Dynamics of the Axially Moving Orthotropic Web

Lecture Notes in Applied and Computational Mechanics Volume 38 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

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Dynamics of the Axially Moving Orthotropic Web Krzysztof Marynowski

With 127 Figures and 13 Tables

Dr. Krzysztof Marynowski Technical University of Lódz Department of Dynamics of Machines 90-924 Lódz ul. Stefanowskiego 1/15 Poland [email protected]

ISBN:

978-3-540-78988-8

e-ISBN:

978-3-540-78989-5

Lecture Notes in Applied and Computational Mechanics ISSN

1613-7736

Library of Congress Control Number: 2008926862 © First Edition 2005. Corrected Second Printing 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 9 8 7 8 6 5 4 3 2 1 0 springer.com

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Identification of Rheological Parameters of the Paper Web . . . 1.1.1 Identification Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Results of the Experimental Identification . . . . . . . . . . . 1.2 Identification of the Corrugated Board as a Composite . . . . . . 1.2.1 Homogenization Method . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Identification Results. . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 5 6 7 9

2

State of Knowledge on Dynamics of Axially Moving Systems . . . . . . . 2.1 String Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Nonlinear String Systems . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Beam Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Nonlinear Beam Systems. . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Plate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Numerical Investigations. . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Experimental Investigations of Axially Moving Plate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 11 13 17 17 19 25 25

Dynamical Analysis of the Undamped Axially Moving Web System . . 3.1 One-Layered Orthotropic Web . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Formulation of Nonlinear Equations of the Web Motion 3.1.2 Solution to the Mathematical Model . . . . . . . . . . . . . . . 3.1.3 Results of Comparative Studies . . . . . . . . . . . . . . . . . . . 3.1.4 Results of Dynamic Investigations of the Moving Paper Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multi-Layered Composite Web . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Mathematical Model of the Axially Moving Multi-Layered Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Solution of the Mathematical Model . . . . . . . . . . . . . . . 3.2.3 Results of the Comparative Studies . . . . . . . . . . . . . . . .

43 43 43 47 54

3

31 38

56 65 65 70 73

vi

Contents

3.2.4

3.3 4

Results of the Dynamic Investigations of the Axially Moving Corrugated Board Web . . . . . . . . . . . . . . . . . . . Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 82

Displacements of the Web in Equilibrium States of the Linearized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Mathematical Models of the Axially Moving Orthotropic Web 4.2 Solution to the Mathematical Model of the Web Loaded with Constant Longitudinal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Displacements of the Web Loaded with Constant Longitudinal Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Web Loaded with a Non-Uniform Longitudinal Force. . . . . . . 4.5 Wrinkling of the Web Loaded with a Non-Uniformly Distributed Longitudinal Force . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98 101

5

Dynamics of the Axially Moving Viscoelastic Web . . . . . . . . . . . . . . . 5.1 Two-Dimensional Rheological Model for Viscoelastic Materials 5.2 Mathematical Model of the Moving Viscoelastic Web . . . . . . . 5.3 Solution to the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results of the Numerical Investigations. . . . . . . . . . . . . . . . . . . 5.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 106 109 110 114

6

Beam Model of the Moving Viscoelastic Web . . . . . . . . . . . . . . . . . . . 6.1 Nonlinear Beam Model of the Viscoelastic Web . . . . . . . . . . . . 6.1.1 Kelvin-Voigt Model of Material . . . . . . . . . . . . . . . . . . . 6.1.2 Poynting-Thompson Model of Material . . . . . . . . . . . . . 6.1.3 Solution to the Problems . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Investigations Results of the Model with the K-V Element. . . . 6.2.1 Linearized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Non-Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Investigations Results of the Model with a P-T Element. . . . . . 6.3.1 Linearized System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Nonlinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 118 119 120 122 123 123 124 129 129 130 135

7

Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

85 85 87 90 94

Fundamental Notations

A b bin c cw D E=Ex Ey F G h J l M x, M y Mxy N x, N y Nxy qi s t u, v, w x, y, z      =  xy  yx

 

cross-section area of the web, width of the web, dimensionless coefficient of internal damping, transport speed of the web, wave speed, flexural stiffness of the plate, Young’s modulus with respect to the longitudinal direction, Young’s modulus with respect to the transverse direction, dimensionless Airy stress function, Kirchhoff’s modulus, thickness of the web, inertial moment of the cross-section, length of the web, sectional flexural moments with respect to the x axis and the y axis, correspondingly, sectional torsion moment, sectional membrane forces, sectional shear force, generalized coordinate, dimensionless transport speed of the web, time, components of displacement of the surface along the directions x, y, z, Cartesian coordinates, coefficient of internal damping, dimensionless plate stiffness, orthotropy coefficient, dimensionless longitudinal displacement of the web, Poisson’s number with respect to the longitudinal direction, Poisson’s number with respect to the transverse direction (the first index denotes the transverse direction, the second one – the longitudinal direction), density of the web material, real part of the eigenvalue,

viii

 x,  y xy ’  !

Fundamental Notations

components of normal stresses along the direction x, y, respectively, dimensionless time, shear stresses in the x – y plane, dimensionless transverse displacement of the web, Airy stress function, imaginary part of the eigenvalue.

