assumes linear creep and indicates the destabilising effect of nonlinear geometry for larger amplitudes. The second case assumes linear geometry and shows the stabilising influence of creep saturation which is enhanced for reduced values of the coefficient of friction. This diagram was derived by numerical integration of the equations of motion, using a Runge-Kutta fourth-order method with interval adjustment in order to achieve satisfactory accuracy. Successive trial solutions, for a given vehicle speed, with a varying initial amplitude make it possible to locate one point on the limit cycle curve. Other points on the limit cycle curve can then be most easily established by varying the vehicle speed in steps and using as initial conditions the vector of state variables corresponding to the limit cycle for the previous speed. Though stability can be investigated by examining the dynamic response to a disturbance using numerical solution of the equations of motion, methods exist for the direct solution of the equations of motion to obtain the steady periodic response to excitation. It is assumed that there is a periodic solution to the equations of motion at a chosen speed V. In the Describing Function method originally introduced by Kochenburger [10] and first applied in the present context by Cooperrider et al [11] it is assumed that the response is sinusoidal so that x = a sin ωt

(1)

Then the nonlinear equations of motion, equations (2.9.8-10), which can be written as x = F(x)

(2)

are replaced by quasi-linear equations

x = G (a, ω)x

(3)

where G the describing function matrix is a function of the amplitudes a and frequency ω of the oscillation. This method is particularly appropriate in cases where the nonlinearity is of the form of a single output dependent on a single input, for example, the rolling radius difference as a function of wheelset lateral displacement. If a typical nonlinear term of F is g(x) then it can be expanded as a Fourier series g ( x ) = A1 sin ωt + A2 sin 2ωt +........

(4)

In the simplest form of the method it is assumed that the frequency response of the rest of the system is limited, and so

DYNAMICS OF THE WHEELSET

87

g ( x ) = A1 sin ωt

(5)

A1 is determined by multiplying (5) by sinωt and integrating over one cycle so that 1 A1 ( a ) = π

2π

∫ g( x) sin ωtd (ωt )

(6)

0

For example, in the case of the rolling radius difference g(y) = rr - rl is replaced by λy where λ is the equivalent conicity and λ = A1(a)/a, as discussed in Section 2.3. Then the feedback model of Section 4 can be extended. The frequency response is given by equation (4.1) in which, if the small terms in f23 and Iy are neglected, λ is in the numerator and can be considered as an amplitude dependent and frequency insensitive describing function. It follows that an approximation to the stability boundary D is given by equation (3.16) where λ is interpreted as the describing function. If the amplitude xi of one state is chosen, the equation

G (a, ω)x = sp x

(7)

gives a set of eigenvalues sp. The eigenvalue with the smallest real part corresponds to an approximation to the limit cycle, the corresponding eigenvector gives the remaining amplitudes and the imaginary part of the eigenvalue the frequency. By iteratively choosing a new value for x, evaluating the describing function, and then repeating the process using the results of the previous step, convergence to the required solution with zero real part is obtained. The stability of the resulting limit cycle follows from the properties of the quasi-linearised system matrix for if the limit cycle is stable, smaller amplitudes will yield eigenvalues with positive real parts and larger amplitudes should produce negative real parts, and conversely for unstable limit cycles, [11]. In the case of strong nonlinearities such as creep saturation, the motion must be approximated by further terms in the Fourier series as the motion is not approximated by a single sinusoidal component. Various methods for the determination of limit cycles exist. For example, de Pater [12, 13] and Van Bommel [14] in their pioneering work on nonlinear oscillations of railway vehicles used the method of Krylov and Bogoljubov. Gasch et al [15, 16] applied Galerkin’s method. It is known that apparently simple dynamical systems with strong nonlinearities can respond to a disturbance in very complex ways. In fact, for certain ranges of parameters no periodic solution may exist [17]. Moreover, systems with large nonlinearities may respond to a disturbance in an apparently random way. In this case, the response is deterministic but is very sensitive to the initial conditions. Such chaotic motions have been studied for railway vehicles by True et al [18, 19].

88

RAIL VEHICLE DYNAMICS

3.6 Energy Balance Insight is given to the mechanism of the instability if the energy balance at the critical speed is considered. Following Duncan [20], and returning to the linear solution for small disturbances, as already discussed in the context of the system eigenvalues, there are two possibilities for instability. If there is a real eigenvalue sp then   q = s p q and q T Cq = spq T Cq = 0 . Thus from (2.10.35) d ( T + U + U G + U F ) + q T Bq / V = 0 dt

(1)

Now q T Bq is positive definite for any assemblage of wheelsets so that d (T + U + U G + U F ) < 0 dt

(2)

As T, U and UG are positive definite, if UF is positive definite, motion will decay. If UF is not positive definite, T + U + UG may increase. There may be conditions for which there is a zero root, then the equations of motion admit a solution in which displacements are not zero so that ( E + E g + C )q = 0

(3)

Such a condition will separate regions of static stability from regions of divergence, in which the system has a real, positive eigenvalue. A system comprising a wheelset which is elastically restrained in an unsymmetric way, fore and aft, can exhibit this behaviour and is considered later. Consider now the motion corresponding to a complex eigenvalue s = µ ± iω so that (4) q = e µt ( a cos ωt + b sin ωt ) where, for the wheelset q is given by (4.6-7). Then an important term in (2.10.35) is   q T Cq = ωe 2 µt b T Ca

(5)

At the speed VB, µ = 0. Integrating (5) over one period τ, and since the kinetic and elastic strain energies have the same values at the end of the cycle as they had at the beginning of the cycle, (2.10.35) reduces to τ

  Vπa Cb = q T Bqdt T

∫ 0

or

(6)

DYNAMICS OF THE WHEELSET

89

( f22a12 + f11a22)ω/V + ( f22 + f11λl/r0 )a1a2sinε = 0

(7)

The first term in (7) represents the dissipation of energy by the creep damping terms and the second term represents the energy input arising from the stiffness coupling terms due to creep and conicity. Note that a phase difference is necessary for sustained oscillation to occur. Though the motion at the critical speed has constant amplitude, energy is being continuously dissipated by the action of creep. The rate of dissipation of energy by the creep forces is given by equation (2.10.41) and is, of course, equal to the energy supplied by the tractive efforts which are given by equations (2.10.43-44). Eliminating the phase angle ε and the yaw angle a2 by means of equations (4.5-6) the energy dissipated in one cycle is

∫ 2Fdt

2

2

2

= (2πVB/ωB){(Kψm - KyI)/( 2f22I + 2f11ml )} ( f22 + f11λl/r0 )a1

(8)

This depends on the separation of the modal frequencies and on the coupling terms in the creep stiffness matrix; if the modal frequencies are equal, the dissipation per cycle is zero, for in this case the mode of instability is the same as the kinematic mode and there is no creepage.

3.7 Dynamic Aspects of Guidance The steady-state behaviour of a wheelset on a uniform curve at low speeds has been discussed above so the dynamic aspects of curve entry at speed will now be considered. The dynamic response of a complex system with nonlinear properties must usually be obtained by step-by-step numerical integration. However, if the system is linear, the response may also be obtained by the method of modal superposition. This involves the calculation of the response of each of the natural modes of the system and then summing to obtain the complete response. As this method gives greater insight into the dynamics of the system it represents a useful starting point for the discussion of the response of a wheelset which is, as discussed above, a nonconservative active system. The method of modal superposition was originally applied to conservative systems, but was extended to non-conservative systems by Halfman [21]. It was applied to the wheelset by Hobbs [22], and the theory has thoroughly discussed by Fawzy and Bishop [23], Collar and Simpson [6] and Newland [7, 24]. The method can be summarised as follows; reference may be made to [6] and [7] for detailed treatments. The linear equations of motion, equations (2.10.51) are x = A x + Y

Defining normal coordinates of the system φ by

(1)

90

RAIL VEHICLE DYNAMICS

x = Uφ

(2)

where U is the eigenvector matrix of the system defined in equation (3.5). It is assumed that the eigenvalues are not repeated which is the usual case for the wheelset. The normal coordinates possess an orthogonality property, for premultiplying (3.5) -1 by U = W yields WA U = Λ

(3)

Substituting from (2) into equation (1), premultiplying by W and using (3) an uncoupled set of 2N equations (s - Λ)φ = WY

(4)

is obtained. For the simple case where the time dependence of Y(t) is a step function, substituting into (2) reduces (4) to a simple expansion in a series of the modal contributions to the response. Taking Laplace Transforms, so that s becomes the Laplacian operator, yields the modal expansion 2N

x =

∑

p =1

u p w pY s − sp

(5)

where up is the column of U corresponding to the pth eigenvalue and wp is the corresponding row of W. Equation (5) can be used as a general basis for the discussion of dynamic response. Consider the dynamic response of a wheelset to entry to a uniform curve of relatively large radius. The wheelset is considered to be elastically restrained to a frame constrained to follow the centreline and take up a radial position on the track. Reference to the simplified equations of motion (2.10.47-48) shows that YT = [ 0

0

-2f11l2H(t)/R0I ]

0

(6)

where H(t) is the unit step function. It has been shown above that, for typical values of the parameters, the wheelset has four eigenvalues, comprising two real roots and a complex conjugate pair. For a complex conjugate pair of eigenvalues, the corresponding rows of U and columns of W will also be complex conjugate, and in (5) pairs of terms may be grouped together to form terms of the form 1 − s 2 + 2 µ1 s + ω1 2 + µ12

(7)

Interpretation of the Laplace Transforms in (5) therefore yields a corresponding term in the modal expansion of the response of the form

DYNAMICS OF THE WHEELSET

91

-4

0

x 10

2000

y

-2

T2l

1000 0

-4 -6

ψ 0

0.02 4

1

t

0.04

-1000

x 10

0.5

0.02

0.04

0.06

100 50

T1l

0

zwW

0

T1l

-0.5 -1

T2r

-2000 0.06 0

0

T2r+T2l

-50 0.02

0.04

0.06

-100 0

0.02

0.04

0.06

Figure 3.8 Response of elastically restrained wheelset to entry to uniform curve for t < 0.06 s. R0 =1000 m. V=10 m/s. 2

x 10

-3

2000

ψ

0

T2r

0 -2000

-2 -4

-4000

y 0

2

t

4

4000

-6000

T2l 0

2

4

4000

zwW

2000

T1r

2000

0 0 -2000

T2r+T2l

-2000 0

T1l

2

4

-4000

0

2

4

Figure 3.9 Response of elastically restrained wheelset to entry to uniform curve, R0 =1000 m. V=10 m/s.

e µ1t ( A cosω1t + B sin ω1t )

(8)

In view of the relative size of the eigenvalues, it is apparent that in general the response will consist of two rapidly damped transients and a damped oscillation at kinematic frequency. This is illustrated by Figure 3.8 which shows the motion of the wheelset immediately after entering a large radius uniform curve, as computed from the full nonlinear equations of motion, equations (2.9.8-10). As might be expected from the eigenvectors, the subsidence in lateral translation is not excited and the lateral displacement is small. At t = 0 the creep couple due to the curvature of the track is reacted by the yaw inertia of the wheelset so that the yaw acceleration is ψ = −2 f11l 2 / R0 I

(9)

As the yaw velocity builds up, the yaw acceleration decays and increasingly the

92

RAIL VEHICLE DYNAMICS

1

x 10

-3

2000

ψ

0 -1 -2

1

t (s)

2

3

400

-4000 0

1

2

3

2000

T 1r

200

1000

0

zyW

0

-200 -400

T 2l

-2000

y (mm) 0

T 2r

0

0

1

T 2r + T 2l

-1000

T 1l 2

3

-2000 0

1

2

3

Figure 3.10 Response of elastically restrained wheelset to entry into curve, R0 = 1000 m and transition length L = 20 m. -3

x 10

0

5000

y (mm)

-2 -4

ψ

-5000

-6 -8

0

2

1 0 -1

0.1 4

x 10

2

T2l

-10000 0

0.1

t (s) 4

-2

T2r

0

T1r

1

T1l

0

0

-1

0.1

x 10

zyW T2r + T2l 0

0.1

Figure 3.11 Response to curve entry with no transition for t < 0.13 s. R0 = 225 m 5

x 10

-3

5000

ψ

T2r

0

0

-5

-5000

T2l

y (mm) -10 0

1

t (s)

2

3

2000

-10000 0 1

T1r

1000 0

0 -0.5

T1l 1

2

1

3

-1

2

3

4

zyW

0.5

-1000 -2000 0

x 10

T2r + T2l 0

1

2

3

Figure 3.12 Response to entry into curve, R0 = 250 m and transition length L = 20 m.

DYNAMICS OF THE WHEELSET

93

couple due to conicity is balanced by the longitudinal creep forces generated by the yaw velocity. The yaw velocity is, approximately

ψ = −

V (1 − e µ4 t ) R0

(10)

so that, after the initial transient, the contribution of this mode to the yaw velocity becomes constant and is equal to -V/R0. The longitudinal creep at t = 0 is equal to ±l/R0 and so for radii of curvature less than about 1000 m there will be slipping. Though transient effects modify the behaviour of the system for small t, this is not of prime interest as far guidance is concerned. The response in the kinematic mode is given approximately, if µ1 << ω1 , for t small y = −(

lr0 )(1 − cos ω1t ) = −V 2 t 2 / 2 R0 λR0

(11)

ψ = −(

lr0 y )(ω1 sin ω1 t ) = = −Vt / R0 V λR0V

(12)

ψ = −(

lr0 )(ω12 cos ω1 t ) = −V / R0 λR0V

(13)

and these results are consistent with the results in Figure 3.8. The response for larger values of t, also computed from the full nonlinear equations of motion, are shown in Figure 3.9. It can be seen that after the initial very fast transient, the response closely matches the above approximation and the system behaves as a simple damped spring-mass system, with stiffness dependent on the conicity and speed, and damping dependent on the suspension stiffnesses, inertia and creep coefficients. Also shown in Figure 3.9 are the creep forces. The lateral creep forces are mainly generated by spin and are largely cancelled out by the gravitational stiffness force. The longitudinal creep forces decay rapidly as the wheelset is able to move out to the equilibrium rolling line. These results indicate one of the deficiencies of the wheelset as a guidance mechanism in that the response time is solely a function of the conicity which is largely out of the control of the designer and which lies within certain practical limits. Figure 3.10 shows comparable results for the wheelset which enters a similar curve as that discussed above but in this case there is a transition of length 20 m over which the curvature is increased linearly from zero to the steady state value. Thus, within the transition, the integration of (2.2.3) yields for the azimuth of the centreline of the track

Ψ = s2/2R0L

(14)

94

RAIL VEHICLE DYNAMICS

and then the integration of (2.2.1-2) gives for the coordinates of the centreline of the track X = s − s 5 / 40 R02 L2 +.....

Y = s 3 / 6 R 0 L − s 7 / 336 R 03 L3 +.......

(15)

Consequently, this form of transition is commonly referred to as a cubic parabola. It can be seen that the excitation of the kinematic oscillation, and the level of the forces acting in the transient, is much reduced. Results for a sharper curve, R0 = 225 m with no transition curve are shown in Figure 3.11. For small t transients similar to those discussed above occur. The wheelset moves in a closely straight line until, after a distance (2R0δf)1/2 into the curve, flange contact takes place. Note that this distance is much less than the kinematic wavelength. When flange contact takes place accurate estimates of the lateral forces require allowance for track flexibility [25]. The provision of a transition curve allows the wheelset to respond in a more satisfactory way. Figure 3.12 shows the dynamic response of the wheelset to a 250 m radius curve with a transition length of 20 m and shows that flange contact is avoided. For small amplitudes of motion and over the frequency range appropriate to the response to track irregularities, the frequency response of a wheelset may be obtained from (5). Using the approximations of equations (3.13), (3.14) and (3.17) and neglecting the contribution of the modes associated with the large real roots the frequency response is found to be H (ω ) =

ω1 2 y (iω ) = y t (iω ) − ω 2 + 2 µ1iω + ω1 2

(16)

as derived by Hobbs [22] . Illingworth [26] carried out model roller rig experiments verifying that the theoretical description of excitation of wheelset motions by track irregularities was substantially correct. In the case where irregularities are distributed continuously along the track the approach offered by stochastic process theory is appropriate. Random process theory was first applied by Hobbs in 1964 [22] to the lateral motions of a wheelset. Subsequently, extensive measurements of the power spectra of irregularities of track have been made and have been used extensively in the assessment of vehicle response.

3.8 Alternative Methods of Guidance The early mining railways employed "peg in the slot" as a guidance system and whilst this was superseded by the coned wheelset early in the 19th Century there is a continuous record of invention of alternative guidance systems. In the past many alternative systems were considered because the running gear of railway vehicles often performed badly; rapid wear of wheels and rails, hunting and derailments occurred as a result of the lack of understanding of how a wheelset should be incorporated in a vehicle. As discussed above, the smallest radius for flange free curving of

DYNAMICS OF THE WHEELSET

95

a wheelset with conventional profiles well adapted to conventional rail profiles is about 150 m, and the longitudinal movement of a coned wheelset in taking up a radial position on sharply curved track can represent a design problem. As a target curve radius for some innovative urban transit systems can be as low as 10 m there is scope for alternative methods of guidance of railway vehicles. For configurations in which the wheels are fixed on a common axle, two conditions, positive centering action for small amplitudes within the flangeway clearance and positive retention or static stability for large displacements, can be made the basis for selection of wheel rail geometry. The possibilities are shown in Figure 3.13. Configurations (c) and (e) have positive conicity and rate of change of contact slope with lateral displacement and hence satisfy the basic requirements. It is interesting that one conforms to the conventional wheel rail combination whilst the other resembles a plateway, a system much used in the later 18th century. For some applications it is possible to separate out the functions of support and guidance completely, and provide separate sets of wheels, exerting only normal forces. There are four basic possibilities as shown in Figure 3.14 and some of these have been used both with steel wheels on steel rails and with pneumatic tyres on beams. However, the multiplicity of wheels and the ponderous nature of the switching arrangements make this approach relatively unattractive. Independently rotating wheels have been frequently proposed as they eliminate the classical hunting problem. Some of the possibilities have been surveyed by Frederich [27]. The essential difference between a conventional wheelset and independent wheels lies in the ability of the two wheels to rotate at different speeds and therefore, strictly, there is an additional degree of freedom. However, the effect in

(a)

ε < 0, λ > 0

(b)

ε > 0, λ < 0

(d)

(e)

ε < 0, λ < 0

ε > 0, λ > 0

(c)

ε > 0, λ > 0 (f)

ε > 0, λ < 0

Figure 3.13 Basic possibilities for wheel-rail geometries. ε (equivalent contact slope) and λ (equivalent conicity) are the linearised geometric parameters defined in equations (2.3.36) and (2.3.37).

96

RAIL VEHICLE DYNAMICS

Figure 3.14 Possible configurations of support and guidance wheels depending on normal forces only.

terms of the linear wheelset model is to render f11 = 0. The kinematic oscillation of a conventional wheelset is therefore eliminated (the feedback loop of Figure 2.19 is broken) but a measure of guidance is then provided by the lateral component of the gravitational stiffness (reduced by the lateral force due to spin creep) which becomes the flange force when the flangeway clearance is taken up, but this leads to slow self centering action. Extensive experimental experience has shown that indeed the kinematic oscillation is absent but that one or other of the wheels runs in continuous flange contact [28]. The detailed analysis of a system embodying independently rotating wheels requires that the calculation of creepages take into account the differing wheel rotational speeds, Ωr and Ωl, and this particularly affects the longitudinal creepage. If there is only one point of contact between each wheel and the rail then, neglecting the rotational acceleration of the wheels, each wheel will be rolling with zero longitudinal force and zero longitudinal creepage in accordance with the simple model. However, if a wheel contacts the rail at two distinct points, as is common with freely rotating wheels, the analysis is more complex. Good agreement between calculation and experiment is demonstrated in [29, 30, 31]. An attempt to increase the effect of the lateral resultant gravitational force but reduce the amount of spin is to incline substantially from the horizontal the axis of rotation of the wheels, Figure 3.15(a), as put forward by Wiesinger [32]. A generic wheelset model including the effect of modest amounts of camber has been studied theoretically and experimentally by Jaschinski and Netter [33]. One approach in applying freely rotating wheels is to provide a pivot ahead of the wheels. The wheelset mounted on a leading or trailing arm, a ‘pony’ axle, was used on many steam locomotives and the lack of fore and aft symmetry has a significant effect on the dynamics of the system. In concept this is closely linked to the peg in the slot type of system to be discussed later, in that stability depends on the lack of foreand-aft symmetry. A similar effect is achieved by a linkage, or the equivalent, to the preceding vehicle in a train. An example of this is the original version of the

DYNAMICS OF THE WHEELSET

97

(b)

(a)

A

C

S

(d)

(c)

Figure 3.15 Some further alternative methods of guidance using independently rotating wheels (a) cambered wheels (b) Talgo train with vehicles guided by vehicle in front (c) Koyanagi’s system with guidance from a central rail ahead of the wheels (d) guidance from a trace using a sensor S, controller C and steering actuator A.

2fψ d 2fλdψ/r0

ψ

2fψ

2fλdψ/r0

ψ

Figure 3.16 Conventional wheelset mounted on a leading or trailing arm.

98

RAIL VEHICLE DYNAMICS

100

O

V (m/s) 50

D 0

-3

-2

d (m) 0

1

2

3

Figure 3.17 Stability diagram for conventional wheelset mounted on trailing or leading arm. Parameters from Table 2.1 except ky = 2 MN/m and kψ =0.

Talgo train [34], Figure 3.15(b). The same effect is achieved by guiding the pivot point along a central rail as in the concept, Figure 3.15(c), analysed by Koyanagi [35]. More complex but related arrangements used in light rail applications have been considered in [36, 37]. A logical extension of these concepts is the form of guidance, using independently rotating wheels, shown in Figure 3.15(d). Not only does the dynamics of this system have some interesting features but a modern implementation using active controls shows considerable promise, [38]. Guidance is achieved by measuring the lateral displacement from a reference trace on the track, and sending a steering command to an actuator through a controller. One aspect of the dynamical behaviour of these unsymmetric fore-and-aft configurations can be illustrated by considering the conventional wheelset mounted on a leading or trailing arm, Figure 3.16. The elastic stiffness matrix becomes ⎡ ky E=⎢ ⎣dk y

dk y ⎤ d 2 k y ⎥⎦

(1)

In the absence of suspension damping, assumed for simplicity, the equations of motion for the wheelset are otherwise unaltered. A stability diagram for this system is shown in Figure 3.17 in which the stability boundaries are plotted as a function of arm length d and speed. Mounting the spring either forward or aft is strongly stabilising as far as the dynamic stability of the system is concerned for the dominant effect is that of increasing the yaw stiffness. As the source of instability is the presence of assymmetric coupling terms in the equations of motion, between lateral translation and yaw, it might be thought that an obvious way of stabilising the system is to negate one or other of these terms by adding lateral stiffness offset either in front of or behind the wheelset. This is similar to the technique known as ‘mass balancing’ much used in Aeroelasticity. Unfortunately the creep stiffness coupling terms are large in relation to the elastic stiffness couplings and impracticably large arm lengths

DYNAMICS OF THE WHEELSET

99

or stiffnesses would be necessary. However, if the spring is moved too far forward static instability or divergence occurs. The condition for static stability is that in the characteristic equation p0 > 0 [9] or ( −d ) =

2 f ⎛ λl / r0 ⎞ ⎜ ⎟ k y ⎝ 1 − λl / r0 ⎠

(2)

The system can only be statically unstable if either λl/r0 > 1 and d > 0 or λl/r0 < 1 and d < 0. Thus, a leading arm with low conicity is prone to divergence. This form of instability has been experienced on locomotives in the past, and is a potential problem for articulated vehicles as discussed in Chapter 8. As mentioned above, the case of freely rotating wheels mounted on a leading or trailing arm may be covered by putting f11 = 0 in the linear equations of motion. In this case the criterion for static stability becomes simply d > 0 so that the wheels must be behind the pivot point for static stability. In fact, more complex arrangements make it possible to provide stability in both directions and various ways of exploiting a lack of fore-and-aft symmetry are discussed in Chapter 9. A simple model of a four wheeled vehicle with independently rotating wheels and guided by peg-in-slot and linkage reveals the basic aspects of the dynamics of such vehicles. These considerations equally apply to the system of Figure 3.15(d). Referring to Figure 3.18, it can be seen that guidance is obtained by steering the wheels (the rear wheels in the opposite sense to those of the front) of the simple vehicle shown by a linkage connected to a reference line (such as a peg in a slot) distance L ahead of the vehicle centre of mass. Assuming a general form of linkage, then for

2a

δ

δ

y

ε

ψ

yt

L

Figure 3.18 Simple model of vehicle with independently rotating wheels guided by peg-inslot and linkage.

100

RAIL VEHICLE DYNAMICS

small displacements, the steering law can be written as δ = −G( y + Lψ − yt )

(3)

where G is the gain dependent on the form and dimensions of the linkage. The linear equations of motion are easily formulated and take the form, writing f for f22, m  y + 4f( y /V - ψ ) = 0

(4)

I z ψ + 4 fa 2ψ / V − 4 faδ = 0

(5)

(ms2 + 4fs/V)y - 4fψ = 0

(6)

Substituting from (3)

2

4fGay + ( I s2+ 4fa s/V + 4fGaL)ψ = 4fGyt z

(7)

The similarity of these equations to the wheelset equations of motion should be noted. The analysis of Section 3 can now be repeated. For the present system, the coefficients of the characteristic polynomial are p4 = mI

p3 = 4f(ma2 + I)/V

p2 = 4fGaLm + 8f 2a2IV2 (8) p0 = 16f 2Ga

p1 = 16f 2GaL/V

A typical locus of the eigenvalues as speed is varied is shown in Figure 3.19 for various values of the gain G. At low speeds, as in the case of the wheelset, the characteristic equation can be approximately factorised in the form 2

2

(s + 2µ s + ω ) (s + µ ) (s + µ ) = 0 1

1

2

3

(9)

where

µ = - 4f/mV 2

2

µ = - 4fa /IV 3

µ1 =

GV { L − (ma 2 + I )V 2 / 4 fa} 2a

ω1 = V G / a

(10) (11) (12) (13)

DYNAMICS OF THE WHEELSET

101 100

100

ω

ω

A 50

A

50

B

B 0 -100

-50

µ

0

50

0

0

100

V (m/s)200

Fig 3.19 Variation of eigenvalues with speed for peg-in-slot system: a = 1.25 m; f = 10 MN.; G = 1; L = 2 m; m = 1250 kg.; Iz = 700 kgm2.

For small values of G, the quadratic factor represents a steering oscillation labelled A in Figure 3.19. This has wavelength a / G independent of speed, in which there is zero creep. For example, if G = 1 rad/m, L = 2 m, a = 1.5 m and V = 10 rn/s then the wavelength of the steering oscillation is 7.69m corresponding to a frequency of 1.3 Hz and the decay rate is 0.81 of critical. The decay rate of this oscillation depends on L being positive; in this simple vehicle, as is intuitively obvious, the peg must be placed forward of the centre of the vehicle. In reverse motion, V negative, such a vehicle would be unstable. The two roots given by (10) and (11), labelled B in Figure 3.19 correspond to subsidences in lateral translation and yaw respectively, exactly analogous to the wheelset. At higher speeds these two real roots coalesce to form a well damped oscillation. For G > 4a/L2 and low speeds the steering oscillation is replaced by two subsidences. Equation (12) shows that as speed is increased, the inertia forces have the effect of reducing the decay rate of the steering oscillation. The damping of the steering oscillation vanishes at an approximate speed VB when one root of the characteristic polynomial is s = iωA. From equation (12) VA2 = 4fa2L/(ma2 + I)

(14)

but this approximate result is only valid for small G. A more exact speed VB is easily obtained by substituting s = iω in the characteristic polynomial and equating real and imaginary parts, to give V B2 = 4 fa 2 L / (ma 2 + I ){1 − GL2 m 2 a 3 / (ma 2 + I ) 2 }

(15)

It can be seen that increasing the gain is stabilising and the vehicle will be stable for all values of V if G > (ma2 + I)2/m2L2a3

(16)

102

RAIL VEHICLE DYNAMICS

Also the vehicle is stable for all values of G if V < 2fa2L(ma2 + I)

(17)

The similarity of the behaviour of the system to that of a railway wheelset is carried over to its curving performance. In steady motion on a uniform curve the equations of motion (4) and (5) reduce to − 4 fψ = − mV 2 / R0

(18)

4 fa 2 / R0 = −4 faδ

(19)

so that the vehicle yaws through an angle

ψ = mV 2 / 4 fR0

(20)

to react the centrifugal force, and the required steer angle is δ = a / R0

(21)

In order to generate these steer angles the vehicle must move laterally through a distance y given by substitution into the steering law, equation (3) from (20) and (21) given by y = L2 / 2 R0 − a / gR0 − LmV 2 / 2 fR0

(23)

It can be seen that at low speeds if G > a/2L2 the vehicle is displaced towards the centre of the curve and if G < a/2L2 it is displaced outwards. As speed increases further outward movement takes place in order to allow the vehicle to yaw whilst generating the correct steer angle. Another approach to the improvement of the wheelset as a guidance element is intermediate between a conventional wheelset and one with independent wheels. This provides a torque connection such as a damper or clutch between the two wheels of a wheelset. Such a scheme was proposed by Benington [39] and a similar scheme has been analysed by Choromanski and Kisilowski [40]. A 'creep controlled' wheelset has been developed [41] which uses a controlled magnetic coupling in the centre of the axle of the wheelset. Various control laws have been used for the control of the coupling, including feedback of creep measurements. Essentially the wheels have a good torque connection at low frequencies so that curving ability is maintained but at high frequencies, typical of wheelset kinematic frequencies, the wheels are more or less uncoupled so that instability does not arise. The stability of wheelsets with independently rotating wheels using a control engineering approach has been considered by Goodall and Li [42].

DYNAMICS OF THE WHEELSET

103

References 1. Matsudaira, T.: Shimmy of axles with pair of wheels (in Japanese). J. of Railway Engineering Research, (1952), pp. 16-26. 2. Matsudaira, T.: Paper awarded prize in the competition sponsored by Office of Research and Experiment (ORE) of the International Unions of Railways (UIC). ORE-Report RP2/SVA-C9, ORE, Utrecht, 1960. 3. Wickens, A.H.: The dynamic stability of railway vehicle wheelsets and bogies having profiled wheels. Int. J. Solids Structures, 1 (1965), pp. 319-341. 4. Wickens, A.H.: The dynamics of railway vehicles on straight track: fundamental considerations of lateral stability. Proc. I. Mech. Engrs. l80, part 3F (1965), pp. l-16. 5. Carter, F.W.: The electric locomotive. Proc. Inst. Civ. Engs. 221 (1916), pp. 221252. 6. Collar, A.R. and Simpson, A.: Matrices and engineering Dynamics. Ellis Horwood, Chichester, 1987. 7. Newland, D.E.: Mechanical vibration analysis and computation. Longman, Harlow, 1989. 8. Culick, F.E.C.: Effects of small changes in the elements Aij on the zeros of |Aij| J. Royal Aeron. Soc. 62 (1958), pp. 898-901. 9. Porter, B.: Stability criteria for linear dynamical systems. Oliver Boyd, Edinburgh, 1967. 10. Kochenburger, R.J.: Frequency-response methods for analysis of a relay servomechanism. Trans. AIEEE, 69 (1950), pp. 270-284. 11. Cooperrider, N.K., Hedrick, J.K., Law, E.H. and Malstrom, C.W.: The application of quasilinearization techniques to the prediction of nonlinear railway vehicle response. In: H.B. Pacejka (Ed.): The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the IUTAM Symposium, Delft, The Netherlands, August 1975. Swets & Zeitlinger, Lisse, 1976, pp. 314-325. 12. de Pater, A.D.: Etude du mouvement de lacet d'un vehicule de chemin de fer. Appl. Sci. Res. A, 6 (1956), pp. 263-316.

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13. de Pater, A.D.: The approximate determination of the hunting movement of a railway vehicle by aid of the method of Krylov and Bogoljubov. Appl. Sci. Res. 10 (1961), pp. 205-228. 14. van Bommel, P.: Application de la theorie des vibrations nonlineaires sur le problem du mouvement de lacet d'un vehicule de chemin de fer. Doctoral dissertation, Technische Hogeschool Delft, 1964. 15. Moelle, D. and Gasch, R.: Nonlinear bogie hunting. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 455-467. 16. Gasch, R., Moelle, D. and Knothe, K.: The effects of non-linearities on the limitcycles of railway vehicles. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 207-224. 17. Thompson, J.M.T. and Stewart, H.B.: Nonlinear dynamics and chaos. John Wiley, Chichester, 1986. 18. True, H.: Dynamics of a rolling wheelset. App. Mech. Reviews, 46 (1993), pp. 438-444. 19. True, H.: Railway vehicle chaos and asymmetric hunting. In: G. Sauvage (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 12th IAVSD Symposium, Linkoping, Sweden, August 1991. Swets and Zeitlinger Publishers, Lisse, 1992, pp. 625-637. 20. Duncan, W.J.: The flutter of systems with many freedoms. Aeronautical Quarterly 1 (1949), p. 59. 21. Halfman, R.L.: Dynamics. Addison-Wesley, Cambridge, Mass., 1959. 22. Hobbs, A.E.W.: The response of a restrained wheelset to variations in the alignment of an ideally straight track. British Railways Research Department Report E542, 1964. 23. Fawzy, I. and Bishop, R.E.D.: On the dynamics of linear non-conservative systems. Proc. R. Soc. Lond. A 352 (1976), pp. 25-40. 24. Newland, D.E. On the modal analysis of non-conservative systems. J. Sound and Vibration 112 (1987), pp. 69-96. 25. Clark, R.A., Eickhoff, B.M. and Hunt, G.A.: Prediction of the dynamic response of vehicles to lateral track irregularities. British Rail Research Tech. Memo. TM DA 42, 1983.

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26. Illingworth, R.: Railway wheelset lateral excitation by track irregularities. In: A. Slibar and H. Springer (Eds.): The Dynamics of Vehicles on Roads and Tracks, Proc. 5th IAVSD -2nd IUTAM Symposium, Vienna, September 1977. Swets and Zeitlinger Publishers, Lisse, 1978, pp. 450-458. 27. Frederich, F.: Possibilities as yet unknown regarding the wheel/rail tracking mechanism. Rail International 16, (1985), pp. 33-40. 28. Becker, P.: On the use of individual free rolling wheels on railway vehicles. Eisenbahn Technische Rundschau No. 11 (1970). 29. Eickhoff, B.M. Harvey, R.F.: (1989) Theoretical and experimental evaluation of independently rotating wheels for railway vehicles. In: R. Anderson (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 11th IAVSD Symposium, Kingston, Ont., August 1989. Swets and Zeitlinger Publishers, Lisse, 1989, pp. 190-202. 30. Eickhoff, B.M.: The application of independently rotating wheels to railway vehicles. Proc. I. Mech. E. 205, Part3F (1991), pp. 43-54. 31. Elkins, J.A.: The performance of three-piece trucks equipped with independently rotating wheels. In: R. Anderson (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 11th IAVSD Symposium, Kingston, Ont., August 1989. Swets and Zeitlinger Publishers, Lisse, 1989, pp. 203-216. 32. de Pater, A.D.: Analytisch en synthetisch ontwerpen. Technische Hogeschool Delft, 1985, pp. 37-38. 33. Jaschinski, A. and Netter, H.: Non-linear dynamical investigations by using simplified wheelset models. In: G. Sauvage (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 12th IAVSD Symposium, Linkoping, Sweden, August 1991, pp. 284-298. Swets and Zeitlinger Publishers, Lisse, 1992. 34. Oriol de, L.M.: El Talgo Pendular. Revista Asociacion de Investigacion del transporte. No. 53 (1983), pp. 1-76. 35. Koyanagi, S.: A new guide system for a wheel-rail vehicle. In: A. Slibar and H. Springer (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 5th IAVSD 2nd IUTAM Symposium, Vienna, September 1977. Swets and Zeitlinger Publishers, Lisse, 1978, pp. 407-415. 36. Bonivert, L., Maes, P. and Samin, J.C.: Lateral dynamics of a guided light transit vehicle. In: R. Anderson (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 11th IAVSD Symposium, Kingston, Ont., August 1989. Swets and Zeitlinger Publishers, Lisse, 1989, pp. 84-96.

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37. Chatelle, Ph., Duponcheel, J. and Samin, J.C.: Investigation on nonconventional railway systems through a generalised multi-body approach. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 43-57. 38. Wickens, A.H.: Dynamics of actively guided vehicles. Vehicle System Dynamics 20 (1991), pp. 219-242. Goodall, R.: Active railway suspensions: implementation status and technological trends. Vehicle System Dynamics 6 (1977), pp. 87-117. 39. Bennington, C.K.: The railway wheelset and suspension unit as a closed loop guidance control system, a method for performance improvement. Journal of Mech. Eng. Sci. 10 (1966), pp. 91-100. 40. Choromanski, W. and Kisilowski, J.: Dynamics of railway trucks with wheelsets with independently rotating wheels and controlled slip. In: R. Anderson (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 11th IAVSD Symposium, Kingston, Ont., August 1989. Swets and Zeitlinger Publishers, Lisse, 1989, pp. 108-125. 41. Geuenich, W., Guenther, C. and Leo, R.: Dynamics of fiber composite bogies with creep-controlled wheelsets. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 225-238. 42 Goodall, R.M. and Li,, H.: Solid axle and independently rotating wheelsets-a control engineering assessment of stability. Vehicle System Dynamics 33 (2000), pp. 57-67.

4 Guidance of the Two-Axle Vehicle 4.1 Introduction As discussed in Chapter 1, guidance is the ability of a vehicle to follow the geometric layout of the track, and is mainly determined by behaviour in curves. It has been seen in Chapter 3 that a wheelset will only be able to move outwards to the rolling line if either the radius of curvature or the flangeway clearance is sufficiently large, otherwise longitudinal creep forces are generated. In addition, the forces of cant deficiency must be reacted by lateral forces acting between wheel and rail so that lateral creep forces are generated. Moreover, for stability the wheelsets in a vehicle must be constrained by the suspension in some way. The forces imposed by the suspension must be reacted by further creep forces. If the vehicle has any dimensional misalignments then these will also be accommodated by further straining of the suspension and additional creep forces. It follows that the way in which the wheelsets are connected together, through the car body or otherwise, is fundamental to the mechanism of guidance. Consequently, the starting point for this Chapter is a general consideration of the way in which two wheelsets may be elastically connected. This is followed by a discussion of the conflict between stability and steering. Finally, the response of the complete vehicle to track curvature, cant deficiency, and misalignments is considered, the attitude of the vehicle and the forces acting being obtained by solving the equations of equilibrium. A two-axle vehicle with a single stage suspension (or a bogie or truck unrestrained by the car body) is considered. The symmetric two-axle vehicle is the simplest form of vehicle which embodies in a practical way wheelsets and their interconnections. The two-axle vehicle is important in its own right, for in addition to its widespread use in the past as both a passenger and freight vehicle, it is increasingly seen as the light weight vehicle of the future. Moreover, its study is an essential step to the understanding of the bogie vehicle and more complex configurations. The pioneering work of Mackenzie [1] was mentioned in Chapter 1. His seminal paper (which was subsequently translated and published in both France and Germany) was suggested by an unintentional experiment in which the springs of the driving wheels of a six wheeled engine were tightened to increase the available adhesion. The leading wheel mounted the rail when the locomotive approached a curve. Mackenzie gave a qualitative but essentially correct description of the forces generated in curving based on sliding friction, and his calculations showed that the outer

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wheel flange exerts against the rail a force sufficient to overcome the friction of the wheel treads. Thus the locomotive’s excessive flange force was not the result of centrifugal force, as was previously thought, but was generated by the friction of the other wheels. He also made the comment that “the vehicle seems to travel in the direction which causes the smallest amount of sliding” which foresaw a later analytical technique. The early development of theories of curving, based on Mackenzie’s ideas, were dominated, understandably, by the paramount need to avoid excessive loads on both vehicle and track caused by steam locomotives with long rigid wheelbases traversing sharp curves. Hence, in these theories the conicity of the wheelsets is ignored, the wheelsets are considered to be rigidly held parallel in the frame and the wheels are assumed to be in the sliding regime. The corresponding forces are then balanced by a resultant flange force, or forces, acting parallel to, and directly below the axis of a wheelset. If equilibrium is maintained by the outer leading flange contacting the rail, this was referred to as ‘free curving’. If two flanges are in contact, this was referred to as ‘constrained curving’. These ideas were developed, refined and applied over the remaining period of steam locomotive design [2], particularly by Heumann [3], who established his minimum principle that equilibrium is attained when the guiding force is at a minimum. Porter [4] gave the final and most complete development of this theory. Complete solutions of the equations of motion show agreement with these classical methods in the extreme conditions postulated. In modern practice, as a wheelset incorporated in a conventional bogie is constrained by the longitudinal and lateral springs connecting it to the rest of the vehicle, it will not be able to take up the radial attitude of perfect steering. A wheelset will then balance a yaw couple applied to it by the suspension by moving further in a radial direction so as to generate equal and opposite longitudinal creep forces, and it will balance a lateral force by yawing more. On a curve of sufficiently large radius the creepages will be small, the flangeway clearance will not be taken up, and linearised conicity may be assumed. The linear theory was first developed by Boocock [5] (who also gives experimental validation with full-scale vehicles) and Newland [6] in 1968, and though restricted in practical application, forms a useful starting point for the examination of the mechanics of curving. On most curves, the analysis of curving of conventional vehicles must involve a consideration of the progressive saturation of the creep forces and realistic wheelrail geometry incorporating flange contact. The first comprehensive non-linear treatment of practical vehicles in curves was given by Elkins and Gostling in 1978 [7], in which numerical solutions were shown to give good agreement with fullscale experiment. Their treatment covers the non-linearities caused by the movement of the contact patch across the rail and its subsequent change in shape, creep saturation and large wheel-rail contact angles. The case of steady motion in a uniform curve is a useful, quasi-static design case but in practice the response to much more complex track geometry must be considered. The problems of dynamic response fall into two major categories. The first concerns the prediction of forces and motions during transients which result from large discrete irregularities, entry into curves or the negotiation of switch and crossing work. The second major category of dynamic response problem lies in the case

GUIDANCE OF THE TWO-AXLE VEHICLE

109

where irregularities are distributed continuously along the track and statistical characteristics of the response are of interest. In the severe responses to the large discrete inputs that are of interest for design purposes, both suspension and wheel-rail contact non-linearities are important and consequently a numerical integration of the equations of motion is necessary. Early studies [8] involved assumptions similar to those employed by Heumann, with the flange force being modelled by a spring. Subsequently, the model of Elkins and Gostling was extended by Clark, Eickhoff and Hunt [9] to cover dynamic response and their simulation was successfully validated by full-scale experiment.

4.2 Properties of the Stiffness Matrix As the elastic connections between the wheelsets and car body have a fundamental effect on curving, and these are described by the elastic stiffness matrix E, it is important to consider the properties of E and the various possibilities for design. Consider the vehicle model of Section (2.11). Without any loss of generality, the symmetric motions of the car body xb, zb, and θb can be dropped from consideration. Moreover, no terms corresponding to the wheelset rotational freedoms appear in E. Consequently, attention can be confined to the system described by 7 generalised coordinates. Let y1, y2 and yb be the lateral displacements, ψ1, ψ2 and ψb be the yaw angles, and φb be the roll angle of the car body, measured from the unstrained position of the vehicle q = [ y1 ψ1 yb φb ψb y2 ψ 2 ]T

(1)

It is evident that if a two-axle vehicle is in steady motion in a uniform curve that the attitude of the vehicle can be defined in terms of the wheelset coordinates alone as these define the position of the car body. Therefore, consider the stiffness matrix E* corresponding to the reduced set of coordinates q = [ y1 ψ1 y2 ψ 2 ]T

(2)

which can be regarded as defining an equivalent inter-wheelset structure. In order to investigate the general form of this stiffness matrix the sum and difference coordinates ϕi defined by, Figure 4.1, y1 = ( ϕ1 + ϕ3 )/2

y2 = ( ϕ3 - ϕ1 )/2

ψ1= ( ϕ2 + ϕ4 )/2

ψ2= ( ϕ4 - ϕ2 )/2

(3)

are introduced. This may be written as q = Tϕ

(4)

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RAIL VEHICLE DYNAMICS

φ2

φ1

φ4

φ3

Figure 4.1 Sum and difference coordinates.

where T is a transformation matrix. Application of the principle of virtual work [10] then gives for the stiffness matrix φE* corresponding to the new set of coordinates φ

E* = TT qE*T

(5)

where qE* corresponds to the original coordinates q. Now, ϕ3 corresponds to a rigid body translation so that φE*i3 = 0, and since φE* is symmetric, φE*3i = 0. Also, in the case considered here, there is a lateral plane of symmetry, so there can be no elements representing coupling between ϕ2 and ϕ1. This is because the state of strain corresponding to ϕ2 is symmetric with respect to the plane of symmetry whereas that corresponding to ϕ1 and ϕ4 is antisymmetric. Therefore, φE*12 = φE*24 = φE*21 = φ E*42 = 0. Moreover, the motions described by ϕ1 and ϕ4 can be combined to give a rigid body rotation, thus

φ = [ h 0 0 1 ]T As the elastic strain energy must be zero in such a motion φT E* φ = 0 which yields φ

E*14 = - (φE*11h2 + φE*44)/2h

Finally, for equilibrium of the structure, φE*44 = φE*11h2 so that φE*14 = -φE*11h. The stiffness matrix, therefore, has only two independent elements, φE*11 = ks the shear stiffness and φE*22 = kb the bending stiffness and is of the form

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111

0 ⎡ ks ⎢ 0 kb φ ∗ Ε = ⎢ ⎢ 0 0 ⎢ − k h 0 ⎣ s

0 − k s h⎤ 0 0 ⎥⎥ 0 0 ⎥ 2⎥ k 0 sh ⎦

(6)

From (5), if U = T--1 E* = UT φE*U

q

(7)

and so reverting to the original coordinates, − ksh ⎡ ks ⎢− k h kb + k s h s q * E = ⎢ ⎢ − ks ksh ⎢ 2 ⎣− k s h − k b + k s h

− ks ksh ks ksh

− k sh ⎤ − k b + k s h 2 ⎥⎥ ⎥ ksh 2 ⎥ kb + k s h ⎦

(8)

This stiffness matrix is quite general and covers any form of elastic inter-wheelset structure. Specifically, it covers the conventional arrangement of two-axle vehicle as shown in Figure 2.22. For, referring to the stiffness matrix given by equation (2.11.28) and considering quasi-static conditions, the coordinates representing the displacements of the car body can be eliminated in favour of the four wheelset coordinates with the result that the reduced stiffness matrix is of the form (8) where

ks = kykψ/(2kyh2 + 2kψ)

kb = kψ/2

(9)

It can be seen from (9) that there is a limitation in the shear stiffness ks that can be achieved as ky is varied as ks ≤ kxb2/h2. Of course, ks and kb can be implemented directly if some form of direct interconnection between the wheelsets is provided, such as cross-bracing, and then there is no limit for the value of the shear stiffness ks. This general approach will be applied to examine the possibilities for different forms of suspension in the next Chapter.

4.3 Steering on Large Radius Curves As mentioned above, the basic mechanisms of guidance can be clarified by considering the case of a vehicle with purely coned wheels on a large radius curve, for in this case the linear equations of motion will be valid. Firstly, the steady-state behaviour on a uniform curve will be considered for the case of zero cant deficiency. In this case, the speed for which the centrifugal force is exactly balanced by the lateral component of the weight due to track cant. In the next Section, the additional input due to cant deficiency will be discussed. The usual definition of the generalised coordinates will be used, so that y1 and y2

112

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are the lateral displacements measured from the centreline of the track, and ψ1 and ψ2 are the yaw angles measured from the radial position. Then for coned wheels and small displacements, the gravitational stiffness and lateral force due to spin creep, together with the other small terms, can be neglected so that the equations of equilibrium, corresponding to the steady-state form of the simplified equations (2.10.4748), reduce to [ C + E* ] q + E* q0 = Qc

(1)

where, assuming that f11 = f22 = f for simplicity, ⎡ 0 ⎢2 fλl / r 0 C = ⎢ ⎢ 0 ⎢ ⎣ 0

−2f 0 0

0 0 0 2 fλl / r0

0

0 ⎤ 0 ⎥⎥ −2f ⎥ ⎥ 0 ⎦

(2)

For small displacements, E* is given by (2.8) and Qc = [ 0 - 2fl2/ R0 0 - 2fl2/ R0]

(3)

and the forces applied to the wheelsets in the reference position due to straining of the suspension in the chosen coordinate system are given by Eq0 where q0 = [ 0 h/R0

0

- h/R0 ]T

(4)

The solution of equations (1) giving the response to track curvature is y*1, y*2 = - h( hr0/λl ± f/ks )/R0(1 + ∆)

(5)

ψ1 = - ψ2 = - h/R0(1 + ∆)

(6)

where y*1, y*2 are the lateral displacements measured from the rolling line so that y*1, y*2 = y1, y2 + lr0/λR0

(7)

∆ = f2λl/ kbksr0

(8)

and where

It can be seen from (8) that the parameter ∆ is indicative of the stiffness of the suspension relative to the elements of the creep stiffness matrix. These solutions indicate that when the wheelsets are restrained elastically further displacements of the wheelsets beyond those for pure rolling are necessary for equilibrium. The leading wheelset moves outwards beyond the rolling line defined by y = - lr0/λR0. If hr0/λl >

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113

f/ks the trailing wheelset also moves outwards but if hr0/ll < f/ks the trailing wheelset moves inwards. It can be seen from equations (5) and (6) that the wheelsets can adopt a radial position and move out to the rolling line if, and only if, kb = 0. The longitudinal creepage on each wheel is given by

γ1ri = -λ y i* /r0

γ1li = λ y i* /r0

(9)

and the lateral creepage is given by

γ2ri = γ2li = - ψ

(10)

The longitudinal creep is largest at the leading wheelset, but the lateral creep on both wheelsets is the same. Dependent on the stiffness of the suspension there are two different regimes to consider, broadly corresponding to a flexible, long wheelbase vehicle (with, typically, a low shear stiffness) and a stiff, short wheelbase vehicle, such as a bogie (with, typically, high bending stiffness). In the case of the flexible vehicle, ∆ >>1 . It can be seen from (6) that the wheelsets adopt a nearly radial attitude and so the lateral creep forces are small. In (5), f/ks > hr0/λl and so the leading wheelset moves outwards beyond the pure rolling line, and the trailing wheelset moves inwards relative to the pure rolling line by a similar amount. These displacements generate longitudinal creep forces which balance the couples due to the yaw stiffness and the radial position of the wheelsets, Figure 4.2. There are two important limitations to the validity of this linear analysis. These are slip or creep saturation and flange contact. The lateral creep is small, so that approximately the limiting creep force is given by

µW/2 = -fγ1 = -kbh(1 + kshr0/fλl)/lR0

(11)

so that slip occurs on all curves of radius less than Rf where Rf ∝ 2kbh/µWl

(12)

when the yaw couples developed by the limiting friction forces are just insufficient to strain the yaw suspension through angles of ± h/R0. If δf is the flangeway clearance then from (5) and (7) flange contact will occur, on the leading outer wheel, on all curves with radius R0 < lr0/λδf (1 + kbh/fl2)

(13)

approximately. Equations (12) and (13) indicate that low bending stiffness, small wheelbase, high conicity and high axle-load promote good response to track curvature.

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kbh/R0l

kbh/R0l

kbh/R0 kbh/R0

kbh/R0l

kbh/R0l

Figure 4.2 Balance of forces for flexible long wheelbase two-axle vehicle on large radius curve.

Applying these considerations to the two-axle vehicle of Table 4.1, from (2.9) kb = 1.25 MN and ks = 50.9 kN/m. The value of λ = 0.1174 corresponding to the worn wheel and rail profiles of Figure 2.2 is retained and regarded as applying to a purely coned wheelset of the same conicity, and f = (f11 + f22)/2. The flexibility parameter ∆ = 154.3. From (5) and (7) the curve radius that will result in the flangeway clearance, assumed to be 8 mm, being taken up by the leading wheelset is R0 = 831 m. Then the lateral displacement of the trailing wheelset is y2 = 0.015 mm compared with the rolling line offset lr0/λR0 = - 3.45 mm. From (6) the wheelsets adopt a nearly radial attitude ψ1 = - ψ2 = - 0.0287 mr and so the lateral creep forces T2r1 = T2l1 = -T2r2 = T2l2 = 204 N and are small. The longitudinal creep forces on the leading wheelset T1r1 = -T1l1 = - 8433 N and on the trailing wheelset T1r2 = - T1l2 = 6408 N. Creep saturation would occur before the flangeway clearance is taken up if µ < 0.226. A comparison between theory and experiment for the curving behaviour of a two-axle research vehicle HSFV-1 was made by Boocock [5] who found excellent agreement over the range of track curvatures for which linear theory is applicable, in this case for R0 > 600 m. For sharper curves, there was considerable scatter as the variable (and unknown) coefficient of friction influences the results, as indicated by equation (12), as creep saturation occurs, but in any case a full nonlinear analysis is necessary as discussed below In the case of the stiff vehicle, ∆ <<1, (5) and (6) show that the wheelsets are prevented from adopting a radial attitude and so outward lateral creep forces are generated at the leading wheelset, and inward lateral creep forces are generated at the trailing wheelset. Equilibrium is provided by yaw moments generated by longitudinal creep forces, both wheelsets moving outwards. With increasing suspension stiffness the behaviour approximates to that of a rigid vehicle. In this case, in (5) f/ks << hr0/λl and so both wheelsets move outwards by the same amount 1,

y2 = - ( 1 + h2/l2 )lr0/λR0

ψ1 = - ψ2 = - h/R0

(14) (15)

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115

Table 4.1 Example parameters for a two-axle vehicle and an isolated bogie. r0 = 0.45 m f11 = 7.44 MN

l = 0.7452 m f22 = 6.79 MN

f23 = 13.7 kNm

W = 78.48 kN

Two-axle vehicle ky = 0.23 MN/m

kψ = 2.5 MNm

kφ =1MNm

cφ =50 kNms

h = 3.7 m

d = 0.2 m

m =1250 kg

I = 700 kgm2

mb=13500 kg

Izb = 170000 kgm2

Iy = 250 kgm2

cy = 50 kNs/m

cψ = 0

Ixb=16000 kgm2

Isolated bogie ky = 40 MN/m

kψ = 40 MNm

kφ =1MNm

cφ =10 kNms

h = 1.25 m

d = 0.2 m

m =1250 kg

I = 700 kgm2

mb=2500 kg

Izb = 3500 kgm2

Iy = 250 kgm2

cy = 0

cψ = 0

Ixb=1000 kgm2

the axles remaining parallel and the displacement from the centreline of the track being ( 1 + h2/l2 ) times the displacement required for pure rolling. Applying these considerations to the bogie of Table 4.1, from (3.5) kb = 20 MN and ks = 7.80 MN/m. Thus, the flexibility parameter ∆ = 0.0629. From (5) and (7) the curve radius that will result in the flangeway clearance, assumed to be 8 mm, being taken up by the leading wheelset is R0 = 1440 m. The lateral displacement of the trailing wheelset y2 = - 6.51 mm compared with the rolling line offset lr0/λR0 = 1.96 mm. From (6) the wheelsets remain almost parallel ψ1 = - ψ2 = - 0.817 mr and so the lateral creep forces T2r1 = -T2l1 = -T2r2 = T2l2 = - 5751 N. The longitudinal creep forces on the leading wheelset T1r1 = -T1l1 = - 11011 N and on the trailing wheelset T1r2 = - T1l2 = 8282 N. Creep saturation would occur before the flangeway clearance is taken up if µ < 0.35. Equation (5) shows that for large values of kb and small values of ks large displacements can occur, the vehicle being unable to react a curvature input. In the limit, if kb = ∞ and ks = 0 the vehicle is in neutral equilibrium on straight and curved track. In this case, in the absence of externally applied forces, the vehicle is able to take up an attitude on straight track defined by y2 = -y1, ψ2 = ψ1 = 0 with the bending moment between the wheelsets being equal to -2fλly1/r0. Such a mode can be termed the ‘anti-bending’ mode. The vehicle is unable to react statically an externally applied yawing moment. Actually there are two limiting cases in which static stability becomes marginal. The other case occurs when ks = ∞ and kb = 0. In the absence of externally applied forces the vehicle is able to take up an attitude on straight track defined by y1 = y2,

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ψ2 = -ψ1 = λly1/r0h with the shear force between the wheelsets being equal to 2fλly1/r0h, the overall balance of forces being in equilibrium. Such a mode can be termed the ‘anti-shear’ mode. The vehicle is unable to react statically an externally applied lateral force.

4.4 Response to Cant Deficiency on a Large Radius Curve Cant deficiency φd is defined as the angle through which the track would have to be canted in order to negate the centrifugal force and is therefore given by

φd = V2/R0g - φ0

(1)

If there is cant deficiency the equations governing the steady state motion in a curve, equations (3.1), now become [ C + E* ] q + Eq0 = Qc + Qd

(2)

and the applied forces due to cant deficiency Qd are Qd = [ -P

0 -P 0 ]

(3)

where P = mgφd + mbgφd/2

(4)

It will be useful to first consider the response of the vehicle to cant deficiency in the absence of the curvature input. The solution of equations (2) is then y1d, y2d = - hP (- hf/kb ± 1 )/2f( 1 + ∆ )

(5)

ψ1d , ψ2d = - P/2f ± λlhP/2kbr0( 1 + ∆ )

(6)

In the absence of elastic restraint, the wheelsets would yaw through an equal positive angle of P/2f in order to generate equilibriating lateral creep forces. When the wheelsets are restrained elastically further displacements of the wheelsets are necessary for equilibrium. In the case of the flexible vehicle, in which ∆ >>1, as hf/kb >> 1 (6) reduces to, approximately,

ψ1d , ψ2d = - P/2f ± hPks/2f2

(7)

so that the wheelsets yaw through almost the same angle P/2f. Similarly, (5) reduces to

GUIDANCE OF THE TWO-AXLE VEHICLE

y1d, y2d = h2P ksr0 / 2f2λl

117

(8)

so that the wheelsets move inwards slightly from the centreline in order to establish equilibrium of the yaw moments. For the two-axle vehicle of Table 4.1, consider the response to a cant deficiency of φd = 0.10. This corresponds to a speed of V = 29.1 m/s2 on track with zero cant for the same radius of R0 = 831 m considered above. From (5) the additional lateral displacement of the leading wheelset is y1d = 0.264 mm and the additional lateral displacement of the trailing wheelset is y2d = 0.290 mm and are therefore small compared with the displacements due to curvature. From (6) the wheelsets additionally yaw through ψ1d = 0.537 mr and ψ2d = 0.566 mr and so the additional lateral creep forces T2r1 = T2l1 = 3821 N and T2r2 = T2l2 = 4027 N. The additional longitudinal creep forces on the leading wheelset T1r1 = -T1l1 = - 488 N and on the trailing wheelset T1r2 = - T1l2 = -537 N and are small. In the case of the stiff vehicle, in which ∆ <<1, (6) reduces to, approximately,

ψ1d , ψ2d = - P/2f ± λlhP/ 2kbr0

(9)

the wheelsets also yaw through almost the same angle P/2f. The trailing wheelset moves outwards and yaws through an angle greater than that required to react the cant deficiency force. The leading wheelset yaws through an angle less than that required to react the cant deficiency force. Consequently, more of the cant deficiency force is reacted at the trailing wheelset than at the leading wheelset. Similarly, (4) reduces to y1d, y2d = - hP (- hf/ kb ± 1 )/ 2f

(10)

so that the leading wheelset moves outwards less than the trailing wheelset and actually moves inwards if hf < kb . With increasing suspension stiffness the behaviour approximates to that of a rigid vehicle, the complete vehicle yawing through the angle P/2f. For the bogie of Table 4.1, the response to a cant deficiency of φd = 0.10 corresponds to a speed of V = 38 m/s2 on track with zero cant for the same radius of R0 = 1440m. considered above. From (5) the additional lateral displacement of the leading wheelset is y1d = 0.360 mm and the additional lateral displacement of the trailing wheelset is y2d = - 0.937 mm and are therefore small compared with the displacements due to curvature. From (6) the wheelsets additionally yaw through ψ1d = 0.506 mr and ψ2d = 0.596 mr and so the additional lateral creep forces are T2r1 = T2l1 = 3606 N and T2r2 = T2l2 = 4242 N. The additional longitudinal creep forces on the leading wheelset T1r1 = -T1l1 = - 667 N and on the trailing wheelset T1r2 = - T1l2 = -1735 N. An optimum response to cant deficiency could be considered to be when the cant deficiency force is equally shared between the wheelsets which can only be satisfied if ks = 0. It can be seen that the requirements for perfect steering and optimum response to cant deficiency are in direct conflict.

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4.5 The Conflict Between Steering and Stability It has been shown that the curving of a vehicle in which two wheelsets are elastically interconnected depends critically on two parameters, the bending and shear stiffnesses. The dynamic characteristics of a two-axle vehicle at low speeds will now be considered in order to derive the influence of the bending and shear stiffness on stability. At low speeds, the form of the equations of motion will be that of equations (2.10.49). Τhe gravitational stiffness and lateral force due to spin creep, together with the other small terms, will be neglected. Hence the free motions of the system, in terms of the sum and difference coordinates defined by (2.3), will be governed by [ B s/V + C + E* ]φ

= 0

(1)

where E* is given by equation (2.6), and after application of the transformation formula (2.5) ⎡f ⎢0 B= ⎢ ⎢0 ⎢ ⎣0

0 fl 2 0 0

0 0 f 0

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ fl 2 ⎦

⎡ 0 ⎢ fλl / r 0 C=⎢ ⎢ 0 ⎢ ⎣ 0

−f 0

0 0

0 0

0 fλl / r0

0 ⎤ 0 ⎥⎥ − f⎥ ⎥ 0 ⎦

(2)

Making a trial solution in which φ, is proportional to est the characteristic equation of (1) becomes fs / V + k s fλl / r0 0 − hk s

−f fl s / V + k b 0 0 2

0 0 fs / V fλl / r0

− hk s 0 =0 −f fl 2 s / V + k s h 2

(3)

Expanding (3) gives p4s4+ p3s3+ p2s2+ p1s+ p0 = 0 where p4 = f 4l 4/V 4 p3 = {ks(l2 + h2 ) + kb}f 3l 2/V 3 p2 = 2 (λl/r0)f 4l 2/V 2 + kskb(l 2 + h2 )f 2/V 2 p1= (λl/r0){ks(l 2 + h2 ) + kb}f 3/V

(4)

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119

π0= (λλ/ρ0)2φ4 + (λλ/ρ0)κσκβφ2 Of major interest is how the eigenvalues of equation (4) depend on the overall suspension stiffnesses ks and kb. When both ks and kb are zero there are two undamped oscillations at wheelset kinematic frequency ω = V(λ/lr0)1/2 . Introduction of small values of the stiffnesses ks and kb results in damping of these oscillations. Putting S = ϕ1/ϕ4 equations (3) can be written as two uncoupled binary equations of the form fs / V + k 1 fλl / r0

−f fl s / V + k 2 2

=0

(5)

where, in one case, k1 = ks (1 - h/S) and k2 = kb and in the other case, k1 = 0 and k2 = ksh(h - S). Equating the two quadratics resulting from equations (5) shows that for small values of ks and kb the frequencies remain unchanged but that the real part is given by

µ = - V{ ks ( 1 - h/S ) + kb}/2f = − V ks h ( h - S )/2f

(6)

so that on solving for, and eliminating, S the eigenvalues are given approximately by s1,2 = ± iV(λ/lr0)1/2- V(ksh2 + kb + ksl2)/4fl2 ± V{(ksh2 + k b + ksl2)2- 4kbksh2}1/2/4fl2 (7) Typically, one of these oscillations can be identified with a steering or bending oscillation whilst the other oscillation can be identified with a shear oscillation. The steering oscillation is lightly damped, but the shear oscillation is heavily damped. If kb were zero, one eigenvalue has a zero real part and the vehicle is capable of undergoing an undamped steering oscillation at kinematic frequency with mode shape shown in Figure 4.3(a). There is no creep and no shear deformation so that ks could be infinitely large without affecting the result, though as discussed previously the system would be neutrally stable. The second root in this case is given by s2 = ± iV(λ/lr0)1/2- ks(h2 + l2)/2fl2

(8)

and the corresponding mode shape is shown in Figure 4.3(b). The motion involves a pure shear deformation. Similarly, if ks were zero, one eigenvalue has a zero real part and the vehicle is capable of undergoing an undamped shear oscillation at kinematic frequency with mode shape shown in Figure 4.3(c). There is no creep and no shear deformation so that kb could be infinitely large without affecting the result, though as discussed in Section 4 the system would be neutrally stable. The second root in this case is given by 2=

± iV(λ/lr0)1/2- kb/2fl2

(9)

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RAIL VEHICLE DYNAMICS

(a) λ = 0.5838i

(b) λ = - 0.3896 + 0.5825i ks = 0.023 MN/m, kb = 0

(c) λ = - 0.5838i

(d) λ = - 0.1480 - 0.5647i ks = 0, kb = 1.25 MNm

Figure 4.3 Mode shapes for steering and shear oscillations showing successive positions of the vehicle during one half period of its motion in the mode.

GUIDANCE OF THE TWO-AXLE VEHICLE

121

and the corresponding mode shape is shown in Figure 4.3(d). The motion involves a pure bending deformation. It may be concluded that for a symmetric two-axle vehicle both bending and shear stiffness are required for stability at low speeds. A symmetric two-axle vehicle which is able to steer perfectly in the sense of adopting a radial attitude within the assumptions discussed above would have zero bending stiffness and therefore a critical speed of zero. Similarly, a symmetric two-axle vehicle which had optimum response to cant deficiency by sharing the lateral force equally between the wheelsets would have zero shear stiffness and therefore a critical speed of zero. It follows that for the symmetric two-axle vehicle there is a compromise between steering (or response to track curvature), cant deficiency and dynamic stability.

4.6 Motion on Sharper Curves The linear analysis given above is useful in clarifying the basic mechanisms of the steady state response of a vehicle to track curvature and cant deficiency and is qualitatively representative of motion on large radius curves where displacements of the wheelsets are small. It can be extended by including the effects of spin and the gravitational stiffness as was done in Boocock’s original work [5]. However, the linear analysis is limited in applicability because on sharper curves, roughly defined as those with curve radius R0 < 1000 m., the displacements of all or some of the wheelsets take up a significant proportion of the flangeway clearance. The wheel-rail geometry cannot be represented by simple linear functions and the actual curvatures, slopes and rolling radii must be taken into account. The contact slopes may be large, giving rise to large amounts of spin, and resulting in the normal forces having significant components in the lateral direction. The variation of the rotational speed of the wheelset may influence the magnitude of the creepages. As the contact patch moves across the rail and changes in shape, the variation of the curvatures influences the magnitude of the creep forces. As the creepages and spin may be large, creep saturation may occur. Hence it is necessary to consider the solutions of the full non-linear equations of motion presented in Section 2.11. As an example, Figure 4.4 summarises the dynamic response of a two-axle vehicle with the parameters of Table 4.1 to entry into a curve of radius 225 m with a cubic parabolic transition of the type discussed in Section 3.7 with a transition length L = 20 m. The vehicle speed V = 15 m/s and the coefficient of friction µ = 0.3. The track has cant φ0 = V2/R0g and therefore the cant deficiency is zero. In the initial stages of curve entry, the vehicle responds by undergoing a complex damped response involving motions at body and kinematic frequency. The steady state motion is then established and the attitude of the vehicle is governed by the quasi-static version of the equations of motion F(q, V) + Eq + Eq0 = 0

(1)

where q0 represents the position from which the displacements are measured relative to the unstrained suspension. Of course, (1) may be solved directly, for example by

122

RAIL VEHICLE DYNAMICS

0.01

y2 yb

0

y1 -0.01 0

2

t

γ1l2 γ1r1 γ1l1

0 -0.01 -0.02 0 2 4 x 10

γ1r2 1

-0.01 -0.02 -0.03 0 0.03

T1r2 T1l1 T1r1 1

ψ2 ψb ψ1 1

γ2l1 γ2r1

0.01 0 -0.01 0 x 10 0

γ2r2 1

2

-2 0

2

T2r2 T2l2 T2r1 T2l1

-1

T1l2

2

0.02

4

0 -2 0

2

0

1

2

-V/r0-Ω2

0 -0.1 -0.2 0 2

-V/r0-Ω1 1

2

ω3l1

0

ω3r2 -2 0 4.2 4 x 10

41

2

T3l1 T3r2 T3l2

4 3.8

T3r1 0

1

2

Figure 4.4 Dynamic response of two-axle vehicle with the parameters of Table 4.1 to entry into a curve of radius R0 = 225 m, V = 15 m/s, µ = 0.3, zero cant deficiency and cubic parabolic transition of length 20 m.

the Newton-Raphson method. The attitude of the vehicle in plan view is shown in Figure 4.5. The leading wheelset has moved outwards and has negative yaw relative to the radial, so that the positive angle of attack generates large values of creep. The outward position of the wheelset also generates a large amount of spin. Reference to equation (2.5.13-14) shows that the variation in the rotational speed of the wheelset, though small, has a significant effect on the magnitude of the longitudinal creepage on both wheels. As the lateral creep is large, creep saturation results in a large proportion of the available friction force being taken by the lateral creep force. Hence only relatively small longitudinal creep forces can be generated and the yaw couple available to strain the yaw suspension is small, and so the wheelset adopts only a small relative angle with the car body. The trailing wheelset is displaced inwards, and yaws positively relative to the radial, but the angle is less than on the leading wheelset so the wheelset is able to generate longitudinal creep forces to react a larger yaw torque of the suspension. The normal force on the outer leading wheel is increased and that on the inner reduced, with only slight variation on the trailing wheels. Most significant is the increase in the lateral components of the normal forces on the leading outer and trailing inner wheels. This is due to the increased slope of the contact plane as the wheelset displacements are large. As a result, the large lateral creep forces on the leading wheelset are reacted by the horizontal com-

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123

Figure 4.5 Attitude of two-axle vehicle on curve of radius R0 = 225 m, V = 15 m/s, µ = 0.3. Relative to the lateral displacements, vehicle dimensions are reduced k times, R0 is reduced k2 times, and yaw rotations are multiplied by k2, where k = 25.

ponent of the normal force on the outer wheel. This latter force is often identified with the ‘flange force’. The corresponding forces applied to the track tend to cause ‘gauge spreading’. Next, the effect of cant deficiency is considered. The influence of cant deficiency on the attitude of the two-axle vehicle traversing the same curve, at the same speed but with varying angles of track cant angle is shown in Figure 4.6. As is obvious, the car body moves in the direction of the centrifugal force on the flexible lateral suspension. The variation of the wheelset yaw angles is small. Figure 4.7 shows the corresponding lateral wheel forces. Cant deficiency causes additional movements of the wheelsets, the resulting change in contact angles providing the necessary lateral components of the normal forces on the leading outer and trailing inner wheels. These latter forces show the strongest variation as the cant deficiency is varied. Figure 4.7 also shows the variation in the lateral wheel forces due to cant deficiency for an exactly similar case of curve entry but with a reduced coefficient of friction µ = 0.1. As might be expected, the magnitude of the forces is reduced, as are the lateral displacements of the wheelsets (Figure 4.6). The yaw angles approach the

0.01

0.02

y2

ψ2

0.01

0.005

0

0

ψb

yb/10 -0.005

-0.01

y1 -0.01 -0.2

0

φd

0.2

-0.02 -0.2

ψ1 0

φd

0.2

Figure 4.6 Variation of attitude of example two-axle vehicle with cant deficiency. Solid lines refer to µ = 0.3, dotted lines refer to µ = 0.1.

124

RAIL VEHICLE DYNAMICS

2000

4 x 10

1000

4

Txl1

2

Tyl1

0 -1000 -2000 -0.2

0

Txr1 0

φd

-2 -0.2

0.2

1

x 10

4

0

φd

0.2

2

0.5

Txr2

4

Txl2

-0.5 0

φd

1

Tyl2

0

Tyr2

x 10

0

-1 -0.2

Tyr1

-1 0.2

-2 -0.2

0

φd

0.2

Figure 4.7 Variation of wheel forces of example two-axle vehicle with cant deficiency. Solid lines refer to µ = 0.3, dotted lines refer to µ = 0.1.

limiting values of ± h/R0. Consider now the corresponding results for the solution of the non-linear equations of motion of an unrestrained bogie, with µ = 0.3, and at the same speed. Approximate allowance has been made for the cant deficiency force acting on the car body which is applied as a lateral force and rolling moment mcV2/2R0 -mcgφ0/2 on the bogie frame. The vertical forces are similarly represented as discussed in Section 2.11. In the steady state motion on a large radius curve, the linear theory showed that the wheelsets of the bogie would move outwards and yaw in opposite directions in attempting to take up radial positions. The nonlinear solution for R0 = 1500 m confirms this and is shown in Figure 4.8(a). As the radius of the curve is reduced, there is increasing yaw of the bogie and the trailing wheelset moves inwards, as shown in Figures 4.8(b)-(e). The leading wheelset moves outwards and has increasing negative yaw, so that the positive angle of attack generates large values of lateral creep and only relatively small longitudinal creep forces can be generated by the leading wheelset. The large lateral creep forces are mainly reacted by the horizontal component of the normal force. The trailing wheelset takes up a more or less radial position, the lateral creep forces are relatively small, so the wheelset is able to generate the longitudinal creep forces necessary for yaw equilibrium. The balance of forces in plan view for R0 = 225 m is shown in Figure 4.9. It can be seen that creep saturation is com-

GUIDANCE OF THE TWO-AXLE VEHICLE

125

plete and the flangeway clearance is taken up at the leading outer and trailing inner wheels. This indicates partial convergence with the classical theory of Porter [4] and the condition referred to as ‘constrained curving’. As the primary suspension is stiff, the relative displacements between bogie frame and wheelsets are small and in Figure 4.8 it can be seen that the movement of the vehicle appears to be as a rigid body. The effect of cant deficiency on the bogie vehicle traversing a curve with R0 = 225 m, at the same speed but with varying angles of track cant angle is indicated in Figures 4.10 and 4.11. Here the coefficient of friction µ = 0.1. Figure 4.10 shows how the bogie yaw angle and outward displacement increase with increasing cant deficiency. Figure 4.11 shows the corresponding lateral wheel forces. The non-linear treatment of practical vehicles in curves was first given by Elkins and Gostling in 1978 [7], in which numerical solutions were shown to give good

(a)

(b)

(c)

(d)

(e)

Figure 4.8 Attitude of bogie with parameters of Table 4.1 on curves of various radii with zero cant deficiency (a) R0 = 1500 m (b) R0 = 800 m (c) R0 = 500 m (d) R0 = 225 m (e) R0 = 100 m.

126

RAIL VEHICLE DYNAMICS

T3l1sinδl1

µW/2

T1l1

T2l1cosδl1

T1l2 Mz2

T2l2cosδl2

T3l2sinδl2

Py1

Mz1

T3r1sinδr1

Py2

T1r1

T3r2sinδr2 T2r1cosδr1

T1r1

T2r1cosδr1

Figure 4.9 Plan-view forces acting on wheels of bogie of Table 4.1 in curve R0 = 225 m, V = 15 m/s, µ = 0.3 and zero cant deficiency.

5

0.01 x 10

0

0.005 0

y2

-5

-0.005

yb

-10

-0.01 -0.2

ψ2

-3

y1 0

φd

0.2

-15 -0.2

ψb ψ1 0

φd

0.2

Figure 4.10 Influence of cant deficiency on attitude of bogie vehicle with parameters of Table 4.1, R0 = 225 m, V =15 m/s., µ = 0.1.

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127

1000

T1l1

500

4

x 10

2

Tyl1 1

0 0

-500 -1000 -0.2

Tyr1

T1r1 0

φd

-1 -0.2

0.2

5000

4

0

1

Tyl2

x 10

T1r2

0.2

φd

Tyr2

0

0 -1

T1l2 -5000 -0.2

0

φd

0.2

-2 -0.2

0

φd

0.2

Figure 4.11 Influence of cant deficiency on wheel forces for bogie vehicle of Table 4.1, R0 = 225 m, V =15 m/s, µ = 0.1.

agreement with full-scale experiments with both bogie and two-axle vehicles. It should be noted that these solutions indicate that a resultant longitudinal force exists on a wheelset in a curve. This force is greater in sharp curves, which explains the greater drag of a train in a curve.

4.7 Response to Misalignments The effects of misalignments may be considered in terms of the configuration of the vehicle when the suspension is in the unstrained state. If y10, ψ10, y20 and ψ20 represent the displacements of the wheelset when the elastic elements are unstrained then the column of initial displacements q0, will be given by q0 = [ - y10

- ψ10

- y20

- ψ20 ]T

(1)

and the effect of the misalignments will be given by the solution of the equations of equilibrium for small displacements (3.1). In the case of the two-axle vehicle or bogie

Ε q 0 = [ - ks δ s

- kb δb + kshδs

ks δs

kb δb + kshδs

]T

(2)

128

RAIL VEHICLE DYNAMICS

where

δs = y10 - hψ10 - y20 - ψ20

(3)

δb = ψ10 - ψ20

(4)

The response to a shear misalignment is accordingly y1s, y2s = δs (- hf/ kb ± 1 )/ 2( 1 + ∆ )

(5)

ψ1s =- ψ2s = - λlf δs/ 2kbr0 ( 1 + ∆ ) }

(6)

In the case of the flexible vehicle, in which ∆ >>1, as hf/kb >> 1 (6) reduces to, approximately,

ψ1s , -ψ2s = - ks δs/2f

(7)

y1d, y2d = - hksr0 δs/2fλl

(8)

Similarly, (5) reduces to

so that the wheelsets move laterally and yaw in opposite senses, the elastic forces induced by a shear misalignment being directly reacted by the corresponding creep forces. In the case of the stiff vehicle, in which ∆ <<1, (7) reduces to, approximately,

ψ1d , ψ2d = - λlfδs/ 2kbr0

(9)

y1d, y2d = δs (- hf/kb ± 1 )/2

(10)

Similarly, (5) reduces to

so that if there is a shear misalignment δs the wheelsets will initially move differentially through ± δs/2. The yaw moments thus generated are balanced by equal and opposite yaw displacements given by (10); overall yaw equilibrium is obtained by further lateral displacements of the wheelsets - hf δs/2kb. The response to a bending misalignment is y1b, y2 b = δb ( hr0/λl ± f/ks )/ 2( 1 + ∆ )

(11)

ψ1b = - ψ2b = δb / 2( 1 + ∆ )

(12)

In the case of the flexible vehicle, ∆ >>1 . It can be seen from (12) that the yaw angles are negligible and so the lateral creep forces are small. In (11) f/ks >> hr0/λl and so, approximately,

GUIDANCE OF THE TWO-AXLE VEHICLE

129

Table 4.2 Response of example vehicles of Table 4.1 to misalignments.

y1 = - 0.323 mm ψ1 = - 0.0178 mr y2 = - 0.355 mm ψ2 = 0.0178 mr

Two-axle vehicle y10 = 5 mm

y1 = 0.7505 mm ψ1 = 0.01637 mr y2 = -0.1259 mm ψ2 = - 0.01637 mr

Two-axle vehicle

ψ10 = 1 mr

Bogie

y1 = 1.306 mm ψ1 = - 0.1623 mr y2 = - 3.398 mm ψ2 = 0.1623 mr

y10 = 2 mm

y1 = 1.567 mm ψ1 = 0.2554 mr y2 = 1.727 mm ψ2 = - 0.2554 mr

Bogie

ψ10 = 0.5 mr

y1b, y2 b = ± kbr0 δb /2fλl

(13)

so that in response to a bending misalignment the wheelsets move laterally and differentially, the elastic moments being directly reacted by the longitudinal creep forces. In the case of the stiff vehicle, ∆ <<1 and (11) and (12) reduce to

ψ1b, ψ2b = δb hr0/2λl

(14)

ψ1b = - ψ2b = δb/2

(15)

so that the bending misalignment is accommodated by equal and opposite yaw displacements, both wheelsets moving laterally to provide overall equilibrium of the yaw moments.

130

RAIL VEHICLE DYNAMICS

Comparison of equations (11) and (12) with (3.5) and (3.6) shows that the response to a bending misalignment is related to the response to curvature so that y*i/(2h/R0) = -yib/ δb

(16)

ψi/(2h/R0) = -ψib/ δb

(17)

which indicates, as might be expected, that the response to a misalignment in bending is the same as the response to curvature if 2h/R0 = - δb. It follows that the response to a typical misalignment in bending will ordinarily be significantly less than the response to curvature. Also comparison of equations (5) and (6) with (4.5) and (4.6) shows that the response to a shear misalignment is related to the response to cant deficiency so that yid/(P/2f) = yis/( δs/2h) ( ψid + P/2f )/(P/2f) = ψis/ (δs/2h)

(18) (19)

which indicates, as might be expected, that the lateral displacement due to a misalignment in shear is the same as that due to cant deficiency if hP/2f = δs. The response to misalignment in shear will ordinarily be significantly less than the response to cant deficiency. Though the displacements caused by typical, practical values of misalignments are small, the creepages and creep forces can be significant and can provide a contribution to creep saturation which should not be overlooked.

4.8 Flange Forces and Derailment The conditions necessary to sustain equilibrium of the forces in flange contact were briefly mentioned in Chapter 1. When a wheel is in flange contact with a rail, small lateral displacements of the wheelset result in large changes in the contact angle. The gravitational stiffness becomes large and may exceed the lateral stiffness of the track. Realistic estimates of the forces acting between wheel and rail in this situation require the inclusion of extra degrees of freedom representing the rail movement at each wheel. For the frequencies under consideration here, usually quite simple spring and damper models will be adequate to represent the dynamics of the track [9]. Derailment is the consequence of wheel climb in which large lateral forces acting on a wheelset of a vehicle cause one wheel to climb up and over the rail. The dynamics of this process was considered by Matsui [11]. Detailed solutions of the equations of motion on irregular track have been compared with experiment [9] and good agreement obtained. The results confirm operating experience in which many derailments occur as a result of the coincidence of high lateral forces (such as those experienced during hunting) with vertical unloading of a wheel, perhaps due to a track defect. Sweet et al. [12] also investigated the dynamics of derailment and obtained good agreement with model experiments.

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References 1. Mackenzie, J.: Resistance on railway Curves as an element of danger. Proc. I. C. Eng. 74 (1883), pp. 1-57. 2. Gilchrist, A.O.: The long road to solution of the railway hunting and curving Problems. Proc. I. Mech. E. 212, Part F (1998), pp. 219-226. 3. Heumann, H.: Grundzuge der Fuhrung der Schienenfahrzeuge. Oldenbourg, Munchen, 1953. 4. Porter, S.R.M.: The mechanics of a locomotive on curved track. The Railway Gazette, London, 1935. 5. Boocock, D.: Steady-state motion of railway vehicles on curved track. Journal Mech. Eng. Sci. 11 (1969), No. 6, pp. 556-566. 6. Newland, D.E.: Steering characteristics of bogies. Railway Gazette, 124 (1968), No.19, pp. 745-750. 7. Elkins, J.A. and Gostling, R.J.: A general quasi-static curving theory for railway vehicles. In: A. Slibar and H. Springer (Eds.): The Dynamics of Vehicles on Roads and Tracks, Proc. 5th IAVSD Symposium, Vienna, September 1977. Swets and Zeitlinger Publishers, Lisse, 1978, pp. 388-406. 8. Muller, C. Th.: Dynamics of railway vehicles on curved track. Proc. I. Mech. E. 180 (1965-66), Part 3F, pp. 45-57. 9. Clark, R.A., Eickhoff, B.M., Hunt, G.A.: Prediction of the dynamic response of vehicles to lateral track irregularities. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 535-548. 10. Collar, A.R. and Simpson, A.: Matrices and engineering Dynamics. Ellis Horwood, Chichester, 1987, p. 482. 11. Matsui, N.: On the derailment quotient Q/P. Railway Technical Research Institute, Japanese National Railways, 1966. 12. Sweet, L.M., Karmel, A. and Fairley, S.R.: Derailment mechanics and safety criteria for complete rail vehicle trucks. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 481-494. Sweet, L.M. and Sivak, J.A.: Nonlinear wheelset forces in flange contact - Part I: Steady state analysis and numerical results. ASME Transactions, J. of Dynamic Systems, Measurement and Control 101, No. 3, September, 1979.

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Sweet, L.M. and Sivak, J.A.: Nonlinear wheelset forces in flange contact - Part II: Measurements using dynamically scaled models. ASME Transactions, J. of Dynamic Systems, Measurement and Control 101, No. 3, September, 1979. Sweet, L.M. and Karmel, A.: Evaluation of time-duration dependent wheel load criteria for wheel climb derailment. ASME Transactions, J. of Dynamic Systems, Measurement and Control 103, No. 3, September 1981, pp. 219-227. 13. Wickens, A.H. and Gilchrist, A.O.: Railway vehicle dynamics − the Emergence of a Practical Theory. Council of Engineering Institutions MacRobert Award Lecture, 1977.

5 Dynamic Stability of the Two-Axle Vehicle 5.1 Introduction In this chapter the dynamic stability of the isolated two-axle vehicle will be considered. Both long wheelbase vehicles and short wheelbase vehicles will be discussed, the former typically representative of freight vehicles and the latter typically representative of bogies which are unrestrained by the secondary suspension connecting the bogie to the car body. The complete bogie vehicle will be considered in Chapter 6. The instability of two-axle vehicles was an accepted and often unremarked occurrence throughout their employment on the railways. For example, the coaches used on the Liverpool and Manchester Railway had a very short wheelbase and were reputed to hunt violently at any speed [1]. One measure employed to control this was to close couple the vehicles. Early bogies had very short wheelbases and were free to swivel without restraint and tended to oscillate violently being the probable cause of many derailments. In the case of locomotives, in the 1850s the wheelbase of the leading bogie was increased which improved stability significantly, [2]. The suspension of the earliest two-axle vehicles was provided by simple leaf springs bearing directly on to ‘shoes’ bolted to the vehicle frame. The lateral suspension was only provided fortuitously as a consequence of the vertical suspension. No lateral restoring force was provided until the lateral clearance of the journal axle-bearing was taken up: thereafter some stiffness was provided by rocking of the heavily cambered spring on its seats. After the axle-guard clearance was taken up, further motion was resisted by bending of the axle-guard. The longitudinal suspension consisted of friction of the shoes until the axle-guard clearance was taken up after which extremely stiff restraint was provided by the axle-guards. Needless to say, the action of such suspensions was highly nonlinear and their modelling has been discussed in [1.8]. In the later ‘eye-bolt’ suspension, more lateral flexibility, within the axle-guard clearances, was provided by swinglinks which were free to move both laterally and longitudinally. In an attempt to improve curving, this type of suspension was later refined in the form of the ‘double-link’ suspension, which became a German standard in 1890, and which provides more flexibility. This form of suspension remains in common use in Europe and Japan. Operation of earlier designs of vehicles at higher speeds in the 1950s and 1960s led to unacceptable rates of derailment which stimulated progress in both theory and design.

134

RAIL VEHICLE DYNAMICS

Matsudaira was the first to investigate the dynamic stability of the two-axle vehicle, using Carter’s theory of creep, and incorporating both the lateral and longitudinal flexibilities between wheelsets and the car body [3]. Subsequently, it was shown in [4] that inclusion of appropriate amounts of suspension damping, neither too small or too large, resulted in regions of clear stability up to relatively high speeds. Moreover, a realistic theory, taking into account the nonlinear flexibility between the wheelsets and the car body or bogie frame, and the forces acting between vehicle and track, yielded results which are consistent with experiment [1.11]. The theory shows that there are two ways of exploiting the suspension stiffnesses in the design of two-axle railway vehicles for stable running at speed. The first is to use a relatively flexible suspension, in which the flexibility parameter ∆ = f 2λl/k bksr0 defined in Section 4.3 is large, with appropriately chosen parameters − this approach is suitable for two-axle vehicles with relatively long wheelbase. The second approach is to use a suspension which has large lateral and yaw stiffness in which the flexibility parameter ∆ is small, − this is appropriate to a short wheelbase vehicle such as a bogie. These concepts were demonstrated in the laboratory on a roller rig [4] and by track tests with a specially designed full-scale variable parameter test vehicle HSFV1[23]. Further experimental and theoretical experience has led to the development of suspensions for two-axle freight and passenger vehicles capable of running at similar speeds to bogie vehicles, such as the widely used Class 143 and 144 passenger vehicles on British Railways. The improvement of curving performance whilst still providing the yaw restraint necessary for stable running can be achieved by frequency sensitive yaw restraint between the wheelsets and car body using a relaxation or yaw damper, as first proposed by Hobbs [5]. The yaw damper provides little restraint at low frequencies so that the wheelsets are able to take up an approximately radial position, but at high frequencies the restraint is sufficient to stabilise the vehicle. The advent of reliable dampers suitable for use in the primary suspension, in the 1970s, made it possible to apply this concept to freight vehicles [6] and articulated passenger vehicles as discussed in Chapter 8. As mentioned in Section 4.2 for conventional bogies in which there are primary longitudinal and lateral springs connecting the wheelsets to a frame there is a limit to the overall shear stiffness which can be provided in relation to the bending stiffness and therefore the stability/curving trade-off in which the bending stiffness must be minimised is constrained. This limitation is removed if the wheelsets are connected directly by diagonal elastic elements or cross-bracing, or interconnections which are structurally equivalent. Such an arrangement is termed a self-steering bogie. Superficially, this arrangement is similar to systems of articulation between axles by means of rigid linkages which have a long history in railway engineering. The first application of cross-bracing appears to have been on the vehicles of the LinzBudweiser Pferdebahn (1827), to be seen in the Vienna Museum of Technology. In the 1970s the self-steering bogie was successfully developed and put into service, notably by Scheffel [7]. The essential feature of cross-bracing in modern practice is that it is elastic. Self-steering bogies are common in current practice and have been applied to locomotives (with benefits to the maximum exploitation of adhesion), passenger vehicles and freight vehicles [8]. It should be noted that inter-

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

135

wheelset connections can be provided by means other than springs and dampers. In [9] the equivalent of cross-bracing is provided by means of a passive hydrostatic circuit which has a number of potential design advantages.