Chapter 1

Introduction

A material continuum moving axially at high speed can be met in numerous different technical applications. These comprise band saws, web papers during manufacturing, processing and printing processes, textile bands during manufacturing and processing, pipes transporting fluids, transmission belts as well as flat objects moving at high speeds in space. In all these so varied technical applications, the maximum transport speed or the transportation speed is aimed at in order to increase ‘efficiency and optimize investment and performance costs of sometimes very expensive and complex machines and installations. The dynamic behavior of axially moving systems very often hinders from reaching these aims. The book is devoted to dynamics of axially moving material objects of low flexural stiffness that are referred to as webs. Problems connected with the dynamic behavior of such objects are clearly visible in paper manufacturing and printing industry. Transport speeds at which the paper web moves during manufacturing and processing can reach even 50 m/s. Under certain circumstances, such high transport speeds can lead to resonance vibrations, instability or web flattering. These behaviors can result in web folding or breaking during its motion. Changes in web tension that follow from vibrations can bring about alternations in thickness of the paper being manufactured. The basic condition to solve these problems is to understand fully the dynamic behavior of an axially moving web in the undercritical and overcritical range of its transport speed. A dynamic analysis of physical and mathematical models of axially moving systems is applied in this recognition. On one hand, these models should describe accurately the phenomena under analysis, and, on the other hand, they should be as simple as possible to enable the engineer who operates the machine to test quickly an influence of individual parameters on the dynamic behavior of the devices. Most often these two aims are contrary to each other. Having computational capabilities of modern hardware at disposal, it is relatively easy to determine – for instance with the finite element method – mathematical models composed of a huge number of differential equations that describe each detail of the process under analysis. However, a solution and analysis of such a model is either impossible or unprofitable. Despite the fact that they are less accurate, simpler models are more useful in recognition and K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_1,  Springer-Verlag Berlin Heidelberg 2008

1

2

1 Introduction

analysis of the phenomena under investigation. This was the author’s conviction while writing this book. A variety of axially moving systems that are met in technical applications would easily lead to scattering the reader’s attention and to an unnecessary increase a volume of this book if all of them were subject to the analysis. Thus, to make tracing the dynamic considerations easier, a paper web is the main object of investigations in the present book. Paper is a very specific material, whose physical properties depend on its structure, raw materials composition, production technology, finishing, processing and hydrothermal state. Different properties of paper that result from its heterogeneity are shown in many works (e.g. [6,7]). Investigation results prove that two main orthotropy directions can be distinguished in the anisotropic paper web, namely: parallel and perpendicular to its longitudinal axis. The longitudinal axis coincides with the machine direction (MD) of the paper web. In opinion of many researchers of paper (e.g. [3,5]), the orthotropic model is correct to describe physical properties of paper because of its fibrous structure. This book is divided into seven chapters. In the next part of the introduction, the results of identification of rheological parameters of two paper webs and a corrugated board web that are subject to dynamic investigations in further sections of the book are to be presented. In Chap. 2, the state of knowledge on dynamic investigations of axially moving systems will be discussed. The third chapter is devoted to the dynamic analysis of the undamped system of the axially moving web. Chapter 4 describes the investigations of web equilibrium states. In the fifth chapter, the results of dynamic analysis of the damped system of the axially moving web are shown, whereas the sixth chapter presents the dynamic analysis of the beam model of the web. The book is concluded with final remarks in the seventh chapter and appendixes.

1.1 Identification of Rheological Parameters of the Paper Web The important problem one can encounter while considering the dynamic behaviour of an axially moving paper web is how to model the web material. Usually paper is considered as a viscoelastic material. This material do not obey the Hooke’s law and its behavior should be modeled by adequate constitutive equations. In the static-type analysis, to model specific paper properties, the fourparameter Bu¨rgers model (Fig. 1.1a) is used. With this model, essential phenomena that appear in paper such as stress relaxation, strain relaxation and plastic strain can be described. However, in the dynamic-type analysis, all these phenomena are not equally essential. For example, during paper production webs are moving at high speed, the paper webs are transported with longitudinal speeds of up to 3000 m/min, and one can assume that during such short time durable strains do not appear.

1.1 Identification of Rheological Parameters of the Paper Web (c)

(b)

(a) E1

E1

γ2

E2

γ

E1 E2

E2

γ

γ1 x

3

x x

Fig. 1.1 Rheological models of paper; (a) Bu¨rgers model, (b) Poynting-Thompson model, (c) Zener model

To model specific paper properties, a three-parameter rheological element such as the Poynting-Thompson model (Fig. 1.1b) and the Zener model (Fig. 1.1c) are used in the dynamic analysis. An advantage of the PoyntingThompson model is a clear division among particular strain kinds. Elastic strains are represented by the elastic element E1 (Fig. 1.1b). Viscoelastic strains are related to the E2 and g parameters (the Kelvin-Voigt model). The sum of all strains represents the global strain eg and it has the following form in the case of a constant velocity of stress increment:   E t   bt b   2 "g ¼ þ e 1 þt : E1 E2 E2

(1:1)

where: b – velocity of stress increment, E1, E2, g – rheological parameters of the Poynting-Thompson model. The first component of sum (1.1) represents the elastic strain. The second component denotes the viscoelastic strain. The identification method is based on a tension test of paper. Rheological parameters of papers are determined according to the procedures which are described below.

1.1.1 Identification Method Individual strain components (elastic and viscoelastic strains) are determined according to the following steps: The first step With a constant velocity of stress increment b, a paper sample is tensioned up to the specified value below the failure stress (about 80% of the failure stress). The velocity of stress increment was selected to obtain the tension time equal to 25 – 2 secs.

4

1 Introduction

The second step When the paper sample reaches the assumed stress level, it is discharged with the velocity equal to 1000 mm/min. From a theoretical point of view, the infinitely high velocity should be applied then because viscoelastic strains would maintain their values from the tension test. In practice, it is impossible to achieve the discharge time equal to zero. An application of very high velocity is possible however, when a sample of typical paper (of elongation about 2%) is discharged within less than 0.5 secs. Then, an error of the viscoelastic strain determination is significantly less than 2%. To illustrate the tension test of paper, Fig. 1.2 shows an example of the strain – stress dependence in the form of a point plot. The E1 parameter of the Poynting-Thompson model was calculated by means of the classical determination method of the Young’s modulus. The slope tangent to the beginning of the tension plot yields the first Young’s modulus (Fig. 1.2). On the basis of the E1 parameter value, the elastic strain for all levels of stresses recorded during the tension test is calculated. Next, the elastic strain is subtracted from the global strain at each measuring point. Thus, the viscoelastic strains "v-el can be obtained. The viscoelastic part of the Poynting-Thompson model has the same form as the Kelvin-Voigt rheological model. The viscoelastic strain – stress relationship for a constant velocity of stress increment in the tension test is calculated on the base of the second component of (1.1): "vel ¼