5.2 Equations of Motion Equations of motion for the two-axle vehicle with a simple suspension have been derived in Section 2.11. Whilst a simple representation of the suspension suffices to discuss the principles of the subject, in design work it is necessary to model the details of the suspension in order to obtain accurate results. Particular examples are the behaviour of springs which carry vertical load and the effect of trailing arm primary suspensions [10, 11, 12] not to mention non-linearities in suspension behaviour. However, a more fundamental aspect is concerned with the options that exist in relation to the arrangement of the suspension components and the bodies that they connect. To discuss this it is necessary to consider the properties of the elastic stiffness matrix. As the rotation of the wheelsets cannot generate any forces in the suspension consider the following set of seven generalised coordinates q =[ y1 ψ1 yb ϕb ψb y2 ψ2 ]T

(1)

(the set that would be considered in a linear analysis). As in Chapter 4, it will be advantageous to use sum and difference coordinates, ϕi, Figure 5.1,

φ1

φ4

φ2

φ5

Figure 5.1 Sum and difference coordinates for two-axle vehicle.

φ3

φ6

136

RAIL VEHICLE DYNAMICS

y1 = ( ϕ1 + ϕ4 )/2

y2 = ( ϕ4 - ϕ1 )/2

ψ1= ( ϕ2 + ϕ5 )/2

ψ2= ( ϕ5 - ϕ2 )/2

yb = ϕ6

ϕb = ϕ7

(2)

ψb= ϕ3/h

Because the vehicle is symmetric and E must be symmetric, in the most general case the stiffness matrix corresponding to the ϕ coordinates will be of the form ⎡ e11 ⎢ 0 ⎢ ⎢ e13 ⎢ E=⎢ 0 ⎢ e15 ⎢ ⎢ 0 ⎢ 0 ⎣

0 e22 0 e 24 0 e 26 e27

e13 0 e 33 0 e 35 0 0

0 e24 0 e 44 0 e 46 e47

e15 0 e 35 0 e55 0 0

0 e26 0 e 46 0 e 66 e 67

0 ⎤ e27 ⎥⎥ 0 ⎥ ⎥ e47 ⎥ 0 ⎥ ⎥ e67 ⎥ e77 ⎥⎦

(3)

and following the procedure of Section 4.2 the condition that no elastic forces can be generated in a rigid body lateral translation Eϕl = 0

(4)

ϕl = [ 0 0 0 2 0 1 0 ]T

(5)

where

yields e26 = - 2e24

e46 = - 2e44

e66 = 4e44

e76 = - 2e74

(6)

Applying the condition that no elastic forces can be generated by a rigid body yaw Eϕy = 0

(7)

ϕy = [ 2h 0 h 0 2 0 1 ]T

(8)

where

yields e13 = - e11 - e33/4 + e55/h2 e15 = - he11/2 + e33h/8 - e55/2h

(9)

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

137

e35 = e11h - e33h/4 - e55/h with the result that there are a maximum of 9 independent parameters. It should be noted that e11 , e33 and e55 are not completely arbitrary as the principal minors of E corresponding to ϕ1, ϕ3 and ϕ5 must be positive definite as the system is conservative. A similar condition applies to e22 and e44. For the two-axle vehicle of Figure 2.22 the matrix of primitive stiffnesses is k = diag [ ky

kφ kψ

ky

kφ

kψ ]

(10)

and the corresponding compatibility matrix, corresponding to the set of coordinates defined by (1), is ⎡ ⎢ ⎢ ⎢ a=⎢ ⎢ ⎢ ⎢ ⎢⎣

1 0 0 0 0 0

0 0 1 0 0 0

−1 0 0 −1 0 0

d 1 0 d 1 0

−h 0 −1 h 0 −1

0 0 0 1 0 0

0 0 0 0 0 1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(11)

Applying (2.11.25) and (4.2.5) and comparing the result with (3), the nine independent elements of E, in terms of the sum and difference coordinates, are e11 = ky/2

e22 = kψ/2

e24 = e27 = 0

e33 = 2ky + 2kψ/h2

e44 = ky/2

e47 = dky

e55 = kψ/2

e77 = 2d2ky+2kφ

(12)

It was discussed in Section 4.2 that at low speeds, where inertia forces can be neglected, the frame and suspension provides only an elastic connection between the wheelsets. This applies not only to the conventional bogie or vehicle, in which two wheelsets are elastically connected to a common frame by means of lateral and longitudinal springs, but applies to other forms of structure, such as direct links between the wheelsets. Whatever form of structure is chosen it can be described by the two independent stiffness parameters; the shear stiffness ks and the bending stiffness kb. When the attitude of the vehicle can be defined in terms of the wheelset coordinates φ1, φ2, φ4 and φ5 alone, the result of eliminating φ3, φ6 and φ7 is that the deflated stiffness matrix E* is defined in terms of the bending stiffness kb and shear stiffness ks by (4.2.6). The discussion of Section 4.5 shows that, for stability at low speeds, kb and ks must be nonzero. For the case in which E is given by equation (3) the shear and bending stiffnesses are given by ks = ( e11 e33 - e132)/e33

(13)

138

RAIL VEHICLE DYNAMICS

(a)

kp/2l12

ky/2

(1+h2/l12)kd/2

2l1

kψ/2l12

ky/2

(b) kψ

kp

ha

kψ

kd

h

h

ha

(c) kq k0

h1

h1

Figure 5.2 Alternative suspension schemes for the two-axle vehicle (a) direct shear and bending connections between wheelsets achieved by cross-bracing (b) and (c) an example generic suspension scheme.

kb = e22 - e242/ e44

(14)

and for the simple configuration ks and kb (13) and (14) reduce to (4.2.9). Written out in full, the equations of motion (in the simplified form of Section 2.10), in terms of the sum and difference coordinates, then become {(m/2)s2 + fs/V + ky/2}φ1 - f φ2 - kyφ3 = 0

(15)

(fλl/r0) φ1 + {(I/2)s2 + fl2s/V + kψ/2}φ2 = 0

(16)

- ky φ1 + {(Izb/h2)s2 + 2ky + 2kψ/h2}φ3- (kψ/h)φ5 = 0

(17)

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

139

{(m/2)s2 + fs/V + ky/2}φ4 - f φ5 - kyφ6 + d1kyφ7 = 0

(18)

(fλl/r0)φ4 + {(I/2)s2 + fl2s/V + kψ/2}φ5 - (kψ/h)φ3 = 0

(19)

- kyφ4 + ( mbs2 + 2 ky )φ6 - 2dkyφ7 = 0

(20)

- 2dky φ6 - kyd1φ4 + (Ixbs2 + 2d2ky + 2kφ)φ7 = 0

(21)

so that with this choice of coordinates, the first set of three equations is coupled to the second set of four equations through only one term proportional to the yaw stiffness kψ and, moreover, the wheelbase h only occurs in conjunction with kψ. These facts can be exploited in deriving solutions to the equations of motion, and are helpful in understanding the dynamics of the system. Now consider an alternative configuration employing cross-bracing. This is often applied in addition to a primary longitudinal and lateral suspension of the conventional kind, Figure 5.2(a). This provides 4 parameters for optimisation. As the overall shear stiffness can be increased beyond the limit for a conventional bogie there are two benefits. Firstly, the reduction of overall bending stiffness improves steering, and, secondly, as discussed later, under certain conditions dynamic stability is improved. In this case the matrix of primitive stiffnesses is k = diag [ ky

kφ kψ

ky

kφ

kψ

kp

kd ]

(22)

and the compatibility matrix is, in terms of the coordinates of (1), ⎡ ⎢ ⎢ ⎢ ⎢ a=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣

1

0

−1

−h

0

d 1

0

0

0

1

0 d

−1

0

0 −1

0 0

0

0

1

0

0

0

0

0 −1

0 1

1 −h

0 0

0 0

0 0

0 h

0⎤ 0 0 ⎥⎥ 0 0⎥ ⎥ 1 0⎥ 0 0⎥ ⎥ 0 1⎥ 0 − 1⎥ ⎥ − 1 − h⎥⎦ 0

(23)

where it is assumed for simplicity that the cross-bracing is in the plane of the axles. Applying (2.11.25) and (4.2.5), the nine independent elements of E are then e11 = ky/2 + kd e44 = ky/2

e22 = kψ/2 + kp e47 = dky

e24 = e27 = 0

e55 = h2kd + kψ/2

e33 = 2ky + 2kψ/h2 e77 = 2d2ky+2kφ

(24)

140

RAIL VEHICLE DYNAMICS

Applying equations (13) and (14) the shear and bending stiffnesses are given by ks = kykψ/(2kyh2 + 2kψ) + kd

(25)

kb = kψ/2 + kp

(26)

and written out in full, the equations of motion (in the simplified form of Section 2.10) then become {(m/2)s2 + fs/V + ky/2 + kd}φ1 - f φ2 - kyφ3 − hkdφ5 = 0

(27)

(fλl/r0) φ1 + {(I/2)s2 + fl2s/V + kψ/2 + kp}φ2 = 0

(28)

- (kψ/h)φ5 - ky φ1 + ( (Izb/h2)s2 + 2ky + 2kψ/h2)φ3 = 0

(29)

{(m/2)s2 + fs/V + ky/2}φ4 - f φ5 - kyφ6 + d1kyφ7 = 0

(30)

(fλl/r0)φ4 + {(I/2)s2 + fl2s/V + kψ/2 + h2kd}φ5 - (kψ/h)φ3 − hkdφ1 = 0

(31)

- kyφ4 + ( mbs2 + 2 ky )φ6 - 2dkyφ7 = 0

(32)

- 2dky φ6 - ky d1φ4 + {Ixbs2 + 2d2ky + 2kφ)}φ7 = 0

(33)

The discussion of Section 4.5 and equations (25) and (26) show that stability at low speeds can be obtained with kψ = 0 and with kp and kd nonzero. In this case the coupling term (kψ/h) between φ2 and φ5 is of course no longer present but a coupling term hkd between φ1 and φ5 has now been introduced. However, a combination of all four stiffnesses is usually exploited. Damper elements may, of course, be disposed in similar ways to the springs so far considered. More generally, complex suspension elements can be exploited, the effect of which is to replace kψ, for example, in the equations of motion by an impedance zψ(s). Figure 5.3(a) indicates the arrangement consisting of a damper in series with a spring. The mass mr is small and so at the frequencies being considered here the effect of the relaxation damper is to replace kψ in the equations of motion by krcrs/(kr + crs). For oscillations at frequency ω the effective stiffness is krcr2ω2/(kr2 + cr2ω2) and the effective damping is kr2crω2/(kr2 + cr2ω2). An alternative arrangement, Figure 5.3(b), is to provide the equivalent of a yaw relaxation spring directly between the wheelsets. It will be noted that in the above schemes e24 and e27 are both zero, so that the full possibilities are not being exploited. Various representations of generic two-axle vehicles or bogies have been considered [13-19]. An example of such a scheme is shown in Figure 5.2(b) and (c). In this case the matrix of primitive stiffnesses is k = diag [ ky

kφ kψ

ky

kφ

kψ

kp

kd kq k0]

(34)

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

141

(a) mr/2l12

2l1

cr/2l12 kr/2l12

(b)

Figure 5.3 The application of relaxation dampers to the two-axle vehicle (a) conventional arrangement (b) direct yaw connection between wheelsets.

and the compatibility matrix is, in terms of the coordinates of (1) ⎡1 ha − 1 d ⎢0 0 0 1 ⎢ ⎢0 1 0 0 ⎢ ⎢0 0 − 1 d ⎢0 0 0 1 a=⎢ 0 0 ⎢0 0 ⎢0 1 0 0 ⎢ 0 ⎢1 − h 0 ⎢1 h − 2 d1 1 ⎢ ⎢⎣1 h1 0 0

− (h + ha ) 0 −1 (h + ha ) 0 −1 0

⎤ ⎥ ⎥ ⎥ ⎥ − ha ⎥ 0 ⎥ ⎥ 1 ⎥ −1 ⎥ ⎥ 0 −1 − h ⎥ 0 1 − h1 ⎥⎥ − 2(h + h1 ) − 1 h1 ⎥⎦ 0 0 0 1 0 0 0

0 0 0

(35)

where d1 is the distance below the car body centre of mass of the spring kq. Applying (2.11.25) and (4.2.5), the nine independent elements of E are then e11 = ky/2 + kd + k0/2(h + h1) e22 = kψ/2 + kp + ha2ky/2 + h12kq

142

RAIL VEHICLE DYNAMICS

e24 = haky/2 + h1kq e27 = dhaky + d1h1kq e33 = 2ky(h + h1)2/h2 + 2kψ/h2 + k0/h2

(36)

e44 = ky/2 + kq e47 = dky + d1kq e55 = h2kd + kψ/2 + ha2ky/2 + h12k0/2(h + h1) e77 = 2d2ky+2kφ + d12kq Applying equations (13) and (14) the shear and bending stiffnesses are given by ⎡ 2 k ψ k y (h + h1 ) 2 + k 0 k ψ + k y k 0 (ha − h1 ) 2 ⎤ ⎥ ks = kd + ⎢ ⎢⎣ 2(h + h1 ) 2 {2 k y (h + ha ) 2 + 2 k ψ + k 0 } ⎥⎦ kb =

kψ 2

+ kp +

k y k q (h1 − ha ) 2 ( k y + 2k q )

(37)

(38)

Note that k0 contributes to ks only and this only in conjunction with kψ or ky (provided that ha ≠ h1); kq contributes to kb only and this only in conjunction with ky. Whilst a limited number of basic stiffnesses are necessary for stability, strictly a full optimisation of the vehicle would involve selecting the optimal values of the independent elements of the stiffness matrix E in order to meet a specified combination of stability and steering performance, and then deriving the basic stiffnesses to achieve these. However, although it is possible to choose at will the maximum number of independent parameters, this usually comes at the price of significant complexity of the mechanical arrangements in order to achieve a marginal advantage.

5.3 Stiff and Flexible Vehicles The influence of speed on the eigenvalues for the two-axle vehicle with the wheel and rail profiles of Figure 2.2 and with the parameters given in Table 4.1 (except that in this Section cy and cφ = 0) is shown in Figure 5.4. The eigenvalues at a specific speed are given in Table 5.1. At a given speed the eigenvalues consist of (a) four relatively large real negative eigenvalues. Two of these eigenvalues are approximately equal to 2f22/mV corresponding to wheelset subsidences in lateral translation, and two eigenvalues are approximately equal to -2f11l2/IV corresponding to wheelset

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

143

Table 5.1 Eigenvalues for the two-axle vehicle of Table 4.1 (seconds-1). At speed, V = 10 m/s

Wheels Fixed

Label

1

-1176

Wheelset subsidence in yaw

2

-1128

Wheelset subsidence in trans.

3

-1128

Wheelset subsidence in trans.

4

-1176

Wheelset subsidence in yaw

5,6

-3.258 ± 10.65i

-3.266 ± 10.65i

Car body upper sway

7,8

-4.711 ± 7.638i

-4.026 ± 7.088i

Car body yaw

9,10

-3.724 ± 4.473i

-3.688 ± 4.555i

Car body lower sway

11,12

-1.617 ± 5.680i

Shear oscillation

13,14

-0.6305 ± 5.246i

Steering oscillation

subsidences in yaw, in accordance with the discussion of section 3.3. At higher speeds these combine to form heavily damped oscillations D and E. (b) two conjugate complex pairs at the kinematic frequency of the wheelsets labelled branches A and B which are initially proportional to speed and which, at low speeds, correspond to the steering and shear oscillations considered in Section 4.5. The real parts of these roots are small and may be positive or negative depending on the speed being considered. (c) three conjugate complex pairs C1, C2 and C3 associated with the vehicle car body modes in which the frequencies are substantially independent of speed and are closely equal to those of the car body oscillating on the suspension with wheels fixed, and are therefore the solution of the following equations { (Izb/h2)s2 + 2ky + 2kψ/h2}φ3 = 0 ( mbs2 + 2 ky )φ6 - 2dkyφ7 = 0

(1)

- 2d kyφ6 + (Ixbs2 + 2d2ky + 2kφ)φ7 = 0 The first equation refers to yaw of the car body and the undamped natural frequency is given by

ω 2 = (2ky h2 + 2kψ)/Izb

(2)

The last two equations yield for the undamped frequencies ΩΙ and ΩΙΙ of the coupled lateral translation and roll motions the characteristic equation

Ω4 − (ωΙ2 + ωΙΙ2)Ω2 + 4kykφ/mbIzb = 0 where

(3)

144

RAIL VEHICLE DYNAMICS

Figure 5.4 Root locus for two-axle vehicle with parameters of Table 4.1 except that cy = 0, cφ = 0, ky = 0.5 MNm.

ωΙ2 = 2ky/mb ωΙΙ2 = (2d2ky + 2kφ)}/Ixb

(4) (5)

The height of the roll centre associated with each undamped natural mode, measured below the plane of the lateral springs follows from the second of equations (1) and is z1 = dωΙ2 /(ωΙ2 − ΩΙ2)

(6)

z2 = dωΙ2 /(ωΙ2 − ΩΙΙ2)

(7)

and in the usual case ωΙΙ > ωΙ, ωΙ > ΩΙ and ΩΙΙ > ωΙΙ so that zI > 0 and zII < 0. Mode I is labelled ‘lower sway’ and mode II ‘upper sway’. Obviously, introduction of sus-

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

145

pension damping complicates the detail but usually the description of the modes remains valid. Reference to Figure 5.4 shows that, at low speeds, the damping of each oscillatory mode is positive. As the suspension damping is zero for this example, the damping in the car body modes must come from the creep forces and is attributable to the interaction between the motion of the wheelsets and the motion of the car body. As the speed is increased, the damping of each mode becomes negative indicating instability. However, the vehicle is unstable at all speeds above VB1 the lowest at which the damping vanishes. As indicated in Figure 5.4, damping is lost in the modes associated with the car body, C1, C2 and C3, at speeds VB1, VB2 and VB3 when the kinematic frequency of a wheelset is rather close to a natural frequency of the car body on the suspension. This mode of instability may be referred to as body instability as the modes of oscillation of the car body are excited by the wheelset oscillation, and the resulting body motions in turn sustain the wheelset oscillation, thus closing the feedback loop. This is illustrated in Figure 5.5 where the representation of the equations of motion as a

hkd wheelset

hkd

wheelset

φ1

φ5

φ4 kψ/h

ky

ky

φ6

body l.t.

ky

ky

φ3

kψ/h

body yaw

Figure 5.5 Block diagram representation of equations of motion. Dotted lines relate to variant with inter-wheelset shear stiffness.

block diagram indicates several feedback loops involving the car body freedoms, in addition to those inherent in the wheelsets as shown in Figure 2.19. It can be seen that at high speeds the branches W1 and W2 are similar to those for the elastically restrained wheelset in which the car body is prevented from lateral movement. The good agreement might be expected as the kinematic frequency is then much greater than the frequency of the car body oscillating on the suspension. At higher speeds, the combined effect of speed, stiffness and inertia modifies the

146

RAIL VEHICLE DYNAMICS

damping in the kinematic oscillations, as discussed in section 3.4, and the damping will vanish at speeds VB4 and VB5 corresponding to wheelset instabilities. It will be noted that in Figure 5.4 the root loci corresponding to the kinematic branches and car body suspension branches either veer or cross-over. When they veer the eigenvectors associated with the eigenvalues on each branch of the loci before veering are interchanged during veering in a rapid but continuous way [20]. Consider the dependence of stability on the plan-view suspension stiffnesses ky and kψ. In accordance with the discussion of Section 4.5, if either ky or kψ are zero, then VB1 is zero. Similarly, if both ky or kψ are infinite, then the solution of equations (2.15 - 21) indicate that VB1 is zero. For this completely rigid vehicle, the solution of the characteristic equation consists of a pair of purely imaginary roots s1,2 = ± iV{(λ/lr0) /(1 + h2/l2)}1/2

(8)

The frequency of the steering oscillation for a rigid vehicle given by equation (8) was first derived by Carter [21]. For zero flexibility the damping of the steering oscillation is zero but unlike the superficially similar kinematic oscillation of a wheelset it is not a motion without creep, as can be easily be verified. The corresponding solution of the equations including the contact stiffness kc = (2Ν ε0∗/ l)( 1 − f /Ν r ) 0

23

0 0

(9)

given by (2.10.45) yields VB 2 =

2 k c ( l 2 + 2h 2 )( l 2 + h 2 ) r0

λl{mb (l 2 + h 2 ) + 2m(l 2 + 2h 2 ) + I b + 2 I }

(10)

which is ordinarily very small. Relatively large values of VB1 can be achieved for the case where both kψ and ky are varied, neither stiffnesses being small. For example, if k = ky = kψ is varied from zero to ∞ the bifurcation speed VB1 rises from zero (as already discussed), reaches a maximum and then decreases to the low value given by (10). The maximum value of VB1 occurs when flexibility parameters similar to those which figured in the discussion of curving behaviour in Section 4.3 are of the order of unity; thus ∆ψ = f2λl3/kψ2r0 ~ 1 and ∆y = f2λl3/ky2r0 ~ 1. In practical terms, these parameters might be thought of as characterising the behaviour of the system when the terms representing creep forces in the equations of motion are commensurate with those representing elastic forces. The vehicle and its behaviour can be distinguished as being “stiff” when both ∆ψ ≤ 1 and ∆y ≤ 1, and is typically descriptive of an isolated short wheelbase bogie. If either ky or kψ is small, then the bifurcation speeds are low. In this case of the ‘flexible’ vehicle one of the flexibility parameters is large; thus ∆ψ > 1 and/or ∆y > 1. In practical terms, this might be thought of as characterising the behaviour of a long wheelbase two-axle vehicle.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

147

5.4 The Flexible Vehicle with Zero Suspension Damping In order to gain insight into the behaviour of the flexible two-axle vehicle it is useful to consider the case where the wheelbase is very long [22]. As already discussed, examination of equations (2.15-21) shows that, for long wheelbase, the choice of the sum and difference coordinates leads to two sets of equations which are uncoupled. Furthermore, if d = 0 or the roll stiffness kφ is very large, the car body roll freedom is uncoupled or suppressed. (The influence of car body roll is considered later). Then the two remaining sets of equations differ only in the value of A33; moreover, if Ιzb/h2 = mb , often a good approximation, they are identical. The solutions of the first set of equations then correspond to the steering oscillation A which is only coupled with yaw of the body whilst the solutions of the second set of equations correspond to the shear oscillation B which is only coupled to lateral translation of the body. These steering and shear oscillations have already been discussed for the low speed case in Section 4.5. In order to investigate dynamic stability one set of equations can be considered as the second set will behave similarly. Hence consider ⎡ms2 + ( 2 f / V + c ) s + k y y ⎢ 2 fλl / r0 ⎢ ⎢ − 2 k y − 2c y s ⎢⎣

−2f Is2 + 2 fl 2 s / V + kψ 0

⎤ ⎡φ ⎤ ⎥⎢ 1 ⎥ 0 ⎥ ⎢φ2 ⎥ = 0 (1) ⎥ 2 m s c s k + + 2 b 4 y 4 y ⎥⎦ ⎢⎣φ3 ⎥⎦ − 2 k y − 2c y s

In this Section, the case in which cy = 0 will be considered. The characteristic equation of (1) is p6s6+ p5s5+ p4s4+ p3s3+ p2s2+ p1s+ p0 = 0

(2)

where p6 = mmbI /4 p5 = mbf(ml2 + I)/2V p4 = mbf2l2/V2 +Iky(m + mb/2)/2 + mmbkψ /4 p3 = fky{(m + mb/2)l2 + I}/V + fmbkψ /2V p2 = 2kyf2l2/V2 + kykψ(m + mb/2)/2 + f2mbλl/r0 p1 = fkykψ/V p0 = 2f2kyλl/r0 An approximation to the root locus can be obtained by applying the perturbation procedure of Section 3.3. In the characteristic polynomial, equation (2), the creep

148

RAIL VEHICLE DYNAMICS

coefficients are numerically large compared with the other coefficients and it can easily be seen that if only the terms in f 2 are retained equation (2) reduces to ( mbs2 + 2 ky )( s2/V2 + λl/r0) = 0

(3)

An approximation to the eigenvalues of equation (2) can be derived by considering that the terms of first order in f are small perturbations of the terms retained in equation (3) so that applying equation (3.3.12) it is found that

λ1 = ± iV(λ/r0l)1/2 - V{kyl2 + kψ - ( ml2 + I )(λ/r0l)V2 - 2ky2l2/(-mb(λ/r0l)V2 + 2ky)}/4fl2 λ2 = ± i (2ky/mb)1/2 - V{ky2/(-mb(λ/r0l)V2 + 2ky)}/f

(4) (5)

These results show that, to this approximation, the real parts of the eigenvalues are inversely proportional to the creep coefficient; at low speeds they are proportional to the suspension stiffness. The damping becomes negative at a speed below that at which frequency coincidence between the kinematic frequency and the suspension frequency occurs. This is the body instability discussed above. In equation (4) the eigenvalue for the elastically restrained wheelset is obtained if mb is made very large and equation (4) therefore predicts the wheelset instability at higher speeds. The exact solutions of equation (1) for the critical conditions in which purely sinusoidal oscillations are possible, thus defining the stability boundary for small displacements, may be obtained by substituting s = iω. Expanding equation (1) and separating real and imaginary parts, results in mb(m + I/l2)ω4 - {2ky( m + I/l2) + mb( ky + kψ/l2)ω2 + 2kykψ/l2 = 0 V2 = 4f2l4ω2/{ 4f2l3λ/r0 - ( kψ − Ιω2)2}

(6) (7)

Equation (6) is independent of speed and creep coefficient f and always has two real roots in ω2. The corresponding bifurcation speeds can then be found by substitution in (7). It is clear that for kψ << 2fl(λl/r0)1/2 instability occurs at the kinematic frequency. For sufficiently large values of kψ > 2fl(λl/r0)1/2 the solutions of (6) and (7) show that stability exists at all speeds; however, in this case it is no longer realistic to assume that the wheelbase is long, and the full set of coupled equations must be considered. If, as is usual, the body mass is much greater than the wheelset mass relevant simple analytical solutions can be obtained by considering two limiting conditions. If mb is large, equations (6) and (7) yield the solution for the elastically restrained wheelset discussed in Section 4.2. If wheelset inertia is neglected so that m = I = 0, then V2 = 2kykψr0l/λmb( kyl2 + kψ)( 1 - kψ2r0/4f2l3λ)

(8)

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

149

or, for kψ << 2fl(λl/r0)1/2 V2 = 2kykψr0l/λmb( kyl2 + kψ)

(9)

yielding the bifurcation speed of the body instability. A similar analysis to the above but including the contact stiffness terms kyc and kψc shows that the bifurcation speed is nonzero when ky and kψ are zero but is very low, [22] and for all practical values of ky and kψ the contact stiffnesses may be neglected. Numerical solutions of the complete system of equations of motion (2.15-21) are displayed in Figures 5.6 and 5.7. Figure 5.6 shows the influence of ky on stability. VB1 increases from zero (or a very small value if the small terms in the equations of motion are retained) to a maximum in accordance with the analysis given above. Figure 5.7 shows the joint influence of kψ and wheelbase h on stability. These results indicate that the strongly stabilising effect of increasing yaw stiffness predicted by equation (8) is only achieved for very long wheelbases as indicated at A in Figure 5.7. Two (or three if roll instability occurs) forms of body instability occur for small values of kψ in accordance with the above discussion of the long wheelbase case. As kψ is increased, the instability involving one mode of oscillation disappears, but the instability involving the other mode occurs for all values of kψ. As the wheelbase is reduced, the stabilising effect of kψ is offset by the destabilising effect of the coupling terms kψ/h. For larger values of kψ, VB1 is independent of kψ , as shown at Β1, Β2 and Β3 in Figure 5.7, the coupling terms kψ/h become dominant and inspection of the equations of motion (2.15-21) shows that an undamped kinematic oscillation of the vehicle is possible where y1 = y2, ψ1 = ψ2 =ψb and yb = 0. The relationship between y1 and ψ1, and y2 and ψ2 is that of the kinematic oscillation, equation (3.2.12), and the frequency is given by

ω2 = 2h2ky/Izb

(10)

Equating this to equation (3.2.11) yields the bifurcation speed VB12 = 2h2kyr0l/λIzb

(11)

and this is in close agreement with the results shown in Figure 5.7 which also shows the associated mode shape. For the ternary sub-system discussed above, as in the case of the elastically restrained wheelset, it was found that reductions in the value of the creep coefficients are stabilising. However, for the complete vehicle, in the absence of suspension damping, owing to the destabilising effect of the coupling terms kψ/h, it is found that the effect on VB of varying a factor on the creep coefficients f11, f22 and f23 is not significant unless the wheelbase is very long. It has been assumed in the above discussion that d = 0. An easy extension of the above analysis would show that there are three possible modes of body instability, two involving the upper and lower sway modes of the car body, and the other involving yaw of the car body.

150

RAIL VEHICLE DYNAMICS 40

V (m/s) 20

0

1

ky (MN/m)

2

Figure 5.6 Variation of VB1 with lateral suspension stiffness ky. Parameters of Table 5.1 except that cy = 0; cφ =0;

(a)

40

V (m/s)

B3 h = 10 B2

20

h=7

A

B1 0

10

h=4 kψ (MNm)

20

(b)

Figure 5.7 (a)Variation of VB1 with yaw suspension stiffness kψ. Parameters of Table 5.1 except that cy = 0; cφ =0; (b) mode shape of oscillation showing one half-cycle at bifurcation speed VB1 for kψ = 20 MNm and h = 10 m.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

151

The above analysis clearly indicates the mechanism of the body instabilities of the two-axle vehicle. It is this form of instability that two-axle vehicles frequently exhibited at quite low speeds [23] though detailed examination of typical designs requires consideration of many significant suspension non-linearities [24]. The results of this Section indicate that improvement of the stability of the flexible vehicle requires the introduction of suspension damping and this is discussed in the next Section.

5.5 Damping and the Long Wheelbase Flexible Vehicle Consider now the effect on the eigenvalues of varying the suspension damping cy indicated in Figure 5.8 for the sub-system of equations (4.1). Figure 5.8(a) corresponds to cy = 0; the root locus has two branches A and B. Branch A starts from the origin when V = 0 and initially has damping proportional to speed. As the speed is increased branch A diverges so as to cross the µ = 0 axis at VB1. With further increases in speed, the locus converges on point C corresponding to the natural frequency of the car body on the suspension. Branch B starts from point C at low speeds, initially has constant frequency and then converges on the root locus of the elastically restrained wheelset. The introduction of cy can have the results shown in Figure 5.8(b), (c) and (d), resulting, under certain conditions, in the elimination of the body instability. These conditions are now investigated.

Figure 5.8 Influence of lateral suspension damping cy on the root locus as speed is varied for the sub-system of equations (4.1). Parameters of Table 4.1 except cφ = 0; kψ = 0.5 MNm; (a) cy = 0 (b) cy = 3500 Ns/m (c) cy = 8000 Ns/m (d) cy = 17000 Ns/m

152

RAIL VEHICLE DYNAMICS

Referring to the equations of motion (4.1) in which the terms in cy are included, φ2 can be eliminated using the second of equations (4.1) so that, making the assumption that the wheelset inertia is negligible, and assuming harmonic oscillations at frequency ω

φ2 = - φ1fλl( kψ/2 - fl2iω/V)/ r0(f2l4ω2/V2 + kψ2/4)

(1)

Assume, consistent with the assumption of suspension flexibility, that f2l4ω2/V2 >> kψ2/4. Then (1) reduces to

φ2 = - V2φ1λ( kψ/2 - fl2iω/V)/ fl3ω2 r0

(2)

φ3 can be eliminated using the third of equations (4.1) , so that on separating real and imaginary parts the first of equations (4.1) becomes λV2 kψ/ω2l3r0 − mbω2{ ky( 2ky - mbω2) + 2cy2ω2}/ {(2ky - mbω2)2 + 4cy2ω2} = 0 2f/V -2λVf/ω2r0l+ mbcyω4/{( 2ky - mbω2)2 + 4cy2ω2} = 0

(3) (4)

From (4) it can be seen that, for f large, if harmonic oscillations can occur they will be at kinematic frequency, ω = V(λ/r0l)1/2. It was assumed in the above discussion that f2l4ω2/V2 >> kψ2/4. If oscillations occur at kinematic frequency this condition reduces to kψ << 2fl(λl/r0)1/2. Equation (3) is a quadratic in ω and has no real roots if kψ/kyl2 > 1/4ζ(1- ζ)

(5)

where ζ = cy/(2mbky)1/2. Thus if this condition is satisfied, body instability is eliminated at all speeds. A necessary condition that (5) is satisfied is that kψ > kyl2. It can be seen that the criterion suggests that stability up to high speeds can be obtained by choosing a sufficiently small lateral suspension stiffness, a sufficiently large yaw stiffness (but not so large as to negate the assumption of long wheelbase, see below), and an optimum value of lateral suspension damping which must be neither too large nor too small. Thus, if the suspension parameters are chosen appropriately, so that the body instabilities are eliminated, stability at speed is then only limited by the wheelset instability and the critical speed is given by equation (3.3.16) and can be made very high. The destabilising effect of wheelset inertia can be seen by extending the analysis leading to equation (5). Retaining the wheelset mass and yaw inertia terms in the first and second equations of equations (4.1) and noting that instability occurs when ω2 = 2ky/mb, roughly, yields kψ/kyl2 > 1/4ζ(1- ζ) + 4m/mb

(6)

which suggests that stability can only be achieved if the wheelset mass is rather less

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

153

than the car body mass. As mentioned above, in reality, the car body is able to roll, and it is found that each of the three modes of body instability (corresponding to the three wheels-fixed natural modes of the car body on the suspension) can be suppressed in turn by an appropriate choice of the parameters as discussed above.

5.6 Stability of the Flexible Vehicle in General Turning to the case of the complete vehicle in which the assumption of a long wheelbase is no longer justified, Figure 5.9 shows the effect on stability of varying the lateral suspension stiffness ky, and Figure 5.10 shows the limit cycle diagrams for three values of ky for the example vehicle of Table 5.1. If the lateral suspension damping cy = 0, the stability boundary would correspond to the body instability. The linear bifurcation speed VB is closely given by (4.9), and is low as it roughly corresponds to the speed for frequency coincidence between one of the wheels- fixed car body modes and the wheelset kinematic mode. For certain values of ky and cy Figures 5.9 and 5.10(a) show that, for small values of ky, body hunting is eliminated and the stability boundary then corresponds to wheelset hunting. Figure 5.10(a) corresponds to point A in Figure 5.9 and may be compared with the limit cycle diagram for the elastically restrained wheelset, Figure 3.6. VB is then closely given by (3.3.16). Figure 5.10(b) corresponds to point B in Figure 5.9. Body hunting occurs for a range of low speeds, and wheelset hunting occurs beyond a higher speed. Figure 5.10(c) corresponds to point C in Figure 5.9 and shows that as ky is increased beyond a certain value, body hunting occurs at a relatively low speed, and for large values of ky, VB is given by (4.9). Figures 5.10(d)-(f) show the corresponding limit cycle diagrams for a reduced coefficient of friction. Limit cycle amplitudes are then restricted but hunting occurs at all speeds above VB in each case. The influence of cy is shown in Figure 5.11. For large values of cy there is instab-

(a)

140

(b)

140 a = 0 mm

VB

100

V (m/s)

VC

3

V (m/s)

5 7

40

40

A 0

100

B

C 3

ky (MN/m)

6

0

3

ky (MN/m)

6

Figure 5.9 Stability of two-axle vehicle with parameters of Table 4.1 as a function of ky. (a) critical speed Vc and bifurcation speed VB. (b) stability boundaries for various assumed amplitudes using equivalent linearisation.

154

RAIL VEHICLE DYNAMICS

8

(a)

(d)

(b)

(e)

(c)

(f)

a (mm) 0

0

V (m/s) 140

Figure 5.10 Limit cycle diagrams for two-axle vehicle with the parameters of Table 4.1 for the three values of ky indicated in Figure 5.9 (a) ky= 0.23 MN/m and µ = 0.3 (b) ky= 2.5 MN/m and µ = 0.3 (c) ky= 5 MN/m and µ = 0.3 (d) ky= 0.23 MN/m and µ = 0.1 (e) ky= 2.5 MN/m and µ = 0.1 (f) ky= 5 MN/m and µ = 0.1.

ility at all speeds above the critical speed for body instability, which is close in value to that for cy = 0. For certain intermediate values of ky , several separate regions of instability can occur, depending on the values of the system parameters. The stabilising effect of cy depends on the values of ky and kψ, as illustrated in figure 5.11. If kψ < kyl2, as in Figure 5.11(a) then no value of cy eliminates body instability. As kψ is progressively increased, as in Figures 5.11(b), (c) and (d) the region of stability is increased and body instability only occurs for extreme values of cy. For very large values of cy the mass of the body will be coupled in with the wheelset mass and the wheelset instability will occur at low speeds. Figure 5.12 illustrates, on a comparative basis with Figure 5.11, the destabilising effect of halving the mass and moments of inertia of the car body. Experimental confirmation of these results was obtained using the full-scale variable parameter test vehicle HSFV-1 which was designed in accordance with the “damped-flexible” approach described above. Roller rig experiments were carried out with this vehicle, and the detailed agreement between theory and experiment is presented in [25]. It was shown that, qualitatively, agreement was excellent, but clearly refinements to the theory to take account of the influence of the rollers are necessary. Such refinements have been considered by a number of writers [26].

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

155

kψ < kyl2

kψ = kyl2

U

V

U

V

S S

S

cy

cy

kψ >> kyl2

kψ > kyl2

U U

V

V

S

S cy

cy

Figure 5.11 Qualitative diagram showing the joint influence of kψ and cy on stability. U = unstable; S = stable.

Following laboratory tests, HSFV-1 was track tested where in spite of the unknown and variable values of conicity and creep coefficients tolerable agreement was obtained. Figure 5.13 shows the joint influence of kψ and wheelbase h on stability, for the case where the suspension dampings cφ and cψ have their nominal values. Figure 5.13(a) shows the stability boundaries for the reduced wheelbase h = 3.3 m. In this case, the body instability involving the bending mode is eliminated for a range of intermediate values of kψ. Figure 5.13(b) shows similar results for a smaller wheelbase of h = 2.7 m and in this case no value of kψ will eliminate the body instability. The boundaries for the wheelset instability are also to be seen in Figures 5.13(a) and (b). As kψ becomes large, as discussed in Section 4, examination of the equations of motion (2.15-21), shows that both wheelsets yaw with the car body and so the effective inertia of the combined mode is large. In addition to reducing the bifurcation speed, the stabilising effect of lateral suspension damping is reduced. In design applications, the emphasis is on the choice of suspension and other parameters which will provide stability under all track conditions. It is a common practice to consider a range of values of equivalent conicity and creep coefficient (often neglecting the stabilising influence of the contact stiffness) and apply a quasilinear analysis. In effect, the dependence on amplitude is replaced by dependence on equivalent conicity. Figure 5.14 shows the effect on the stability boundaries of varying the equivalent conicity. As expected, reduction in conicity increases the critical speeds but also reduces the range of k ψ for which the body instability is elimin-

156

RAIL VEHICLE DYNAMICS

kψ < kyl2 V

kψ = kyl2 V

U

U

S S

S

cy

cy

kψ >> kyl2

kψ > kyl2 V

S

U

V

U

S

S cy

cy

Figure 5.12 Qualitative diagram showing the effect of halving the car body mass and moments of inertia on the joint influence of kψ and cy on stability. U = unstable; S = stable.

ated. Thus, it can be seen that body instability is promoted by low conicity, short wheelbase and either low or high yaw stiffness. It can be inferred from these results that for certain parameter combinations increasing conicity is actually stabilising. This is also indicated in Figure 5.15 which shows the influence of reductions in the magnitude of the creep coefficients and variations of conicity. In Figure 5.15(a) and (b) where the parameters relate to the two-axle vehicle of Table 4.1 reduction in the creep coefficients is stabilising, the

(a)

h = 3.3 m

(b)

U

h = 2.7 m

U

V

S

U

V

S U

U

S kψ

kψ

Figure 5.13 Qualitative diagram showing the joint influence of kψ and h on stability. U = unstable. S = stable.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

157

λ = 0.4

λ = 0.15

U

U

U

V

V

S

S

U

U kψ

kψ

λ = 0.05

λ = 0.1

S U

U

V

V

U

S

S kψ

kψ

Figure 5.14 Qualitative diagram showing the joint influence of kψ and λ. U = unstable; S = stable.

(a)

(b)

λ = 0.3

λ = 0.05

U B S

U V

C

0

0.5

(c)

U D

S

S 1



 (d)

λ = 0.3

U V

V

A

G

λ = 0.05

U E

V

F S 

S 

Figure 5.15 Stability charts showing the joint influence of reducing the creep coefficients f11, f22 and f23 by a factor F and conicity (a) and (b) parameters of Table 4.1 except cy = 8kNs/m (c) and (d) parameters of Table 4.1 except ky = 2 MN/m, kψ = 6 MNm and h = 3.3 m. U = unstable, S = stable.

RAIL VEHICLE DYNAMICS

158

λ = 0.4

λ = 0.15

U

U V

V

S

S

kp

kp

λ = 0.05

λ = 0.1

S

S V

V

U U kp

kp

Figure 5.16 Qualitative diagram showing the joint influence of kp and λ, when kψ = 0. This is drawn on a strictly comparitive basis as Figure 5.14 and shows the elimination of the yaw body instability for most values of kp. U = unstable; S = stable.

U

U

V

U

S

S cy

ky

U S

S U

kr

U cr

Figure 5.17 Qualitative stability diagrams for two-axle vehicle with yaw dampers between wheelsets and car body with kψ = 0.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

159

critical speed associated with the wheelset instability rising broadly in accordance with the discussion of Section 3.4, as shown at A and B. In the high conicity case, the body instability is eliminated as shown at C, but in the low conicity case the body instability occurs over a wider range of speeds, as shown at D. In Figure 5.15(c) and (d) the parameters relate to a two-axle vehicle with the parameters of Table 4.1 except that kψ = 6 MNm and ky = 2 MN/m. Consistent with Figure 5.14, body instability occurs for λ = 0.05 and reduction in the magnitude of the creep coefficients is destabilising as shown at E. However, for the high conicity case, the body instability is eliminated for the nominal values of the creep coefficients, but reappears when they are reduced, as shown at F. The critical speed of the wheelset instability increases with reduction of the creep coefficients as shown at G.

5.7 The Application of Cross-Bracing and Yaw Relaxation Reducing the wheelbase is strongly destabilising for the conventional configuration of two-axle vehicle, and this has been traced to the coupling term (kψ/h) between φ2 and φ5 in the equations of motion, equations (2.15-21). However, consider the application of cross bracing to the flexible two axle vehicle so that the wheelsets are connected by parallel and diagonal springs of stiffness kp and kd respectively. The equations of motion are given by (2.27-33). For clarity, the yaw stiffness kψ will be equated to zero, and then from (2.25-26) the shear stiffness ks = kd, and the bending stiffness kb = kp. Figure 5.16 shows the stability boundaries in the plane of V and kp for the case where kψ = 0 and h = 3 m. These diagrams should be compared with Figure 5.14 and, as kp provides a stiffness to ground without the coupling in yaw between the wheelsets and car body there is no body instability, and the boundary shown corresponds to the wheelset instability. The critical speed is quite insensitive to reductions in wheelbase, and reductions in conicity simply increase the critical speed in accordance with (3.3.16). The modifications to the equations of motion necessary to cover the application of primary yaw dampers have been discussed in Section 2. A repetition of the analysis in Section 4 for the long wheelbase vehicle shows that the eigenvalue corresponding to the kinematic oscillation, equation (4.4), is now given by

λ1 = ± iV(λ/r0l)1/2 - V{kyl2 + (kr cr2λV2)/(cr2λV2 + kr2r0l) - ( ml2 + I )(λ/r0l)V2 - 2ky2l2/(-mb(λ/r0l)V2 + 2ky)}/4fl2

(7)

and this becomes at low speeds

λ1 = ± iV(λ/r0l)1/2 - V3λ( cr2/ kr - ml2 - I -mbl2/2)/4fr0l3

(8)

as at low frequencies the effective stiffness is approximately ω2cr2/ kr. The analysis of this simple system suggests that for small values of cr the effective stiffness will be inadequate to overcome the destabilising effect of the inertia forces. Similarly,

160

RAIL VEHICLE DYNAMICS

very large values of kr will be destabilising, as body yaw instability is introduced. This is illustrated by Figure 5.17 which shows the stability boundaries for a two-axle vehicle with yaw dampers connecting the wheelsets to the car body, instead of yaw springs, as a function of the principal parameters. With zero yaw stiffness, the stability boundaries as ky is varied are broadly similar to those of the standard vehicle shown in Figures 5.9 and 5.11. Provided ky is sufficiently small, stability is only limited by the occurrence of wheelset instability at high speed. In this case the stability boundary is given by (3.3.16) with kψ replaced by kr. As discussed in Section 2 an alternative arrangement is to provide the equivalent of a yaw relaxation spring directly between the wheelsets, and as the arrangement provides a stiffness to ground without the coupling in yaw between the wheelsets and car body there is improved stability.