  E t   b   2 e  1 þt : E2 E2

(1:2)

40 35 30 25 Stress 20 [N/mm2]

15 10 5 0 0

0,001

0,002

0,003

0,004

0,005 Strain [–]

Fig. 1.2 Tension plot

0,006

0,007

0,008

0,009

0,01

1.1 Identification of Rheological Parameters of the Paper Web

5

To determine the E2 and g parameters of the Poynting-Thompson model, the central part of (1.2) has been presented as a function, i.e.: y ¼ ea  1:

(1:3)

Because for ascending a values, y tends to –1, above a certain a value it can be assumed that the function value equals –1. On this assumption, the error is less than 2% for a = 4, and the error is less than 1% for a = 5. Substituting a ¼ E2 t= into (1.3) under assumption that a for a certain time tg is sufficiently large (e.g. a > 4) one can assume y  1. Thus, for the sufficient long time of stretching the (1.2) can be written in the following form:   b   ð1Þ þ t : E2 E2

"vel

(1:4)

For t > tg one can approximate the stress – viscoelastic strains ev-el plot with the following linear form: "vel ¼ d1 t þ d2 :

(1:5)

Thus, the E2 and g parameters of the Poynting-Thompson model can be calculated as follows: E2 ¼

b d2  E22 ; ¼ d1 b

(1:6)

Both the Poynting-Thompson rheological model and the Zener model possess three parameters and are equivalent [8]. Mathematical conditions of their equivalence have the following form: E1Z þ E2Z ¼ E1  E1Z  E2Z ¼ Z E1  E2

(1:7)

E1Z ðE1 þ E2 Þ ¼ E1 E2 where: E1Z, E2Z, gZ – parameters of the Zener model.

1.1.2 Results of the Experimental Identification

9

The experimental investigations were carried out at the Papermaking and ´ ´ , Poland. Two kinds of papers used in the corrugated Printing Institute in Lodz board, namely the liner paper and the fluting paper, were employed in those investigations. The properties of these papers are as follows:

6

1 Introduction

Table 1.1 Rheological parameters Paper Poynting-Thompson model Zener model E1Z, N/mm2 E2Z, N/mm2 gZ, Ns/mm2 E1, N/mm2 E2, Ns/mm2 g, Ns/mm2 Machine direction Liner 5354 4504 26729 2446.2 2907.8 7884.3 Fluting 3373 2084 12863 1288.1 2084.9 4914.4 Cross direction Liner Fluting

2285 1863

1042 968

6370 5704

715.6 637.0

1569.3 1226.0

3004.7 2470.2

– basic weights: 220 g/m2 (liner paper), 123 g/m2 (fluting paper); – thicknesses: 0.347 mm (liner paper), 0.241 mm (fluting paper). It is worth noting that in the earlier identifications with the four-parameter Burgers rheological model of material, the same liner paper was taken into account [6]. Paper samples (dimensions: 180 mm  15 mm) were conditioned according to the Polish Standard PN–EN 20187. Rheological parameters were determined in the machine direction1 and in the cross direction2 on the basis of 10 repetitive tension tests performed in each direction. A Zwick universal tensile test machine equipped with a measurement head of a range up to 500 N that allows for recording load changes with accuracy of 0.001 N and displacements with accuracy of 0.001 mm was used in the tests. The final results in the form of the Poynting-Thompson model parameters and the equivalent Zener model parameters are presented in Table 1.1.

1.2 Identification of the Corrugated Board as a Composite The corrugated board was treated as a multilayer orthotropic structure. The corrugated layer (i.e. a core – according to the classification of thin plates) was subject to the homogenization process [1], and afterwards that layer was treated as a thin orthotropic plate (Fig. 1.3). Flat covers were also treated as thin orthotropic layers because of two-dimensional properties of paper. As a result of combining these two component elements, as it happens in the corrugated board, a multilayer orthotropic structure with a rigid core is obtained. Basic assumptions of multilayer thin plates are satisfied, including the KirchhoffLove hypothesis about the straight normal line. In the corrugated board, main directions of orthotropy coincide with the machine direction (MD) and the cross direction (CD) of the paper it is made from. 1

The direction in the paper plane in which the web is made, referred to as MD in abbreviation The cross direction is the direction perpendicular to the machine direction, lying in the paper plane, referred to as CD in abbreviation.

2

1.2 Identification of the Corrugated Board as a Composite

7

Fig. 1.3 Homogenization of the corrugated layer

x (CD)

s p

Mx

My

y (MD)

My

Mx

1.2.1 Homogenization Method When we analyze a corrugated plate with a large number of segments, we can take into account an individual segment assuming the symmetry conditions on the longitudinal edges. In this case, the boundary conditions on the longitudinal edges are neglected, i.e. an infinitely wide plate is assumed. Then, the plate stiffnesses has the following forms: 0   0 0 Ey  3 p Ex Iy  G 3 p c ; H¼ þ ; Dy ¼ DMD ¼ (1:8) Dx ¼ DCD ¼ p 12 c 12 c p where: p = 2 l ; c = 2 s ; s – length of the sinusoidal half-wave (Fig. 1.4),  – thickness of the fluting paper, Iy – moment of inertia. For the sinusoidal core on the basis of Fig. 1.4, one can determine:   p2 f 2 s¼‘ 1þ 2 4‘

(1:9)

The moment of inertia Iy has been determined in an approximate way: " # Iy f 2  0:81 1 ¼ ; 2 p 1 þ 2:5 ð f Þ2 2‘

where: f ¼

h2  2

(1:10)

p f h2 s l

Fig. 1.4 Corrugated layer

z

y(MD)

8

1 Introduction

In the homogenization process, the substitute stiffnesses of the core has been determined on the following assumptions: height of the core – h2, wideness – p, mass of the corrugated plate and the homogenized orthotropic plate are the same. From the orthotropic plate theory, the plate stiffnesses has been obtained in the form: ECD h32 EMD h32 ; DMD ¼ ; 12ð1  CDMD MDCD Þ 12ð1  CDMD MDCD Þ   EMD CDMD G  þ H ¼ h32 6 12ð1  CDMD MDCD Þ

DCD ¼

(1:11)

The equivalent quantities EMD ; MDCD ; ECD ; CDMD ; G can be determined on the basis of a comparison of (1.8) and (1.11). To obtain EMD and ECD , the following assumption has been taken: MDCD CDMD 550:

(1:12)

Finally, one receives: 0

12ECD Iy ECD ¼ ; ph32

0

E 3 p EMD ¼ MD3 h2 c

(1:13)

The substitute Poisson ratio CDMD has been determined from the law of mixtures:  cCDMD ¼ h2 p CDMD

(1:14)

Thus, taking into account (1.13), one receives: CDMD ¼

c CDMD h2 p

EMD MDCD ¼  CDMD ECD

(1:15)

To obtain the Kirchhoff modulus G value, the assumption (1.12) has been omitted:   0   EMD h32 MDCD 6 G 3 c p þ G ¼ 3  12ð1  CDMD MDCD Þ h2 12 p c

(1:16)

The mass equality condition of the core and the substitute plate (that means:  c ¼  h2 p) enables us to determine the substitute density:  ¼

c  h2 p

(1:17)

According to the classical multilayer plate theory [2], the plate stiffnesses of the three-layered corrugated board have the form:

1.2 Identification of the Corrugated Board as a Composite

Dx ¼

Ex h32 Ex ðh3  h32 Þ þ ; 12ð1  xy yx Þ 12ð1  xy yx Þ

Dy ¼

Ey h32 Ey ðh3  h32 Þ þ 12ð1  xy yx Þ 12ð1  xy yx Þ

9

(1:18)

 3 Ey xy ðh3  h3 Þ Ey xy h32  Gðh3  h32 Þ Gh 2 þ 2þ þ H¼ 6 12ð1  xy yx Þ 12ð1  xy yx Þ 6 To determine the Poisson ratio MDCD and CDMD of the corrugated board, the following assumption is proposed: the Poisson ratios have the approximate values assumed usually for paper. Even considerable differences in these values do not cause significant changes in the remaining material constants. For orthotropic plates, the following relation has to be satisfied: EMD CDMD ¼ ECD MDCD

(1:19)

One can assume either MDCD or CDMD and the second value should be determined from (1.19). Both the Poisson ratio values have to be at most equal to 0.5. The Kirchhoff’s module G has been determined from the approximate relation: G¼

Ex Ey ECD EMD ¼ Ex þ Ey þ 2xy Ey ECD þ EMD þ 2CDMD EMD

(1:20)

1.2.2 Identification Results Two kinds of papers used in the corrugated board, namely the liner paper and the fluting paper, were employed in the investigations of web displacements. Values of parameters describing the physical properties of the liner paper and the fluting paper are shown in Table 1.2. The results of the investigations, which Table 1.2 Physical parameters of papers Parameter, notation Liner paper Thickness, h [mm] 0.347 634.01 (220) Density,  [kg/m3] ([g/m2]) 5354106 Young’s modulus, EMD [N/m2] Young’s modulus, ECD [N/m2] 2285106 2 Kirchhoff’s modulus, G [N/m ] 1186106 Poisson’s number, [–] 0.25 0.11 Poisson’s number,  yx [–] 0.427 Orthotropy coef.,  = ECD/EMD [–] Plate stiffness, DMD [Nm] 19.1510–3 Plate stiffness, DCD [Nm] 8.1810–3

Fluting paper 0.241 510.37 (123) 3373106 1863106 908106 0.25 0.14 0.552 4.0710–3 2.2510–3

Homogenization 3.5 49.2 (12) 0.79106 37.8106 2.24106 0.0241 0.0005 0.021 2.8210–3 135.0510–3

10

1 Introduction

Fig. 1.5 Composite structure of the corrugated board

liner paper

h1= 0.347 mm

fluting paper (homogenized layer) liner paper

h2= 3.5 mm

Table 1.3 Flexural stiffnesses of the corrugated board D12= 1.51 Nm D11= 14.16 Nm D22= 6.17 Nm D21= 1.51 Nm D62= 0 D61= 0

h = 4.194 mm

h1= 0.347 mm

D16= 0 D26= 0 D66= 1.87 Nm

9

´ ´ , were used to were carried out at the Papermaking and Printing Institute in Lodz determine these parameters. To find parameters of the three-layered corrugated board, which consists of two papers under investigation, the fluting paper, forming the middle layer of the C–type board, was homogenized. The homogenization results are shown in the last column of Table 1.2. To determine flexural stiffness, the three-layered corrugated board was treated as a composite structure [4]. Figure 1.5 shows geometrical dimensions of the structure under consideration. Table 1.3 shows investigation results of flexural matrix coefficients of the composite structure representing the corrugated board. The identification results shown in the Tables 1.1, 1.2 and 1.3 yield the data for dynamic analyses presented in further parts of the book.

References

9

1. Bhannic N, Cartrand P, Quensel T (2003) Homogenization of corrugated core sandwich panels. Composite Structures 59: 299–312 2. Jones R M (1975) Mechanics of Composite Materials, International Student Edition, McGraw-Hill Kogakusha Ltd, Tokyo 3. Mann R W, Baum G A, Habeger C C (1979) Elastic wave propagation in paper. Tappi Journal 62(8): 115–118 4. Marynowski K, Kol akowski Z (2003) Dynamic behaviour of an axially moving multilayered web. Journal of Theoretical and Applied Mechanics 41(2): 323–340 5. Seo Y B, Castagnede B, Mark R E (1992) An optimization approach for the determination of in-plane elastic constants of paper. Tappi Journal 75(11): 209–214 6. Szewczyk W, Marynowski K, Tarnawski W (2006) The analysis of Young’s modulus distribution in paper plane. Fibers & Textiles in Eastern Europe 58: 91–94 7. Uesaka T, Marakami K, Imamura R (1979) Biaxial tensile behaviour of paper. Tappi Journal 62(8): 111–114 8. Zener C (1937) Internal friction in solids. Physics Review 52: 230–240