5.8 The Stiff Vehicle or Bogie For the stiff vehicle, it is found that suspension damping has little effect on stability and consequently, in this Section, suspension damping will be assumed to be zero. To start with, the dynamics of the stiff vehicle are considered at low speeds. The completely rigid vehicle has already been discussed in Section 3. The equations of motion of a two-axle vehicle at low speeds were derived in Section 4.5. Inspection of the characteristic polynomial (4.5.4) shows that it is a symmetric function of kb and ks(h2 + l2). In general, its eigenvalues consist of two real roots s1 = -kbV/fl2

(1)

s2 = - ksV(h2 + l2)/fl2

(2)

corresponding to subsidences in pure bending and pure shear respectively, and a conjugate complex pair µs ± iωs corresponding to the steering oscillation. For certain impracticably large values of the wheelbase all the roots of the characteristic equation are real. Numerical solutions, shown in Figure 5.18, show that for small values of ks and kb the eigenvalues are consistent with the discussion of Section 4.5. For intermediate values of ks and kb, and ordinary values of the wheelbase h, the real part of the eigenvalue µs has absolute maxima for ks = ∞ (occurring for kb = fl(λl/r0)1/2) and kb = ∞ (occurring for ks (l2 + h2) = fl(λl/r0)1/2) and a saddle point for ks(l2 + h2) = kb. For either ks = ∞ or kb = ∞, and ordinary values of the wheelbase, the maximum value of the real part of the eigenvalue µs increases as wheelbase is increased. Whitman [27] applied a perturbation procedure to establish expressions for µs and ωs. Similar results may be established by applying the perturbation technique of Section 3.3 to equation (4.5.4). If h = 0, a solution of equation (4.5.4) is s = ± iω1 =± iV( λ/r0l)1/

(3)

The condition in which the wheelbase is non-zero may now be considered by

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

161

Figure 5.18 The real part of the eigenvalue corresponding to the steering oscillation plotted in the plane of kd(h2 + l2) and kp.

regarding the terms in h2 as small perturbations δpn of the terms pn that exist when h = 0. Equation (3.3.12) then gives the corresponding perturbation in the eigenvalue ss = µs ± iωs provided that there are no repeated roots, which is the case for the system under consideration. This yields as rough, but useful, approximations

µs = -V(λl/r0)fh2 ke / 2(l2 + h2){f2l2(λl/r0) + ke2}

(4)

where ke is a weighted series combination of the stiffnesses 1/ke = 1/ks(l2 + h2) + 1/kb

(5)

and

ωs = V(λ/lr0)1/2 [1 - h2 ke2 /2 (l2 + h2){f2l2(λl/r0) + ke2}]

(6)

Regarding the terms in parantheses as the first two terms of a binomial expansion, (6) can be replaced by

ωs = V(λ/lr0)1/2 [1 - h2 ke2 / (l2 + h2){f2l2(λl/r0) + ke2}]1/2

(7)

which tends to the correct limits as ks and kb are varied, so that the frequency of the steering oscillation varies from that of the kinematic oscillation for small values of the stiffnesses ks and kb to that of the rigid vehicle, given by equation (3.8), for large values of the stiffnesses ks and kb. These results give a useful qualitative impression of the behaviour of the system though their accuracy is limited. The damping increases with wheelbase and reduces with conicity particularly at low values of ke. The maximum value of µs is

162

RAIL VEHICLE DYNAMICS

µsmax = - (λ/lr0)1/2 h2/4l(l2 + h2)

(8)

and this occurs when the flexibility parameter f 2λl/ke2r0 = 1 and is independent of the value of the creep coefficient. Whitman [27] and de Pater [28] have given more accurate solutions, and Scheffel [29] has given a complicated factorisation, but in all these cases the complexity of the results is not justified by the gross assumptions made in formulating the model. The limiting case of small flexibility may now be considered by applying the perturbation technique of Section 3.3 to equation (4.5.4). In this case, the stiffnesses ks and kb in the equations of motion are much larger than the other coefficients. If k now represents the typical stiffness ks and kb an inspection of the characteristic equation shows that if all terms of order less than k2 are neglected the solution for the rigid vehicle is recovered. The condition that all terms of order less than k2 be neglected may now be relaxed by regarding terms of order k as small perturbations δpn of the terms pn of order k2. As there are no repeated roots in the case of the rigid vehicle, equation (3.3.12) then gives the corresponding perturbation in the eigenvalue ss. The eigenvalue corresponding to the steering oscillation is then given by ss = ± iV{(λ/lr0) / (1 + h2/l2)}1/2 -Vf(λl/2r0){h2/(l2 + h2)ke}

(9)

so that the frequency is unchanged from the case of complete rigidity. Even small flexibility introduces significant damping into the steering oscillation so that the assumption of complete rigidity does not accord with practical reality. In contrast with the flexible vehicle, for large values of ke the damping increases with f.

Table 5.2 Eigenvalues for the isolated bogie of Table 4.1 at low speed (seconds-1). Νο.

V = 10 m/s

Wheels Fixed

Label

1

-1129

Wheelset subsidence in yaw

2

-1058

Wheelset subsidence in trans.

3

-1058

Wheelset subsidence in trans.

4

-1129

Wheelset subsidence in yaw

5,6

-16.60 ± 189.9i

-1.941 ± 187.7i

Bogie frame upper sway

7,8

-18.20 ± 245.1i

± 242.0i

Bogie frame yaw

9,10

-18.08 ± 38.51i

-18.06 ± 38.60i

Bogie frame lower sway

11

-44.07

Bogie shear A

12

-47.92

Bogie shear B

13,14

-0.626 ± 3.183i

Bogie steering

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

163

Figure 5.19 Root locus for stiff bogie with parameters of Table 4.1.

Now consider the influence of speed on the eigenvalues. For the typical stiff bogie, such as one with the parameters of Table 4.1, the eigenvalues at low speed are given in Table 5.2 and a schematic variation of the eigenvalues with speed is given in Figure 5.19. At a given speed the eigenvalues consist of (a) a relatively lightly damped conjugate complex pair corresponding to the steering oscillation labelled branch A. The real part of this root is negative at low speeds, and positive at high speeds, the system losing stability at V = VB. (b) at low speeds, a pair of real roots, B1 and B2 corresponding to the shear and bending subsidences, equations (1) and (2). At higher speeds these coalesce to form a heavily damped oscillation (c) three complex pairs corresponding to the oscillations of the bogie frame on the primary suspension (not shown in Figure 5.19) and (d) four large real roots or two complex pairs corresponding to the wheelset subsidences. Some insight into the influence of various parameters on stability is obtained by considering the following model. As mentioned above the stiff vehicle characterises

164

RAIL VEHICLE DYNAMICS

the bogie where d, the vertical distance between the plane of the lateral springs and centre of mass of the bogie frame, is small. If it is assumed that d = 0, the bogie frame roll coordinate is uncoupled from the rest of the system. For a vehicle with sufficiently stiff lateral and yaw primary suspension the natural frequencies of the bogie frame oscillating in lateral translation or yaw on the suspension will be much higher than the steering frequency, see for example Table 5.2. Consequently, consider a model in which all the mass is concentrated at the wheelsets, and the wheelsets are connected by a massless suspension with shear and bending stiffness. Then, consistent with equations (2.27-33) and (4.5.1), {(m/2)s2 + fs/V + ks}φ1 - f φ2 − hksφ5 = 0

(10)

(fλl/r0) φ1 + {(I/2)s2 + fl2s/V + kb}φ2 = 0

(11)

{(m/2)s2 + fs/V }φ4 - f φ5 = 0

(12)

(fλl/r0)φ4 + {(I/2)s2 + fl2s/V + h2ks}φ5 − hkdφ1 = 0

(13)

where, in this case, m and I are equivalent values representative of the complete bogie. Making the assumption that I = ml2, it follows from the form of these equations that the characteristic equation of (4.5.4) for speed V1, with eigenvalues σi, will be the same as that of (10-13) for speed V2, with eigenvalues si, if fσi/V1 = msi2/2 + fsi/V2

(14)

Equation (14) can be regarded as a quadratic in si. For each eigenvalue σi of equation (4.5.4) found for low speeds there will be two roots si found from equation (14) which are roots of the characteristic equation corresponding to the equations of motion with inertia included. The combined effect of speed and inertia is therefore to modify the damping of the steering oscillation, modify the shear and bending subsidences, and to introduce the usual subsidences in lateral translation and yaw of the wheelsets. It can be seen that in the case of a steering oscillation which is stable for low speeds, so that the eigenvalue σi = µ ± iω where µ is negative, the damping will vanish at a speed VB with corresponding eigenvalue si = ± iΩ. It can be seen that by separating real and imaginary parts (Ω /VB) = (ω /V)0

(15)

VB2 = −2f(µ/V)0 /m(ω/V)02

(16)

The instability is an inertia driven instability of the steering oscillation. The mode of instability is analogous to that of the elastically restrained wheelset in that the steering oscillation is stabilised by elastic forces generated by the primary suspension at low speeds but at higher speeds destabilising inertia forces dominate. From (4), (6) and (16) an estimate for the bifurcation speed (derived by Whitman [27]) may be

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

165

obtained. As (ω /V)0 < (λ/lr0)1/2, a conservative estimate of VB is given by putting (ω /V)0 = (λ/lr0)1/2 resulting in VB2 = 2f2l2h2ke / m(l2 + h2){f2l2(λl/r0) + ke2}

(17)

The maximum bifurcation speed obtained from (17) is VBmax2 = flh2ke / 2m(λl/r0)1/2(l2 + h2)

(18)

and this occurs at ke = 2fl(λl/r0)1/2. Figure 5.20 shows the results of a numerical eigenvalue analysis of equations (10)-(13) in which contours of constant bifurcation speed VB are plotted in the ks(l2 + h2)-kb plane. As is evident from equations (4.5.4) and (16), ks(l2 + h2) and kb have similar effects on stability and so the diagram has a line of symmetry. For large values of ks(l2 + h2) stability is independent of ks and similarly for large values of kb, stability is independent of kb. As the stiffness is increased, the bifurcation speed VB for small oscillations increases, reaches a maximum and the decreases to zero as complete rigidity is reached. The corresponding critical frequency is progressively reduced from that of the kinematic oscillation of a single wheelset to that of the steering oscillation of a rigid vehicle. The effect of small flexibility can be derived by substituting the results of (9) into (16) yielding VB2 = f2l4/m(l2 + h2)ke

(19)

independent of the conicity λ. This follows from (4) as the damping in the steering oscillation is proportional to conicity for large values of ke. For extreme values of stiffness the critical speeds are low. The inclusion of the small terms in the equations, such as the contact stiffness, would be stabilising, though calculations show that VB is very low. As indicated in Figure 5.19 the steering oscillation of the example bogie loses its stability at small amplitudes at VB = 63 m/s. The eigenvalues are given in Table 5.3 Table 5.3 Eigenvalues for the isolated bogie of Table 4.1 at speed VB (seconds-1). No.

V = VB = 63.1 m/s

Label

1,2

-35.95 ± 304.6i

Wheelset yaw

3.4

-36.18 ± 242.8i

Wheelset lateral

5,6

-88.28 ± 198.3i

Bogie frame upper sway

7,8

-93.31 ± 215.4i

Bogie frame yaw

9,10

-18.57 ± 38.19i

Bogie frame lower sway

11,12

-107.0 ± 20.94i

Bogie shear

13,14

± 19.98i

Bogie steering

166

RAIL VEHICLE DYNAMICS

25

VB (m/s)

20

55 15 kb (MNm/rad)

60

10

65

5

70 75 80 85

0

5

10 15 ks(l2 + h2) (MNm/rad)

20

25

Figure 5.20 Contours of constant bifurcation speed VB in the plane of ks(l2 + h2) and kb for a bogie with the parameters of Table 4.1.

and the eigenvector corresponding to the mode in which µ = 0 is given in Table 5.4. This solution is a sub-critical bifurcation leading to a limit cycle with amplitude 7 mm at V=60 m/s, the details of which are shown in Figure 5.21. This shows the numerical solution of the nonlinear equations of motion for the bogie with the parameters of Table 4.1 and the wheel rail geometry of Figure 2.2. Figure 5.21(a) and (b) indicate that the limit cycle is in the form of a steering oscillation very similar to that predicted by the small amplitude solution in which yb lags y1 by approximately h, y2 lags yb by approximately h, and ψ1, ψb and ψ2 roughly lag y1, yb and y2 by 90°. The

Table 5.4 Eigenvector for mode 13 of the isolated bogie of Table 4.1 at speed, V = VB .

y1 ψ1 yb φb ψb y2 ψ2

1

0.03613 + 0.2701i 0.8825 - 0.3395i 0.07077 - 0.07804i 0.08869 + 0.2718i 0.7148 - 0.6392i 0.05866 + 0.3142i

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

167

relative displacements between bogie frame and wheelsets are small. The frequency of the oscillation is, however, increased as the equivalent conicity is increased. Figure 5.21(c) shows the variation of the rate of rotation of the leading wheelset which is about 0.1% of V/r0. This fluctuates at twice the frequency of the oscillation as a major component of the torque applied to the wheelset is the product of the longitudinal creep force and the rolling radius both of which are varying at the frequency of the oscillation. Figures 5.21(d) and (e) show the longitudinal and lateral creepages for the right hand wheel of the leading wheelset. As the variation in the wheelset rotational speed is small, reference to equation 2.5.13 shows that the longitudinal creep of the right hand leading wheel is approximately given by rr/r0 + 1 − lψ / V and so the waveform reflects the variation of rolling radius with lateral displacement. Similarly, reference to equation 2.5.15 shows that the lateral creep of the right hand leading wheel is approximately given by y / V − ψ and the waveform indicates the true departure of the response from a sinusoid. Figure 5.21(f) shows the spin of the leading wheelset, and reference to equation 2.5.17 shows that the spin behaves like δr/r0 and so the waveform reflects the geometrical nonlinearity of the contact slope variation with lateral displacement. Figures 5.21(g) and (h) shows the longitudinal and lateral creep forces for the right hand wheel of the leading wheelset, and consideration of their resultant and the normal force indicate that slipping is occurring over a significant part of the cycle of the oscillation. Figure 5.21(i) shows the variation of the normal force acting on the right hand wheel which is in phase with the lateral displacement. Figure 5.22 shows the variation of stability, as measured by the critical speed derived by the method of describing functions, as the lateral and yaw stiffnesses 0.01

(a)

.005

0

0

(d)

t (s) 1

.002 γ2r1 0

-0.01

-.002

(g)

5

(c) Ω −V/r0

0

-5

0

5

0.2

ψ

y (m)

-0.01 0 0.01 γ 1r1

(b)

-0.2

(e)

0

(f)

ω3r1 -0.5 -1

(h)

50

T1r1 (kN) 0

T2r1 (kN) 0

Nr1 (kN) 40

-5

-5

30

Figure 5.21 Limit cycle of example bogie of Table 4.1, V = 60 m/s.

(i)

168

RAIL VEHICLE DYNAMICS

100

V (m/s)

λ = 0.05 λ = 0.1143

50

λ = 0.5

0

20

λ = 0.05, f x 0.5

ky = kψ/l2 (MN/m) 60

80

Figure 5.22 The joint influence of variations of conicity and creep coefficient on VB as the lateral and yaw stiffnesses are varied, where ky = kψ/l2.

(assuming that ky = kψ/l2) are varied. The critical speed VC is given by the solution of the full linear equations of motion, for various values of the equivalent conicity and creep coefficient. The maximum critical speed occurs at lower values of the stiffnesses for lower conicities, in accordance with the above discussion. For large values of the stiffnesses, the critical speed is proportional to the creep coefficient and is largely independent of conicity. The practical significance of these results for the conventional bogie is as follows. Because as ky is increased the shear stiffness is limited to a maximum value of ks = kb / h2 (equation (4.2.6)) high critical speeds can more easily be achieved by increasing yaw or bending stiffness, with consequent adverse effects on steering. However, stability is dependent on the range of equivalent conicity and creep coefficient likely to be met in service. It can be seen from Figure 5.22 that if the typical design specification is 0.5 > λ > 0.05 and fnom > f > fnom/2 where fnom is the value derived from Kalker’s analysis (Section 2.4.4.3), and reasonable margins are made for both speed and stiffness, the maximum design speed is limited and the highest critical speeds cannot be exploited. For this reason, high speed bogies depend on restraint from the car body. See Chapter 6, also [30]. Equivalent levels of stability can be achieved with lower values of bending stiffness by adopting cross-bracing as discussed above, which implements directly the shear stiffness ks. There is then no limit in the values of shear stiffness that can be achieved and a better trade-off between stability and curving can be obtained.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

169

5.9 The Three-Piece Bogie The three-piece bogie has evolved over the last hundred years mainly for freight vehicles in North America, but enjoys wide application throughout the world. In the three-piece bogie, the bogie frame of a conventional bogie is replaced by two separate side-frames which rest directly on the axle boxes through adaptors which allow only rotational freedom. A bolster supports the car body, with a centre-plate which allows relative yaw between bogie and body. The bolster is connected to the sideframes by a suspension which allows some relative motion in all senses except longitudinal. Thus, the side frames have additional degrees of freedom of longitudinal and pitching motions. This results in warping or differential rolling movements of the wheelsets (useful on twisted track) and lozenging of the side-frames with respect to the wheelsets in plan-view. It follows that the form of construction yields a very low overall shear stiffness. If the warp stiffness is very large then the bogie behaves like a conventional bogie, [31, 32]. Quite apart from the low shear stiffness of the configuration, there are many rubbing surfaces and consequently the behaviour tends to be highly nonlinear with the result that crabwise motions are common, with resultant asymmetric wear of wheel profiles. The application of cross-bracing to the three-piece bogie is particularly useful because the anti-lozenging function of the cross-bracing allows the designer to omit the usual constraints between the bolster and the side-frames and provide a proper lateral suspension at that point. Moreover, if controlled flexibility is provided, by means of a elastomeric pad mounted between the side-frames and bearing adaptors for example, then in conjunction with cross-bracing specified suspension stiffnesses can be provided enabling optimisation of the design. Reduced bending stiffness between the wheelsets improves curving and the improvement of the primary and secondary lateral suspension enhances ride quality and reduces loads. Bogies incorporating these ideas were produced in the 1970's in Britain, Pollard [33], South Africa, Scheffel [7],and in North America, List [34], though widespread application has been retarded by the increment in first cost.

References 1. Hamilton Ellis, C.: Nineteenth Century Railway Carriages. Modern Transport Publishing, London, 1949, p. 13. 2. White, J.H.: A History of the American Locomotive. The John Hopkins Press, Baltimore, 1968, pp. 169-174. 3. Matsudaira, T.: in ORE Committee C9. Problems of interaction of vehicles and track-essays awarded prizes. Report 2, Part 2, Office for Research and Experiments, Utrecht, June 1960. 4. Wickens, A.H.: The dynamics of railway vehicles on straight track: fundamental considerations of lateral stability. Proc. I. Mech. E. 180 (1965), Part 3F, pp. 29-44.

170

RAIL VEHICLE DYNAMICS

5. Hobbs, A.E.W.: Improvements in or relating to railway vehicles. British Patent Specification 1261896, 1972. 6. Wolf, E.J. and Stec, F.F.: Single axle suspensions for intermodal freight cars. Conference Proceedings: The Economics and Performance of Freight Car Trucks, Montreal, October, 1983. 7. Scheffel, H.: A new design approach for railway vehicle suspensions. Rail International 5 (1974), pp. 638-651. 8. Scheffel, H.: Unconventional bogie designs − their practical basis and historical background. Vehicle System Dynamics 24 (1995), No. 6-7, pp. 497-524. 9. Wickens, A.H.: British Patent 1179723, 1967. 10. Evans, J.R.: The modelling of railway passenger vehicles. In: G. Sauvage (Ed.): The Dynamics of Vehicles on Roads and on Railway Tracks, Proc.12th IAVSD Symposium, Lyon, August 1991. Swets and Zeitlinger Publishers, Lisse, 1992, pp. 144156. 11. Eickhoff, B.M., Evans, J.R. and Minnis. A.J.: A review of modelling methods for railway vehicle suspension components. Vehicle System Dynamics 24, No. 6-7, 1995, pp. 469-496. 12. Sauvage, G.: Determining the characteristics of helical springs for applications in suspensions of railway vehicles. Vehicle System Dynamics 13, 1984, pp. 13-41. 13. Wickens, A.H.: Steering and dynamic stability of railway vehicles. Vehicle System Dynamics 5, 1978, pp.15-46. 14. Horak, D. et al.: A comparison of the stability performance of radial and conventional rail vehicle trucks. ASME J. Dynamic Systems, Measurement and Control 103, 1981, p. 181. 15. Kar, A.K., Wormley, D.N, Hedrick, J.K.: Generic rail truck characteristics. In: H.-P. Willumeit (Ed.): The Dynamics of Vehicles on Roads and on Railway Tracks, Proc.6th IAVSD Symposium, Berlin, September 1979. Swets and Zeitlinger Publishers, Lisse, 1980, pp. 198-210. 16. Kar, A.K. and Wormley, D.N.: Generic properties and performance characteristics of passenger rail vehicles. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and on Railway Tracks, Proc.7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 329-341.

DYNAMIC STABILITY OF THE TWO-AXLE VEHICLE

171

17. Fujioka, T.: Generic representation of primary suspensions of rail vehicles. In A.H. Wickens (Ed.) The Dynamics of Vehicles on Roads and on Railway Tracks, Proc.11th IAVSD Symposium, Kingston, August 1989. Swets and Zeitlinger Publishers, Lisse, 1989, pp. 233-247. 18. Fujioka, T., Suda,Y. and Iguchi, M.: Representation of primary suspensions of rail vehicles and performance of radial trucks. Bulletin of JSME 27 (1984), No.232, pp. 2249-2257. 19. Hedrick, J.K. Wormley, D.N. Kim, A.K. Kar, A.K. and Baum, W.: Performance limits of rail passenger vehicles: conventional, radial and innovative trucks. U.S. Department of Transportation Report DOT/RSPA/DPB-50/81/28, 1982. 20. Perkins, N.C. and Mote, C.D.: Comments on curve veering in eigenvalue problems. J. Sound and Vibration 106 (1986), pp. 451-463. 21. Carter, F.W.: Railway Electric Traction. London, Edward Arnold, 1922. 22. Wickens, A.H.: The dynamic stability of a simplified four-wheeled railway vehicle having profiled wheels. Int. J. Solids and Structures 1 (1965), pp. 385-406. 23. Wickens, A.H. and Gilchrist, A.O.: Railway vehicle dynamics-the emergence of a practical theory. Council of Engineering Institutions MacRobert Award Lecture, 1977. 24. Gilchrist, A.O., Hobbs, A.E.W., King, B.L. and Washby, V.: The riding of two particular designs of four wheeled vehicle. Proc.I.Mech.E. 180, Part 3F (1965), pp. 99-113. 25. Hobbs, A.E.W.: Lateral stability experiments with HSFV-1. British Railways Tech. Note DYN 53, 1967. 26. Jaschinski, A. et al.: The application of roller rigs to railway vehicle dynamics. Vehicle System Dynamics 31 (1999), pp. 345-392. 27. Whitman, A.M.: On the lateral stability of a flexible truck. ASME Journal of Dynamic Systems, Measurement and Control 105 (1983), pp. 120-125. 28. De Pater, A.D.: The lateral behaviour of railway vehicles. CISM Courses and lectures no.274, Springer, Vienna New York, 1982, pp. 223-279.

29. Scheffel, H.: The influence of the suspension on the hunting stability of railway vehicles. Rail International 10 (1979), p. 662. See also: Scheffel, H.: The dynamic stability of two railway wheelsets coupled to each other in the lateral plane by elastic and viscous constraints. In A.H. Wickens (Ed.) The Dynamics of Vehicles on Roads and on Railway Tracks, Proc.7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 385-400.

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30. No, M., Hedrick, J.K.: High speed stability for railway vehicles considering varying conicity and creep coefficients. Vehicle System Dynamics 13 (1984), pp. 299-313. 31. Tuten, J.M., Law, E.H. and Cooperrider, N.K.: Lateral stability of freight cars with axles having different wheel profiles and asymmetric loading. ASME J. Engineering for Industry 101 (1979), pp. 1-16. 32. Whitman, A.M. and Khaskia, A.M.: Freight car lateral dynamics-an asymptotic sketch. ASME Trans. Journal of Dynamic Systems, Measurement and Control 106 (1984), pp. 107-113. 33. Pollard, M.G.: The development of cross-braced freight bogies. Rail International, September 1979, pp. 736-758. 34. List, H.A.: An evaluation of recent developments in railway truck design. ASME Paper 71-RR-1, April 1971.

6 The Bogie Vehicle 6.1 Introduction Long two-axle vehicles, and other vehicles with several wheelsets incorporated in a single frame (a so-called rigid wheelbase), like locomotives, have obvious limitations in curves. The first known proposal for the bogie was made in Britain by William Chapman in 1797 [1]. It was, however, in the United States that the concept was first employed to a significant extent. Dissatisfied with the performance of the rigid wheelbase British locomotives on the lightly built and curvaceous American track, John B. Jervis with advice from Horatio Allen designed the first locomotive with a leading swivelling bogie in 1832. In this, two wheelsets were mounted in a frame which was free to swivel without restraint relative to the main body of the locomotive. This radically improved curving behaviour [2]. In addition to its application to locomotives, passenger coaches employed bogies in North America from the 1840s. These early bogies had very short wheelbases and tended to oscillate violently being the probable cause of many derailments. In the 1850s the wheelbase of the leading truck of locomotives was increased which improved stability significantly [3]. In Britain with relatively straight track there was little need for the use of bogies until trains increased in size in the 1860s, and in any case the few occasions when bogies had been used gave them a bad reputation with British railway engineers. Fernihough pointed out the danger of bogie oscillation in his evidence before the Gauge Commission in 1845 [4] and suggested that it might be controlled by the frictional resistance of a bearing ring of large diameter supporting the car body on the bogie, an idea subsequently adopted widely. The centre friction plate, or alternatively, friction at the side bearers, provided yaw restraint for small relative motions thus preventing bogie hunting on straight track at currently prevailing speeds. On sharp curves, at low speeds, the friction was overcome and the bogie was able to take up a more radial position on the track. In the case of bogie passenger coaches it was appreciated that isolation of the car body from motions of the bogie required some form of secondary suspension. So, in addition to a secondary vertical suspension, the swing bolster which provided lateral flexibility between bogie and car body, was invented by Davenport in 1841 [5]. In Britain, though the use of bogies in locomotives was exceptional until the 1870s, those that were built often incorporated lateral movement of the bogie pivot restrained by some form of spring, called a centring spring [6].

174

RAIL VEHICLE DYNAMICS

In the period before adequate mathematical models became available, evolution of the bogie vehicle had been based on rather general ideas. Much later, when calculations were attempted, the motions in the lateral plane were often assumed to be kinematic with amplitude equal to the flange-way clearance. Hence emphasis was given to the avoidance of coincidence between the natural frequencies of the car body on the suspension and the kinematic frequencies. Based on these ideas, and empirical development, surprisingly good results had been obtained before 1960, providing that conicities were kept low by re-turning wheel treads and speeds were moderate, for example below 160 km/h. The discussion of Chapter 5 shows that for an isolated bogie, in which there is no longitudinal or lateral suspension damping, for both small and large values of the longitudinal and lateral suspension stiffnesses critical speeds are relatively low. However, for an intermediate range of stiffnesses, the critical speed reaches a maximum, and this maximum can be quite high. The results suggest that for many low or moderate speed applications careful choice of the longitudinal and lateral stiffnesses will give adequate stability without the use of secondary yaw restraint between the bogie and the body. With the advent of higher speeds, the consequences of bad riding became more serious and Matsudaira [7] carried out the first mathematical modelling embracing primary and secondary suspension stiffnesses, conicities appropriate to worn wheels and creep. This was applied to the bogie developed for the Shin Kansen in the early 1960s which was tested to speeds of 246 km/h in 1964, and used in service operation subsequently at speeds of 210 km/h. Generally, high speed trains still follow a similar prescription, in which the stabilising action of the primary suspension is augmented by the restraint offered by the secondary suspension. For high speed trains the emphasis is on stability. For this application typical bogies are relatively stiff with primary longitudinal stiffness in the range 30-60 MN/m reflecting the requirement for stability at high speed. Operation is on relatively straight lines and considerations of curving are secondary. Careful choice of bogie wheelbase, primary suspension parameters and secondary yaw restraint provide a substantial margin of stability, [8]. Stable operation has been demonstrated at speeds of over 500 km/h. The models discussed here are the simplest that reveal the basic principles of guidance and stability of the conventional bogie vehicle. In practice, for engineering design realistic modelling must take account of much important detail [9, 10]. Many efforts have been made to improve performance in curves by making bogie vehicles more flexible in plan view, thus permitting the axles to take up a radial position in curves. There is no doubt that many of these configurations exhibited an even wider spectrum of various hunting instabilities than more conventional designs. Schemes for so-called body steered railway vehicles, in which provision is made for the radial steering alignment of the wheelsets by means of linkages or levers connecting the wheelsets with the car body, appeared early in the 19th century [11, 12]. Earlier in the 20th century significant and successful development was carried out by Liechty [13, 14]. Schwanck [15] reported on service experience with a particular design of body-steered bogie and its advantages of reduced wheel and rail wear, reduced energy consumption and increased safety against derailment. Many examples of body steering are in current use.

THE BOGIE VEHICLE

175

In this Chapter, a discussion of the dynamics of the conventional bogie vehicle is followed by a general analysis of the conflict between steering and stability of multiaxle vehicles. This is then applied to a variety of configurations with body steered bogies.

6.2 Equations of Motion Consider the idealised four-axle vehicle with a body and two bogies, shown in Figure 6.1. Each bogie has a frame and two wheelsets and this sub-system has been discussed in Chapters 4 and 5. The bogies support a car body by means of a secondary suspension, but there is no direct connection between the bogies. As in the case of the twoaxle vehicle, the wheelsets, bogie frames and car body are all assumed to be rigid and connected by massless elastic structures and massless damper elements. The vehicle is symmetric about a longitudinal plane of symmetry but the bogies are not assumed to be symmetric. Therefore, for small displacements, motions which are symmetric about this plane of symmetry are decoupled from those which are anti-symmetric and will not be considered here. Hence in addition to the nine degrees of freedom for the leading bogie with generalised coordinates bq, nine degrees of freedom for the trailing bogie with generalised coordinates dq, the car body has three degrees of freedom, lateral translation, roll and yaw with generalised coordinates cq.

yd

ψ4

y4

yb

ψd

ψ3

h

h

ψc

y3

yc

ψ2

c

yc

φc φb

Figure 6.1 Generalised coordinates for bogie vehicle.

yb

y2 ψb

ψ1

y1

RAIL VEHICLE DYNAMICS

176

Thus the motion of the vehicle, Figure 6.1, is defined by the following set of twentyone generalised coordinates q = [bq | cq | dq ]T

(1)

In the discussion of stability it will usually be possible to neglect the variation of the rotational speed of the wheelsets in which case q = [y1 ψ1 yb φb ψb y2 ψ2 yc φc ψc y3 ψ3 yd φd ψd y4 ψ4 ]T

(2)

so that yb, yd and yc refer to lateral translation of the bogie frames and vehicle body and ψb , ψd , and ψc refer to yaw of the bogie frames and vehicle body. The other yi, ψi are standard wheelset coordinates. The equations of motion will be of the form of (2.11.23) where F = block diagonal[ F1 O33 F2 O33 F3 O33 F4 ]

(3)

and O33 is the 3 × 3 null matrix, and Fi refers to the ith wheelset. The inertia matrix is A = diag [m I mb Ixb Izb m I mc Ixc Izc m I mb Ixb Izb m I]

(4)

Figure 6.2 shows a basic form of secondary suspension for a conventional bogie vehicle. The component stiffness and compatibility matrices are k = diag[ ky1 kφ1 kψ2 ky2 kφ2 kψ2 kyb kφb kψb kyb kφb kψb ky2 kφ2 kψ2 ky1 kφ1 kψ2]

⎡1 h1 ⎢0 0 ⎢ ⎢0 1 ⎢ ⎢0 0 ⎢0 0 ⎢ ⎢0 0 ⎢0 0 ⎢ ⎢0 0 ⎢0 0 a=⎢ ⎢0 0 ⎢ ⎢0 0 ⎢0 0 ⎢ ⎢0 0 ⎢0 0 ⎢ ⎢0 0 ⎢ ⎢0 0 ⎢0 0 ⎢ ⎣0 0

− 1 d1 0 1 0

0

− 1 d1 0 1

− h − h1 0 0 0 −1 h + h2 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0 1 − h2 0 0

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0 0

0 0

0

0

1 0 0

0

0

1

0

−1

1 0

d2 1

h5 0

0 0

0 0

0

0

1

0

0

0

0

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

−1 0

0 1

0

0

0

0

0

0

0

0

0 0 1 h2

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 1

0 0

0 0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

− 1 d 3 − c − h5 0 −1 0 0 0 −1 c + h5 − 1 d3

0 0

0

0

0

0 0

0 0

0 0

0

0

0

0

0

0

0 0

0 0

0 0

0

0

0

0 d2

0 − h5

1

0

0 0 0

0 − 1 d1 1 0

−1 − h − h2 0

0 0

0 0 − 1 d1 0 1

−1 h + h1

0 1

0

0

1

0

0

0

0

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ − h1 ⎥ 0 ⎥ ⎥ −1⎦

(5)

(6)

THE BOGIE VEHICLE

177

h5 kyb/2 ky1/2

ky2/2

kψ1

kψ 2

kψb h1

h2 h

h c

ϕc

yc

d3

d2

kyb

kφb yb

d1 ky1

kφ 1

Figure 6.2 Arrangement of bogie vehicle showing suspension stiffnesses.

It should be noted that in practice the formation of E using equation (2.11.25) is unnecessarily lengthy as there are a large number of zeros in a. Consequently, a simpler process of assembly may be used which depends on the topology of the connections between the bodies of the system, and which is described in any text on the Finite Element Method.

6.3 Dynamics of the Conventional Bogie Vehicle A set of example parameters is given in Table 6.1 for a conventional bogie vehicle in which h1 = h2 = h3 = 0 so the bogies themselves are symmetric fore-and-aft. The example vehicle has bogies which are the same as the isolated bogie of Chapter 4 and defined in Table 4.1. It should be noted that this vehicle has no secondary yaw restraint. Figure 6.3 shows the locus of eigenvalues of the equations of motion discussed in the previous Section as the speed is varied for the example bogie vehicle. Figure 6.3 should be compared with Figure 5.19 which gives corresponding results for the isolated bogie. In addition to the eigenvalues corresponding to the modes already seen in the isolated bogie there are eigenvalues approximately corresponding to motions of the car body as if the wheelsets were fixed and which are substantially independent of speed.

RAIL VEHICLE DYNAMICS

178 Table 6.1 Example parameters for conventional bogie vehicle. primary suspension

kψ = 40 MNm ky = 40 N/m cφ =10 kNms secondary suspension kyb = 0.45 MN/m kψb= 0 cφb = 60 kNms

kφ =1MNm

cy = 0

cψ = 0

kφb=1 MNm

cyb = 60 kNs

cψb = 0

creep coefficients f22 = 6.79 MN f11 = 7.44 MN vehicle geometry

f23 = 13.7 kNm

h = 1.25 m c = 8.75 m

h2 = 0 d2 = 0.4 m

h1= 0 d1= 0.2 m

h3 = h4 =1.25 m d3 =1 m

h5 = 0

inertia

m =1250 kg mc = 22000 kg

I = 700 kgm2 mb=2500 kg Ixb=1000 kgm2 Izb = 3500 kgm2 2 6 2 2 Ixc = 30000 kgm Izc=1 10 kgm Iy = 250 kgm

The eigenvalues of the complete system at one value of the forward speed and those of the system with either the wheelsets fixed or the car body fixed are given in Table 6.2 where the various modes are labelled. At speeds away from that for frequency coincidence between the car body modes and the bogie modes it is a good approximation to consider the car body fixed. It can be seen from Figure 6.3 that for the chosen set of parameters the bifurcation speed VB is little changed from that for the isolated bogie.

Figure 6.3 Root locus as speed is varied for conventional bogie vehicle.

THE BOGIE VEHICLE

179

Table 6.2 Eigenvalues for the conventional bogie vehicle (seconds-1). No.

1 2 3 4 5 6 7 8 9,10 11,12 13,14 15,16 17,18 19,20 21 22 23 24 25,26 27,28 29,30 31,32 33,34

V = 10 m/s

-1130 -1130 -1058 -1058 -1058 -1058 -1130 -1130 -25.33 ± 190.8i -25.36 ± 190.5i -18.20 ± 245.1i -18.20 ± 245.1i -46.55 ± 25.42i -46.45 ± 24.01i -44.09 -44.09 -47.91 -47.91 -5.770 ± 8.443i -4.304 ± 6.96i -0.563 ± 3.998i -0.612 ± 3.189i -0.582 ± 3.212i

Wheels Fixed

Car Body Fixed

Label

-1129 -1129 -1060 -1060

-10.74 ± 187.9i -10.73 ± 187.6i ± 242.0i ± 242.0i -46.47 ± 25.40i -46.35 ±23.95i

-5.826 ± 8.534i -4.301 ± 7.081i -4.990 ± 4.032i

Wheelset subsidence in yaw Wheelset subsidence in yaw Wheelset subsidence in trans. Wheelset subsidence in trans. Wheelset subsidence in trans. Wheelset subsidence in trans. Wheelset subsidence in yaw Wheelset subsidence in yaw -25.22 ±191.1i Bogie upper sway, anti-phase Bogie upper sway, in-phase -18.19 ±245.2i Bogie yaw, anti-phase Bogie yaw, in-phase -46.16 ± 27.2i Bogie lower sway, anti-phase Bogie lower sway, in-phase Bogie shear A -43.83 Bogie shear A Bogie shear B -48.02 Bogie shear B Car body upper sway Car body yaw Car body lower sway -0.276 ± 3.12i Bogie steering Bogie steering

At speeds for which the bogie steering frequencies approach the natural frequencies of the vehicle car body on the suspension, there is the possibility of considerable interaction, leading to instabilities in which the amplitude of the car body is large relative to that of the wheelsets, though for the example set of parameters such body instabilities do not occur. Figure 6.4 shows the stability boundaries for a conventional bogie vehicle showing VB as a function of the secondary yaw stiffness kψb, secondary lateral stiffness kyb, secondary yaw damping cψb and secondary lateral damping cyb. A denotes bogie instability, and B denotes body instability. For the smaller values of kyb VB is associated with the bogie instability, labelled A in Figure 6.4, discussed in Chapter 5 and increasing kψb is strongly stabilising. Larger values of kψb lead to body instability involving relatively large amplitudes of car body yaw, labelled B in Figure 6.4. This is analogous to the behaviour of the two-axle vehicle discussed in Chapter 5. Low values of conicity and creep coefficient promote this mode of instability. Similarly, kyb and cyb must not exceed certain limits if body instability is to be avoided, and cyb must exceed a certain limit. It was shown in Chapter 5 that with an appropriate choice of suspension parameters and a limited range of conicity, the isolated conventional bogie can possess stability up to moderately high speeds but if a wide range of conicity and

RAIL VEHICLE DYNAMICS

180

200

200

A

V

V

A

B 100

100 B

0

0

1

kψb

200 V

2

kyb

200 V

A

100

A

100

B

B

0

1

cψb

0

2

1

cyb

2

Figure 6.4 Stability boundaries for bogie vehicle showing VB (m/s) as a function of the secondary yaw stiffness kψb (108 Nm), secondary lateral stiffness kyb (106 N/m), secondary yaw damping cψb (106 Nms) and secondary lateral damping cyb (104 Ns/m). A denotes bogie instability, and B denotes body instability.

200

λ=0.05

λ=0.05 f/2 λ=0.1143

λ=0.05 f/2

Vc 100

λ=0.5

S

0

4

kψb

8

Figure 6.5 Influence of secondary yaw stiffness kψb (107 MNm) on critical speed Vc (m/s) for various values of equivalent conicity and creep coefficient for the conventional bogie vehicle. S denotes the stable region.

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181

x 10

0.02

ψ 0

c/10

0.01

2

y 0 b

1

3

x 10

d 1

b

3

-10

d

2

c 4

2

-5 4

-0.01 0 2

-3

t

4

-15

0

2

4

4

5000 T2

1

4r

0

T1 0

-5000

-1 -2

1r 0

4r

3l

-10000

2l

3l 1r

2l 2

4

-15000 0

2

4

Figure 6.6 Dynamic response of bogie vehicle to curve entry. V = 15 m/s, R0 = 225 m, length of transition L = 20 m.

creep coefficients are to be catered for, stability cannot be guaranteed. Reduction in bogie mass and increases in wheelbase make a contribution to the improvement of stability as shown in the discussion of the isolated bogie, but the major design measure is, as discussed in the introduction, to exploit a suitably high value of secondary yaw restraint, provided in the form of springs, dampers or relaxation dampers. Figure 6.5 shows the influence of secondary yaw stiffness kψb on critical speed Vc for various values of equivalent conicity and creep coefficient for the conventional bogie vehicle. It can be seen that the stable region of operation, denoted by S in Figure 6.5, is much enlarged for larger values of kψb. As noted above, for low conicity and creep coefficients very large values of secondary yaw stiffness are destabilising, analogous to the behaviour of the two-axle vehicle. Figure 6.6 shows the lateral and longitudinal creep forces generated during curve entry, for the example bogie vehicle without secondary yaw stiffness and zero cant deficiency. The response is very similar to that of the isolated bogie shown in Figures 4.8 and 4.9 and discussed in Chapter 4. Clearly, provision of secondary yaw stiffness in order to promote stability would degrade curving performance further, and so it is appropriate to consider means by which the trade-off between curving and stability might be improved.

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182

6.4 Steering and Stability of Multi-Axle Vehicles in General It has been shown in Chapter 3 that for two-axle vehicles there is a conflict between steering and stability. The approach taken there is now generalised to vehicles with any number of axles and it will be shown that it is possible to achieve a better resolution of this conflict if there are more than two axles [16, 17]. Consider the elastic stiffness matrix E. For stability and steering only motions which are anti-symmetric with respect to the longitudinal plane of symmetry are considered, and the vehicle is assumed to move forward at constant speed. As in the case of the two-axle vehicle, two rigid body motions are possible, lateral translation and yaw, described by the column matrices ql and qy. No elastic forces can be generated in a rigid body motion so that Eql = 0

(1)

Eqy = 0

(2)

For a system of rigid bodies with R degrees of freedom connected by M massless elastic elements, imposition of the conditions (1) and (2) means that E will have degeneracy P = 2 and with a suitable choice of coordinates will be of the partitioned form O2 , R − 2 ⎤ ⎡ O22 E=⎢ (3) E + ⎥⎦ ⎣O R − 2 ,2 where E+ is of order (R - 2). It follows that there are (R2 - 3R + 2)/2 disposable elements. As there can only be (R - 2) independent rows in a, the M × R compatibility matrix defined by equation (2.11.5), the number of corresponding disposable stiffness elements will be equal to (R - 2).The remaining (R - 3)(R - 2)/2 disposable elements are geometrical and are contained in a. The most general choice of elastic stiffness matrix is therefore possible if the number of physical parameters, element stiffnesses and their positions, equals or exceeds this number. There is no limit to the number of element stiffnesses and hence to the number of rows in the compatibility matrix a or the order of k, the primitive stiffness matrix of the elastic elements. If the vehicle has fore-and-aft symmetry, choice of coordinates which describe either motions which are symmetric or anti-symmetric with respect to the plane of symmetry results in the following partitioned form for E ⎡ O22 ⎢ E = ⎢ OS 2 ⎢⎣OU 2

O2 S E1 OUS

O2U ⎤ ⎥ OSU ⎥ E 2 ⎥⎦

(4)

where E1 is of order S × S where S = (R-2)/2 if R is even and S = (R-3)/2 if R is odd; E2 is of order U × U where U = (R-2)/2 if R is even and U = (R-1)/2 if R is odd. Therefore the number of independent elements in E for a symmetric vehicle is (R2-2R)/4 if R is even and is (R2-2R+1)/4 if R is odd.