Chapter 2

State of Knowledge on Dynamics of Axially Moving Systems

One of the most important phenomena that are analyzed during investigations of the dynamic behavior of the system are vibrations and dynamic stability of the system motion. Due to high transport speeds that are attained by webs in various industrial applications, investigations of motion stability are crucial as they allow to avoid folding or even breaking of the web. Stability is the feature of the dynamic system that decides about its susceptibility to changes in the quantities the system reactions depend on. A stable system is characterized by low susceptibility to changes in the quantities that are decisive as far as system reactions are concerned. Strictly speaking, a dynamic system can be defined as stable when the disturbed solution to the set of motion equations does not withdraw from the undisturbed solution, independently of the fact whether the undisturbed solution has a form of the equilibrium position or the limit cycle. Dynamics of axially moving systems has been investigated for almost 60 years already. In the early 1950s, first publications including results of investigations of transverse vibrations of an axially moving string are to be found. Those papers were authored by Sack [49] and Archibald and Emslie [3]. Since then, a vivid interest of researchers in axially moving system can be observed. This interest supported by newer and newer fields of applications (e.g. in outer-space) throughout years remains firm on a high level till today.

2.1 String Systems 2.1.1 Linear Models Among earliest studies devoted to the linear analysis of vibrations of a moving string, the most frequently quoted is Archibald and Emslie [3]. It presents the results of investigations on transverse vibrations of a string moving with a constant speed c along the longitudinal direction, denoted by x. Starting from the Hamilton’s principle, the already classical differential equation with partial derivatives has been expressed as: K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_2, Ó Springer-Verlag Berlin Heidelberg 2008

11

12

2 State of Knowledge on Dynamics of Axially Moving Systems

@2y @2y  þ 2c 2 @t @t @x



T   l c2 l



@2y ¼0 @x2

(2:1)

where l – mass of the string unit length [kg/m], T – longitudinal tension force [N]. A solution to equation (2.1) has been presented in the form of a sum of harmonic functions as follows: pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi      cl þ l T cl  l T y ¼ C1 cos ! t þ cos ! t þ x þ C x 2 T   l c2 T   l c2

(2:2)

where C1, C2 – constants, ! – angular frequency of transverse vibrations. Assuming the simple boundary conditions yjx¼0 ¼ 0; yjx¼l ¼ 0 on the basis of solution (2.2), an analytical dependence of the transverse vibration frequency of the string fn on its transport speed c has been determined: fn ¼

! n ðT  l c2 Þ pffiffiffiffiffiffiffiffi ; ¼ 2p 2 l l T

n ¼ 1; 2; 3 ::::

(2:3)

Relationship (2.3) has been shown graphically in the form of a linear plot of the change in string transverse vibrations fn as a function of the dimensionless ratio lc2/T (Fig. 2.1). In the same study [3], the authors have made one more important discovery. By introducing the string wave propagation speed cw =(T/l)1/2 into the string motion equation (2.1), they have shown that a solution to the motion equation can be expressed in the general form as: y ¼ F1 ½x  ðcw þ cÞt þ F2 ½x þ ðcw  cÞt

(2:4)

where F1 and F2 are functions describing propagation of waves in opposite directions along the axis x. The discussed study, as well as later works showing the results of investigations of transverse vibrations of axially moving string systems within the linear theory, allow us to draw the following conclusions:

Fig. 2.1 Transverse frequency fn versus the dimensionless transport speed [3]

2.1 String Systems

13

(a) if cw is the speed of wave propagation measured by an observer moving together with the string, and c is the speed of displacement of the string, then disturbances propagate with the speed cw + c along the direction the string moves and with the speed cw – c along the direction opposite to the string motion (both the speeds are determined with respect to the still observer); (b) a difference in the speeds causes that the motion of the string is characterized by a variable angle of the phase displacement; the distortions that displace along the direction opposite to the string motion are in a phase lag with respect to the distortions that propagate along the string motion direction; (c) frequency of natural vibrations decreases with an increase in the string transport speed; when c = cw, a divergent-type instability of motion occurs. An influence of damping on the dynamic behavior of the moving string within the linear theory has been investigated in several studies. In Mote [39], there is a linear damping force in the form of w,t, where  denotes a viscous damping coefficient, in the equation of transverse vibrations. In [30,52] damping in the form of  (w,t+cw,x) has been considered. The quantity (w,t + cw,x) represents a transverse speed of the string element, measured by a still observer, w,t is an analogue speed measured by an observer moving together with the string. In all these works, the authors have investigated changes in complex natural frequencies and amplitudes of system vibrations in relation to the string transport speed. The system motion stability has been found dependent on the way the dissipative force is modeled and on the value of the damping coefficient. In several studies, string systems moving with a variable axial speed have been analyzed (e.g. [4,27,44]). Positive acceleration of the transport motion of the string model has a stabilizing effect on its transverse vibrations. Negative acceleration of the string transport motion acts just the opposite. Then, the occurring destabilization of transverse vibrations can be compared to the effect of negative damping.