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183

If (2.11.25) is substituted into (1) and (2) then kaql = 0

(5)

kaqy = 0

(6)

An additional condition on E is provided by consideration of steering on curves. Following the discussion on the motion of an unrestrained wheelset on a curve in Chapter 2, assume that the lateral creep force due to spin is exactly negated by the gravitational stiffness. Then the equilibrium of the vehicle in a curve being traversed at equilibrium speed and hence zero cant deficiency will be determined by the simplified form of equations of equilibrium [ C + E ]qc = Qc

(7)

where qc represents a column of the values of the generalised coordinates consistent with steady state motion on a uniform curve, and Qc are the corresponding forces applied by the track. Zero lateral and longitudinal creep, or perfect steering, will occur when Cqc = Qc

(8)

Eqc = 0

(9)

and so

Substitution of equation (2.11.25) into equation (9) shows that for perfect steering kaqc = 0

(10)

Thus, equations (1), (2) and (10) imply that the elastic stiffness matrix for a vehicle with perfect steering will have degeneracy P > 3. Repeating the arguments used above it is found that the number of independent elements in E is equal to (R2 - 5R + 6)/2 for an unsymmetric vehicle and to (R2 - 4R + 4)/4 (R even) or to (R2 - 4R + 3)/4 (R odd) for a symmetric vehicle. Dynamic stability requires that all the real parts of the eigenvalues of | As2 + (B/V + D)s + C + E | = 0

(11)

are negative. As discussed in Chapters 3 and 5, in general, as speed is increased, stability is lost at a bifurcation speed VB at which at least one eigenvalue becomes purely imaginary. A necessary condition for the existence of a nonzero critical speed is that the eigenvalues of the system at low speeds |(B/V + D)s + C + E | = 0

(12)

all have negative real parts, as discussed in Chapter 3. If all N wheelsets are identical,

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184

and if there were no structural connections between the various rigid bodies of the system, E and D would be null, and equations (12) would reduce to N uncoupled sets of binary equations where N is the number of wheelsets. There would be N identical eigenvalues s = ± iV(λ/lr0)1/2 corresponding to a kinematic oscillation of each of the identical wheelsets. The elements of the corresponding eigenvectors qk will be yi =βi

1/2

ψi = iβi(λ/lr0)

all other qi = 0

(13)

where the βi are arbitrary in magnitude so that N quantities remain undetermined. Now if Eqk =0

(14)

equation (12) will be satisfied, and there will be at least one undamped mode at low speeds involving a kinematic oscillation of the wheelsets. The condition that equation (14) is satisfied is that P > N where P is the degeneracy of the E matrix. If P ≤ N then no solution corresponding to a kinematic oscillation can exist, and thus P ≤ N is a necessary but not sufficient criterion for dynamic stability. A necessary condition for a vehicle with wheelsets of equal conicity to be dynamically stable and capable of perfect steering in the case of zero cant deficiency is therefore 3≤P≤N

(15)

The case where the wheelsets have unequal conicities is discussed in Chapter 9 where it is shown that a small margin of stability can be obtained but is generally of little practical importance. Now consider further the behaviour of the system at low speeds. Denoting the generalised coordinates associated with the wheelsets by qs and the remaining coordinates by qv so that for the bogie vehicle qv = [ yb φb ψb yc φc ψc yd φd ψd ]T

(16)

qs = [ y1 ψ1 y2 ψ2 y3 ψ3 y4 ψ4 ]T

(17)

the equations of motion can be written in the partitioned form ⎡ Ars s 2 + ( Brs / V + Drs ) s + Crs + Ers ⎢ Dus s + Eus ⎣⎢

⎤ ⎡ q s ⎤ ⎡0 ⎤ ⎥⎢ ⎥ = ⎢ ⎥ Auv s 2 + Duv s + Euv ⎦⎥ ⎣qv ⎦ ⎣0⎦ Drv s + Erv

(18)

Making the substitution s = VD and letting V tend to 0 yields the two sets of uncoupled equations governing the motion at low speeds [BrsD + Crs + Ers - ErvEuv-1Eus ] qs = 0 or

(19)

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185

+ Crs + Ers* ] qs = 0

[BrsD

(20)

where Ers* = Ers - ErvEuv-1Eus is the deflated stiffness matrix involving only the wheelset coordinates and [ Auvs2 + Duvs + Euv ] qv = 0

(21)

indicating that at low speeds the eigenvalues of the system fall into two distinct sets. Firstly, a set associated with the motions of the wheelsets, modified by the influence of the quasi-static elastic interaction between the wheelsets through the elastic connections to the bogie frames and car body. Secondly, a set associated with the natural modes of the vehicle body oscillating on the suspension as if the wheelsets were fixed. This generalises the results already found for the two-axle and bogie vehicles. The eigenvalues of (20) are the same as those of (12) and so the condition for stability at low speeds is dependent on E*. Similarly, the equations governing steady motion in curves, equations (7), can be replaced by [ Crs + Ers* ] qs = Qrc

(22)

The conditions for stability and perfect curving discussed above therefore also apply to the reduced system involving only the wheelset freedoms, and it follows that the number of independent elements in E* is correspondingly reduced. A similar approach has been followed by de Pater [18, 19] with some significantly different details.

6.5 Steering and Stability of a Generic Bogie Vehicle This general approach is now applied to bogie vehicles in which rather general forms of suspension are incorporated so as to meet the dual requirements of steering and stability. As the complete vehicle is assumed to have a transverse plane of symmetry the structure of the stiffness matrices is simplified. Clearly, reversal of the direction of motion should result in identical equations of motion if account is taken of the sign convention chosen for the generalised coordinates. The complete stiffness matrix takes the form bb

E

E= (1)

cb

E

db

E

bc

E

cc

E

dc

E

bd

E

cd

E

dd

E

It will be assumed that there is no direct connection between the leading and trailing bogies, so that bdE and dbE are null. The relationship between the coordinates of the leading bogie and the trailing bogie is therefore defined by

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186

qb = Tqd

(2)

where T = 0 except that T16 = T33 = T44 = T61 = -1 and T27 = T55 = T72 = 1. Hence e66 -e67 e77

e63

e61

-e62

-e73 -e74 e75 -e71

e72

e33 dd

E =

e64 -e65

-e32

e34 -e35

e31

e44 -e45

e41 -e42

(3)

(sym) e55 -e51

e52

e11

-e12 e22

e86 -e87

e83

e84 -e85

e81

-e82

e96 -e97

e93

e94 -e95

e91

-e92

cd

E =

(4)

-e10,6 e10,7 -e10,3 -e10,4 e10,5 -e10,1 e10,2

From the above discussion, if there were direct connections between the leading and trailing bogies and their wheelsets the total number of parameters would be 64 assuming that the rigid body conditions were satisfied and the vehicle is symmetric. This number is much reduced if there are no direct connections between the leading and trailing bogies. Further reductions in the number of disposable parameters will occur if it is assumed that the bogies are themselves symmetric and also if the curving condition is imposed. Fully generic schemes, such as those discussed in Chapter 4, could be the starting point for an analysis of the bogie vehicle. However, a more direct approach is to derive the simplest arrangements which are capable of both perfect steering and stability as fully generic schemes are unnecessarily complex. The spirit of the discussion so far, concerning the resolution of the conflict between steering and stability, is that the vehicle should be made as flexible as possible in plan view, consistent with the need for stability. Whilst the range of configurations with the maximum number of parameters are of interest for two and three-axle vehicles, the situation for a four-axle vehicle is quite different. As discussed above, in the case of the vehicle with four axles under consideration here, a fully generic configuration meeting the criterion expressed by equation (3.15) would have a very large number of parameters, far more than is needed for practical configurations which should be as simple and as flexible as possible.

THE BOGIE VEHICLE

187

If, however, attention is confined to low speeds or quasi-static conditions then the equations of motion can be reduced to the form of (4.20) and (4.22), as discussed in Section 3, involving only 8 wheelset coordinates. If the curving condition, equation (4.10), is applied in addition to the rigid body conditions, equations (4.5) and (4.6), the stiffness matrix E* can be expressed in the form of equation (4.4). There will therefore be a maximum of 9 and a minimum of 5 independent parameters in E*. It will be shown below how this suggests an approach to the generation of generic configurations for the steered bogie vehicle. Noting that fully generic arrangements may be useful in generating design variants, the further development of the argument takes as its starting point the simpler form of arrangement shown in Figure 6.2. The arrangement is similar to conventional practice, except that provision is made for asymmetry in both geometry and in the magnitude of the stiffnesses. The solution of equations (4.10) will initially be discussed for this simplified generic arrangement. Assuming zero cant deficiency, equations (4.10) become y1{ y1 + h1ψ1 - yb + d1ϕb - (h + h1)ψb } = 0

(5)

kψ1( ψ1 - ψb) = 0

(6)

ky2{- yb + d1ϕb + (h + h2)ψb + y2 - h2ψ2} = 0

(7)

kψ2( ψb - ψ1 ) = 0

(8)

kyb{ yb + d2ϕb + h5ψb - yc + d3ϕc - (c + h5)ψc} = 0

(9)

kψb( ψb- ψc ) = 0

(10)

kϕ1 ϕb = 0

(11)

kϕb( ϕb - ϕc ) = 0

(12)

kϕ2 ϕb = 0

(13)

ky1{ y4 - h1ψ4 - yd + d1ϕd + (h + h1)ψd } = 0

(14)

kψ1( ψ4 - ψd ) = 0

(15)

ky2{- yd + d1ϕd - (h + h2)ψd + y3 + h2ψ3} = 0

(16)

kψ2( ψd - ψ3 ) = 0

(17)

kyb{ yd + d2ϕd - h5ψd - yc + d3ϕc + (c + h5)ψc} = 0

(18)

kψb( ψd - ψc ) = 0

(19)

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188

kϕ1 ϕd = 0

(20)

kϕb( ϕd - ϕc) = 0

(21)

kϕ2 ϕd = 0

(22)

From equations (11) to (13) and (20) to (22) ϕb = ϕc = ϕd = 0. From equations (6), (8), (15) and (17) the wheelsets can only take up a radial position if kψ1= kψ2=0. Similarly, from equations (10) and (19) kψb =kψd = 0. Therefore for zero creep all inter-body yaw stiffnesses must be zero. If h1 and h2 are fixed then equations (5) and (7) define yb and ψb, and equations (14) and (16) define yd and ψd. If h5 is fixed then equations (9) and (18) define yc and ψc. If h1, h2 and h5 are zero then yb = ( y1 + y2 )/2

ψb = ( y1 - y2 )/2h

(23)

yd = ( y3 + y4 )/2

ψd = ( y3 - y4 )/2h

(24)

yc = ( yb+ yd)/2

ψc = ( yb - yd )/2c

(25)

so that in this case the attitude of the vehicle is completely defined by the lateral positions of the wheelsets yi. In the case of motion on a uniform curve, the analysis of Chapter 4 shows that in order to achieve zero lateral and longitudinal creep each wheelset moves outwards and aligns itself radially. Because of symmetry yd = yb, ϕd = ϕb, ψd = -ψb, y4 = y1, y3 = y2, ψ4 = -ψ1, ψ3 =-ψ2 and ψc = 0. For example, in the case of linear conicity and the outer and inner wheelsets are identical, the lateral displacements are y1 = y4 = - lr0/λR + (c + h)2/2R

y2 = y3 = - lr0/λR + (c - h)2/2R

(26)

and the yaw displacements are

ψ1 = -ψ4 = (c + h)/R

ψ2 = -ψ3 = (c - h)/R

(27)

and the displacements of the car body and bogie frames are given from (23-25) as yb = yc = yd = -lr0/λR + (c2 + h2)/2R

ψb = − ψd = c/R

ψc = 0

(28)

In the general case, where h1, h2 and h5 are not zero, Figure 6.7 indicates the geometry in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the frames and car body take up the positions connecting the wheelsets without strain of the suspension elements. Thus, under the conditions considered here, perfect steering is obtained because of the combined action of creep and conicity which is the primary mechanism of guidance. Additional elastic restraint is needed in order to stabilise the kinematic oscillations of the wheelsets. For the system with the elastic stiffness matrix E as derived above it

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189

Figure 6.7 Geometry of basic bogie vehicle on curve: a bogie frame, b unstrained spring connecting bogie frame to car body (corresponding to kyb ), c and d unstrained springs corresponding to primary lateral suspensions ky1 and ky2.

may be easily verified that there are 4 pairs of imaginary eigenvalues at low speeds corresponding to undamped kinematic wheelset oscillations. This is consistent with the necessary criterion for stability, equation (4.15), derived above. As the degeneracy of E is 8, additional stiffnesses are required to reduce the degeneracy to 4 so that the criterion may be satisfied. Four additional stiffnesses must be provided, and the corresponding rows in the compatibility matrix a must satisfy the rigid body conditions expressed by equations (4.5) and (4.6) and the steering condition expressed by equation (4.10). In addition they must be linearly independent from the existing rows of a. Since the corresponding rows of the compatibility matrix will involve the wheelset yaw angles they can be considered as defining "steering laws", a term used in the literature. These sub-matrices of k and a relating to the additional elastic elements will be denoted by ks and as for convenience. Various possibilities are considered in the next section. These results are consistent with the above discussion concerning the number of independent parameters in E. The basic system, without the additional stiffnesses corresponding to the steering laws, has 3 independent stiffnesses ky1, ky2 and kyb, and a minimum of 5 are required. The two additional stiffness parameters will each appear twice in the 4 additional rows of as, two rows for the leading bogie and two rows for the trailing bogie. (The roll stiffnesses do not affect the degeneracy of E). Since no elastic forces can be generated in perfect steering, as expressed by equations (4.10), it is clear that the above solutions hold even if all the elastic stiffnesses are very large. For the configuration under discussion, in addition to equations (4.1), (4.2) and (4.9) being satisfied, symmetry implies that there is a fourth solution Eqd = 0 where qd is antisymmetric so that yd = -yb, ϕd =− ϕb, ψd =ψb, y4 = -y1, y3 = -y2, ψ4 =ψ1, ψ3 =ψ2 and yc = ϕc = 0. It can be seen that when the elastic stiffnesses are all large the system acting as a free body would have four degrees of freedom, rigid body lateral translation, rigid body yaw and symmetric and antisymmetric bending as a mechanism, corresponding to degeneracy of four in the elastic stiffness matrix, Figure 6.8.

190

RAIL VEHICLE DYNAMICS

Figure 6.8 Mechanism modes for basic bogie vehicle: a rigid body lateral translation, b rigid body yaw, c symmetric bending, d antisymmetric bending.

Though the application of direct structural connections between the bogies is not considered here, it is worth noting that the antisymmetric bending mode can be suppressed, without degrading the ability of the vehicle to steer properly, by providing a direct shear connection between the bogies. Such inter-couplers are used in practice with the objective of reducing forces in curves, [20].

6.6 Application to Specific Configurations Analytical studies of body-steered bogie vehicles were initiated by Bell and Hedrick [21] and Gilmore [22] who identified various instabilities which were promoted by low conicities and reduced creep coefficients. A considerable body of work by Anderson and Smith and colleagues is reported in [23] to [28], covering the analysis of a vehicle with bogies having separately steered wheelsets. Weeks [29] has described dynamic modelling and track testing of vehicles with steered bogies, noting the enhanced sensitivity of this type of configuration to constructional misalignments. The theoretical considerations of the previous Section are illustrated by the scheme shown in Figure 6.9. Two additional stiffnesses ka and kb , representing linear springs between the wheelsets and the car body, are added to the basic scheme

THE BOGIE VEHICLE

191

h7

ka

h9

kc

h3

k

kb

h

h

c

ϕb

yb dc7

ϕb

ka

db7

yb

ks = [ ka kb kc ] as =

1 - h7 0 0 0 0 0 -1 dc7 -(c + h - h7) 0 0 0 0 0 0 0 0 0 0 0 0 1 h9 -1 dc9 -(c - h +h9) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 - h3 0 0 0 - 1 -h4 0 0

Figure 6.9 Simplified generic bogie with additional stiffnesses: top, plan view, bottom, end view looking forward at the leading bogie and outer wheelset. (h7 = - ch/(c + h); h9 = - ch/(c - h); h4 = 2h - h3).

discussed above. Then the matrices ks and as for the leading bogie are also shown in Figure 6.9, and there will be similar matrices for the trailing bogie. The first two rows relate to the stiffnesses ka and kb, and satisfy equations (4.5), (4.6) and (4.10) as do the corresponding rows for the trailing bogie. In accordance with the above discussion only four additional stiffnesses are required to decrease the degeneracy of the stiffness matrix to the maximum of four that is required. The addition of interwheelset shear stiffness kc such as might be provided by cross-bracing is covered by the third row in as which relates to kc and is also shown in Figure 6.9 for the leading bogie. There is a corresponding row for the trailing bogie and both rows satisfy equations (4.5), (4.6) and (4.10). This element could replace either ka and kb and the criteria for stability and perfect steering would still be satisfied. Thus a

RAIL VEHICLE DYNAMICS

192

f

c

b

d

a

g

Figure 6.10 Wiesinger bogie: a and d pivots connecting wheelsets to bogie frame g (ky1 and ky2 very large), b pivot connecting bogie frame to car body (kyb very large), c pivot connecting steering arm e to car body (ka very large), f linkage providing shear connection between the wheelsets (kc very large).

e4

e3 e1

c5 c1

e2 c6 c4 c3

c2

ks = [ ka kb kc ] 0 - 1 0 0 1 + h/c 0 0 0 0 - h/c 0 0 0 0 0 0 0 as =

0 0 0 0 1 - h/c 0 -1 0 0 h/c 0 0 0 0 0 0 0 1 - h3 0 0 0 -1 -h4 0 0 0

0

0 0 0 0 0 0 0

ka = e1e3c22(c5 - c1)2/{e1(c2 - c1)2 + e3(c5 - c1)2} kb = e2e4c32(c6 - c4)2/{e2(c4 - c3)2 + e4(c6 - c4)2} c2(c5 - c1)/c1(c5 - c2) = c/(c + h) c3(c6 - c4)/c4(c6 - c3) = c/(c - h) Figure 6.11 Linkage steered bogie with separately steered wheelsets.

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193

practical choice for design is to steer the outer wheelset from the car body and steer the inner wheelset by means of a shear connection from the outer wheelset. This is exemplified by designs of steered bogie due to Robinson (1881) (cited by Smith [12]) and Wiesinger (1932) [11], Figure 6.10. The need for elastic elements in order to achieve a margin of stability and satisfactory response was not then recognised; so in these schemes ka and kc are very large, leading to the use of the pivots a, b, c and d. However, the analysis above suggests that it is possible to dispense with pivots and accommodate the necessary relative displacements by the use of suitably disposed elastic elements. A serious drawback of this configuration is that the direct coupling of the wheelset motions to the car body degrades stability and ride quality as shown in [30]. In the configuration shown in Figure 6.11 each wheelset is separately steered through a linkage connected to the car body, and is the basis of the steered bogie developed by UTDC. Such arrangements have been analysed by Smith and Anderson [23-27] and Shen [31]. It can be seen that the effective stiffnesses ka and kb are functions of the linkage stiffnesses e1, e2 and e3. Expressions for ka and kb are derived by considering the equilibrium of the system in terms of the generalised coordinates and the angular position of the link, and then eliminating the latter as inertia effects in the links are negligible. As in the previous example, alternative configurations can be derived by providing inter-wheelset shear stiffness kc. Then any one of ka, kb and kc, or the corresponding element stiffnesses, could be set to zero and the condition for stability satisfied. Alternative configurations could therefore have either the outer or the inner wheelsets steered together with an interwheelset shear connection. This could represent a useful simplification, which does not appear to have been studied in modern times, though it is the equivalent of the schemes by Robinson and Weissinger mentioned above. Careful choice of the dimensions of the linkages is necessary in order to keep within practical sizes. For this reason schemes have been developed in which the linkage is in the vertical plane. A more common variant of body steered bogie in which the wheelsets are jointly steered by one lever is shown in Figure 6.12. To represent this configuration 3 additional rows of the compatibility matrix are required for the leading bogie, corresponding to stiffnesses ka, kb and kc which are functions of the element stiffnesses e1, e2 and e3 as shown in Figure 6.12. The matrices ks and as for the leading bogie are also shown in Figure 6.12, and there will be similar matrices for the trailing bogie. Elements of this design go back as far as 1841, and there are many modern examples. The third row of a is equal to the sum of the first and second rows and thus only two of the three rows in the compatibility matrix for the linkage are independent. kc is equivalent to an inter-wheelset shear stiffness, and thus the arrangement not only provides the steering action defining the yaw angles of the wheelsets relative to the body, but also imposes inter-wheelset shear restraint. Figure 6.13 shows a scheme which received intensive development by Liechty [11]. A further example is provided by the configuration shown in Figure 6.14 in which the wheelsets are steered by a linkage pivoted on a member joining the wheelsets. This is clearly derived from the scheme of Figure 6.11 and provides a useful model for the discussion of stability below.

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194

c1

ks = diag[ ka kb kc ] as =

0 - 1 0 0 1 + h/c 0 0 0 0 - h/c 0 0 0 0 0 0 0 0 0 0 0 1 - h/c 0 -1 0 0 h/c 0 0 0 0 0 0 0 0 -1 0 0 2 0 -1 0 0 0 0 0 0 0 0 0 0

ka = e1e3c22(c5 - c1)2/ ∆

kb = e2e3c12(c5 - c3)2/ ∆

kc = e1e2c22(c3 - c1)2/ ∆

∆ = {e1(c2 - c1)2 + e2(c3 - c1)2+ e3(c5 - c1)2} c2(c5 - c1)/c1(c5 - c2) = c/(c + h) c3(c5 - c1)/c1(c5 - c3) = c/(c - h) Figure 6.12 Steered bogie with shared wheelset linkage.

Figure 6.15 shows the dynamic response of a vehicle with the configuration of Figure 6.14 and parameters given in Table 6.1 to entry into a curve consisting of a linearly increasing curvature over a distance of 20 m and the constant curvature of 225 m. The track is assumed to be canted so cant deficiency is zero. Figure 6.15 should be compared with Figure 6.6 which gives the corresponding results for the conventional bogie vehicle. When the vehicle is completely on the uniform section of the curve, the wheelsets move outwards and adopt a radial position with negligible longitudinal creep and creep forces. The spin on the outer wheels is relatively large and results in corresponding lateral creep forces but the lateral forces on the inner wheels are small. These results are consistent with equations (3.2.8-9). The bogie frames and car body also adopt radial positions consistent with the wheelset positions so that in the steady state the suspensions are not strained. However, during the transition quite large longitudinal and lateral creep forces are exerted, as on the transition the suspension is strained and the resulting forces must be reacted by inertia and creep forces. These forces reduce to zero as the vehicle

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e3 c1

e1 e2 h c2 ks = diag[ ka kb kc ]

⎡1 − h − 1 0 c 2 a = ⎢⎢0 0 − 1 0 c2 ⎢⎣1 − h 0 0 0 s

0 0 − c2 1 h 0 0 − c2 0 −1 − h 0 0 0

0

kb = e2e3c12/∆

ka = e1e3c12/∆

kc = e1e2c22/∆

0 0 0 0 0 0 0⎤ 0 0 0 0 0 0 0⎥⎥ 0 0 0 0 0 0 0⎥⎦ 2 2 2 ∆ = e1c2 + e2c2 + e3c1

Figure 6.13 Liechty’s bogie. c2 = h2/c.

e3 c5 c1

e1

e4 a

e2

kc

c6 c4 c3

c2

ks = diag[ ka kb kc ] ⎡(1 + h / c) / 2h − 1 0 0 0 − (1 + h / c) / 2h 0 0 0 − h / c 0 0 0 0 0 0 0⎤ a s = ⎢⎢(1 − h / c) / 2h 0 0 0 0 − (1 − h / c) / 2h − 1 0 0 h / c 0 0 0 0 0 0 0⎥⎥ ⎢⎣ 1 0 0 0 0 0 0 0 0⎥⎦ −h 0 0 0 −1 − h 0 −1

Figure 6.14 Bogie with wheelsets separately steered from an inter-wheelset member. Expressions for ka and kb and the relationship between the ci are the same as the scheme of Figure 6.11.

196

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leaves the transition. This is consistent with the results given in Figure 6.15. A compensating steering arrangement that would account for transition geometry has been proposed by Smith [32].

Figure 6.15 Dynamic response of steered bogie vehicle of Figure 6.14 to curve entry. V = 15 m/s, R0 = 225 m, length of transition L = 20 m.

The concept of the body-steered bogie has been applied to freight vehicles with three piece bogies by Scales [33], List [34] and Shen [31] shows that dynamic stability as well as curving performance is significantly improved in comparison with a conventional design. So far it has been shown that a general formulation of the equations of motion of a symmetric railway vehicle with two unsymmetric bogies, capable of perfect steering, makes it possible to consider a wide variety of possible configurations on a common basis. Though there is a large number of disposable parameters theoretically available, only a limited number are needed to derive practical configurations which are simple and yet are capable of perfect steering. The configurations discussed satisfy the necessary conditions for dynamic stability at low speeds, and so the actual range of values of parameters necessary to achieve both static and dynamic stability must be discussed next.

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6.7 Stability of Bogie Vehicles with Steered Wheelsets In this section the dynamic behaviour, as exemplified by the root locus as speed and suspension stiffness is varied, of the body-steered four-axle bogie vehicles is considered. The configuration of steered bogie vehicle chosen for discussion is that of Figure 6.14. Consider firstly the behaviour at low speeds, and how the eigenvalues of the system vary with suspension stiffness. Figure 6.16 shows the variation of the eigenvalues with k, a factor applied to ky, ka, kb and kc. When k = 0 there are four undamped oscillations at wheelset kinematic frequency ω =V(λ/lr0)1/2. Introduction of stiffness results in damping of these oscillations, the real part of the eigenvalue being initially proportional to k. Two of these oscillations can be identified with the steering or bending oscillation A and D of the leading and the trailing bogies respectively. The other two oscillations can be identified with the shear oscillation B and C of the leading and the trailing bogies respectively. In both cases the steering oscillation is lightly damped. The shear oscillations are heavily damped, splitting into four subsidences B1, B2, C1 and C2 as k increases. It can be seen that if these results are compared with those for the isolated bogie discussed in Section 5.8, the eigenvalues corresponding to the steering and shear oscillations are significantly changed in character. In fact, both oscillatory instability and divergence occur for certain values of k, and the conditions for this are discussed below in detail. Not shown in Figure 6.16 are eigenvalues which are largely independent of k. These eigenvalues, of which there are 9 complex conjugate pairs, refer to modes

Figure 6.16 Variation of the eigenvalues for the body-steered vehicle at low speeds as a function of a factor on the stiffnesses ka and kc (eigenvalues of large modulus not shown).

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Figure 6.17 Root locus as speed is varied for body-steered bogie vehicle of Figure 6.11. The left hand plot shows the root locus in the complex plane and the right hand plot is the variation of the imaginary parts with speed. Parameters of Table 6.1 with ka = 40 MNm, kb = 40 MNm, kc = 0, ky = 0.

which are substantially oscillations of the vehicle body and the bogie frames on the primary and secondary suspension as if the wheels were fixed. The influence of speed on the eigenvalues for the steered bogie vehicle is shown in Figure 6.17, for values of ky, ka, kb and kc which give stability at low speeds. Comparison with Figure 6.3 shows that the results are very similar to those for a conventional bogie vehicle, with the notable exception that the branches A1 and A2, initially proportional to speed, corresponding to the bogie steering modes, differ. Inspection of the eigenvectors shows that one branch refers to motions mainly involving the leading bogie and the other branch refers to the trailing bogie. The eigenvalues associated with the vehicle body and bogie frame modes, substantially independent of speed, are broadly similar to those of the conventional bogie vehicle, as are the 8 relatively large real negative eigenvalues corresponding to wheelset subsidences in lateral translation and yaw. As in the case of the conventional bogie vehicle, at speeds for which the steering frequencies approach the natural frequencies of the vehicle body on the suspension, there is the possibility of body instability. Such body instability can be eliminated by a suitable choice of secondary suspension parameters as discussed for the conventional bogie vehicle. As speed is increased, the combined effect of speed and inertia modifies the damping in the steering oscillations. For each steering oscillation which is stable for low speeds, the damping will vanish at a bifurcation speed VB. The instability corresponds to a form of bogie hunting, and unlike the conventional bogie vehicle where the two bifurcation speeds are approximately the same, as Figure 6.17 shows those for the body steered vehicle can be quite different.

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6.8 Simple Bogie Model Examination of the reduced order stiffness matrix E* defined in equations (4.19) and (4.20) shows that, for practical vehicles, E* is of the form E*= E*0 + ∆E where ∆E is small compared with E0*, and

e11

e12

-e11

e14

0

0

0

0

e12

e22

-e12

e24

0

0

0

0

-e11

-e12

e11

-e14

0

0

0

0

e14

e24

-e14

e44

0

0

0

0

0

0

0

0

e11

e14

-e11

e12

0

0

0

0

e14

e44

-e14

e24

0

0

0

0

-e11

-e14

e11

-e12

0

0

0

0

e12

e24

-e12

e22

E0*=

where

(1)

e11 = kaρ2/4h2 + kbσ2/4h2 + kc e12 = - kaρ/2h - kch e14 = - kbσ/2h - kch e22 = ka + kch2 e24 = kch2 e44 = kb + kch2

where ρ = 1 + h/c, σ = 1 - h/c. E0* is the stiffness matrix derived by assuming that the car body is restrained from deviating from a uniform forward motion. As in the case of the conventional bogie vehicle, for the purposes of trend studies a good approximation to the eigenvalues associated with bogie motions can be obtained by treating the vehicle body as an inertial mass. Accordingly, referring to the equations of motion put yc, φc, ψc = 0. Furthermore, if it assumed that ky is large and remembering that kψ = 0, the bogie frame co-ordinates are given by yb = ( y1 + y2)/2 and ψb = ( y1 - y2)/2h. If d1 = 0 then the bogie frame roll coordinate is laterally uncoupled from the rest of the vehicle. The system is thus reduced to two uncoupled sets of four degrees of freedom each, involving y1, ψ1, y2, ψ2

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and y3, ψ3, y4, ψ4. Finally the secondary lateral suspension stiffness kyb has only a small influence on the stability of the bogies and will be neglected. Introducing sum and difference coordinates ϕi defined by equation (4.2.3)

and neglecting all suspension damping terms, the equations of motion will be {(m/2 + mb/4)s2 + fs/V + e11}φ1 - f φ2 ± e12 φ2+ e14φ4 = 0 (fλl/r ± e12) φ1 + {(I/2)s2 + fl2s/V + e22}φ2 ± e24φ4 = 0

(3) (4)

{(m/2 + mb/4)s2 + fs/V }φ3 - f φ4 = 0

(5)

e14φ1 ± e24φ2 + (fλl/r)φ3 + {( I/2)s2 + fl2s/V + e44}φ4 = 0

(6)

where now e11 = kaρ2/4h2 + kbσ2/4h2 + kc e12 = - kaρ/4h + kbσ/4h e14 = - kaρ/4h - kbσ/4h - kch e22 = ka /4 + kb /4 e24 = ka /4 - kb /4 e44 = ka /4 + kb /4 + kch2 The upper sign refers to the leading bogie and the lower sign refers to the trailing bogie. A particularly simple case arises when the bogie frame mass mb = 0, corresponding to the configuration shown in Figure 6.14 with a light inter-wheelset member, for then if it is assumed that I = ml2 it is possible to write

D = ms2 /2f + s/V

(7)

then a trial solution q = aeD t leads to the characteristic equation p4D4 + p3D3 + p2D2 + p1D + p0 = 0 where p4 = f 4l 4

(8)

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p3 = f 3l 2{ ka (1/2 +ρ2l2/4h2) + kb(1/2 +σ2l2/4h2) + kc( l2 + h2 )} p2 = 2f 4l 2α + f 2kakc{l2 + h2 + l2(1-ρ)2 }/4 + f 2kakb{1+ l2/h 2 + l2( ρ - σ )2/4h2}/4 + f 2kbkc{l2 + h2 + l2(1-σ)2 }/4 ± f 3l2(kbσ - kaρ )(1 - α )/4h p1 = f 3α{ ka(1/2 + ρ2 l2/4h2) + kb(1/2 + σ2 l2/4h2) + kc(l2 + h2)} ± f 2(1 - α){-kakc(ρ -1)h + kbkc(σ - 1)h + kakb( σ - ρ )/2h}/4 p0 = f 4α2 + f 2α (kakb/h2 + kakc + kbkc)/4 ± f3α(kbσ - kaρ)(1 - α )/4h and α = λl/r0. 6.9 Stability of Simple Bogie Model First consider stability at low speeds so that D becomes equal to s/V. In this case the results are applicable to the configurations shown in Figures 6.11 and 6.14. Routh’s criterion for oscillatory stability, equation (3.4.3), will be applied. It can be shown that if any two of the stiffnesses ki are zero, T3 = 0 and an undamped oscillation at wheelset kinematic frequency occurs in accordance with the discussion of Section 4. Consequently, it is of interest to discuss the three cases, each for the leading and trailing bogie, firstly where the outer wheelset is steered and there is shear stiffness between the wheelsets, secondly where the inner wheelsets are steered and there is shear stiffness between the wheelsets and thirdly where both outer and inner wheelsets are steered. Firstly consider the possibility of oscillatory instability at very low speeds. This can occur only for very large values of the stiffnesses ki and then the sign of T3 is determined by the sign of p1 so that on substitution for r and s the condition for oscillatory stability is ± (1 - α){kakc + kbkc + kakb/h2} < 0

(1)

where the upper sign refers to the leading bogie and the lower to the trailing bogie. Thus the stability depends on ± (1-α ).For the leading bogie, oscillatory instability occurs if α <1, but not if α >1. For the trailing bogie, oscillatory instability occurs if α >1, but not if α <1. The mechanics of the instability is illustrated by examination of the case where kb = 0 and both ka and kc are made very large, which implies that -(ρ/h)ϕ1 + ϕ2 + ϕ4 = 0

ϕ1 - h ϕ4 = 0

(2) (3)

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202

Therefore defining new generalised coordinates θ1 and θ2 so that

or

ϕ = Τθ

(5)

and performing the transformation of equations (3) to (6) using (4.2.5), the equations of motion at low speeds become fsθ1/V - fθ 2 = 0 (fλl/r)θ1 + [f{(l2 + h2 + l2(ρ - 1)2}s/V + fh(1 - ρ )(1 - α )]θ2 = 0

(6) (7)

The characteristic equation then reduces to the quadratic {l2 + h2 + l2(ρ - 1)2}D2 + h(1 - ρ )(1 - α )D + (λl/r) = 0

(8)

Equations (4) and (5) indicate that if ρ =1 the bogie behaves as a rigid two-axle vehicle. Note the similarity of the frequency of the steering oscillation 2 2 2 2 1/2 ω = (λl/r)V/{l + h + l (ρ - 1) }

(9)

to that of a rigid bogie with the same wheelbase, ω = (λl/r)V/(l2 + h2 )1/2. Thus the mechanism of the oscillatory instability lies in the modification of this oscillation by the action of the steering linkage. The oscillation is illustrated in Figure 6.18.

Figure 6.18 Mode shape of low speed oscillatory instability.

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203

Figure 6.19 Stability diagrams showing the bifurcation speed VB (m/s) for the body-steered vehicle with separately steered wheelsets of Figure 6.11 as a function of ka (107 MNm) and kb (107 MNm). Complete vehicle (left hand), trailing bogie with car body fixed (centre) and leading bogie with car body fixed (right hand). Coned wheels, λ = 0.05.

Figure 6.19 gives the stability boundaries as ka and kb are varied for the case where kc = 0 as given by the solution of the equations of motion for the complete vehicle with coned wheels and λ = 0.05, the trailing bogie with car body fixed and the leading bogie with car body fixed. It can be seen that the stability diagram for the complete vehicle is closely approximated by the superposition of the diagrams for the bogies with car body fixed in accordance with the discussion in Section 7. The low speed oscillatory instability just discussed occurs only for very large values of ka and kb for ordinary values of the system parameters, but static instabilities do occur within the practical range of ka and kb. A close approximation to these results follows from equations (8.8) because for static stability p0 > 0 or 4f2α + kakb/h2 + kakc + kbkc ± f(kbσ - kaρ)(1 - α )/h > 0

(10)

Assuming the usual case where α <1 , examination of equation (10) shows that • for the case where the outer wheelset is steered (kb = 0) the trailing bogie is statically stable for all values of ka and kc and the leading bogie is statically unstable for sufficiently large values of ka and sufficiently small values of kc. Instability occurs for kc = 0 if ka > 4fhα /ρ ( 1 - α ) and for ka very large if kc < fρ ( 1 - α )/h. if the inner wheelset is steered the leading bogie is statically stable for all • values of kb and kc and the trailing bogie is statically unstable for sufficiently large values of kb and sufficiently small values of kc. Instability occurs for kc = 0 if kb > 4fhα /σ ( 1 - α ) and for kb very large if kc < fσ ( 1 - α )/h. • if both wheelsets are steered, and kc = 0, the leading bogie is statically unstable for sufficiently large values of ka and for sufficiently small values of kb. Instability will occur for kb = 0 if ka > 4fhα /ρ ( 1 - α ) and for ka very large if kb < fρh( 1 - α ). The trailing bogie is statically unstable for sufficiently large values of kb and for sufficiently small values of ka. Instability occurs for ka = 0 if kb > 4fhα /σ ( 1 - α ) and for kb very large if ka < fσ h( 1 α ). In all cases low values of conicity promotes divergence and increasing conicity in-

204

RAIL VEHICLE DYNAMICS

creases static stability. The divergence boundaries shown in Figure 6.19 occur for values of the stiffnesses closely in accordance with the simple criteria discussed above. Note that these static instabilities all occur if ρ =1. They are the result of

Figure 6.20 Stability boundaries showing contours of the bifurcation speed VB for body steered bogies with wheelsets separately steered from an interwheelset member (Figure 6.14) with car body fixed: outer wheelset steered through stiffness ka with inter-wheelset shear connection kc, trailing bogie (a) and leading bogie (b); inner wheelset steered kb with inter-wheelset shear connection kc, trailing bogie (c) and leading bogie (d); both inner and outer wheelsets steered with no inter-wheelset shear connection, trailing bogie (e) and leading bogie (f). Coned wheels, λ = 0.05. D refers to divergence, O refers to low speed oscillatory instability.

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205

assymmetry rather than steering as such. For if ρ =1 then the bogie behaves as an unsymmetric vehicle. The stability of unsymmetric vehicles is discussed in Chapter 9 but the salient feature of the static instability is revealed by considering a wheelset mounted on a pivoted arm as discussed in Chapter 3. Inspection of equation (10) shows that static instability can be eliminated completely if the linkage stiffnesses ka and kb are chosen to be different such that kb/ka = ρ/σ. The form of these results is very similar to those obtained by Bell and Hedrick [21] for a bogie in which the relative angular displacement between the wheelsets was proportional to the relative motion between the bogie frame and car body, a configuration similar in some respects to that of Figure 6.10. The results given here may also be compared with those given by Smith and Anderson [25] for a bogie in which there is a fixed relationship between ka and kb. Figure 6.19 also shows contours of the bifurcation speed VB. The instability corresponds to a form of bogie hunting which is an inertia driven instability of the steering oscillation. It is similar to the instability of a conventional bogie, but as the linkage restraining the bogie can be made stiff without degrading curving performance, quite high critical speeds can be achieved when parameters are chosen appropriately. Figure 6.20 indicates the stability as a function of the stiffnesses ki for the leading and trailing bogies for each of the cases where one of the three ki are zero. For a complete vehicle values of the ki must be chosen to give a margin of stability for both leading and trailing bogies. Moreover, as a range of conicities and creep coefficients must be considered in practice, values of the stiffnesses ki must be chosen accordingly. Figure 6.21 shows an example of the effect of varying equivalent conicity and creep coefficient on stability. In this case, the usual bogie instability is promoted by large conicities and the instability of the leading bogie discussed above is promoted by low conicities and creep coefficients. The result is a restricted range of speeds for stable operation unless the range of conicities is restricted.

Figure 6.21 Stability of the body-steered vehicle of Figure 6.11 showing the effect of varying conicity and creep coefficient on the bifurcation speed VB (m/s) as ka (107 MNm) and kb (107 MNm) are varied. f/2 indicates that f11, f22 and f23 are halved. ky = 6 MN/m.

206

RAIL VEHICLE DYNAMICS

In general, detailed optimisation of the parameters is necessary considering not only the linkage stiffnesses but the lateral primary suspension stiffness. Whilst it has been shown that arrangements providing only two of the stiffnesses ka, kb and kc are sufficient it is possible that schemes in which all three stiffnesses are implemented could provide better performance. More generally, the approach followed in this Chapter makes it possible to consider many possible configurations on a common basis. Though there are many disposable parameters theoretically available, only a limited number are needed to derive practical configurations which are relatively simple and which improve the resolution of the conflict between steering and stability. Full assessment of a given design must depend on design detail and performance in sharp curves, a situation in which these schemes are likely to be applied, obviously requires a nonlinear analysis. Furthermore, it has been shown that the introduction of more refined suspension arrangements results in more forms of potential static and dynamic stability.

References 1. Lewis, M.J.T.:Early Wooden Railways. Routledge and Kegan Paul, London, 1974, p. 291. 2. Stover, J.F.: American Railroads. University of Chicago Press, Chicago, 1961, p. 25. 3. White, J.H.: A History of the American Locomotive. The John Hopkins Press, Baltimore, 1968, pp. 169-174. 4. Ahrons, E.L.: The British Steam Railway Locomotive 1825-1925. The Locomotive Publishing Co., London, 1927, p. 62. 5. White, J.H.: The American Railroad Passenger Car. The John Hopkins Press, Baltimore, 1978, p. 497. 6. Ahrons, E.L. p. 157. 7. Matsudaira, T.: Hunting problem of high-speed railway vehicles with special reference to bogie design for the New Tokaido Line. Inst. Mech. Eng. Proc. 180, Part 3F (1965), pp. 58-66. 8. Sauvage, G.: Running quality at high speeds. In: O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, Sweden, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 496-508. 9. Evans, J.R.: The modelling of railway passenger vehicles. In: G. Sauvage (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 12th IAVSD Symposium, Linkoping, Sweden, August 1991. Swets and Zeitlinger Publishers, Lisse, 1992, pp. 144-156.

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10. Eickhoff, B.M., Evans, J.R. and Minnis, A.J.: A review of modelling methods for railway vehicle suspension components. Vehicle System Dynamics 24 (1995), pp. 469496. 11. Liechty, R.: Das Bogenlaufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934. 12. Smith, R.E.: Steering rail vehicle axles − a historical review. Proc. CSME Canadian Engineering Centennial Convention, 1987. 13. Liechty, R.: Studie uber die Spurfuhrung von Eisenbahnfahrzeugen. Schweizer Archiv f. Angewandte Wissenschaft und Technik, 3 (1937), pp. 81-100. 14. Liechty, R.: Die Bewegungen der Eisenbahnfahrzeuge auf den schienen und die dabei auftretenden Kräfte. Elektrische Bahnen, 16 (1940), pp. 17-27. 15. Schwanck, U.: Wheelset steering for bogies of railway vehicles. Rail Engineering International 4 (1974), pp. 352-359. 16. Wickens, A.H.: Steering and dynamic stability of railway vehicles. Vehicle System Dynamics, 5 (1975), pp. 15-46. 17. Wickens, A.H.: Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics, 7 (1979), pp. 33-48. 18. de Pater, A.D.: Optimal design of a railway vehicle with regard to cant deficiency forces and stability behaviour. Delft University of Technology Lab. for Eng. Mech. Report 751, 1984. 19. de Pater, A.D.: Optimal design of railway vehicles. Ingenieur-Archiv 57 (1987), pp. 25-38. 20. Topham, W.L.: Methods of reducing flange wear on diesel and electric locomotives. J. Inst. Loco. Engineers 49 (1959), pp. 771-825. 21. Bell, C.E. and Hedrick, J.K.: Forced steering of rail vehicles: stability and curving mechanics. Vehicle System Dynamics 10 (1981), pp. 357-385. 22. Gilmore, D.C.: The application of linear modelling to the development of a light steerable transit truck. In: A.H. Wickens (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 7th IAVSD Symposium, Cambridge, September 1981. Swets and Zeitlinger Publishers, Lisse, 1982, pp. 371-384. 23. Fortin, J.A. and Anderson, R.J.: Steady-state and dynamic predictions of the curving performance of forced-steering rail vehicles. In: J.K. Hedrick (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass., August 1983. Swets and Zeitlinger Publishers, Lisse, 1984, pp. 179-192.

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24. Fortin, J.A.C., Anderson, R.J. and Gilmore, D.C.: Validation of a computer simulation forced-steering rail vehicles. In: O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 100-111. 25. Smith, R.E and Anderson, R.J.: Characteristics of guided-steering railway trucks. Vehicle System Dynamics, 17 (1988), pp. 1-36. 26. Anderson, R.J., Fortin, C.: Low conicity instabilities in forced-steering railway vehicles. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 17-28. 27. Smith, R.E.: Forced-steered truck and vehicle dynamic modes-resonance effects due to car geometry. In: A. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 423-424. 28. Smith, R.E.: Dynamic characteristics of steered railway vehicles and implications for design. Vehicle System Dynamics, 18 (1989), pp. 45-69. 29. Weeks, R.: The design and testing of a bogie with a mechanical steering linkage. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 497508. 30. Li, W.: The dynamics of perfect steering bogie vehicles and its improvement with a reconfigurable mechanism. Doctoral Dissertation, Loughborough University, 1995. 31. Shen, Z.Y., Yan, J.M., Zen, J., and Liu, J.X.: Dynamical behaviour of a forced-steering three-piece freight car truck. In: M. Apetaur (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 10th IAVSD Symposium, Prague, August 1987. Swets and Zeitlinger Publishers, Lisse, 1988, pp. 407-418. 32. Smith, R.E.: Multiaxle steered articulated railway vehicle with compensation for transitional spirals. U. S. Patent Application 157565, 1989. 33. Scales, B.T.: Behaviour of bogies on curves, Railway Engineering, December, 1972. 34. List, H.A.: Means for improving the steering behaviour of railway vehicles. Annual Meeting of the Transportation Research Board, Washington, D. C. January, 1976.

7 The Three-Axle Vehicle 7.1 Introduction Various forms of three-axle vehicle have been used widely in the past. In most of these designs the wheelsets were connected to the car body by a conventional suspension similar to that used in two-axle vehicles. Negotiation of curved track was catered for by allowing greater flexibility or clearances for the central wheelset. However, there is also a long history of inventions which attempt to ensure that wheelsets are steered so that they adopt a more or less radial position on curves. It was argued that three axles, connected by suitable linkages, would assume a radial position on curves and then re-align themselves correctly on straight track. As will be shown below, a wide range of new potential instabilities are introduced. In fact, the three-axle configuration is also important because it gives considerable insight into the dynamic behaviour of articulated vehicles discussed in Chapter 8. Three-axle vehicles were in use from an early date. According to Liechty [1] a three-axle vehicle in which the lateral displacement of the central axle steered the outer axles through a linkage was tried out on the Linz-Budweis railway in 1826. Germain patented a design in 1837 in which radial steering was provided [2], and in 1844 Themor built a similar vehicle which was operated for some time [1]. Fidler also patented a similar arrangement in 1868 [3], Figure 7.1(a). In these last three schemes the outer wheelsets were pivoted to the car body. In 1889, Robinson’s arrangement [4] introduced the refinements of guides for the central wheelset, the body pivots was placed slightly inboard of the outer wheelsets, and the central wheelset had a much smaller radius than the outer wheelsets. Faye’s 1898 patent [5] removed the guides for the central wheelset in order to avoid reported difficulties with Robinson’s design on reverse curves. There were, of course, many different ways of providing inter-wheelset steering, such as complex linkages, and this is exemplified by the variety of designs produced since. Fidler introduced direct shear connection between the outer wheelsets in 1868, [3]. The central wheelset was mounted without lateral freedom in the car body, Figure 7.1(b). A similar arrangement was invented by Grover in 1880 [6]. All these developments were based on very simple ideas about the mechanics of vehicles in curves. In this Chapter, the basic instabilities of three-axle vehicles with a single car body will be considered, and how they are related to the natural steering properties of the three-axle vehicle. Particular emphasis will be given to the various

210

RAIL VEHICLE DYNAMICS

Figure 7.1 Three-axle vehicles with steered wheelsets due to Fidler (a) outer wheelsets pivoted on car body (b) central wheelset pivoted on car body, direct shear connection between outer wheelsets.

possibilities for the connections between the wheelsets and the car body in order to meet the conflicting requirements of stability and curving. The three-axle vehicle was first examined in this context in a series of papers [7]-[14]. A similar approach, though with slightly different assumptions, has been followed by de Pater [15, 16] and Keizer [17].