2.1.2 Nonlinear String Systems Since the end of the 1960s, works analyzing vibrations of the moving string within the nonlinear theory have begun to appear. Originally, limitations of the linear theory have been pointed out in numerous studies [1,2,36,41]. Although in some of them the beam model has been analyzed mainly, the results of analysis of the string model have been presented as well. The paper by Wickert published in 1992 [58] can be treated as a sort of the summary of the initial stage in nonlinear investigations. While analyzing transverse-longitudinal beam vibrations, including geometrical nonlinearity, the author published the results of analysis conducted for the string model in the form of asymptotic solutions. To obtain an asymptotic solution to the differential-integral equation of motion, Wickert employed the modal theory of perturbations based on the

14

2 State of Knowledge on Dynamics of Axially Moving Systems

Krylov-Bogolubov method. The angular frequency of nonlinear transverse vibrations of the string moving with a speed c for n-th mode has been defined as follows: (   ) 1 1 sinðnp cÞ 2 2 3 2 2 2 (2:5) ! ¼ np ð1  c Þ þ ð" a0 v1 Þ ðnpÞ ð1 þ c Þ þ 8 2 np c wherepffiffiffiffiffiffiffiffiffiffiffiffi " – ffi small parameter; v1 – dimensionless longitudinal stiffness, v1 ¼ EA=T; ("a0v1) – dimensionless parameter describing the amplitude of string nonlinear vibrations. A comparison of the results obtained on the basis of (2.5) for the first mode of vibrations with the results from the linear theory, according to Wickert [58], is presented in Fig. 2.2. The plot shows that a difference in the results from the linear and nonlinear theory grows with the string transport speed. For instance, for the amplitude of ea0v1 = 0.2, the difference is equal to 7% for the still string and it increases up to 20% at the critical speed obtained from the linear model. Figure 2.2 illustrates also an increasing effect of nonlinear stiffness with speed, which is strictly connected to a gradual decrease in the modal linear stiffness. A comparison of calculation results of the fundamental frequency of transverse vibrations of the string obtained by Wickert [58] for ea0v1 = 0.25 to the results from the earlier, most important nonlinear analyses is presented in Table 2.1. Values of the string transport speed have been referred to the critical speed determined for the linear model v(1), whereas values of the angular frequency of vibrations have been referred to the linear solution !1(v=0) = p. The presented results show distinctly how significant it is to take into account an influence of transverse motion in the analysis, because it was performed only by Mote and Thurman [41] among all the works compared. The results of the works under discussion point out to the fact that the linear theory can be applied only in the case of high stresses of the string, low

Fig. 2.2 Linear and nonlinear frequencies versus the transport speed of the string [58]

2.1 String Systems

15

Table 2.1 Comparison of linear and non-linear fundamental frequency calculations for the traveling string at subcritical speeds; ea0v1 = 0.25. Speed and frequency values are normalized with respect to the critical speed v(1)=1.0 and the linear solution !1(v= 0) = p [58] (License No. 1870791412125) Dimensionless transport speed c/c(1) Linear model Asymptotic solution (Wickert [58]) Bapat [5], Korde [22] Mote, Thurman [41] Mote [38]

0.0

0.2

0.4

0.6

0.8

1.0

1.000 1.112

0.960 1.077

0.840 0.966

0.640 0.792

0.360 0.569

0.000 0.308

1.161 1.103 1.127

1.143 1.067 1.034

1.089 0.955 0.997

0.993 0.770 –

0.841 – –

0.589 – –

transverse displacements and within low transport speeds. If nonlinear stiffness of the string is not taken into account within the transport speed below the critical speed, then the values of critical speeds corresponding to individual modes of vibrations are substantially underrated. Moreover, when an influence of longitudinal displacements is neglected, then in connection to the fact of faster propagation of stress waves than transverse waves, it is followed by the fact that it is not possible to define precisely the position of the propagation front of perturbations. The publications issued in the recent years and devoted to dynamic investigations of string systems present mainly results of nonlinear analyses of these systems. Various factors that introduce nonlinearity in the mathematical description of the system are taken into consideration. Among studies devoted to this topic that were published in the 1990s, one of the most interesting is the study by Cheng and Perkins [11] that analyzes vibrations and stability of the motion of a moving guide bar with dry friction. Using the perturbation solution, regions of divergent and flatter type instability have been determined. In [14] a viscoelastic rheological model of the string material and a harmonically variable transport speed of the string was analyzed for the first time. As a result of the dynamic analysis of the nonlinear model of the system, an influence of model parameters on the amplitude and frequency of nonlinear vibrations have been determined. In [7,9,15,18], the authors investigated the dynamic behavior of the moving string with a harmonic alternation in the tension force, defining thus instability regions of parametric vibrations. Among works published recently and devoted to nonlinear dynamics of the axially moving string, in which equations of motion have been integrated, study [67] is characteristic. To describe rheological properties of the string material, a Kelvin-Voigt model was employed there. The string was loaded along the longitudinal direction with a force T that varies harmonically around a constant loading T0. An amplitude of changes in variable loading is defined by T1. The

16

2 State of Knowledge on Dynamics of Axially Moving Systems

nonlinear equation of the transverse motion of a string in the dimensionless form, considered in [67], takes then the following form:  2 2  2  @2v 3 @2v @2v @v @ v þ  E þ 2   1   cos !  e 2 2 @ @@ @ 2 @  @ 2     1 @ @v 2 @ 2 v @v @ @v @ 2 v E  Ev þ ¼0 v 2 @ @  @ 2 @ @ @ @2

(2:6)

where g – dimensionless transport speed,  = T1/ T0 – amplitude of the periodic load, ! – dimensionless frequency of the periodic load, Ee – nondimensional elastic modules, Ev – nondimensional dynamic viscosity. On the basis of eigenfunctions of the moving string and employing the Galerkin method, a differential equation with partial derivatives (2.6) was discretized to obtain a system of ordinary differential equations. The equations were then subjected to numerical investigations, whose results are presented in the form of bifurcation diagrams, phase trajectories and Poincare maps. Some of the results obtained in [67] are depicted in Figs. 2.3 and 2.4. The bifurcation diagram in Fig. 2.3 shows a dependence of the dimensionless transverse displacement of the string v on the amplitude of parametric excitation. For low amplitudes, first a double-period bifurcation can be seen, then along with an increase in the amplitude, a tangential bifurcation is observed, whereas for high amplitudes a region of the chaotic motion occurs. Figure 2.4 presents a bifurcation diagram of the transverse displacement of the string v as a function of the magnitude of the dimensionless internal damping coefficient Ev. A bifurcation cascade can be seen, starting from a chaotic motion at low values of damping to a regular motion at high damping.