7.2 Steering and Stability of Three Axle Vehicles Consider the idealised three-axle vehicle shown in Figure 7.2. As the rotation of the wheels cannot generate any forces in the suspension consider the set of nine generalised coordinates q = [ y1 ψ1 y2 ψ 2 yb φb ψb y3 ψ 3 ]T

(1)

as indicated in Figure 7.2. The conditions governing steering and stability of multi-axle vehicles have been discussed in Section 6.4 and it can be seen that the criterion expressed by equation (6.4.15) can be satisfied by a three-axle vehicle. As in the case of the bogie vehicle, the total number of disposable parameters is very large. Fully generic schemes could

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211

(a)

ψ3

y3

ψb

yb

ψ2

y2

ψ1

y1

2h*

(b)

ϕb

yb

Figure 7.2 Generalised coordinates for three-axle vehicle.

be the starting point for an analysis of the three-axle vehicle. However, it is more important to derive the simplest arrangements which are capable of both perfect steering and stability. The main thrust will be that the vehicle should be made as flexible as possible in plan view, consistent with the need for stability. Noting that fully generic arrangements may be useful in generating design variants, the further development of the argument is based on a simpler form of arrangement shown in Figure 7.3. Firstly, consider the case where all wheelsets are similar and each wheelset is connected to the car body in the conventional manner. Then, the element stiffness and compatibility matrices are k = diag [ ky1 kφ1 kψ1 ky2 kφ2 kψ2 ky1 ⎡1 ⎢ ⎢0 ⎢0 ⎢ ⎢0 a = ⎢0 ⎢ ⎢0 ⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

0 0 1 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0

0 − 1 d1 0 0 1 0 0 0 0 −1 d4 0 0 1 1 0 0 0 − 1 d1 0 0 1 0 0 0

− 2h * 0 −1 0 0 −1 2h * 0 1

kφ1 kψ1 ]

(2)

0 0⎤ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0⎥ 0 0⎥ ⎥ 0 0⎥ ⎥ 1 0⎥ 0 0⎥ ⎥ 0 − 1⎥⎦

(3)

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RAIL VEHICLE DYNAMICS

2h* ky1/2

kψ2/2

ky2/2

kψ 1

ky1/2 k ψ1

kyd kt kyd

ky1/2

ky2/2

kψ2/2

ky1/2

h1 Figure 7.3 Schematic showing suspension stiffnesses for three-axle vehicle.

The solution of equations (6.4.10) will initially be discussed for this simplified generic arrangement. Equations (6.4.10) become ky1(y1 - yb + d1ϕb - 2h*ψb ) = 0

(4)

kϕ1 ϕb = 0

(5)

kψ1( ψ1 - ψb ) = 0

(6)

ky2(- yb + d4ϕc + y2 ) = 0

(7)

kϕ2 ϕb = 0

(8)

kψ2( ψb - ψ2 ) = 0

(9)

ky1( y3 - yb + d1ϕb + 2h*ψb ) = 0

(10)

kϕ1 ϕb = 0

(11)

kψ1(ψ3 - ψb) = 0

(12)

From equations (5), (8) and (11) ϕb = 0. From equations (6), (9), (12), either ψ1 = ψb or kψ1 = 0

(13)

either ψ2 = ψb or kψ2 = 0

(14)

either ψ3 = ψb or kψ1 = 0

(15)

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213

Figure 7.4 Geometry of two alternative configurations on uniform curve: top, ky1 = 0 and yb = y2; bottom, ky2 = 0 and yb = y1.

For steering without lateral or longitudinal creep on a uniform curve, within the assumptions of Section 6.4, and relative to the unstrained vehicle y1 = y3 = - lr0/λR0 + 2h*2/R0

(16)

y2 = - lr0/λR0

(17)

ψ1 = -ψ3 = 2h*/R0

(18)

ψ2 =

(19)

so that from (19) and (14) ψb = 0 and the value of kψ2 can be arbitrarily chosen. From (4), (7) and (10), either ky2 = 0 and yb = y1 = y3 or ky1 = 0 and yb = y2. Figure 7.4 indicates the geometry for each of these configurations in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the car body takes up the positions connecting the wheelsets without strain of the suspension elements. These two configurations satisfy the conditions for steering so it is now necessary to consider stability. For the first configuration, in which ky2 = 0 and ky1, kφ1, kφ2, and kψ2 are non-zero, and the degeneracy of E is 5, and the stability criterion, equation (6.4.15), is not satisfied and there are two pairs of imaginary eigenvalues at low speeds corresponding to undamped kinematic wheelset oscillations. At least two additional stiffnesses are required to reduce the degeneracy to three as required by the stability criterion. For the second configuration, in which ky1 = 0 and ky2, kφ1, kφ2, and kψ2 are nonzero, the degeneracy of E is 6, and again the stability criterion, equation (6.4.15), is not satisfied and there are three pairs of imaginary eigenvalues at low speeds corresponding to three undamped kinematic wheelset oscillations. In this case at least

214

RAIL VEHICLE DYNAMICS

three additional stiffnesses are required to reduce the degeneracy to three as required by the stability criterion. As in the case of the bogie vehicle, for these additional stiffnesses, the corresponding rows in the compatibility matrix a must satisfy the rigid body conditions expressed by equations (6.4.5) and (6.4.6), and the steering condition expressed by equation (6.4.10). In addition they must be linearly independent from the existing rows of a. (Of course, the ki can be repeated as long as the corresponding row in a is linearly independent, thus increasing the rank of a ). The sub-matrices of k and a relating to the additional elastic elements will be denoted by ks and as for convenience. Figure 7.3 shows that further elastic connections corresponding to direct shear connections between adjacent wheelsets (of stiffness kyd) and the outer wheelsets (of stiffness kt) have been introduced. As in the case of the body-steered bogie, in schemes of the early period, pivots were used to provide lateral restraint and yaw freedom. For example, in Robinson’s scheme ky1 and kψ2 were very large though the principles discussed above remain valid. As mentioned above, such a vehicle would be unstable at low speeds. The sub-matrices of ks and as relating to these additional elastic elements are (20) ks = diag [ kyd kyd kt ] ⎡1 − h1 a = ⎢⎢0 0 ⎢⎣1 − 2h * s

− 1 − h2 1 − h2 0 0

0 0 0 0 0 ⎤ 0 0 0 − 1 − h1 ⎥⎥ 0 0 0 − 1 − 2h * ⎥⎦

(21)

where h2 = 2h* - h1. The condition for stability at low speeds has been shown, in Section 6.4, to depend on the stiffness matrix E* involving only the wheelset coordinates. In the present case the number of disposable parameters will be (R2 - 4R + 3)/4 = 4 and so E* may be expressed in terms of the given elements e13*, e15*, e16*, and e22* and then e11*= - e13*- e15* e12*= e13*h*+ e16* e14*= 4e15*h* - 2e16* + e13*h* e23*= - e13*h* e24*= - 2e22* - 3e13*h*2 - 4e16*h* e25*= - e16* e26*= e22* + e13*h*2 e33*= - 2e13*

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215

e34*= 0 e35*= e13*

(22)

e36*= e13*h* e44*= 4e22* + 16e16*h* - 16e15*h*2 + 2e13*h*2 e45*= - e14* e46*= e24* e55*= e11* e56*= - e12* e66*= e22* For the enhanced configuration, E* has the following four independent elements e13* = - kyd e15* = - kt - ky1kψ2/2(8ky1h*2 + kψ2)

(23)

e16* = - 2kth* e22* = h12kyd + 4kth*2 From equation (6.4.15), for stability, the degeneracy of E* must be not more than three and this is satisfied if either (a) ky1 = 0 and kyd, kt, ky2 and kψ2 are non-zero, (b) ky2 = kt = 0 and ky1, kψ2 and kt are non-zero, or (c) ky2 = kψ2 = 0 and ky1, kyd and kt are non-zero.

7.3 Steering with Unequal Conicities The above discussion of curving requires modification to account for unequal conicities of the wheelsets. There is a geometric requirement that a bending displacement enabling all three wheelsets to take up radial alignment is compatible with the lateral displacements required, by their various equivalent conicities, for pure rolling. Figure 7.5 shows the geometry when such a vehicle negotiates a curve. For zero longitudinal and lateral creep, the wheelsets must adopt a radial position, and, assuming for simplicity purely coned wheels, move outwards lr0/λR0 (for the outer wheelsets) and lr1/λ1R0 (for the central wheelset). If the radial displacements are small compared with the radius of the curve, the geometry of Figure 7.5 then yields the position of an effective pivot, measured from the central points between the

216

RAIL VEHICLE DYNAMICS

lr1/R0λ1

h1 lr0/R0λ

Figure 7.5 Geometry of three-axle vehicle for zero lateral and longitudinal creep on uniform curve when conicities are not uniform.

wheelsets, necessary for perfect steering h*- h1= l( r0/λ - r1/λ1 )/2h*

(1)

The possible range of configurations which satisfy this equation is constrained by two practical considerations. Firstly, large values of conicity are not achievable in practice. Secondly, assuming that the difference in conicity between outer and central wheelsets is achieved by a difference in wheel diameters, it is unlikely that in a good mechanical design the ratio of diameters could exceed about two, or exceptionally three. Figure 7.6 then summarises the possible range of configurations which steer perfectly on curves for the case where h* = 1.25 m. Lines A and B give the relationship between λ1 and λ for the virtual pivot position being at the central and outer wheelsets respectively. For longer wheelbases the position of the effective pivot lies nearer the centre of the wheelbase. The lines C and D represent the limits of a 2:1 ratio in conicity between central and outer wheelsets. A smaller ratio in conicity leads to a pivot position between the wheelsets as shown at E, and large ratios in conicity could require pivot positions outside the wheelbase.

0.3

A

0.2

λ1l/r0

E C

0.1

B D

0

0

0.1

λ1l/r0

0.2

0.3

Figure 7.6 Relationship between conicities and virtual pivot position for negotiation of uniform curve with zero longitudinal and lateral creep. h* = 1.25 m.

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217

7.4 Stability of Vehicle with Uniform Conicity In order to discuss stability attention will be concentrated on option (c) of Section 2 in which ky2 = kψ2 = 0 and ky1 and the inter-wheelset stiffnesses kyd and kt are nonzero. Moreover in this Section, the case where h1 = h2 = h* and λ1 = λ will be considered, deferring discussion of non-uniform conicity to the following Section. It is convenient to consider, firstly, the behaviour at low speeds. As already discussed, at low speeds, the car body degrees of freedom can be eliminated and the equations of motion will be of the form of (6.4.20). Neglecting the small terms associated with the contact stiffness, and assuming for simplicity that all the creep coefficients are equal, the non-zero terms of B and C are B11 = B33 = B55 = 2f C12 = C34 = C56 = -2f

B22 = B44 = B66 = 2fl2

C21 = C65 = 2flλ/r0

C43 = 2flλ1/r1

and the characteristic equation becomes p6D6 + p5D5 + p4D4 + p3D3 + p2D2 + p1D + p0 = 0

(1)

where D = d/Vdt, α = λl/r0 and α1 = λ1l/r0 and p6 = 1 p5 = (l2 + 4h*2)kt/fl2 + (2l2 + h12 + h22)kyd/fl2 p4 = (2α + α1 )/ l2 + {2(l2 + 4h*2)(2l2 + h12 + h22) - (l2 + 2h*h1)2}kydkt/2f 2l 4 + {(2l2 + h12 + h22)2 - (l2 - h22)2}kyd2/4f 2l 4 p3 = (l2 + 4h*2)(α + α1)kt/f l4 + (α1l2 + 3αl2 + h12α1 + h12α + 2h22α)kyd/f l4 + (3l2 + 8h*2)(3l2 + h12)kyd2 kth22/4f 3l 6 p2 = α(α + 2α1)/l4 + {α1 ( l2 + h12)2 + 8h22 l2α + 2α( l2 + h12)( l2 + h22) - h12 l2(α - 1)2 - 2h1h2 l2(α - 1)(1 - α1)}kyd2/4f 2l 6 + [(2h* - αh1)(2h*α - h1)l2 + 2(l2 + 4h*2){α1(l2 + h12 ) + α(l2 + h22)} - α1(l2 + 2h*h1)2]kydkt/2f 2l 6 p1 = α(α l2 + α1 l2 + h12α1 + h22α)kyd/f l6 + αα1(l2 + 4h*2)kt/f l 6 + {24h*2α + 4h*h1α(α1 - α) + (α1 + 2α)(3l2 + h12)}kyd2kth22/4f 3l 6 p0 = α2α1/ l6 + {4α2h22 - α1h12(α - 1)2 - 2h1h2α(α - 1)(1 - α1)}kyd2/4f 2l 6 + α1(2h* - h1α)(2h*α - h1)kydkt/2f 2l 6

218

RAIL VEHICLE DYNAMICS

Figure 7.7 Root locus for three-axle vehicle at low speeds as a factor k on the stiffnesses kt and kyd is varied, for the parameters of Table 7.1.

Figure 7.7 shows the variation of the eigenvalues with k, a factor applied to kyd and kt. When k = 0 there are three undamped oscillations A, B and C at wheelset kinematic frequency ω = V(λ/lr0)1/2 as discussed above. Introduction of stiffness results in damping of these oscillations, the real part of the eigenvalue being initially proportional to k. One of these oscillations can be identified with a steering or bending oscillation A of the complete vehicle, in which there is very little interwheelset shear. As k is varied from zero to very large values the steering oscillation remains lightly damped. As k increases, the frequency of the steering oscillation decreases, tending towards a constant value for large values of k. The other two oscillations B and C involve much more shear of the inter-wheelset structure and are heavily damped, splitting into four subsidences B1, B2, C1 and C2 as k increases. The damping of one of these subsidences, C2, eventually decreases as k is increased so that at large values of k it vanishes. An important feature of these results is the small damping in the steering oscill-

THE THREE-AXLE VEHICLE

219

Table 7.1 Example parameters for three-axle vehicle. suspension ky1 = 0.23 MN/m cy1 = 100 kNs/m kyd = 0.70 MN/m creep coefficients f11 = 8.83 MN vehicle geometry h* = 4.125 m inertia m =1250 kg Izb = 269000 kgm2

kψ1 = kψ2 = 0 cψ1 = cψ2 = 0 kt = 0.20 MN/m

kφ1 = kφ2 = 1MNm cφ1 = cφ2 = 50 kNms ct = cyd = 0

f22 = 8.06 MN

f23 =17.7 kNm

W = 100.6 kN

d1 = d4 = 0.2 m

r0 = 0.45 m

l = 0.7452 m

I = 700 kgm2 Iy = 250 kgm2

mc=27000 kg

Ixc=32400 kgm2

ky2 = 0 dy2 = 0

ation A and in the subsidence C2 at large values of k. In fact, both oscillatory instabilities and divergence occur for certain values of the parameters, and the conditions for stability will now be discussed in detail. If k is very large, it is appropriate to neglect terms in k of order less than two in (1) which then reduces to p4D4 + p3D3 + p2D2 + p1D + p0 = 0

(2)

where p4 = {2(l2 + 4h*2)(2l2 + h12 + h22) - (l2 + 2h*h1)2}kydkt/2f 2l 4 + {(2l2 + h12 + h22)2 - (l2 - h22)2}kyd2/4f 2l 4 p3 = (3l2 + 8h*2)(3l2 + h12)kyd2 kth22/4f 3l 6 p2 = {α1 ( l2 + h12)2 + 8h22 l2α + 2α( l2 + h12)( l2 + h22) - h12 l2(α - 1)2 - 2h1h2 l2(α - 1)(1 - α1)}kyd2/4f 2l 6 + [(2h* - αh1)(2h*α - h1)l2 + 2(l2 + 4h*2){α1(l2 + h12 ) + α(l2 + h22)} - α1(l2 + 2h*h1)2]kydkt/2f 2l 6 p1 = {24h*2α + 4h*h1α(α1 - α) + (α1 + 2α)(3l2 + h12)}kyd2kth22/4f 3l 6 p0 = {4α2h22 - α1h12(α - 1)2 - 2h1h2α(α - 1)(1 - α1)}kyd2/4f 2l 6 + α1(2h* - h1α)(2h*α - h1)kydkt/2f 2l 6 If α = α1 and h1 = h2 and k is taken to the limit then this reduces to p 3D 3 + p 1D = 0

(3)

the solutions of which consist of a purely imaginary root corresponding to an un-

220

RAIL VEHICLE DYNAMICS

(a)

(b)

Figure 7.8 Mode shapes for three-axle vehicle, kyd = kt = ∞ (a) corresponding to zero root (b) one cycle corresponding to si = iωI where ωI is given by equation (4).

damped oscillation frequency given by

ωI2 = 9αV2(l2 + 3h*2)/(8h*2 + 3l2)( h*2 + 3l2)

(4)

and a zero root corresponding to a mode of neutral stability. This zero root implies that the vehicle is capable of quasi-static misalignment as discussed in connection with two-axle vehicles, Section 4.3, and the corresponding mode shape is shown in Figure 7.8(a). The oscillation at frequency ωI is one in which the creep forces are in overall equilibrium even though the motion of individual wheelsets is not one of pure rolling. This is analogous to the behaviour of a rigid two-axle vehicle, Section 5.3. Figure 7.8(b) shows that the mode shape of this oscillation resembles the steering oscillation mentioned above. Instability at low speeds can occur either by the real part of a complex conjugate pair of eigenvalues becoming positive (oscillatory instability) or by a real eigenvalue becoming positive (divergence). As the instabilities occur for large values of the shear stiffnesses it is most instructive to use the perturbation method introduced in Section 3.3. The terms in the expressions for the coefficients which are of order k2 can be considered as small perturbations δpi of the initial values pi* which are of order k3. The corresponding perturbation in the ith eigenvalue then follows from equation (3.3.12). Thus, from (2), the initial values pi* are

THE THREE-AXLE VEHICLE

221

p 4* = 0 p3* = (3l2 + 8h*2)(3l2 + h*2)kyd2 kth*2/4f 3l 6 p 2* = 0

(5)

p1* = {24h*2α + 3α(3l2 + h*2)}kyd2kth*2/4f 3l 6 p 0* = 0 and

δp4 = (3l4 + 16h*2l2 + 12h*4)kydkt/2f 2l 4 + (3l2 + h*2)(l2 + 3h*2)kyd 2/4f 2l 4 δ p3 = 0 δp2 = {α (3l2 + h*2)(l2 + 3h*2) + h*2l2(1 + α)2}kyd2/4f 2l 6 + {(2 - α)(2α - 1)h*2l2 + α(3l4+ 16h*2l2 + 12h*4)}kydkt/2f 2l 6

(6)

δ p1 = 0 δp0 = (1 + α)2αh*2kyd2/4f2l6 + (2 - α)(2α - 1)αh*2kydkt/2f 2l 6 The approximations for the eigenvalues found using equation (3.3.12) are sI = µΙ ± ωI where ωI is given by equation (3) and

µΙ = - V(δp4 ωI4 - δp2 ωI2 + δp0 )/(- 3p3 ωI2 + p1 )

(7)

and sII = µΙΙ = - Vp0/p1. Equation (7) can be expressed in the form

µΙ = Vp4 (ωI2 - ω12)( ωI2 - ω22)/2p1

(8)

where ω1 and ω2 are the roots of

δp4s4 + δp2s2 + δp0 = 0

(9)

which are given by

ω12 = V2{(1 + α)2αh*2kyd + 2(2 - α)(2α - 1)αh*2kt}/ {2(3l4 + 16h*2l2 + 12h*4)kt + (3l2 + h*2)(l2 + 3h*2)kyd} (10)

ω22 = V2α

222

RAIL VEHICLE DYNAMICS

Equation (7) shows that for stability either

ω1 < ωI < ω2

or

ω1 > ωI > ω2

(11)

This is identical to Leonhard’s criterion [18] and is, of course fully equivalent to Routh’s criterion; it provides a simple way of discussing the stability of the system. From (4) and (10) it can be seen that ωI < ω2 for all values of h* and l, independent of α. Therefore for stability ω1 < ωI which for given kyd and kt is a quadratic in α. For practical values of the parameters this has solutions for small values of α, which can occur in practice, and very large values which are not of interest. It follows from (4) and (10) that for very small values of α stability cannot be achieved if kyd > 4kt. For large values of kyd and h* >> l the condition that ω1 < ωI implies that oscillatory instability will only occur for low values of conicity, such as α < 1/10. The mode shape of the unstable mode is similar to that shown in Figure 7.8(a) and thus instability of the steering oscillation which occurs for large values of kyd and kt is promoted by low conicity, inter-wheelset flexibility, and short wheelbase. Even where it occurs, it should be noted that the real part of the corresponding pair of eigenvalues, though positive, remains small. The criterion for static stability is p0 > 0 or

α3 + α(1+ α)2kyd2 h*2/4f2 + α(2 -α )(2α - 1)ktkyd h*2/2f 2 > 0

(12)

Divergence can only occur if α <1/2 (or α >2 which is beyond the practical range of values), and for kt > kyd/4. As divergence only occurs when α is small (12) reduces to kyd/ kt > 2(2 - α)(1 - 2α )/ (1 + α)2

(13)

independent of wheelbase and creep coefficient. The mode of neutral static stability which exists for very large values of kyd and kt is destabilised by the introduction of flexibility between the wheelsets. It is now possible to consider the influence of speed on the eigenvalues and this is illustrated by the root locus shown in Figure 7.9. As speed is increased the eigenvalues corresponding to branches A, B and C1and C2 are initially proportional to speed and are closely equal to the solutions of (1). For the chosen set of parameters there is stability at low speeds. Similar to the case of the two-axle vehicle, the eigenvalues associated with the vehicle body modes, D1, D2 and D3 are substantially independent of speed and are closely equal to the wheels-fixed eigenvalues as illustrated in Table 7.2. For the three-axle vehicle, like other configurations, at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension, there is the possibility of body instabilities. As in the case of the two-axle vehicle, stability can be obtained by choosing suitable values for ky1, ky2, dy1, and dy2. As a result, for the chosen set of parameters there is little interaction between the wheelset modes and the modes involving the car body oscillating on the suspension so body instabilities do not arise. In addition there are the usual 6 relatively large eigenvalues corresponding to the wheelset subsidences in lateral translation and yaw,

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223

Figure 7.9 Root locus as speed is varied for three-axle vehicle with parameters of Table 7.1 except that kt = kyd = 0.1 MN/m. Table 7.2 Eigenvalues for three-axle vehicle with parameters of Table 7.1 except that kt = kyd = 0.1 MN/m.

1 2 3,4 5 6 7,8 9 10 11,12 13 14 15,16 17,18

At Speed, V = 10 m/s -1290 -1395 -1377 ± 6.066i -1377 -1397 -2.406 ± 9.254i -45.30 -2.415 -3.468 ± 2.019i -14.18 -2.500 -1.796 ± 5.722i -0.7354 ± 5.770i

Wheels - Fixed

-2.407 ± 9.244i -48.19 -2.415 -3.735 ± 1.828i

Label Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Car body upper sway Car body yaw Car body yaw Car body lower sway Shear subsidence C1 Shear subsidence C2 Shear oscillation B Steering oscillation A

224

RAIL VEHICLE DYNAMICS

two of which appear at E and X in Figure 7.9. As the speed is increased, the steering oscillation A loses stability at a bifurcation speed VB = 48.7 m/s. Figure 7.10 summarises stability as a function of kyd and kt, for the extreme cases of λ = 0.5 and λ = 0.05 with the creep coefficients halved. Analogous to the case of the two-axle vehicle, the behaviour is different depending on the value of kyd/2flα1/2 and kt/2flα1/2. The ‘stiffness to ground’ provided by the yaw stiffness for the twoaxle vehicle is provided by kyd and kt for the three axle vehicle. As already discussed, stability depends on non-zero values of kyd and kt; if kyd = 0, there will be two undamped eigenvalues at wheelset kinematic frequency at low speed, one mode involving motion of the central wheelset, and the other mode involving motion of the outer wheelsets such that there is zero shear between the outer wheelsets. If kt = 0 there will be one undamped eigenvalue at wheelset kinematic frequency at low speed, the motion involving zero shear between adjacent wheelsets. As mentioned above, a suitable choice of parameters has been made so that the body instabilities do not occur. If both kyd and kt are non-zero, examination of the root locus has shown that it is the steering oscillation, branch A, that becomes unstable above the bifurcation speed VB. VB is increased if kyd and kt are increased. This region is labelled O1 in Figure 7.10. This instability is analogous to wheelset instability in which the inertia forces induced by the steering oscillation drive the oscilla-

λ = 0.5

λ = 0.05 f x 0.5

150

150 V (m/s)

V (m/s) 100

O1

50

0 150 V (m/s)

50

S 0.2

kyd (MN/m)

S 0.8

1

O1

0

0

100

S 0.2

kt (MN/m) 0.8

50

10

D

S

O1

150 V (m/s)

100 50

O2

100

0

0.2

kyd (MN/m) 0.8

1

O1 O2

S

D 0.2 kt (MN/m) 0.8

1

Figure 7.10 Joint effect of conicity and creep coefficient on stability as a function of kt and kyd for three-axle vehicle with the parameters of Table 7.1. D = divergence, O = oscillatory instability and S = stable.

THE THREE-AXLE VEHICLE

225

tion, there being little involvement of the car body. Low conicity and creep coefficients, stabilise the instability O1, but a separate instability O2 is introduced where the critical speed reduces with increasing kyd and kt. The mode shape of this oscillation is similar to that of the low speed steering oscillation shown in Figure 7.8(b) and involves significant yaw of the car body. In addition, the combination of low conicity and creep coefficient introduces divergence over a range of small values of kyd and above a limiting value of kt in accordance with equation (13). For very large values of kyd and kt, beyond the range of values in Figure 7.10, there is a further region of oscillatory instability which occurs at low speeds in accordance with equation (11). In the light of these results, it might be thought that the selection of values for kyd and kt to obtain the maximum critical speed for a range of conicity and creep coefficient is a difficult compromise. However, it should be borne in mind that a number of the instabilities, both dynamic and static, discussed above are associated with eigenvalues which small in magnitude. This is particularly true when the system has low values of equivalent conicity. In these circumstances, judgement of parameter values should be informed by nonlinear response studies.

7.5 Stability with Unequal Conicities The preceding results apply to a configuration in which the wheelset conicities are uniform and the inter-wheelset shear structure is symmetric so that the shear spring kyd is located midway between the wheelsets. As discussed above, it is necessary to vary the spring position h1(measured from the leading wheelset) if the conicities are not uniform in order to achieve perfect steering. However, it is convenient to consider the influence of h1 and conicity on stability separately. Figure 7.11 shows the effects on stability of variations of h1, α and α1 for the vehicle with parameters of Table 7.1, based on the linearised equations of motion. Both oscillatory and static instabilities occur. From equation (4.1), for static stability p0 > 0 so in the case where α = α1 then

α2 + {4αh22 + h1(α - 1)2(2h2 - h1)}kyd2/4f 2 + (2h* - h1α)(2h*α - h1)kydkt/2f 2 > 0

(1)

Without discussing (1) in detail, it can be seen in general terms that movement of the pivot towards the central wheelset and increasing kt all promote divergence. This is illustrated by the results in Figure 7.11(c). Two regions of oscillatory instability occur as h1 is varied and are shown in Figure 7.11(c). Movement of the position of the spring kyd towards either the outer or inner wheelsets is destabilising and oscillatory instability occurs at all speeds when either h1 = 0 or h1 = 2h*. This is as expected as s = ± iV(λ/r0l)1/2 is then a solution of the characteristic equation for all values of λ, λ1, kyd and kt, an undamped kinematic oscillation occurring because of the lack of elastic restraint.

226

RAIL VEHICLE DYNAMICS (a)

300 V (m/s)

0

O

V (m/s)

O

200 100

(b)

300

O

200 D

S

0.1

100

0.3

λ 0.5

0

0.1

0.3 λ 0.5 1

(c)

300 V (m/s)

S

O

O

200 100

0

S 4

D

h1 (m)

8

Figure 7.11 The influence of unequal conicities and effective pivot point on stability for the three-axle vehicle with parameters of Table 7.1. D = divergence; O = oscillatory instability; S = stable.

As indicated in Figure 7.11(c) the steering oscillation of the example three-axle vehicle, with h3 = 3 m., loses its stability at small amplitudes at V =168.4 m/s. The eigenvalues are given in Table 7.3 and the eigenvector corresponding to the mode in which µ = 0 is given in Table 7.4. This solution is a sub-critical bifurcation leading to a limit cycle in which the lateral displacement of all three wheelsets have a similar amplitude of 5 mm at V= 178 m/s, the details of which are shown in Figure 7.12. This shows the numerical solution of the nonlinear equations of motion for the vehicle with the parameters of Table 7.1 and the wheel rail geometry of Figure 2.2. Figures 7.12(a) and (b) indicate that the limit cycle is in the form of a steering oscillation very similar to that predicted by the small amplitude solution and there are many points of similarity with the corresponding results for the unrestrained bogie shown in Figure 5.21. The displacements of the car body are small. The frequency of the oscillation is 7.46 Hz and is closely equal to that given by the small amplitude eigenvalues. Figure 7.12(c) shows the variation of the rate of rotation of the leading wheelset which is about 0.1% of V/r0. This fluctuates at twice the frequency of the oscillation as a major component of the torque applied to the wheelset is the product of the longitudinal creep force and the rolling radius both of which are varying at the frequency of the oscillation. Figures 7.12(d) and (e) show the longitudinal and lateral creepages for the right hand wheel of the leading wheelset. As the variation in the wheelset rotational speed is small, reference to equation 2.5.13 shows that the largest

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227

Table 7.3 Eigenvalues for the three-axle vehicle of Table 7.1, h1 = 3 m, V = VB = 168.4 m/s. Eigenvalue(s-1)

No. 1,2 3.4 5 6 7,8 9 10,11 12 13,14 15,16 17 18

-41.04± 269.1i -38.29 ± 159.7i -176.7 -147.8 -36.03 ± 83.43i -79.52 ± 47.50i -46.98 -2.44 ± 9.32i -4.30 ± 3.68i -4.64 -2.42

Label Large wheelset roots Large wheelset roots Large wheelset root Large wheelset root Shear oscillation Shear subsidence Steering Car body yaw Car body upper sway Car body lower sway Shear subsidence Car body yaw

Table 7.4 Eigenvector for mode 10 of the isolated bogie of Table 7.1 , V = VB = 168.4 m/s.

y1 ψ1 y2 ψ2 yb φb ψb y2 ψ2

1

-0.2454 + 0.3839i -0.2901 - 0.8140i 0.1594 + 0.09210i -0.01508 + 0.001898i 0.002628 - 0.00006271i 0.07224 - 0.05877i -1.0462 - 0.19032i 0.4814 - 0.4148i

contribution to the longitudinal creep is made by the variation in rolling radius with lateral displacement and this is reflected in the waveform. Similarly, reference to equation 2.5.15 shows that the largest contribution to the lateral creep is made by the angle of yaw, as indicated by the waveform. Figure 7.12(f) shows the spin of the leading wheelset, and reference to equation 2.5.17 shows that the spin behaves like δr/r0 and so the waveform reflects the geometrical nonlinearity of the contact slope variation with lateral displacement. Figures 7.12(g) and (h) shows the longitudinal and lateral creep forces for the right hand wheel of the leading wheelset, and consideration of their resultant and the normal force indicate that slipping is occurring over part of the cycle of the oscillation. Figure 7.12(i) shows the variation of the normal force acting on the right hand wheel which is in phase with the lateral displacement and is relatively large in magnitude.

228

RAIL VEHICLE DYNAMICS 0.01

(a)

.005

yb

0

ψ1

(b)

0

-0.01 0 .005

y1 y2

y3

-.005 0.1 t (s) 0.2 0 .005

(d)

γ1r1

γ2r1

0

(c)

1

ψ3

-V/r0-Ω1

0

ψ2

ψb 0.1

0.2

(e)

0

-1 0 0.5 0

0.1

0.2

(f) ω3l1

ω3r1 -5 0 20

0.1

(g)

-.005 0.2 0 20

T1r1 (kN)

T2r1

0

0

-20

0

0.1

0.2

-20 0

0.1

(h)

0.2

-0.5 0 100

-T3r1

0.1

0.2

(i)

50

0.1

0.2

0

0

0.1

0.2

Figure 7.12 Displacements, creepages and forces in the limit cycle of three-axle vehicle of Table 7.1 (except that h3 = 3 m) at V = 178 m/s.

Figures 7.11(a) and (b) show the effect of variation of λ and λ1 on stability. For static stability p0 > 0 so if h = h1 then from equation (4.1)

α2α1 + (1 + α){α(1 + α1) + α - α1}kyd2h2/4f2 + α1(2 - α)(2α - 1)kydkth2/2f2 > 0

(2)

Reduction of λ promotes divergence as shown in Figure 7.11(a) In order to get a feel for the behaviour of the system, it is useful to consider the simple case of a vehicle in which kt = 0 and kyd is very large. Figure 7.13 shows the root locus as h1 and λ1 are varied. Solution of (4.1) for this case shows that there are two purely imaginary eigenvalues when α = α1 and h1 = h2. One of these, A in Figure 7.13, corresponds to kinematic oscillations of the wheelsets in which the phase between the wheelset motions is determined by the shear connection between the wheelsets so that the shear spring is unstressed. The other, B in Figure 7.13, refers to a steering oscillation in which creep occurs. The frequency of this steering oscillation, when α = α1 and h1 = h2, is given by

ω12 = (1 + α)2V2h*2/(3l2 + h*2)(l2 + 3h*2)

(3)

It can be seen that even if the conicity is zero, a steering oscillation occurs in which the wavelength is solely determined by the geometry of the vehicle. Consider the variation of the conicities, but maintaining h1 = h2. As α1 is reduced in

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229

Figure 7.13 Root locus as h1 and α1 are varied for three-axle vehicle of Table 7.1 with α = 0.025 and kt =0.

value, the two eigenvalues approach each other and, at a critical value, frequency coalescence takes place, further reductions in α1 leading to eigenvalues of the form ±µ ± iω. This form of instability may be termed ‘flutter’ as it is closely related to instabilities experienced with other non-conservative systems such as those arising in Aeroelasticity [20]. It can be shown [11] that the equations of motion of the system with kt = 0 and kyd very large are of the same form as those of undamped aeroelastic systems which have been extensively studied [20]. Increasing α1 beyond the value α1 = α results in the natural frequencies diverging in value. The frequency of the steering oscillation, branch B, reduces in value until

α1 = 2α(1 + α )/(1 - 2α2 )

(4)

consistent with (3), when it becomes zero. Further reductions in the value of α1 result in eigenvalues of the form ±µ and divergence occurs. Now, in this simple case of a vehicle in which kt = 0 and kyd is very large, consider the variation of the position of the inter-wheelset shear spring kyd, but maintaining α = α1. Solution of (4.1) for this case shows that there are two purely imaginary eigenvalues. Irrespective of the position of the pivot, one of these corresponds to kinematic oscillations of the wheelsets, and the other refers to a steering oscillation. The frequency of this steering oscillation is given by

230

RAIL VEHICLE DYNAMICS ω2 =

V 2 {h2 2 (1 + α ) 2 − (1 − α ) 2 (h2 − h1 ) 2 } {(2l 2 + h1 2 + h2 2 ) 2 − (l 2 − h2 2 ) 2 }

(5)

When the effective pivots of the shear springs move towards the central axle, the frequency of the steering oscillation reduces in value until h1/2h = 2/(3 - α )

(6)

when it becomes zero. Further increases in h1 result in roots of the form ±µ and divergence occurs. As the pivots move toward the outer axles, the frequency of the steering oscillation reaches a maximum and then decreases in value until it vanishes at a point outside the wheelbase when h1/2h = -2α /(1 - 3α )

(7)

Further outward movement of the pivots result in roots of the form ±µ and divergence occurs. These results show that for the three-axle vehicle there are many potential modes of instability. However, the lateral stiffness and damping between the car body and wheelsets may be chosen appropriately to eliminate the body instabilities, in a similar way to that prescribed for the two-axle vehicle. Then stability may be obtained up to high speeds providing that extreme values of conicity and inter-wheelset stiffnesses are avoided. The historical difficulties with the stability of three-axle vehicles with inter-wheelset linkages can be ascribed to the linkages either being deficient in that the criteria of equation (6.4.15) are not satisfied or that linkages were too stiff.

7.6 Dynamic Response In addition to problems with stability, experience with three-axle vehicles has shown that whilst steady-state curving may be improved, the form of the inter-wheelset connections may give rise to large steering errors on non-uniform curves [5]. Accordingly, it is important to consider the dynamic response of three-axle vehicles to track geometry and the following examples illustrate the principles involved. Figures 7.14 and 7.15 show the dynamic response of the three-axle vehicle of Table 7.1 (except that kyd = kt = 0.1 MN) to entry into a curve of radius R0 = 225 m, with a cubic parabolic transition of length L0 = 20 m as computed from the full nonlinear equations of motion. The vehicle speed V = 15 m/s and the coefficient of friction µ = 0.3. The track is canted so that the cant deficiency is zero throughout. The results may be compared with the results for the elastically restrained wheelset in Section 3.7 and the results for the two-axle vehicle in Section 4.6. Figure 7.14 shows the motion of the wheelsets which after an initial transient on the transition take up a radial attitude with an outwards lateral displacement closely equal to that necessary to negate longitudinal creep as given by equation (3.2.23). In the transient, the wheelsets attempt to follow the curve, and the steering errors are quite small.

THE THREE-AXLE VEHICLE

231

0.005 0

ψ1

ψ2

ψ3

y1

y2

y3

-0.005 0

2

t (s)

4

0

2

4

0

2

4

Figure 7.14 Dynamic response of the wheelsets of the three-axle vehicle with the parameters of Table 7.1 (except that kyd = kt = 0.1 MN) to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m. 2

5

ψ1 (mr)

0

T2r1 (kN)

0

-2

y1 (mm)

-4 -6 0

2

t (s)

4

T1r1 (kN)

10 0

-5 -10

T2l1 (kN) 0

2

4

zyW

10 0

T1l1 (kN)

-10 0

2

4

T2r1+ T2l1 -10 0

2

4

Figure 7.15 Dynamic response of the leading axle of the three-axle vehicle with the parameters of Table 7.1 (except that kyd = kt = 0.1 MN) to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m.

However, on very sharp curves flange contact readily takes place, and steering errors can be large. This effect can be pronounced on reverse curves as has been found in the past. Figure 7.15 shows the time history of the creep forces acting on the leading wheelset. As in the case of a single wheelset, once the steady-state motion is established, the lateral creep forces are mainly generated by spin and are largely cancelled out by the gravitational stiffness force, and the longitudinal creep forces are small. However, during the transition the deviations of the wheelsets from a radial attitude induce elastic forces generated by the inter-wheelset suspension which have to be

232

RAIL VEHICLE DYNAMICS

5 0

ψ1 (mr)

-5

-5

-10

y1 (mm) -10 0

T2r1 (kN)

0

2

t (s)

4

-15

T2l1 (kN) 0

2

4

20

T1r1 (kN)

10 0

zwW (kN)

0

-10 -20

2

T1l1 (kN) 0

2

4

-2 0

T2r1 + T2l1 2

4

Figure 7.16 Dynamic response of the leading axle of the three-axle vehicle with the parameters of Table 7.1 to curve entry with a cubic parabolic transition of length 20 m, V = 15 m/s and R0 = 225 m.

reacted by longitudinal creep forces, which therefore reach a peak and then subside to their steady-state value. Figure 7.16 presents the corresponding results for the same vehicle but with the nominal values of the stiffnesses kyd and kt ( kyd = 0.7 MN, kt = 0.2 MN). The increased stiffness of the inter-wheelset structure results in larger longitudinal creep forces both in the steady-state and the transition. It is clear from these indicative results that the three-axle vehicle offers the possibility of improved steady-state curving in which the wheelsets take up a closely radial position, but the dynamic response in transitions and reverse curves requires careful consideration. Thus, the trade-off between stability and curving is not completely eliminated.

References 1. Liechty, R.: Das Bogenlaufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934, p. 23. 2. White, J.H.: The American Railroad Freight Car. The John Hopkins Press, Baltimore, 1993, p. 168. 3. Fidler, C.: British Patents 2399, 3825, 1868.

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233

4. Elsner, H.: Three-Axle Streetcars. N.J. International, Hicksville, 1994, Vol. 1, Chapter 1. 5. Elsner, H.: Three-Axle Streetcars. N.J. International, Hicksville, 1994, Vol. 1, p. 60. 6. Liechty, R.: Das Bogenlaeufige Eisenbahn-Fahrzeug. Schulthess, Zurich, 1934, p. 31. 7 Wickens, A.H.: Steering and dynamic stability of railway vehicles. Vehicle System Dynamics 5, No. 1-2 (1975), pp. 15-46. 8. Wickens, A.H.: Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics 7, No.1 (1979), pp. 33-48. 9. Wickens, A.H.: Static and dynamic stability of a class of three-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 6, No.1 (1977), pp. 1-19. 10. Wickens, A.H.: Flutter and divergence instabilities in systems of railway vehicles with semi-rigid articulation. Vehicle System Dynamics 8, No.1 (1979), pp. 3348. 11. Wickens, A.H.: The stability of a class of multi-axle railway vehicles possessing perfect steering. In: K. Magnus (Ed.): Proc. IUTAM Symposium on Dynamics of Multibody Systems, Munich, August-September 1977, pp. 345-356. SpringerVerlag, Berlin, 1978. 12. Wickens, A.H.: Static and dynamic stability of unsymmetric two-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 11, No. 2 (1982), pp. 89-106. 13. Wickens, A.H.: Stability optimisation of multi-axle railway vehicles possessing perfect steering. ASME Journal of Dynamic Systems Measurement and Control 110, No.1 (1988), pp. 1-7. 14. Wickens, A.H.: Static and dynamic stability of a generalised symmetric threeaxle railway vehicle possessing perfect steering. Archives of Transport Quarterly 1, No.2 (1989), pp. 139-160. 15. de Pater, A.D.: Optimal design of a railway vehicle with regard to cant deficiency forces and stability behaviour. Delft University of Technology, Laboratory for Engineering Mechanics, Report 751, 1984. 16. de Pater, A.D.: Optimal design of railway vehicles. Ingenieur-Archiv 57, No.1 (1987), pp. 25-38.

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17. Keizer, C.P.: A theory on multi-wheelset systems applied to three wheelsets. In O. Nordstrom (Ed.): The Dynamics of Vehicles on Roads and Tracks. Proc. 9th IAVSD Symposium, Linkoping, June 1985. Swets and Zeitlinger Publishers, Lisse, 1986, pp. 233-249. 18. Porter, B.: Stability criteria for linear dynamical systems. Oliver and Boyd, Edinburgh, 1967, p. 37. 19. Bolotin, V.V.: Non-conservative problems of the theory of elastic stability. Pergamon, Oxford, 1963. 20. Done, G.T.S.: The flutter and stability of undamped systems. British Aeronautical Research Council Reports and Memorandum No.3553, 1966.

8 Articulated Vehicles 8.1 Introduction The economics of railways in which expensive infrastructure is justified by large traffic flows requires the operation of either many vehicles with short headways or very large vehicles to provide the required capacity. Hence the concept of the train. In many cases the interaction between the vehicles in a train is minimised by the form of coupling between the vehicles, so that longitudinal forces can be transmitted between car bodies, but the coupler is capable of transmitting little or no lateral force or yaw couple. In this case it is a good approximation to treat each vehicle as if it were isolated and the lateral dynamics of each vehicle can be considered to be largely independent of that of the rest of the train. However, the need to improve curving performance, maximise the use of the clearance gauge, minimise axle loads, reduce mass, aerodynamic drag and cost has led to many designs in which there is articulation of the car bodies of a vehicle or train, so that the connections between vehicles form an essential part of the running gear. In this Chapter articulated vehicles, in which the relative motion between the car bodies is used to influence the stability and guidance of the vehicle, are considered. The first articulated locomotive was designed by Horatio Allen in 1832 [1]. Though this had a short career, it probably stimulated several of the articulated designs for the Semmering Contest in 1851. Thereafter, there was a succession of articulated locomotives the development of which is described by Weiner [2], and which sought to resolve the conflict between the long wheelbase made necessary by high power and the large curvature of many railway lines. Most of these had unsymmetric fore-and-aft configurations which are considered in Chapter 9. In the early days of the railways, it had become customary to link together two and three axle vehicles not only by couplings but also by side chains to provide yaw restraint between adjacent car bodies in order to stabilise lateral motions [3]. The need to lengthen vehicles stimulated measures which allowed bending in plan view. As an alternative to the use of the bogie, in 1837 W.B. Adams proposed an articulated two-axle carriage [4]. The first three-axle vehicle with articulated car body was proposed by Fidler in 1868 [5]. In Fidler’s patent the wheelset is mounted on the car body at the point where the car body is tangential to the curve, and the central wheelset is mounted on a steering beam, Figure 8.1(a). Machlachan [6] proposed a similar configuration in 1878 but made the significant addition of a shear connection

236

RAIL VEHICLE DYNAMICS

(a)

(b)

(c)

(d)

(e)

Figure 8.1 Historical articulated railway vehicle configurations. (a) Fidler, 1868. (b) Machlachan, 1878. (c) Barber, 1907. (d) Liechty, 1931. (e) Configuration using steering linkage.

or pivot between the car bodies, Figure 8.1(b). In Barber’s 1907 patent [7] the outer bodies are pivoted together but instead of the wheelset being mounted on the body at the point of tangency, the outer wheelsets are freely pivoted and are connected to the central wheelset by cross-bracing, Figure 8.1(c). The steering beam has disappeared. A similar objective is achieved by Liechty’s 1931 patent [8] in which the outer wheelset is mounted on an arm pivoted on the car body and actuated by the steering beam, Figure 8.1(d). A similar scheme has been used recently in the Boa design [9]. An alternative approach is to use a linkage, Figure 8.1(e), similar to the arrangement used in body-steered bogies, but driven by the angle between the car bodies. In trains of bogie vehicles, the use of the Jacob bogie which is shared by adjacent car bodies reduced mass (and cost and drag) of an articulated rake. However, the resulting fixed consist was disliked by some operators, and individual body lengths had to be shorter to maintain limits on the ‘throwover’ (lateral displacements of the car bodies) on curves. Nevertheless, articulation was successfully exploited by Gresley [10] in a number of carriage designs between 1900-1930. In 1939, Sta-

ARTICULATED VEHICLES

237

(a) (b) (c) (d) (e)

Figure 8.2 Examples of some of the many configurations of vehicles with articulated car bodies and single-axles. Equivalent bogie versions are common.

nier employed a double point articulation, one at each end of the bogie, which allowed longer car bodies [11]. Application to a new generation of passenger trains in the U.S. in 1930-50 was less successful, and generally articulated trains acquired a reputation for bad riding. More recently, articulation of bogie vehicles has been used, most successfully, in high speed trains such as the TGV [12]. Another important modern development is the use of articulation on vehicles with single axle running gear, as discussed in Chapters 4 and 5.The dynamic behaviour of the Talgo train is influenced by its unsymmetric configuration and the problems arising will be considered in Chapter 9. Current examples of trains which have single-axle running gear such as the Copenhagen S-Tog [13] embody forced steering of the wheelsets through mechanical linkages or hydraulic actuators driven by the angle between adjacent car bodies. Extensive design calculations were carried on this train [13] and its lateral stability is also discussed in [14]. Another recent example is provided by the Wien trams [15]. A wide range of articulated configurations have been used in practice, particularly for trams, and some of the variations are shown in Figure 8.2. This figure is

238

RAIL VEHICLE DYNAMICS

based in part on [16]. The simplest form of articulated vehicle is one with two car bodies supported on three or four axles. Not only has this been a common configuration, particularly for trams, but its study reveals many of the dynamic characteristics of articulated trains with many axles. For a vehicle with three or more axles it was shown in Chapter 7 that stability could be achieved without compromising steering on uniform curves, the stiffness necessary for stability being provided in a way that does not impede radial steering. This provides a theoretical basis for the discussion of articulated vehicles exploiting single-axles in this Chapter. As in the case of the bogie vehicle considered in Chapter 6, two broad approaches to configurations can be distinguished. The first may be termed forced steering (steering derived from relative motions between car bodies) and the second self steering (steering derived from wheelset motions only). Various possible future stages in the development of single-axle suspensions may be envisaged, in which mechanical linkages are progressively displaced by systems with actuators and sensors and either passive or active control. 8.2 Steering and Stability As a wide variety of vehicles with articulated car bodies exist, it will be necessary to consider a limited number of variants, so the behaviour of a number of representative configurations with three and four axles will be discussed. Figure 8.2(a) shows two two-axle vehicles coupled together; Figure 8.2(b) and 8.2(c) show similar con-

\3

yd y3 \d h

\c

yc

\2

y2

y3

\2

\b

h1 c yd \4

y4

h

\d

\3

h

yb

y2

\b \1 y 1

c

Figure 8.3 Generalised coordinates for articulated vehicles.

yb y1 \1

ARTICULATED VEHICLES

239

k\b (a)

ky2/2

k\1

kyb

k\2

ky1/2

ky2/2

ky1/2 kyb

k\b

ky2/2

k\1

ky1/2

(b) k\1

k\2

k\2 kyb

k\1

Figure 8.4 Simplified arrangement of suspension showing basic stiffnesses for articulated vehicles.

figurations but with one axle removed; Figure 8.2(d) and (e) show symmetric threeaxle vehicles with either two or three car bodies, including vehicles with two car bodies carrying a steering beam on which the central wheelset is mounted. Comparison of the behaviour of these various configurations will give insight into the dynamics of articulated vehicles in general. Consider the equations of motion of these configurations. It is clear that the configurations of Figure 8.2(b) and (c) are derivable from that of Figure 8.2(a) by the deletion of a wheelset, and such unsymmetric configurations will be discussed in Chapter 9. Moreover, the configurations of Figure 8.2(d) and (e) are equivalent in terms of their degrees of freedom. Consequently, making the same assumptions that have been made in the case of the two-axle vehicle and other configurations, the motions of these vehicles may be defined by the generalised coordinates shown in Figure 8.3. The form of the elastic stiffness matrix will be determined by applying the criteria for stability of a vehicle capable of perfect curving given in Chapter 6. The solution of equations (6.4.10) will initially be discussed for the three-axle vehicle with three car bodies of Figure 8.2(d) so that neglecting the variation of the rotational speed of the wheelsets q >y\ybMb\by\ycMc\cy\ydMd\d@T

(1)

so that yb, yd and yc refer to lateral translation of the vehicle bodies and \b ,\d , and \c refer to yaw of vehicle bodies. The other yi, \i are standard wheelset coordinates. The equations of motion will be of the form of (2.11.23) where

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RAIL VEHICLE DYNAMICS

F = block diagonal[ F1 O33 F2 O33 F3 O33 ]

(2)

and O33 is the 3 u 3 null matrix, and Fi refers to the ith wheelset. The inertia matrix is A = diag [m I mb Ixb Izb m I mc Ixc Izc m I mb Ixb Izb ]

(3)

The component stiffness and compatibility matrices for the basic form of secondary suspension shown in Figure 8.4(a) are k = diag[ ky1 kI1 k\1 kyb kIb k\b ky2 kI2 k\2 kyb kIb k\b ky1 kI1 k\1]

a

ª1 «0 « «0 « «0 «0 « «0 «0 « «0 «0 « «0 « «0 «0 « «0 «0 « «¬0

0 1

d1

h

0 0

0

0

0

0 0

0

0

0

0

1

0

0 0

0

0

0

0 0

0

0

1 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0

0  d2 1 0 0 0 0 0 0 0 0 0 0

1  h1 0 1 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0  1 d 3 h1  c 0 0 1 0 0 0 0 1 0 1 d4 0 0 0 1 0 1 0 0 1 0  1 d 3 c  h1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  d2 0 0 1 0 0 0 0  1 d1 0 0 1 1 0 0

(4)

0º 0 »» 0» » 0» 0» » 0» 0» » 0 » (5) 0 »» h1 » » 0»  1» » h» 0» »  1»¼

Then equations (6.4.10) become ky1( y1 - yb + d1Mb - h\b ) = 0

(6)

kM1Mb = 0

(7)

k\1( \1 - \b ) = 0

(8)

kyb{ yb - d2Mb - h1\b - yc + d3Mc - (c - h1)\c} = 0

(9)

kIb( Mb - Mc ) = 0

(10)

k\b( \b - \c ) = 0

(11)

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241

ky2(- yc + d4Mc + y2 ) = 0

(12)

kM2Mc = 0

(13)

k\2( \c - \2 ) = 0

(14)

kyb{ yd - d2Md + h1\d - yc + d3Mc + (c - h1)\c} = 0

(15)

kIb(Md - Mc ) = 0

(16)

k\b(\d - \c ) = 0

(17)

ky1(y3 - yd +d1Md + h\d) = 0

(18)

kM1Md = 0

(19)

k\1(\3 - \d) = 0

(20)

From equations (7), (13) and (19) Mb = Mc = Md = 0. From equations (8), (11), (14), (17), and (20) either \1 = \b or k\1 = 0

(21)

either \c = \b or k\b = 0

(22)

either \2 = \c or k\2 = 0

(23)

either \d = \c or k\b = 0

(24)

either \3 = \d or k\1 = 0

(25)

For the case where the outer wheelsets have radius r0 and conicityO and the inner wheelset has radius r1 and conicityO1, for steering on a uniform curve without lateral and longitudinal creep, the displacements measured from the unstrained position are y1 = y3 = - lr0/OR0 + (c + h )2/2R0

(26)

y2 = - lr1/O1R0

(27)

\1 = -\3 = (c + h )/R0

(28)

\2 = 0

(29)

Because of symmetry yd = yb, \d = -\b , and \c = 0 so that from (23) the value of

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RAIL VEHICLE DYNAMICS

k\2 can be arbitrarily chosen. From (12), either ky2 = 0 or yc = y2. Choosing the latter case, from (6), (9), (26) and (27) yb = {- h1lr0/O- hlr1/O1+ h1(c + h )2/2}/R0( h + h1)

(30)

\b = {- lr0/O+ lr1/O1+ (c + h )2/2}/R0( h + h1)

(31)

If the condition (21) is satisfied by putting \1 = \b then, from (28) and (31), - lr0/O + lr1/O1 + (c + h )( c - h - 2h1)/2 = 0

(32)

or, if the inner and outer wheelsets have the same radius and conicity h1 = (c - h)/2

(33)

If this condition is satisfied the conditions for perfect steering can be satisfied with a nonzero yaw stiffness k\1. The remaining conditions (22) and (24) requires that k\b = 0. This implies that there are two main possibilities for symmetric articulated three-axle vehicles. The first has geometry which satisfies the geometric requirement (32) or (33) and the second does not satisfy this requirement so that k\1 = 0. In addition, there is, of course, the further possibility of the vehicle with a single car body which has been discussed in Chapter 7. An analogous analysis can be carried out for the configurations of Figure 8.2(a), where the basic stiffnesses are shown in Figure 8.4(b). As this vehicle consists of two-axle vehicles coupled together with simple suspension elements, it is not necessary to reiterate the equations of motion. The conditions for steering give the simple result y1 = y4 = - lr0/OR0 + (c + h )2/2R0

(34)

y2 = y3 = - lr1/O1R0 + (c - h )2/2R0

(35)

\1 = -\4 = (c + h )/R0

(36)

\2 = -\3 = (c - h )/R0

(37)

yb = yd = - lr0/2OR0 - lr1/2O1R0 + (c2 + h2)/2R0

(38)

\b = -\d = - lr0/2OhR0 + lr1/2O1hR0 + c/R0

(39)

Similar results, of course, apply to the configurations of Figures 8.2(b) and 8.2(c). Figure 8.5 indicates the geometry for each of these configurations in which the wheelsets adopt a radial position with radial displacements appropriate to their conicity, and the car bodies take up the positions connecting the wheelsets without

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243

strain of the suspension elements. Thus, for small displacements and large radius curves, steering without lateral and longitudinal creep is obtained because of the combined action of creep and conicity which is the primary mechanism of guidance. It is evident that on curves with moderate and large curvature, a similar analysis may be carried out based on the full nonlinear equations of motion, cf. Section 3.2. These configurations satisfy the conditions for steering so it is now necessary to consider stability. For simplicity it will be assumed that all wheelsets have the same radius and conicity. The necessary condition for the vehicle to be able to steer without lateral and longitudinal creep and be dynamically stable at low speeds is given by (6.4.15). For the three-axle configuration of Figure 8.4(a) if the geometrical condition (33) is not satisfied and the yaw stiffness k\1 is zero, the degeneracy of the basic stiffness matrix E is 7, so that in this case three additional stiffnesses are required to

(a)

c h1

h

(b)

\b y b

(c + h)2/2R0

y1

Ol/R 0r0

c (c)

h

\b y

b

(c + h)2 /2R 0

y1



Figure 8.5 Attitude on curve of various articulated configurations.

Ol/R 0 r0

244

RAIL VEHICLE DYNAMICS

reduce the degeneracy to 3 so that the criterion may be satisfied. For configurations where the geometrical condition (33) is satisfied and there are nonzero yaw stiffnesses k\1, the degeneracy of E is 4, so that in this case only one additional stiffness is required to satisfy the criterion. The basic stiffness matrix E for the four-axle configuration of Figure 8.4(b) has degeneracy 7 and therefore requires 4 additional stiffnesses in order to satisfy the criterion. Thus additional stiffnesses must be provided, and this is considered next. 8.3 Application to Specific Configurations Based on the configurations discussed in Section 1, Figure 8.6 shows an extension of Figure 8.4 in which further elastic connections have been introduced. Figure 8.6(a) and (b) show the body-steered and self-steered versions of the three-axle vehicle and Figure 8.6(c) and (d) show the corresponding versions of the four-axle vehicle. Firstly, it is useful to consider the early schemes shown in Figure 8.1 in the light of the above criteria. As mentioned in Chapter 7, pivots were originally used to provide lateral restraint and yaw freedom, instead of elastic elements. For example, in Fidler’s patent for the three-axle vehicle, Figure 8.1(a), the geometric condition (2.33) is satisfied and ky1, ky2, kyb, k\1, and k\2, were very large though the principles discussed above remain valid. As mentioned above, E has degeneracy of 4 and the vehicle would be unstable at low speeds. The steering beam is equivalent to the central car body of Figure 8.2(d). In Machlachan’s scheme, Figure 8.1(b), there is a shear connection or pivot between the outer car bodies, which may be regarded as an additional stiffness kyc. From the above discussion, and assuming that the geometrical condition (2.33) is satisfied, the addition of an additional row and corresponding stiffness kyc would have the effect of stabilising the vehicle so that all eigenvalues had negative real parts at low speeds. In Barber’s scheme, Figure 8.1(c), the outer bodies are connected by a shear stiffness kyc but the geometrical condition (2.33) is not satisfied. Instead the outer wheelsets are freely pivoted and are connected to the central wheelset by cross-bracing, with shear stiffness kyd. In this case the three stiffnesses kyc and kyd (leading and trailing bay) would ensure that the degeneracy of E would be reduced to 3 as required by the stability criterion. A similar objective is achieved by Liechty’s scheme, Figure 8.1(d) in which the outer wheelset is controlled by a steering beam. Again, the steering beam is the central car body of Figure 8.2(d) steered through springs of stiffness kd and ke, but if the steering beam is small its independent coordinates can be eliminated and its effect represented as shown in Figure 8.7. It is therefore an equivalent arrangement to a steering linkage connecting the yaw angle of the leading wheelset with the angle between adjacent car bodies. Thus, these additional stiffnesses are provided in two distinct ways, consisting of the basic configuration as defined in Section 3 with, either body steering or self steering. In the former case the outer wheelsets are steered by a linkage, of stiffness kst, from the angle between the outer and central car body, the third additional

ARTICULATED VEHICLES

(a) e1

245

kyc

c1 c2

c3 e2 ks =diag [ kst

as

kst kyc ]

0 0 0 º ª0 c  h 0 0  2(h  h1 ) 0 0 0 0  c  h  2h1 0 0 » «0 0 0 0 0 0 0 0 0 2 0 0 0 2 2  c  h  h c  h  h  h 1 1 » « «¬0 0 1  d5 0 0 0 0 0 0 0  1 d5 c  c »¼

kst = e1e2c22(c3 - c1)2/{e1(c2 - c1)2 + e2(c3 - c1)2}

(b)

c2(c3 - c1)/c3(c2 - c1) = (c + h)/(c - h -2h1)

kt

kyd

kyd

ks = diag[ kyd kyd kt]

as

0 0 0  1  c  h  h3 h3 ª0 «0 0 0 0 0 1  c  h  h3 « «¬1  c  h 0 0 0 0 0

0 0 0 0 0 0 0 0º 0 0 0  1  h3 0 0 0»» 0 0 0  1  c  h 0 0 0»¼

k st2 (c)

k st1

k st1 k st2

ks = [ kst1 kst2 kst2 kst1 ]

ª0 «0 « «0 « ¬0

as

1 0 0  1  h / 2c 0 0 0 0 0 0 0 0 0  1  h / 2c 0 1 0 0 0 0 0 0 0 0 0 0

 h / 2c h / 2c

kyf

(d) kyd

0 0º 0 0»» 0 0 0 1 0 0  1  h / 2c 0 0» » 0 0 0 0 0 0  1  h / 2c 0 0¼

kt

h / 2c  h / 2c

kyd

kyf

kye

ks = [ kyd kyd kt kye kyf kyf]

as

h ª1 «0 0 « «1  c  h « 0 «0 «1 c « 0 «¬0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

1 0

0 0 0

1

0 0 0 0 0 º 0 0 0 1  h »» 0 0 0  1  c  h» » 0 »  c  h 1  c  h 0 0 0 0 0 0 0 0 0 0 » 1 c »  c »¼ c 0 0 0 0 0 1 h 0 0

0

0

1 0

h 0

Figure 8.6 Additional stiffnesses for articulated vehicles.

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RAIL VEHICLE DYNAMICS

c1 \d

O

c-h

c1

R P

Q

S

\b

\2

c2 h 2c

c1  c2  c  h 2 c2

as = [ 0 0 0 0 -1+ h/2c 0 1 0 0 0 0 -h/2c 0 0 ]

Figure 8.7 Steering beam applied to rear wheelset of leading vehicle showing the equivalence to a steering linkage.

stiffness required for stability is provided by kc, as shown in Figure 8.6(a). In the latter case the adjacent wheelsets connected by a shear spring kyd and the outer wheelsets connected by a shear spring kt, as shown in Figure 8.6(b). Similarly, there are equivalent schemes for the four-axle vehicles with two car bodies as shown in Figures 8.6(c) and (d). For long articulated vehicles direct connection of the wheelsets may seem difficult to implement, but it is possible to utilise passive sensors and actuators replicating the stiffness and damping characteristics of mechanical linkages and arranged so as to be de-coupled from the motions of the car bodies. 8.4 Stability of an Articulated Three-Axle Vehicle As might be expected, there is considerable similarity between the behaviour of the articulated three-axle vehicle and the three-axle vehicle with single car body considered in Chapter 7. Firstly, the scheme in which the adjacent wheelsets connected by a shear spring kyd and the outer wheelsets connected by a shear spring kt, as shown in Figure 8.6(b) will be considered in detail. Consider the influence of speed on the eigenvalues of the configuration of Figure 8.6(b) with the nominal set of parameters given in Table 8.1. The root locus as speed is varied is shown in Figure 8.8, and values for the eigenvalues at V = 10 m/s are given in Table 8.2. It was shown in Section 6.4 that, at low speeds, the equations of motion can be separated into uncoupled sets involving either the wheelset coordinates or the car body coordinates. The resulting system of equations involving the wheelset coordinates is of exactly the same form, at low speeds, as those derived in Section 7.2 for the three-axle vehicle with single car body and, in particular, the form of the stiffness matrix E* is as given in (7.2.22-23). It follows that the system has six eigenvalues proportional to speed consisting of a steering oscillation A, a

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247

Table 8.1 Example parameters for three-axle articulated vehicle. suspension of basic system ky1 = ky2 = 0.23 MN/m k\ = 0 kI = kI = 1MNm cy2 = 50 kNs/m cy1 = 200 kNs/m c\= c\= 0 cI = cI = 50 kNms kyb = 1 MN/m k\b = 0 kIb = 0 cyb = 0 cIb = 0 c\b = 0 additional suspension for body-steered system kst = 30 kNm kyc = 1 MN/m k\ = 2 MNm c\b = 0.1 MNms cst = cyc = c\2 = 0 additional suspension for self-steered system kyd = 0.1 MN/m kt = 0.1 MN/m ct = cyd = c\2 = c\b = 0 k\ = 5 MNm vehicle geometry r0 = 0.45 m l = 0.7452 m c = 6.25 m d = 0.2 m h=2m h1 = 3.75 m creep coefficients f11 = 8.83 MN f22 = 8.06 MN f23 =17.7 kNm inertia m =1250 kg I = 700 kgm2 mb=10000 kg Ixc=12000 kgm2 Izb = 130000 kgm2 mc=7000 kg Ixc=8400 kgm2 Izb = 9000 kgm2 2 Iy = 250 kgm

shear oscillation B and two shear subsidences C1 and C2 which are closely equal to the solutions of equation (7.4.1). For the chosen set of parameters there is stability at low speeds. A further set of eigenvalues is associated with wheelset motions and consist of six large subsidences (three roots are equal to -2f/mV and three roots are equal to -2fl2/IV) and these are not shown in Figure 8.8. The eigenvalues associated

Figure 8.8 Root locus of the three-axle configuration of Figure 8.6(b) with the nomi-

nal set of parameters given in Table 8.3.

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Table 8.2 Eigenvalues for articulated three-axle vehicle with parameters of Table 8.1.

1 2 3 4 5 6 7,8 9,10 11 12,13 14,15 16,17 18,19 20,21 22,23 24 25,26 27,28 29,30

At Speed, V = 10 m/s -1398 -1392 -1383 -1331 -1331 -1331 -0.6423 ± 45.74i -2.368 ± 22.60i -14.18 -2.909 ± 10.40i -2.259 ± 8.833i -2.162 ± 8.801i -3.820 ± 7.333i 0.2710 ± 5.210i -0.8677 ± 5.601i -2.542 -3.199 ± 4.351i -1.303 ± 3.329i -3.162 ± 4.389i

Wheels - Fixed

-0.02164 ± 45.83i -2.445 ± 22.60i -2.909 ± 10.40i -2.181± 8.927i -2.165 ± 8.797i -0.2211± 7.246i

-3.266 ± 4.448i -1.317 ± 3.305i -3.012 ± 4.471i

Label Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence Wheelset subsidence D9 D8 Shear subsidence C2 D7 D6 D5 D4 Steering oscillation A Shear oscillation B Shear subsidence C1 D2 D1 D3

Figure 8.9 Natural modes of articulated vehicle with parameters of Table 8.3 with wheels fixed and zero suspension damping, D = 0.

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249

V= 8.5

Z 4.829

73.0

37.36

78.9

42.26

Figure 8.10 Mode shapes of the critical modes at the bifurcation speeds of articulated vehicle with parameters of Table 8.3, showing one cycle of the oscillation.

with the vehicle body modes, D1-D9, are substantially independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension. They are closely equal to the wheels-fixed eigenvalues, as illustrated in Table 8.2. The mode shapes of these wheels-fixed modes are shown in Figure 8.9 for the case where the suspension damping is zero. It is noteworthy that many of the wheels-fixed modes are completely uncoupled from the wheelset motions but in other cases there is considerable interaction, leading to body instability. For the given parameters, body instability occurs at a bifurcation speed VB1 = 8.5 m/s and the associated mode shape is shown in Figure 8.10(a). As in the case of the two-axle vehicle, stability can be obtained by choosing suitable values for ky1, ky2, dy1, and dy2, but in addition inter-car body dampers may be effective. At higher speeds, the shear oscillation B loses stability at a bifurcation speed VB2 = 73.0 m/s. As shown in Figure 8.10(b), this is analogous to the wheelset instability, there is little motion of the car bodies and is predicted accurately if the car bodies are assumed fixed. A further instability at a bifurcation speed VB3 = 78.9 m/s involves yaw of the central car body and the mode shape is shown in Figure 8.10(c).

250

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Figure 8.11 Variation of the imaginary part of the eigenvalues for the three-axle configuration of Figure 8.6(b) with the nominal set of parameters given in Table 8.1.

O = 0.5

80

O2

S

V (m/s)

80

O = 0.05, f x 0.5

O1 D S

O1 0 80

kyd (MN/m) 1 O2

O2 S

V (m/s)

O1 0 kt (MN/m) 0.5 80 V (m/s)

0 kyd (MN/m) 1 80

O1

S

0 k\2 (MNm) 0.5

S

D

0 kt (MN/m) 0.5 80 S

0 k\2 (MNm) 0.5

Figure 8.12 Joint influence of conicity, creep coefficient and stiffnesses kyd, kt and k\2 on stability for articulated three-axle vehicle of Fig. 8.6(b) with parameters of Table 8.1 except that cy1 = cy2 = cyb = 0.1 MNs/m, kyd = 0.7 MN/m, kt = 0.2 MN/m and k\2 = 5 MNm. S denotes stability, D divergence and O1 and O2 oscillatory instabilities.

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251

O 

80

O

V (m/s)

0 80

C

S

S

kst (MNm) 0.4 O

V (m/s)

O

V (m/s)

B A

O= 0.5 f/2

80

0 80

kst (MNm) 0.4

V (m/s)

S

S 0 80 V (m/s)

kyc (MN/m) 1

O

S O

S 0

ky2 (MNm)

0 kyc(MN/m) 1 80 O V (m/s)

1

0 k (MNm) 1 y2

Figure 8.13 Joint influence of conicity, creep coefficient and stiffnesses kst, kyc and k\2 on stability for articulated three-axle vehicle of Figure 8.6(a) with forced steering with parameters of Table 8.3. S denotes stability and O oscillatory instability.

Figure 8.11 shows the variation of the imaginary parts of the eigenvalues with speed for a range of low speeds. Frequency coincidence occurs between the steering oscillation A and shear oscillation B and the car body on suspension modes, but in only one case in the chosen speed range does instability occur. Clearly, coupling of the wheelset modes and the car body modes depends not only on a degree of frequency coincidence but also on compatibility of mode shape for the energy transfer that would sustain the amplitude of the oscillation. In this connection, the zig-zag mode shape of the car body mode D4 is particularly likely to couple with the steering oscillation, and this illustrates the tendency of articulated vehicles to suffer this form of body instability. The joint influence of the stiffnesses kyd, k\2 and kt, conicity and creep coefficient is shown in Figure 8.12. This may be compared with Figure 7.10 which shows a similar plot for the three-axle vehicle with a single car body. If any of the stiffnesses kyd, kt and k\2 are zero then the critical speed is zero in accordance with the discussion in the preceding Section. For large values of the conicity, instability is confined to smaller values of kyd, k\2 and kt. The instability, labelled O1 in Figure 8.12, is the body instability discussed above. The instability O2 is analogous to the wheelset instability. For low values of the conicity and creep coefficients, divergence occurs for a range of values of kyd and for larger values of kt. Moreover, the instability O2 occurs at lower speeds.

252

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Turning to the scheme of Figure 8.6(a) in which the wheelsets are connected by a steering linkage to the car bodies, it is found that the root locus is qualitatively similar to the root locus for the variant with direct connections between the wheelsets, but there is a very lightly damped steering oscillation at low speeds. Three additional points arise. Firstly, as might be expected, in this case there is more interaction between the car body modes and the wheelset modes. Secondly, the addition of yaw damping, c\b, between the car bodies is effective in eliminating the body instability. Thirdly, very large values of c\b are effective in stabilising the system at low speeds, and increasing the damping in the steering mode, even when kyc = 0 and the stability criteria discussed in Section 3 are not satisfied. However, the bifurcation speeds associated with the wheelset modes are little affectedҠ. Hence Figure 8.13 shows the influence of the stiffnesses kst, kyc and k\2 on stability for extreme values of the conicity and creep coefficient for the scheme of Figure 8.6(a) and the parameters of Table 8.1. In accordance with the discussion above if the contact stiffnesses were neglected then for stability at low speeds either kyc or k\2 is needed in addition to kst. If both kyc and k\2 are zero, the critical speed is determined by the contact stiffnesses and would be low. Initially, increases in kst increase the critical speed, A in Figure 8.13(a). Beyond a certain value of kst the critical speed is not influenced by kst, B in Figure 8.13(a), because the mode of instability involves mainly kinematic motion of the central wheelset. For large values of kst, an instability involving interaction with the lateral bending mode of the car bodies occurs. This instability occurs for smaller values of kst in the case of small values of conicity and reduced creep coefficient as shown in Figure 8.13(b). Thus, in this case as in others that have already been discussed, there is a conflict between the requirement that kst must be sufficiently large in order to achieve a large margin of stability at high conicity, but this will tend to encourage instability for low values of conicity and creep coefficient. Figures 8.13(c)-(f) show that both k\2 and kyc are effective in stabilising the system for low conicities and creep coefficients, but not at high conicities. 8.5 Stability and Response of an Articulated Four-Axle Vehicle For the configuration of Figure 8.6(c) and the nominal set of parameters given in Table 8.3, the root locus as speed is varied is shown in Figure 8.14. As might be expected, in this case there are eight eigenvalues proportional to speed representing four oscillations at kinematic frequency O. These oscillations are stable at low speeds provided that elastic stiffness has been provided in accordance with the prescription of the Section 3. In addition there are the usual set of eigenvalues also associated with wheelset motions and consisting of eight large subsidences (at low speeds, four roots are equal to -2f/mV and four roots are equal to -2fl2/IV) which become heavily damped oscillations at higher speeds as shown at S in Figure 8.14. As usual, the eigenvalues associated with the vehicle body modes, D, are more or less independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension. They are closely equal to the wheels-fixed eigenvalues. As in the case of the threeaxle vehicle,

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Table 8.3 Example parameters for four-axle articulated vehicle suspension of basic system ky1 = ky2 = 0.23 MN/m k\ = k\ = 0 kI = kI = 1MNm cy1 = 50 kNs/m c\= c\= 0 cI = cI = 50 kNms additional suspension for body-steered system kst1 = 2.5MN/m kst2 = 2.5MN/m dst1 = dst2 = 0 additional suspension for self-steered system kyd = 1MN/m kt = 0.1MN/m kyf = 0.1MN/m ct = cyd = cyf = 0 vehicle geometry l = 0.7452 m r0 = 0.45 m h = 3.5 m c=6m d = 0.2 m creep coefficients f11 = 7.44 MN f22 = 6.79 MN f23 =13.7 kNm inertia m =1250 kg I = 700 kgm2 mb=10000 kg Ixc=12000 kgm2 Izb = 130000 kgm2 2 mc=7000 kg Ixc=8400 kgm Izb = 9000 kgm2 2 Iy = 250 kgm

kyb = 40 MN/m cy2 = cyb = 0

kye = 1MN/m

Figure 8.14 Root locus of the four-axle configuration of Figure 8.6(c) with the nominal set of parameters given in Table 8.3.

254

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it is noteworthy that many of the wheels-fixed modes are completely uncoupled from the wheelset motions but in other cases there is major interaction, potentially leading to body instability. One of the oscillations at kinematic frequency, shown at O1 in Figure 8.14, has very low damping and is associated with near coincidence between one of the wheels-fixed modes and a steering mode. At higher speeds, all four steering modes lose stability at bifurcation speeds between 85.5 and 89.5 m/s. This is analogous to the wheelset instability, there is little motion of the car bodies and is predicted accurately if the car bodies are assumed fixed. Then application of equation (3.3.15) shows that in this case it is the stiffness of the steering linkage kst1 which determines the bifurcation speed. A full survey of the stability boundaries will not be given here, but as an example Figure 8.15 shows stability as a function of speed and the stiffness in the steering linkage kst (= kst1 =kst2), for three values of the equivalent conicity and a case of reduced creep coefficient. If kst is zero then the system is unstable at low speed as already discussed. For small values of kst instability O1 occurs above a certain speed. For large values of kst another form of instability O2, which involves relatively large amounts of yaw of the car bodies, occurs for which low conicity and creep coefficient is destabilising. This again illustrates the difficult trade-off for many configurations of railway vehicle, for if a design has to cater for a wide range

100

O = 0.5

O = 0.2

100

O1 V (m/s)

V (m/s)

S

S

O2 0 100

k st (MNm)

20

O = 0.05

0 100

V (m/s)

V (m/s) S

0

20 kst (MNm) O = 0.05 f x 0.5

k st (MNm)

S 20

0

kst (MNm)

20

Figure 8.15 The effect of variations of conicity and creep coefficient on stability as a function of the stiffness of the steering linkage kst and speed for the four-axle vehicle with the parameters of Table 8.5. S = stability, O1 and O2 instability.

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of conicities, comparison of the boundaries for high and low conicity in Figure 8.15 shows that the speed range for stability is severely reduced. These results can be compared with those for the configuration of Figure 8.6(d), for a set of nominal parameters, given in Table 8.3, comparable with those used in the discussion of the configuration of Figure 8.6(c), in which the wheelsets are connected directly and not through the car bodies. Figure 8.16 shows the root locus as speed is varied, and can be compared with Figure 8.14. In this case the eight eigenvalues which are proportional to speed, as suggested by the analysis of Section 6.4, consist of two conjugate complex pairs corresponding to steering oscillations, shown as O1 and O2 in Figure 8.16 and four real roots, shown at R. One of the latter is very small indicating a condition of marginal static stability, though for the chosen set of parameters there is stability at low speeds in accordance with the discussion of Section 3. The lightly damped kinematic oscillation of the body steered configuration is absent. The usual set of eigenvalues associated with wheelset motions consists of six large subsidences (three roots are equal to -2f/mV and three roots are equal to -2fl2/IV) and these are not shown in Figure 8.16. The six pairs of eigenvalues associated with the vehicle body modes are substantially independent of speed except at speeds for which the kinematic or steering frequencies approach the natural frequencies of the vehicle body on the suspension and they are closely equal to the wheels-fixed eigenvalues. These are shown as D1-D5 in Figure 8.16. (D6 is off the scale). It is noteworthy that in the present case the wheels-fixed modes are completely uncoupled from the wheelset motions. motions.re 8.17 shows stability as a function of speed and the stiffnesses kyd (= kye), kyf and kt for two extreme values of the equivalent conicity and creep coefficient. Note that in accordance with the discussion of Section 3, stability at low speeds is obtained if either kt or kyf are zero. However, an appropriate choice of stiffnesses

Figure 8.16 Root locus of the four-axle configuration of Figure 8.6(d) with the nominal set of parameters given in Table 8.3.

256

RAIL VEHICLE DYNAMICS

additional to the minimum required for stability at low speeds enhances stability. Compared with the body steered vehicle, the range of stability is much increased for larger values of conicity and there is considerable scope for optimisation. However, divergence occurs for smaller values of kyd and larger values of kyf and kt when the conicity is low and the creep coefficient is reduced. It was seen above that this behaviour is characteristic of some articulated vehicles having three axles. The conditions for static stability depend in a complicated way on the inter-wheelset stiffnesses, as indicated for three-axle vehicles. In order to assess the practical significance of these results it is necessary to consider a typical solution of the complete nonlinear equations of motion for the directly steered system of Table 8.3. Figures 8.18(a) and (b) show, for the set of parameters indicated, the dynamic response to an initial condition y1(0) = 4 mm applied to the lateral displacement of the leading wheelset. The response is dominated by a lightly damped oscillation, following this transient, by the vehicle taking up a steady-state attitude in which overall balance of the creep forces is achieved. Figure 8.18(d) shows the stability boundaries, consistent with those shown in Figure 8.17, in the kyd (= kye) - O plane. As O is a function of the wheelset lateral displacement, it can be

O =

O = 0.05 f /2 O

V

S

D S kyd

kyd

O V

S D

S kyf

V

S

kyf

O D S

kt

kt

Figure 8.17 Qualitative diagram showing the joint influence of conicity, creep coefficient and stiffnesses kyd (= kye), kyf and kt on stability for a directly steered four-axle articulated vehicle. S denotes stability, D divergence and O oscillatory instability.

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257

(a)

4

1

2 y (mm)

4 3

0

-4 -6

2 0

2

4

6

b 3 1

0

b d

-2

(b)

1 \(mr) 0.5

-0.5

d

-1

2

-1.5 0

8

4

2

t (s)

4 t (s)

6

8

(c)

(d)

0.2

S

C

O A

D

0.1

B O1 0

O2 C1

A1 D 1

kyd = kye (MN)

2

Figure 8.18 Dynamic response and stability of vehicle with low conicity subjected to initial condition of 4 mm applied at leading wheelset. (a) and (b) lateral and yaw displacements (c) steady-state attitude of vehicle (d) stability chart. Parameters of Table 8.3 except that kyd = kye = 1 kN/m, kt = 2 MN/m, kf = 1 MN/m, creep coefficients halved and wheel-rail geometry consistent with O= 0.065.

258

RAIL VEHICLE DYNAMICS

0

2

\1 (mr)

0 y1 (mm) -10

3

t (s)

-5

10

5 T1l

T2r

(kN)

(kN) T1r

T2l

-10

-15

Figure 8.19 Dynamic response of linkage steered vehicle of Figure 8.6(d) to curve entry. Speed 15 m/s, length of transition 20 m., curve radius 225 m cant deficiency zero. y1 lateral displacement, \1 yaw angle, T1r and T1l longitudinal creep forces on right and left, T2r and T2l lateral creep forces, for leading wheelset.

2

0.5 0

0

\(mr)

y1 (mm) -10 0

t (s)

3

1.5 (kN)

2 T1l

0 T1r -1.5

-1.5

0

T2r

(kN)

T2l

-10

Figure 8.20 Dynamic response of self-steered vehicle of Figure 8.6(c) to curve entry. Speed 15 m/s, length of transition 20 m., curve radius 225 m cant deficiency zero.

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seen that if A corresponded to the starting point of the transient, the motion would decay towards the stability boundary at B, and in this case this point corresponds to a zero real root. Reference to equation (3.7.5) shows that for the equivalent linear system with a zero real root the response to an impulse tends to a motion in the mode corresponding to the zero real root. In fact, Figure 8.18(c) shows that the attitude of the vehicle is closely similar to that indicated by the corresponding linear system. If C were the starting point, the motion would decay to point D and a limit cycle. If A1 and C1 were the starting points then the motions would grow until the steady state is established at B and D respectively. It is clear that because, in this case, the instabilities O2 and D are associated with low conicity, displacements and creep forces will be small. This is not the case with the instability O1. The self-steered configuration therefore allows considerable freedom in the selection of the parameters. While both configurations can be expected to perform well on curves of constant radius, the forces generated in a transition or reverse curve will be influenced by the suspension parameters and there is conflict between the requirements for stability and dynamic response. This can be illustrated by comparing the dynamic response in curve entry for examples of the two configurations. Figure 8.19 shows a typical dynamic response of the linkage-steered vehicle of Figure 8.6(d) on entry to a curve in which the transition has linearly increasing curvature. The track is canted and is being traversed at the speed for zero cant deficiency. On the transition, the varying angles between the wheelsets and the car bodies induces forces in the steering linkage so that the leading wheelset fails to steer and moves out beyond the rolling line, large lateral and longitudinal creep forces being induced. These forces are significantly reduced when running on the uniform part of the curve as the wheelsets take up a radial position and attempt to move out to the rolling line. For a much smaller radius curve, the wheelsets are still able to take up a radial position but because of the restricted flange-way clearance, cannot move out to the rolling line and hence significant creep forces are generated. If the cant deficiency is not zero then further yaw movements of the wheelsets are necessary and the creep forces are increased. In the case of the self-steered configuration of Figure 8.6(c) it is possible to select suitably low values of the stiffnesses which, whilst maintaining stability for a range of conicities, also give low forces in the transition to a uniform curve. This is illustrated in Figure 8.20. References 1. White, J.H.: American Locomotives, Revised Ed., The John Hopkins Press, Baltimore, 1997, p. 509. 2. Weiner, L.: Articulated Locomotives, Constable, London, 1930. 3. Fryer, C.E.J.: A History of Slipping and Slip Carriages, Oakwood, Oxford, 1997 p. 7.

260

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4. White, J.H.: The American Railroad Passenger Car, The John Hopkins Press, Baltimore, 1978, p. 627. 5. Fidler, C.: British Patents 2399, 3825. 1868. 6. Liechty, R.: Das Bogenlaeufige Eisenbahn-Fahrzeug, Schulthess, Zurich, 1934, p. 31. 7. Barber, T.W.: British Patent 24632, 1907. 8. Liechty, R.: British Patent 390036, 1933. 9. Anon.: International Railway J. September, 1988. p. 26. 10. Jenkinson, D.: British Railway Carriages of the 20th Century, Volume 1: The End of an Era 1901-22. Patrick Stephens, Wellingborough, 1988, p. 155. 11. Jenkinson, D.: British Railway Carriages of the 20th Century, Volume 2: The Years of Consolidation 1923-53. Patrick Stephens, Wellingborough, 1990, p. 197. 12. Tachet, P. and Boutonnet, J.-C.: The Structure and Fitting Out of the TGV vehicle Bodies. French Railway Techniques 21, No. 1 (1978), pp. 91-99. 13. Rose, R.D.: Lenkung und Selbstlenkung von Einzelradsatzfahrwerken am Beispiel des KERF im S-Tog Kopenhagen. Proc. 4th International Conference on Railway Bogies and Running Gears, Budapest, 1998, pp. 123-132. 14. Slivsgaard, E. and Jensen, J.C.: On the dynamics of a railway vehicle with a single-axle bogie, Proc. 4th Mini Conference on Vehicle System Dynamics, Identification and Anomalies, Budapest, 1994, pp. 197-207. 15. Anon.: Wien Light Rail Vehicles by Duwag-Bombardier. Railway Gazette International, September, 1992, p. 586. 16. Charlton, E.H.: Articulated Cars of North America. Light Railway Transport League, London, 1966, Figure 34.

9 Unsymmetric Vehicles

1 Introduction Though early railway passenger and freight vehicles were generally symmetric, locomotives early adopted an unsymmetric configuration in order to maximise the axle-load on driving wheels and make use of the available adhesion. Additional smaller wheelsets were soon provided to improve the steadiness of running. It has already been mentioned, in Chapter 6, how the introduction of the leading swivelling bogie improved both stability and curving behaviour, and it was a matter of experience that the running behaviour of these unsymmetric configurations was strongly dependent on the direction of motion [1]. A later example of an unsymmetric vehicle design, used in trams, is provided by “maximum traction trucks” devised in the 1890’s and in which the driving wheels were followed or preceded by pony wheels which had a diameter about two-thirds of the driving wheels [2]. In this case each bogie was unsymmetric but the complete vehicle was usually symmetric. The classic work on the stability of unsymmetric railway vehicles is by Carter [3] which was directed toward the configurations then current in railway practice. Carter applied Routh's stability theory, not only to electric bogie locomotives, then exhibiting many problems of instability, but also to a variety of steam locomotives. In his mathematical models, a bogie consists of two wheelsets rigidly mounted in a frame, and locomotives comprise wheelsets rigidly mounted in one or more frames. Following Carter’s first paper of 1916 the theory was elaborated in a chapter of his 1922 book [4]. Carter’s next paper [5] gave a comprehensive analysis of stability within the assumptions mentioned above. As he was concerned with locomotives the emphasis of his analyses was on the lack of fore-and-aft symmetry characteristic of the configurations he was dealing with, and he derived both specific results and design criteria. Carter's work expressed, in scientific terms, what railway engineers had learnt by hard experience, that stability at speed required rigid-framed locomotives be unsymmetric and uni-directional. His analysis of the 0-6-0 locomotive found that such locomotives were unstable at all speeds if completely symmetric and he comments that this class of locomotive is “much used in working freight trains; but is not employed for high speed running on account of the proclivities indicated in the previous discussion.” Carter analysed the 4-6-0 locomotive both in forward and reverse motion and found that in forward motion beyond a sufficiently high speed or sufficiently stiff

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262

(a)

(b) 60

60

O

V (m/s)

O

V (m/s) 40

40

D 20

20

S

S 0

1

2 3 ky (MN/m)

4

0

1

2 3 ky (MN/m)

4

Figure 9.1 Carter’s stability diagram for the 4-6-0 locomotive in (a) forward motion and (b) reverse motion. ky is the centring stiffness. (Recalculated in modern units from [5]). S = stable; O = oscillatory instability; D = divergence.

bogie centring spring (laterally connecting the bogie to the locomotive body) oscillatory instability occurs, but as the mass of the bogie is small compared with the main mass of the locomotive, the resulting oscillation was unlikely to be dangerous at ordinary speeds. Carter’s stability diagrams, the first of their kind in the railway field, are shown in Figure 9.1(a) and 9.1(b). In reverse motion, Figure 9.1(b), Carter found that beyond a certain value of the centring spring stiffness buckling of the wheelbase occurred which would tend to cause derailment at the leading wheelset. As this wheelset is incorporated in the main frame of the locomotive, the lateral force acting between wheel and rail would be proportional to the mass of the main frame and would be correspondingly large, and potentially dangerous. This was the explanation of a number of derailments at speed of tank engines such as the derailment of the Lincoln to Tamworth mail train at Swinderby on 6 June 1928, as discussed in his final paper [6]. Carter’s analysis of the 2-8-0 with a leading Bissel, or single-axle bogie, similarly explained the need for a very strong aligning couple for stability at high speed, whilst noting that in reverse motion a trailing Bissel has a stabilising effect for a large and useful range of values of aligning couple. Rocard [7,8] also considered unsymmetric configurations and proposed the use of different conicities fore-and-aft. A general theory for the stability of unsymmetric vehicles and the derivation of theorems relating the stability characteristics in forward motion with those in reverse motion was given in [9,10]. In the case of articulated two-axle vehicles at low speeds it was shown that a suitable choice of elastic restraint in the inter-wheelset connections results in static and dynamic stability in forward and reverse motion and which will steer perfectly, without any modification dependent on the direction of motion. However, the margin of stability is small. A further practical result of Carter’s work was a series of design measures, the subject of various patents [11], for the stabilisation of symmetric electric bogie locomotives, because the introduction of the symmetric electric locomotive had been accompanied by a more or less common experience of lateral instability at high speed. In particular, Carter suggested an arrangement of the running gear in which

UNSYMMETRIC VEHICLES

263

the configuration was altered depending on the direction of motion. This makes it easier to resolve the conflict of stability and steering with an adequate margin of stability. Such re-configurable systems have been studied by Li [12]. Chapter 3 discusses the use of the pony axle (a wheelset mounted on a leading or trailing arm) in conjunction with freely rotating wheels, and the example of the Talgo train was cited, in which the wheelsets are mounted on the car bodies in an unsymmetric way. A further source of lack of symmetry in railway vehicles can occur when a configuration which is intended to be symmetric suffers from badly distributed loading or unequal wheel wear, thus giving rise to asymmetry about a transverse plane. Asymmetry about a longitudinal plane can, of course, also occur for the same reasons; however, this tends to merely alter the equilibrium position of the vehicle, and, while it should be taken into account in detailed numerical calculations, no new phenomena are introduced. Moreover, there is evidence to suggest that asymmetry about a transverse plane due to unequal wheel wear is more important for vehicles which have inherently poor curving ability, such as the three-piece freight truck. In this case, calculations have been described by Tuten, Law, and Cooperrider [13]. Illingworth [14] suggested the use of unsymmetric stiffness in steering bogies. Elkins [15] showed both by calculation and experiment that a configuration of bogie, with the trailing axle having independently rotating wheels and the leading axle conventional, significantly improved stability and curving performance and reduced rolling resistance. Suda et al [16-17] have studied bogies with unsymmetric stiffnesses and symmetric conicity, and their development work has led to application in service [18]. The concept has been extended to include lack of symmetry of the wheelsets by equipping the trailing axle with freely rotating wheels. This provides a practical example of a re-configurable design as the wheelsets are provided with a lock which is released on the trailing wheelset (allowing free rotation of the wheels) and locked on the leading wheelset (providing a solid axle). The lock is switched depending on the direction of motion. In modern vehicles the provision of secondary suspension between bogies and car body has the result that lack of symmetry in the car body rarely introduces new phenenoma. However, lack of symmetry in the bogie itself can be exploited to improve performance and this is discussed below.