Fig. 2.3 (a) Displacement, (b) velocity. Bifurcation of the center of string versus the nondimensional amplitude of the periodic perturbation [67]

2.2 Beam Systems

17

Fig. 2.4 (a) Displacement, (b) velocity. Bifurcation of the center of string versus the nondimensional dynamic viscosity [67]

2.2 Beam Systems 2.2.1 Linear Models In first works devoted to axially moving beam systems, transverse vibrations of the Bernoulli beam rigidly supported were analyzed. Similarly as in the case of the moving string, the transverse motion of the beam is characterized by a variable angle of the phase displacement. Due to dissipation of energy in the beam, divergent-type instability occurs for individual modes of vibrations at various critical transport speeds. Beam support flexibility and the normal component of acceleration of the beam system in its motion around guide rolls cause that stresses in the axially moving beam are a function of its transport speed [35,40]. The Timoshenko equations for the axially moving beam were generated for the first time in the study by Simpson [50]. Then, Patula [45] investigated vibrations of the moving beam band under the transverse point loading. The critical speed at which the system losses its stability was determined. A comparison of the results obtained for both the models, i.e.: the Bernoulli beam and the Timoshenko beam shows their good convergence only for low transport speeds. The value of critical speed determined with the Timoshenko beam model is lower than with the Bernoulli beam model. In the initial period, only operation of band saws inspired researchers in their investigations of the beam model. While cutting wood or metal with band saws, the saw blade is subjected to edge loading forces. These forces generate both a transverse motion as well as a torsional motion of the blade and can lead to the band buckling. In early studies, torsional vibrations of the axially moving beam loaded with a force focused on the edge were investigated, determining thus a dependence between the torsional vibration frequency and the load. However,

18

2 State of Knowledge on Dynamics of Axially Moving Systems

later in [36,37] a significant error was found that is made while neglecting a coupling between torsional and transverse vibrations of the beam. An analysis of the uncoupled system leads to overrated values of critical buckling load, which finally yields wrong results in analysis of the system motion stability. In the early 1990s, Wickert and Mote [59] published their study, in which the conclusions resulting directly from the previously issued works on string systems and beam systems within the linear theory were employed. Treating an object as a gyroscopic system and not taking any simplified assumptions, the authors investigated free and forced vibrations with the modal analysis introduced by Meirowitch [42] for discrete gyroscopic systems and by D’Eleuterio and Huges [12] for continuous gyroscopic systems. To solve the linear problem, Green functions were used. The authors showed that each eigenfrequency of the system decreased with an increase in the transport speed. Eigenfrequencies, despite their conservative nature, have a complex form due to that fact that there is a transport speed. A very characteristic diagram that illustrates the dynamic behavior of linear axially moving systems in the form of a dependence of eigenfrequencies of transverse vibrations on the transport speed, which can be found in Wickert and Mote [59], is presented in Fig. 2.5.

Fig. 2.5 Frequency spectrum of a simply supported traveling beam [59]

2.2 Beam Systems

19

A solution to the linear problem of the axially moving string and beam presented in [59] has become a traditional approach to which all references on the topic, issued after its publications, refer to. The same authors Wickert and Mote have shown the periodicity of the energy transfer of a uniform traveling string and beam systems. In the work [61] they reported that the total mechanical energy of an axially moving string or beam that travels between two supports is not constant. The work [61] has been extended in [13] to a study of boundary control of an axially moving string via Lyapunov method. The optimal strength for the gain of boundary feedback control of moving string under which the transverse vibrations of the system can be driven to zero has been presented in [19].

2.2.2 Nonlinear Beam Systems Works devoted to investigations of nonlinear vibrations of moving beams constitute a very important chapter in the studies on the dynamic behavior of axially moving systems. Nonlinear investigations of the moving beam under tension are motivated by two clearly visible drawbacks of the linear theory. Each of them can be noticed when the transport speed increases. The first drawback can be seen at velocities lower than the first (undercritical) wave speed of the beam. Much the same as in the case of the moving string, a participation of nonlinearity in beam displacements increases with an increase in the transport speed, thus imposing some limitations of an application of the linear analysis with respect to a low amplitude and a low transport speed. The earliest calculations of the fundamental period of nonlinear transverse vibrations were made by Mote [38] for a limited case of vanishing flexural stiffness. Calculation problems in integration of the motion equation restricted the solution to the speed below 40% of the critical speed (Table 2.1), but the results were sufficient to show that an effect of changes in the tension force during vibrations increased significantly with displacement. The second drawback of the linear theory can be seen in the range of overcritical transport speeds of the moving beam. Then, the nontrivial equilibrium position bifurcates from a simple configuration and either a local motion around the equilibrium position or a global motion between the coexisting equilibrium positions can occur. In the system of the initially straight pipe with a moving fluid, Holmes [17] investigated precisely the symmetrical saddlenode bifurcation, which occurs at the critical speed. The motion equation was discretized with the Galerkin method and a low-dimensional model was defined on the central manifold near the bifurcation point in order to describe the global dynamic behavior qualitatively and geometrically. For transport speeds higher than the critical speed, a stable limit cycle does not occur when nonlinearity is taken into consideration, and solutions disappear in time up to the stable nonsinusoidal equilibrium position. A considerably different dynamic behavior

20

2 State of Knowledge on Dynamics of Axially Moving Systems

in the overcritical range occurs in the case of a cantilever pipe with a moving fluid. At the critical speed, the trivial solution bifurcates into a periodic orbit and is subject to a Hopf bifurcation. The results of the conducted dynamic analyses have allowed early enough for drawing a conclusion that it is impossible to obtain an analytical description of nonlinear vibrations of axially moving systems that can be employed within the whole range of transport speeds. In the literature, the motion limitations were defined and simplified assumptions that tended to cover only most significant features of the system began to appear in the mathematical description of the systems under analysis. These tendencies are well illustrated in the already quoted publication by Wickert [58]. The author deals with transverse and longitudinal displacements of the beam moving at a constant speed. A schematic view of the system under consideration is depicted in Fig. 2.6. Two coupled, nonlinear differential equations of the beam motion along the longitudinal and transverse directions were formulated with the use of the Hamilton’s principle:   1 ðu0 tt þ 2 v u0 xt þ v2 u0 xx Þ  v2l u0 x þ w20 x ¼ 0 2 0x   