9.2 Stability Theorems for Rigid and Semi-Rigid Vehicles Important insights are obtained by considering, first of all, the static and dynamic stability of vehicles in which the wheelsets are incorporated rigidly in one or more frames, which are themselves connected by joints imposing non-elastic constraints. Thus, in these cases the elastic stiffness matrix E is null. A rigid vehicle is one in which all the interwheelset elastic stiffnesses are infinite, so that the number of degrees of freedom M = 2, corresponding to lateral translation and yaw. A semi-rigid vehicle is defined as one in which E is null but 2N > M > 2, where N is the number of wheelsets, so that in addition to possessing rigid body freedoms, the vehicle is able to articulate like a mechanism. Now referring to the simple form of the equa-

RAIL VEHICLE DYNAMICS

264

tions of motion of a single wheelset, because the wheelset is symmetrical about a transverse plane through its centre, for a motion in the reverse direction, while retaining the same definition of generalised coordinates, the equations of motion (2.10.47-48) become y + 2f22 y /V + kyy + 2f22ψ = Qy m  (1) 2

-2f11λly/r0 + Iz ψ + 2f11l ψ /V + kψψ = Qψ

(2)

and in general the equations of motion for reversed motion of a complete vehicle are [ As2 + Bs/V - C + E] q

= Q

(3)

Thus, the equations of reversed motion are obtained by reversing the sign of the creep stiffness matrix. In the case of the semi-rigid or rigid vehicle the equations of motion reduce to [As2 + (B/V)s ± C ] q

= Q

(4)

where + refers to forward motion and - refers to reversed motion. The trial solution q α est leads to the determinantal equation

⏐As2 + (B/V)s ± C⏐ = 0

(5)

On making the substitution s = VD this becomes

⏐AV2 D2 + BD ± C⏐ = 0

(6)

and for low speeds, this reduces to

⏐BD ± C⏐ = 0

(7)

which expands to, for forwards motion, pMDM + pM-1DM-1 +..........+ p1D + p0 = 0

(8)

For reversed motion, since all the elements of C change sign, some of the pi change sign. In particular, p0 and p1 will have opposite signs. The condition for static stability is that p0 > 0 and a necessary (but not sufficient) condition for dynamic stability is that all the pi have the same sign. Hence, if an unsymmetric rigid or semi-rigid vehicle is statically and dynamically stable in forward motion, it must be dynamically unstable in reverse motion. If the number of degrees of freedom M is even, p0 does not change sign in reversed motion and if the vehicle is statically stable in both directions, it will be dynamically unstable in one direction. If M is odd, p0 does change sign in reversed motion and the vehicle will be statically unstable in one direction of motion.

UNSYMMETRIC VEHICLES

265

Hence, an unsymmetric semi-rigid or rigid vehicle can possess a margin of stability in one direction only. This margin of stability is derived from the lack of symmetry, for if the vehicle were symmetric since (8) must be invariant with the direction of motion it follows that pM-1, pM-3,...... must all be zero. Then, in many cases (8) can be factorised in the form (for M even) pM(s2 + ω12) (s2 + ω22)........ (s2 + ωP2) = 0

(9)

where ω1, ω2,......... ωP are a set of P = M/2 steering frequencies, or (for M odd) pM(s2 + ω12) (s2 + ω22)........ (s2 + ωP2)s = 0

(10)

involving a set of P = (M-1)/2 steering frequencies, with the addition of a zero root. This zero root indicates that the vehicle is capable of quasi-static misalignment similar to that discussed for two-axle vehicles in Section (4.3). In other cases, roots coalesce to form a pair of roots ±µ ± iω indicating oscillatory instability or alternatively roots of the form ±µ occur indicating divergence. The assumption of low speeds has made it possible to neglect the inertia terms in the equations of motion. For semi-rigid vehicles in the more general case, (6) leads to a characteristic equation of order 2M. However, the coefficients p1 and p0 will remain unchanged, and it follows that irrespective of speed, an unsymmetric semirigid or rigid vehicle will be unstable in at least one direction of motion. This suggests that the examination of stability at low speeds is likely to yield useful information on the behaviour of various configurations, as is done in the following.

9.3 Unsymmetric Rigid Vehicle The general theory is exemplified by the unsymmetric rigid vehicle as considered by Carter [5]. Whereas Carter considered rigid assemblies of an arbitrary number of wheelsets, it will be sufficient for present purposes to consider only the case of a two-axle vehicle or bogie. If y and ψ represent the lateral displacement and yaw of the vehicle then the standard form of the simplified equations of motion of Section 2.10 applies where ⎡ m0 m 0 c ⎤ A=⎢ ⎥ ⎣ m0 c I 0 ⎦ ⎡4 f B=⎢ ⎣0

(1)

⎤ 4 f (h + l ) ⎥⎦ 0

2

2

(2)

RAIL VEHICLE DYNAMICS

266

−4f 0 ⎤ ⎡ C=⎢ ⎥ ⎣2 f (α1 + α 2 ) 2 fh(α1 − α 2 ) ⎦

(3)

where m0 = 2m + mb and I0 = Izb + 2mh2 and c is the distance of the centre of mass ahead of the mid point, and α1 = λ1l/r0, α2 = λ2 l/r0. The characteristic equation of this system is p4s2 + p3 s3 + p2 s2 + p1 s + p0 = 0

(4)

At low speeds, this reduces to p2 s2 + p1 s + p0 = 0

(5)

and then p2 = 2(h2 + l2)/V2 p1 = h(α1 − α2)/V p0 = (α1 + α2) In accordance with the general theory discussed in the previous section, since the number of degrees of freedom is even, p0 is invariant with change in direction of motion and is essentially positive. On the other hand, the sign of p1 changes when the direction of motion changes. Thus, the distribution of conicity can be arranged to give stability in one direction of motion, but not in both. This was suggested as a means of stabilising a vehicle by Rocard [7] who states that a successful experiment was made by French National Railways in 1936. Equation (5) yields an eigenvalue µ ± iω where, if p12 << 4p0p2,

µ=−

ω2 =

h(α1 − α 2 )V 4( h 2 + l 2 )

(α1 + α 2 )V 2 2( h 2 + l 2 )

(6)

(7)

This steering oscillation was discussed in Section 5.3 for a symmetric vehicle and is the analogue for a rigid vehicle of the kinematic oscillation of a single wheelset. If the vehicle is symmetric, the oscillation is undamped, though it may be readily verified that creep occurs throughout the motion. If the vehicle is unsymmetric, the damping in the steering oscillation will be positive if the conicity of the leading wheelset is larger than that of the trailing wheelset. Consider now the stability of the vehicle at speed, firstly assuming that c = 0. Making the reasonable assumption that Izb = m0 (h2 + l2), it follows from the form of the equations of motion that the characteristic equation (5) for speed V1, with eigen-

UNSYMMETRIC VEHICLES

267

values σi, will be the same as (4) for speed V2, with eigenvalues si, if 4fσι/V1 = m0si2/2 + 4fsi /V2

(8)

Equation (8) can be regarded as a quadratic in s. Typically, the solutions of (4) will consist of a complex eigenvalue together with two large real roots s1 = -4f/m0V

(9)

s2 = -4f(h2 + l2)/I0V

(10)

The combined effect of speed and inertia is therefore to modify the damping of the steering oscillation, and to introduce subsidences in lateral translation and yaw of the vehicle. It can be seen that in the case of a steering oscillation which is stable for low speeds, so that the corresponding real part of the eigenvalue σ = µ ± iω is negative, the damping will vanish at a speed VB and the corresponding eigenvalue will be si = ± iΩ. By separating real and imaginary parts (Ω /VB) = (ω /V)0

(11)

VB2 = −4f(µ/V)0 /m0(ω/V)02

(12)

The instability is an inertia driven instability of the steering oscillation. Substituting from (6) and (7), (12) yields V B2 =

2(α1 − α 2 ) fh m0 (α1 + α 2 )

(13)

For a bogie with the parameters of Table 4.1 with λ1 = 0.15 and λ2 = 0.075 equation (13) gives a bifurcation speed VB = 34.4 m/s indicating that lack of symmetry offers scope for useful enhancement of stability. Numerical solutions of (4) show that variation of the longitudinal position c of the centre of mass of the vehicle have only small influence on the bifurcation speed.

9.4 Steering of a Vehicle with Unsymmetric Inter-Wheelset Structure It was shown in Chapter 5 that the behaviour of a symmetric vehicle is dominated by the elastic connections between the wheelsets, and it can therefore be expected that lack of symmetry in that respect may be important. In considering the properties of the stiffness matrix for a general unsymmetric inter-wheelset structure, the argument of Section 4.2 may be followed but in the present case fore-and-aft symmetry is not assumed. In terms of the sum-and-difference coordinates of Section 4.2 the stiffness matrix then takes the form [10]

RAIL VEHICLE DYNAMICS

268

k sb ⎡ ks ⎢ k kb sb E=⎢ ⎢ 0 0 ⎢ ⎣− k s h − k sb h

0 − k sh ⎤ 0 − k sb h ⎥⎥ 0 0 ⎥ ⎥ 0 k sh2 ⎦

(1)

and there are only three independent elements. In addition to the bending and shear stiffnesses kb and ks already defined, there is a coupling term ksb indicative of lack of fore-and-aft symmetry. For a symmetric vehicle ksb = 0 and E reduces to the form given in Section 4.2. Reverting to the usual definition of the generalised coordinates, using equations (4.2.3), so that y1 and y2 are the lateral displacements measured from the centreline of the track, and ψ1 and ψ2 are the yaw angles, E becomes ks k sb − k s h ⎡ ⎢ k − k h k − 2k h + k h 2 sb s b sb s E=⎢ ⎢ − ks − k sb + k s h ⎢ − kb + k sh 2 ⎣− k sb − k s h

− ks − k sb + k s h ks k sb + k s h

− k sb − k s h ⎤ − k b + k s h 2 ⎥⎥ ⎥ k sb + k s h 2⎥ k b + 2 k sb h + k s h ⎦

(2)

Consider the steady-state behaviour on a uniform curve for the case of zero cant deficiency. Then for coned wheels and small displacements, the gravitational stiffness and lateral force due to spin creep, together with the other small terms, can be neglected so that the equations of equilibrium, corresponding to the steady-state form of equations (2.10.47-48), reduce to [ C + E ] q + Eq0 = Qc

(3)

where, assuming that f11 = f22 = f for simplicity, 0 ⎡ ⎢2 fλ l / r 1 0 C = ⎢ ⎢ 0 ⎢ 0 ⎣

−2f 0 0 0

0 0 0 2 fλ2 l / r0

0 ⎤ 0 ⎥⎥ −2f ⎥ ⎥ 0 ⎦

(4)

E is given by (2) and Qc = [ 0 - 2fl2/R0 0 - 2fl2/R0]T

(5)

and the forces applied to the wheelset due to straining of the suspension to achieve the reference position in the chosen coordinate system are given by Eq0 where q0 = [ 0 h/R0

0

-h/R0 ] T

(6)

UNSYMMETRIC VEHICLES

269

For motion on a curve with zero lateral and longitudinal creep

[

q = − lr0 / λ1 R0

0 − lr0 / λ2 R0

]

0

T

(7)

The solution of equations (3) is satisfied by (7) if both k sb lr0 ⎛ 1 1⎞ = ⎜ − ⎟ 2h ⎝ λ1 λ2 ⎠ ks

(8)

kskb - ksb2 = 0

(9)

and

These are the conditions for steering on a uniform curve with zero creep and are the same for forward and reverse motion. It can be seen that (9) is the generalisation, applicable to the present case of unsymmetric elasticity, of the requirement that for perfect steering in a uniform curve the bending stiffness must be zero. Similarly, (8) is equivalent to a geometric requirement that a "bending displacement" about the point of articulation is compatible with lateral displacements of the wheelsets necessary, by their respective conicities, to ensure pure rolling. Figure 9.2 indicates the geometry when a vehicle with a structural arrangement of the kind meeting the criteria (8) and (9) negotiates a curve. For zero creep, both wheelsets must adopt a radial position with outward displacements lr0/R0λ1 for the leading wheelset and lr0/R0λ2 for the trailing wheelset. If the radial displacements are small compared with the radius of the curve, the geometry of Figure 9.2 then yields (8).

h + ksb/ks

lr0/R0λ2

h - ksb/ks

lr0/R0λ1

Figure 9.2 Attitude of unsymmetric vehicle negotiating a uniform curve with zero longitudinal and lateral creep

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(a) 2a

k

θ

2b

ks = k(b - a)sin2θ ksb= k(b - a)sinθcosθ/2 kb = k(h2sin2θ − abcosθ)

(b) h1

h2

k

h

h

ks = k ksb = k(h2 - h1)/2 kb = k(h2 - h1)2/4

Figure 9.3 Examples of unsymmetric inter-wheelset structures which satisfy the condition given by equation (9).

An alternative approach due to Suda [16,17] imposes the steering condition that ψ1 and ψ2 are zero, so that the lateral creep is zero in a curve. It is possible to meet this criterion using only the unsymmetric inter-wheelset structure, with the wheelset conicities being equal. However, the wheelsets do not move out to the rolling line and longitudinal creep forces are not zero. For the wheelsets to take up a radial position and move outwards to the rolling line on a curve both conditions (8) and (9) must be satisfied and this requires lack of symmetry in both the distribution of conicity and elasticity. These considerations may be illustrated with reference to the example unsymmetric inter-wheelset structures shown in Figure 9.3 which obey (9). Figure 9.3(a) shows an arrangement based on inclined tension members which provide a virtual pivot, and Figure 9.3(b) shows an arrangement embodying a single shear spring at the articulation point. In both cases a pivot position (virtual or real) can be provided at any point within (or, theoretically, beyond) the wheelbase. Another way in which (8) or (9) can be implemented is by offsetting the lateral stiffnesses ky1 and ky2 as shown in Figure 9.4 (a). Suda [16,17] has studied and experimented with the scheme shown in Figure 9.4(b) (applying the condition for zero lateral creep described above) and this is the basis for a design of bogie actually put into commercial service.

UNSYMMETRIC VEHICLES

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(a)

ha

hb

k y1/2

k y2/2

kψ

kψ

h

h

ks = 2ky1ky2kψ/ ∆ ksb = ky1ky2kψ(h a - h b)/ ∆ kb = [(ky1 + ky2)kψ2 + ky1ky2kψ{(h + h a)2 + (h + hb )2}/ ∆ ∆ = 2(ky1 + ky2)kψ + ky1ky2(2h + h a + h b)2} (b) k y/2

kψ1

k y/2

kψ2

ks = (kψ1 + kψ2)ky/ ∆ ksb = hky (k ψ2 - kψ1)/ ∆ kb = {2kψ1 kψ2 + h 2ky(kψ2 + kψ1)}/∆ ∆ = 2(kψ1 + kψ2) + 4h 2 ky Figure 9.4 Two examples of unsymmetric structures connecting wheelsets to bogie frame.

The possible range of configurations of vehicle which satisfy (8) and (9) is constrained by the practical considerations mentioned in connection with the three-axle vehicle. It is assumed that that values of the non-dimensional equivalent conicities α1 and α2 are always less than unity, and the ratio of conicity between leading and trailing wheelsets is will not exceed about two, or exceptionally, three. Then Figure 9.5 summarises the possible range of configurations which can steer without lateral and longitudinal creep on curves. It can be seen that the pivot position falls within the wheelbase for those configurations which meet these practical constraints. This is indicated by the lines A and B giving the relationship between α1 and α2 for the pivot position at the front and rear axles respectively. The lines C and D represents

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272

0.5

A

G

0.4

E

α1

F

C

0.3

0.2

D 0.1

0

B 0

0.1

0.2

H

0.3

0.4

0.5

α2 Figure 9.5 Configurations of unsymmetric vehicles capable of steering with zero longitudinal and lateral creep.

the limits of a 2:1 ratio in conicity between the axles. A smaller ratio in conicity leads to a pivot position between the wheelsets as shown at E and F. Conversely, configurations which do not meet these practical constraints require pivot positions well outside the wheelbase as shown at G and H.

9.5 Stability of a Two-Axle Articulated Vehicle Consider the stability at low speeds of an unsymmetric two-axle vehicle. For simplicity, it is assumed that the creep coefficients for each wheelset are equal. Equations of motion of the form (2.7) are obtained where ⎡2 f ⎢0 B=⎢ ⎢0 ⎢ ⎣0

0 2 fl 2

⎡ 0 ⎢2 fα 1 C=⎢ ⎢ 0 ⎢ ⎣ 0

−2f

0

0 0 0

0 0 2 fα 2

0 0

0 0 2f 0

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 2 fl 2 ⎦ 0 ⎤ 0 ⎥⎥ −2f ⎥ ⎥ 0 ⎦

(1)

(2)

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273

and E is given by (4.2). The trial solution q α est leads to a characteristic quartic polynominal with coefficients given by p4 = 1 p3 = (ksl2 + ksh2 + kb)V/fl2 p2 = {2(kskb - ksb2)(l2 + h2) + 2f 2l2(α1 +α2) ± ksfhl2(α1 -α2) ± ksbfl2(2- α1 -α2)}V2/2f2l4 (3) p1 = {± (kskb - ksb2)(α1 -α2)h + 2ksbfh(α1 -α2) + f(ksl2 + ksh2 + kb) (α1 +α2)}V3/2f2l4 p0 = {(kskb - ksb2)(α1 +α2) + 2f2α1α2 ± ksfh(α1 -α2) ± ksbf(α1 +α2 - 2 α1α2)}V4/2f2l4 where ± refers to forward (+) or reversed (-) motion, respectively, and αi = λil/ri. The semi-rigid case will be considered first so that for the vehicle shown in Figure 9.3(b) k = ∞. Then (3) reduces to the cubic polynominal with coefficients given by p3 = (2l2+h12+h22)l2 p 2 = ±h1 (α1 − 1)Vl 2 ± h2 (1 − α 2 )Vl 2 (4) p1 = α 2 ( l

2

+ h12 )V 2

+ α1 ( l

2

+ h22 )V 2

p 0 = ±α 2 h1 (α1 − 1)V 3 ± α1h2 (1 − α 2 )V 3 where the ± signs refer to forward and reverse motion respectively. Figure 9.6 shows how the eigenvalues vary as the equivalent conicity of the trailing wheelset varies for the bogie with the parameters of Table 4.1 except that α1 = 0.22 and h1 = 0.75 m. It can be seen that the eigenvalues when α2 = α1 comprise a zero root corresponding to the quasi-static misalignment discussed in Section 4.3 and an oscillatory root µ ± iω corresponding to the kinematic oscillation of a single wheelset. However, for the unsymmetric vehicle, when α2 < α1 there is dynamic instability and static stability, and for α2 > α1 there is dynamic stability. When α2 is sufficiently greater than α1 there is divergence. In reverse motion, the frequency of the steering oscillation remains unaltered but the real parts of the eigenvalues change sign. This is, of course, in accordance with the discussion in Section 2.

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1

ω1

0.5

ω 0

ω2 0.2 0.4 0.6 α2

0

0.4 0.2

µ1

0

µ

-0.2 -0.4

µ2 0

0.2

0.4 α2

0.6

Figure 9.6 Behaviour of eigenvalues of semi-rigid unsymmetric vehicle at low speeds as conicity of rear wheelset is varied. Parameters of Table 4.1 except that α1 = 0.22, h1 = 0.75.

If µ1 is small, the characteristic polynomial can be factorised approximately, yielding a real root µ1

µ1 = −

p0 {α h (α − 1) + α1h2 (1 − α 2 )}V = − 2 1 21 p1 α 2 (l + h12 ) + α1 (l 2 + h22 )

(5)

and a complex pair λ = µ ± iω where

ω2 =

p1 {α 2 ( l 2 + h12 ) + α1 (l 2 + h22 )}V 2 = p3 ( 2l 2 + h12 + h22 )l 2

µ=−

=−

(6)

( p 2 p1 − p 3 p 0 ) 2 p 3 p1

(α 2 − α1 ){h1 ( l 2 + h22 )(1 − α1 ) + h2 ( l 2 + h12 )(1 − α 2 )}V 2l 2 ( 2l 2 + h12 + h22 ){α 2 ( l 2 + h22 ) + α1 ( l 2 + h12 )}

(7)

UNSYMMETRIC VEHICLES

275

h1 = 0

1

1

h1 = h/3, h2 = 2h/3 D

S

S

α2

α2

O

O

0

0

1

α1

1

α1

h1 = h2 = h

1

D

α2 O

0

1

1

α1

h1 = 2h/3, h2 = h/3 D

α2

h2 = 0

1 D

α2

D,O

D,O

O 0

α1

1

0

α1

1

Figure 9.7 Regions of stability in the α1, α2 plane as a function of h1 and h2 for the semi-rigid vehicle at low speeds; S = stability, D = divergence and O = oscillatory instability.

Though these expressions are based on an approximate solution of the characteristic equation, they do describe accurately enough the behaviour of the system as the conicities and pivot position vary. The real root represents a subsidence or divergence depending on the relative magnitudes of h1 and h2 and α1 and α2. In fact, the exact condition for static stability is p0 > 0 or

α 2 h1 (α1 − 1) + α1 h2 (1 − α 2 ) > 0

(8)

α 2 < 1 / {1 + h1 (1 − α1 ) / α1 h2 }

(9)

or

The oscillatory root represents an oscillation with frequency roughly equal to the mean frequency of the kinematic frequencies of the two wheelsets, and is influenced

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276

by joint position. The damping in this oscillation is strongly dependent on the difference in equivalent conicities and the pivot position exercises a less important influence. The condition for stability is simply

α 2 > α1

(10)

The regions of stability, as given by (9) and (10) are plotted in the α1, α2 plane for various spring positions in Figure 9.7. It is possible to select parameters which will ensure both static and oscillatory stability for one direction of motion, but these parameters would lead to both static and oscillatory instability in reverse motion. Since, in most practical cases, α1 <1 and α2 < 1 a necessary condition for static stability is a1h2 > a2h1. This criterion can be combined with (10) to give the necessary but not sufficient condition h2/h1 > α2/α1 >1

(11)

Thus, for stability in forward motion, the pivot must be nearer the front wheelset than the back wheelset and the rear wheelset must have larger conicity (e.g., smaller radius) than the front wheelset. These results contrast with those obtained for the rigid vehicle.

9.6 The Influence of Elastic Stiffness on Stability The influence of the stiffness on the stability of the two-axle vehicle shown in Figure 9.3(b) is now considered. The analysis will show that the configuration studied is not only stable for a range of parameters in both forward and reverse motion, but it is also capable of perfect steering. Consider the behaviour of the eigenvalues as a factor k on the stiffnesses ks, kb and ksb is varied, and the steering conditions (4.8) and (4.9) are applied. There are two regimes in which the behaviour is quite distinct. The first of these is for small values of kl/2f and the second is for all other values of kl/2f. As will be explained, the former regime is one of small interaction between the wheelsets and the latter regime is one of intensive interaction. When kl/2f = 0 the eigenvalues are purely imaginary and distinct, corresponding to undamped kinematic oscillations. As the eigenvalues are distinct it is possible to employ the perturbation analysis of Section (3.3), equation (3.3.12). Therefore, for small values of kl/2f , the eigenvalues are µ1 ± iω1 and µ2 ± iω2 where ω12 =

V 2 λ V 2 ( k sb − k s h)(1 − λ1l / r0 ) ± lr0 2 fl 2

(1) µ1 = −V ( k s l 2 + k s h 2 + k b − 2 k sb h) / 4 fl 2

UNSYMMETRIC VEHICLES

277

ω 22 =

V 2 λ V 2 ( k sb + k s h)(1 − λ2 l / r0 ) ± lr0 2 fl 2

(2) µ2 = −V ( k s l + k s h + k b + 2 k sb h) / 4 fl 2

2

2

the ± signs referring to forward and reverse motion respectively. Both µ1 and µ2 are negative for all values of ks and kb and the system is stable. These results are consistent with a wheelset oscillating under asymmetric elastic restraint, the other wheelset being restrained from lateral motion. In fact, (1) can be derived by the solution of the binary set of equations of motion involving y1 and ψ1, whilst (2) similarly can be derived from the sub-system involving y2 and ψ2. The analysis is then a simple extension of that for a single symmetric wheelset given in Chapter 3, to which the present analysis is reducible. These two wheelset subsystems (leading and trailing wheelsets) are, for the small values of kl/2f under consideration here, weakly-coupled in the sense of Milne [19], and his analysis could be applied to give more complete rigour. The validity of the perturbation method applied here depends on the eigenvalues of the sub-systems being sufficiently separated. This requirement is, of course, not satisfied for the symmetric vehicle with equal conicity on both wheelsets, and a different approach is needed as employed in Chapter 4. As kl/2f is increased the interaction between the wheelsets dominates the behaviour of the vehicle. Some insight into the pattern of behaviour is given by the results for the symmetric vehicle discussed in Chapter 4, where it was shown that two modes of oscillation exist; a lightly damped steering oscillation which takes place at the kinematic frequency and a "shear" oscillation which for smaller values of kl/2f takes place at the kinematic frequency. As the real part of the eigenvalue increases rapidly as kl/2f increases this oscillation is replaced by two subsidences. Similar results may be expected for the unsymmetric vehicle, but there is the possibility of static and oscillatory instability. The condition for static stability is that p0 > 0 and if the steering conditions (4.8) and (4.9) are applied, this becomes 2fα1α2 ± ksh(α1 -α2) ± ksb(α1 +α2 - 2 α1α2) > 0

(3)

which reduces to (5.8) as kl/2f becomes large. Routh’s condition for oscillatory stability is that the test function of equation (3.4.3) should be positive or, on substitution ±(α1 -α2)[-A2{± f(α1 -α2) - ksh(2 −α1 - α2) - ksb(α1-α2)} + Α ksbh{2ksh(α1 -α2) - 2ksb(α1+α2) + 4ksb} ± 4ksb2h2f(α2 -α1)] > 0

(4)

where A = k s l 2 + k s h 2 + k b . Equation (4) reduces to (3.10) as kl/2f becomes large. For small values of kl/2f, for the vehicle shown in Figure 9.3(b), (4) is satisfied

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278

for all values of the parameters consistent with (1) and (2), and the vehicle is oscillatory stable for both forward and reverse motion, except when α1 = α2 when it is of course marginally stable, irrespective of the position of the spring. Figure 9.8 summarises these results in the form of a stability diagram in the ks, h1 plane, for both forward and reverse motion at low speeds, where the position of the spring and the conicities are chosen so that (4.8) and (4.9) are satisfied. As the position of the spring moves aft from the central position, there is the possibility of both oscillatory and static instability. If the spring were placed at the trailing axle, then from (5) for oscillatory stability k<

f {(l 2 + 4h 2 ) 2 + 4h 4 }(α1 − α 2 )

(5)

4h 3 (l 2 + 2h 2 )(1 − α 2 )

and from (3) for static stability k<

fα 1 h(1 − α1 )

(6)

As the position of the spring moves forward from the central position neither oscillatory or static instability occurs. Thus, if the criteria (5), and (6) are satisfied, the vehicle possesses static and oscillatory stability, at low speeds, for either direction of motion and the ability to steer perfectly on curves. This result may be contrasted with the conflict between steering and stability exhibited by the symmetric two-axle vehicle discussed in Chapter 4.

(a)

2

2

O,D

1.5 ks (MN/m) 1

D

1.5 ks (MN/m) 1

O

S

(b)

0.5

O

S

0.5

S 0

1.25 0.1 0.08

0.06

h1 (m) 0.05

λ2

S 2.5

0.04

0

h1 (m) 2.5

1.25 0.1 0.08

0.06

0.05

λ2

0.04

Figure 9.8 Stability diagram for unsymmetric vehicle at low speeds for the case where equations (4.8) and (4.9) are satisfied. α1 = 0.1. (a) forward motion (b) reverse motion.

UNSYMMETRIC VEHICLES

279 (a)

10

O

V (m/s) 6

2 0

(b)

10

O

V (m/s) 6

S

S 1.25

h1 (m) 2.5

2 0

S

S 1.25

h1 (m)

2.5

Figure 9.9 Stability diagram for unsymmetric vehicle for the case where equations (4.6) and (4.7) are satisfied. (a) forward motion (b) reverse motion. ks = 10 kN/m.

Figure 9.9 shows the stability boundaries for the vehicle in both forward and reverse motion as a function of the effective pivot position and vehicle speed. It can be seen from these results that the choice of parameters in order to achieve static and dynamic stability and steering without creep, in both directions of motion, is limited, as only moderate values of the shear stiffness are destabilising. For physically realisable masses the critical speed associated with instability in the steering mode (driven by the inertia forces) is quite low. Even if stability at low speeds is achieved, the margin of stability is very small resulting in low damping in the steering oscillation at low speeds. The significance of this simple example lies not in its direct practical use but in the fact that it demonstrates that two-axle configurations exist that resolve the conflict between steering and stability.

9.7 Applications of Unsymmetry Considering the model analysed in the previous Section, suppose that, in addition to the shear stiffness k1 at the pivot point, yaw restraint k2 is also provided. This could be provided by a relaxation damper so that steering at low speeds is not compromised, the conditions for steering without creep, equations (4.8-9) will be satisfied, and stability at speed will be enhanced. Then Figure 9.10 shows the effect on stability of separately introducing different conicities on the leading and trailing wheelsets and elastic unsymmetry in the interwheelset connections. The bifurcation speed is shown as a function of the shear and bending stiffnesses and these results can be compared with those for the symmetric bogie shown in Figure 5.20. If the conicity is greater on the trailing wheelset than on the leading wheelset, and if the effective pivot point is towards the rear of the vehicle, then stability is higher than the corre-

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280

λ1 = 0.075, λ2 = 0.15 25 k2 (MNm)

h1 = 0.625, h2 = 1.875 25

VBMAX = 2.08V0

k2 (MNm)

forward VBMAX = 0.86V0 0

k1 (N/m)

25

25

0

k1 (N/m)

25

2.5 VBMAX = 1.77V0 k2 (MNm)

k2 (MNm)

reverse VBMAX = 0.91V0

0

k1 (N/m)

25

0

k1 (N/m)

25

Figure 9.10 Stability diagrams for unsymmetric vehicle with both shear stiffness k1 and yaw stiffness k2 at joint showing the effects of differing conicities (left hand) and joint position (right hand) in forward motion (top row) and reverse motion (bottom row). V0 is the maximum bifurcation speed when λ1 = λ2 = 0.1 and h1 = h2 = 1.25.

sponding symmetric vehicle for a range of stiffnesses. But, in each case, operation in reverse results in lower bifurcation speeds. As the margin of stability is adequate for one direction of motion if appropriate parameters are chosen this suggests the application of a variable configuration which changes with the direction of motion. For the system discussed above, in order to preserve the conditions for steering without creep at low speeds it would be necessary to vary both the position of the effective pivot point and the ratio of the conicities. Obviously, this would be very difficult to achieve in practice. A practical application of similar ideas has been described by Suda [16,17] in which depending on the direction of travel, elastic stiffness parameters are switched fore-and-aft. Three configurations are considered (a) conventional or radial bogies with unsymmetric interwheelset connections (b) bogies with symmetric interwheelset structures but using independently rotating wheels on the trailing axle (c) bogies with a combination of unsymmetric inter-wheelset connections and independently

UNSYMMETRIC VEHICLES

281

rotating wheels on the trailing axle. Suda proposes that the change from independent rotation of the wheels to a solid axle when the direction of motion is changed is effected using an electromagnetic clutch. In (a) it is found that even though the steering condition is relaxed stability is low. A practical method of increasing stability in this case is to switch the parameters when the direction of motion is changed is by using switched primary longitudinal dampers in the conventional configuration of bogie. Yaw restraint is large for the trailing wheelset and low for the leading wheelset. In both (b) and (c) steering performance must be relaxed in order to achieve adequate stability. In actual application to the bogie vehicles of the JR-Central Series 383 train [18], the longitudinal stiffness of the wheelsets in each vehicle was originally configured in a soft-hard-soft-hard arrangement, with the first soft wheelset leading and the configuration reversed when the train changed direction. However, experiments showed that there was no need to have soft restraint on the third wheelset, and the configuration was modified to soft-hard-hard-soft removing the need to switch the configuration when the train reverses direction. The dynamic behaviour of symmetric bogie vehicles with two unsymmetric bogies has been considered in Chapter 6.

References 1. Ahrons, E.L.: The British Steam Railway Locomotive 1825-1925. Bracken Books, London, 1987. 2. Goodwin, A.M.: The evolution of the British electric tramcar truck. Tramway and Light Railway Society, London, 1977. 3. Carter, F.W.: The electric locomotive. Proc. Inst. Civil. Engrs. 221 (1916), pp. 221-252. 4. Carter, F.W.: Railway Electric Traction. Edward Arnold, London, 1922. 5. Carter, F.W.: On the stability of running of locomotives. Proc. Roy. Soc. 121, Series A (1928), pp. 585-611. 6. Carter, F.W.: The running of locomotives, with reference to their tendency to derail. Institution of Civil Engineers, Selected Engineering Paper, No.81, 1930. 7. Rocard, Y.: La stabilite de route des locomotives. Actual. Sci. Ind. 234 (1935), Part 1. 8. Rocard, Y.: General Dynamics of Vibrations. Crosby Lockwood, London, 1960. (trans. from French, 1st pub. 1943) 9. Wickens, A.H.: Steering and stability of unsymmetric articulated railway vehicles. Trans. A.S.M.E. Journal of Dynamic Systems, Measurement and Control. 101 (1979) pp. 256-262.

282

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10. Wickens, A.H.: Static and dynamic stability of unsymmetric two-axle railway possessing perfect steering. Vehicle System Dynamics 11 (1982) pp. 89-106. 11. Carter, F.W.: Improvements In and Relating to High Speed Electric or Other Locomotives, British Patent Specifications 128,106 (1918), 155,038 (1919), and 163,185 (1920). 12. Li, W.: The Dynamics of Perfect Steering Bogie Vehicles and its Improvement with a Re-Configurable Mechanism. Doctoral Dissertation, Loughborough University, 1995. 13. Tuten, J.M., Law, E.H., and Cooperrider, N.K.: Lateral stability of freight cars with axles having different wheel profiles and asymmetric loading, ASME Paper No. 78-RT-3, 1978. 14. Illingworth, R.: The use of unsymmetric plan view suspension in rapid transit steering bogies. Hedrick, J. K. (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 8th IAVSD Symposium, Cambridge, Mass. August 1983, pp. 252-265. Swets and Zeitlinger Publishers, Lisse, 1984. 15. Elkins, J.A.: The performance of three-piece trucks equpped with independently rotating wheels. Anderson, R. (Ed.): The Dynamics of Vehicles on Roads and Tracks, Proc. 11th IAVSD Symposium, Kingston, Ontario, August 1989, pp. 203-216. Swets and Zeitlinger Publishers, Lisse, 1984. 16. Suda, Y.: Improvement of high speed stability and curving performance by parameter control of trucks for rail vehicles considering independently rotating wheelsets and unsymmetric Structure. JSME International Journal, Series III, 33-2 (1990), pp. 176-182. 17. Suda, Y.: High speed stability and curving performance of longitudinally unsymmetric trucks with semi-active control, Vehicle System Dynamics, 23 (1994), pp. 29-52. 18. Yoshikawa, K.: Faster through the curves on JR-Central, Railway Gazette International, August 1997, pp. 531-532. 19. Milne, R.D.: The analysis of weakly coupled dynamical systems. Int. J. Control, 2 (1st series) (1965), pp. 171-199.

Index actuators, 246 alternative guidance systems, 94-102 Amontons law of friction, 32 anti-bending mode, 115 anti-shear mode, 116 antisymmetric degrees of freedom, 65 articulation, 1, 16, 99, 235 asymptotic stability, 76, 84 axle-guard clearance, 133 axle load, 64,71 bending oscillation, 119 bending stiffness, 110, 137 bifurcation, 84 Bissel, 262 block diagram, 59, 81, 145 body instability, 145-154, 179, 254 body steering, 174, 190 bogie: conventional bogie: dynamic response, 181 eigenvalues, 177-179 equations of motion, 175-176 generalised coordinates, 175 parameters, 178 stability, 179 steered bogie: dynamic response, 194 eigenvalues, 197-198 equations of motion, 185-186 stability, 197-205 unrestrained bogie: curving, 124-127 eigenvalues, 160-163 equations of motion, 61-66 generalised coordinates, 61 limit cycles, 166-167 parameters, 115 stability, 160-160, 168 buckling, 262 braking or tractive torque, 50 Brunel, 5 cant, 15, 19 cant deficiency, 76

optimum response to, 117 response to, on large radius curve, 116 centre friction plate, 173 centrifugal forces, 47 centring spring, 173, 262 chaotic motions, 87 characteristic polynomial, 77 Class 143/144 passenger vehicles, 134 close coupling of vehicles, 133 conicity, 21-31 coning, 3-8, 11, 28 constrained curving, 108 constrained motion, 48 constraints, 19 contact forces, 42 contact mechanics, 32 contact patch, 6, 7, 32-40 contact slope difference, 21-31 contact stiffness, 13, 57-59 Coulombs laws of friction, 32 creep, 6, 33-36, 40-42 creep coefficients, 36, 58, 59 'creep controlled' wheelset, 102 creep damping matrix, 55 creep forces, 7-8, 36-37, 47, 50, 54-60 creep saturation, 84, 113 creep stiffness matrix, 55 critical speed, 85 cross-bracing, 111, 134, 138, 139, 159 cross-level, 19 cylindrical treads, 3 degrees of freedom, 1, 19 derailment, 13-14 describing functions, 30, 86 divergence, 99, 222, 229, 265 double-link suspension, 133 dynamic response: conventional bogie, 181 four-axle artic. vehicle, 256- 257 steered bogie, 194 three-axle vehicle, 230-232 two-axle vehicle, 121 effective conicity, 28

RAIL VEHICLE DYNAMICS

285 elastic stiffness matrix, 55 eigenvalue problem, 76-77 eigenvalues: conventional bogie vehicle, 177-179 four-axle articulated vehicle, 252-255 steered bogie, 197-198 three-axle vehicle, 217-223 three-axle artic. vehicle, 246-250 two-axle vehicle, 143-145 unrestrained bogie, 160-162 wheelset, 76-81 energy balance, 50-52, 56-58, 88 equation of compatibility, 62 equation of energy balance, 52, 56-57 equations of motion, 46-66 conventional bogie vehicle, 175 four-axle articulated vehicle, 242 steered bogie, 185-187 three-axle vehicle, 210-212 three-axle artic. vehicle, 239-240 two-axle vehicle, 61-66 unrestrained bogie, 61-66 wheelset, 46-60 equivalent conicity 11, 13, 30-31 effective conicity, 28 eye-bolt suspension, 133 FASTSIM, 37 feedback loop for wheelset, 8, 51, 81 feedback loops for two-axle vehicle, 145 flange, 2-5, 9, 11, 13-15, 94, 123, 130 flange contact, 2-4, 13-15, 113 flange wear, 15 flange-way clearance, 2, 3, 75 flexibility parameter, 112, 134, 146 flexible, long wheelbase vehicle, 113, 146 flutter, 229 forced steering, 16, 174, 238 four-axle articulated vehicle: dynamic response, 259 eigenvalues, 252-255 equations of motion, 242 generalised coordinates, 238 parameters, 253 stability, 252 free curving, 108 freedoms, 19 Galerkin’s method, 87

gauge spreading, 123 generalised applied forces, 48-50 generalised coordinates: conventional bogie vehicle, 175 steered bogie, 175 three-axle vehicle, 210 three-axle articulated vehicle, 239 two-axle vehicle, 61 unrestrained bogie, 61 wheelset, 49-65 gravitational forces, 47 gravitational stiffness, 13, 21, 50 gravitational stiffness matrix, 55 guidance, 10, 107 gyroscopic forces, 47 gyroscopic matrix, 55 Hertz stiffness, 48 Heumann minimum principle, 108 high speed trains, 174, 237 HSFV-1 vehicle, 114, 134, 155 hunting, 9, 10, 85 impedance of yaw spring, 140 independently rotating wheels, 95, 263 inertia matrix, 55, 65 instability, 8, 9, 10, 16, 145 inter-couplers, 190 Jacob bogie, 236 kinematic oscillation, 5-6, 9, 71-74 Klingel’s formula, 5-6 Krylov and Bogoljubov method, 87 lateral creepage, 6-7, 34, 40 lateral creep force, 6-7, 36-39, 42 laws of friction, 32 Liechty’s bogie,195 limit cycles, 84-86, 153 linkage-steered bogie, 192-195 locomotives, 261-262 longitudinal creepage, 6-7, 34, 40 longitudinal creep force, 6-7, 36-39, 42 lower sway mode, 144 misalignment, response of two-axle vehicle to, 127 modal expansion, 90 modal superposition, 89

INDEX model roller rig experiments, 94 normal coordinates, 89 Nyquist criterion, 81 open loop transfer function, 81 orthogonality, 90 parameters: conventional bogie vehicle, 178 four-axle articulated vehicle, 253 three-axle vehicle, 219 three-axle articulated vehicle, 247 two-axle vehicle, 115 unrestrained bogie, 115 wheelset, 59 path coordinates for two-axle vehicle, 61 peg-in-slot, 94, 99-102 perturbation procedure, 79 plane of symmetry, 61, 65 plateway, 95 ‘pony’ axle, 96-98, 263 potential energy, 52, 56 quasi-linear equations, 86, 155 radial steering, 16, 182 Rayleigh’s dissipation function, 57 re-configurable systems, 263, 280-281 Redtenbacher’s formula, 3-4, 74-75 relaxation damper, 134, 140, 159 reversed motion, equations of motion, 264 ride, 15 rigid two-axle vehicle: curving, 114 eigenvalues, 146 rotation matrix, 45 roller rig, 71, 154 Routh’s criterion for stability, 81 self-steering, 16, 134, 238 semi-rigid vehicle, 263 sensors, 246 shear oscillation, 119, 143 shear stiffness, 110, 137 Shin Kansen, 174 side bearers, 173 simplified equations of motion, 60 spin, 6-8, 13, 18, 34-40 static instability, 10, 99, 115, 220, 256 steering, 117, 118, 182, 210, 238-242

286 steering oscillation, 119 steering beam, 244 steering laws, 189 Stephenson, 5 stiff, short wheelbase vehicle, 113 stiffness matrix: conventional bogie vehicle, 176 deflated, 137 degeneracy, 183 four-axle articulated vehicle, 245 properties, 109, 135, 182, 185 steered bogie vehicle, 185-186, 191195 three-axle vehicle, 211 three-axle articulated vehicle, 240 two-axle vehicle, 62, 65 unrestrained bogie, 62, 65 unsymmetric structure, 268 wheelset, 55 S-Tog, 237 subsidences, 80 sum and difference coordinates, 109 suspension for simple two-axle vehicle, 62 swing bolster, 173 switching, 3, 95 system matrix, 60 Talgo, 98, 237 tangential tractions, 32-35 three-dimensional geometry, 31 three-piece bogie, 169, 263 throwover, 236 track curvature, response to, on large radius curves, 111 track flexibility, 94 track irregularities, 1, 15 transient effects, 40, 81 transition, 15, 93-94 two-point contact, 31, 39, 49 upper sway mode, 144 veering of eigenvalues, 146 Wiesinger bogie, 192 worn profiles, 13, 28, 39 yaw relaxation, 134, 140, 159 yaw damper, 134, 140, 159