1 ðw0 tt þ 2 v w0 xt þ v2 w0 xx Þ  1 þ v2l u0 x þ w20 x w0 x þv2f w0 xxxx ¼ 0 2 0x

(2:7)

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ c  A=P ; vl ¼ EA=P ; vf ¼ EI=P l 2 Next, the author introduced a simplification that consists in expressing the dynamic component of the longitudinal stress in second equation (2.7) as a function of time only into the system of equations. He refers here to the results of experiments on the basis of which it was stated that longitudinal perturbations propagated much faster than the transverse ones within the technologically usable range of beam model parameters [35]. A steel band saw with a blade of the length of l = 2 m and a cross-section of 300 mm  2 mm was analyzed. This blade subject to tensioning with a force of P = 26 kN is characterized by the longitudinal phase speed of 5100 m/s, whereas the speed of transverse vibrations corresponding to the lowest mode is equal to 75 m/s. The typical working speed of such a saw equals 50 m/s. On the time diagrams of low transverse modes of vibrations, changes in stresses are almost instantaneous as the effect of longitudinal inertia is low. Taking this into account and having

Fig. 2.6 Model of the moving beam [58]

2.2 Beam Systems

21

neglected time derivatives and the term v2u,xx as well, the author integrated the first equation of motion (2.7). The obtained solution points out to the fact that the longitudinal displacement increases within a limited range for finite transverse vibrations. Thus, a simplified form of the differential-integral equation of the beam transverse motion has been defined in the following form: 2

ðw0 tt þ 2 v w0 xt þ v w0 xx Þ  w0 xx þ

v2w w0 xxxx

v2 ¼ l w0 xx 2

Z1

w20 x dx

(2:8)

0

To obtain an asymptotic solution to the differential-integral equation of motion (2.8), a modal theory of perturbations with the Krylov-Bogolubov method has been employed. The assigned points of motion equation (2.8) have been examined analytically on the assumption of the following bifurcation configuration: ffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðkÞ ¼  v2  vðkÞ2 sinðkp xÞ ; k ¼ 1; 2; ::: (2:9) k p v1 Each configuration w(k) exists only if v exceeds the k-th critical speed defined by the following relationship: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi (2:10) vðkÞ ¼ 1 þ kp vw In this solution Wickert has shown that for the range of undercritical velocities, the linear theory underestimates an occurrence of the instability region, whereas for overcritical transport velocities, there is an overestimation. Figure 2.7, where the same notations as in Fig. 2.2 are used, illustrates this regularity. The results presented in the study by Wickerta point out to the fact that it is the most essential to include the geometrical nonlinearity in the investigations of the moving material continuum in the neighborhood of critical speeds. In these regions, low modal stiffness is dominated by nonlinear longitudinal stiffness.

Fig. 2.7 Linear and nonlinear fundamental frequencies of the moving beam [58]

22

2 State of Knowledge on Dynamics of Axially Moving Systems

The ideas of Wickert have found their continuation in publications by Pellicano and Vestroni [46,47,48], where the nonlinear beam model (2.7) together with the conditions of simple support at the ends was discretized with an application of the Galerkin method. In discretization, the usage of eigenfunctions is most often the best choice for a description of the system displacement. However, in the case of axially moving objects, eigenfunctions are complex and they depend on the transport speed. Moreover, the eigenfunction basis depends on the equilibrium position under consideration in the nonlinear system after bifurcation. Thus, the authors approximated the solution with a series of the sinus function, not limiting the number of terms in this series. wðx; tÞ ¼

N X

qn ðtÞ sinðnpxÞ

(2:11)

n¼1

The number of terms N in (2.11) depends on the convergence of the series and it is related to the smoothness of the approximated function of the beam displacement. When solution (2.11) is introduced into (2.8) and the Galerkin procedure is applied, we obtain N differential ordinary equations in the following form: # h i   ð1Þnþk 1 ð1Þnk 1 þ k q€n  2 v q_ k þ v2f n4 p4  v2  1 n2 p2 qn ¼ nþk nk k¼1;k6¼n (2:12) N v21 n2 p2 X 2 2 ¼ qn k qk 4 k¼1 N X

"

wherepffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi vf ¼ EI=Pl2 ; v1 ¼ EA=P As a result of the dynamic analysis conducted later on in [47], the authors showed that the same equilibrium positions (2.9), assumed by Wickert in [58], could be obtained from the discrete model (2.12). It confirms that the assumed solution (2.11) is sufficient to render the essential properties of the system under analysis. The linear analysis of stability of equilibrium positions, based on the potential energy, showed these features in the overcritical range of transport velocities. In this range, only the first equilibrium position is stable, whereas the simple equilibrium position and further bifurcation positions are unstable in general. Numerous works in which the dynamic behavior of beam systems with various kinds of nonlinearity were studied have been issued. In these works, the sources of nonlinear reactions were as follows: dry friction, whose source was located along the beam band [57], periodically variable loading of the beam end [36,62], and periodically variable speed of the beam [10] and periodically variable beam loading [31,33,66].

2.2 Beam Systems

23

A special position among the works devoted to dynamics of systems described with the beam model is occupied by studies analyzing the dynamic behavior of a coupled system of the moving band (belt) in the form of a loop and guide belt pulleys. In such a system, two band spans are considered between the pulleys: a primary span and a secondary span. In the beginning, the works devoted to dynamic analyses of the band-pulleys system were focused only on the primary band dynamics. It was assumed that the secondary band motion and the pulley motion were uncoupled. The existence of such a coupling in the form of ‘‘beating’’ effects of the secondary band was found during the experimental investigations. An accurate dynamic model of the moving band, together with guide belt pulleys located on the elastic structure, was investigated by Wang and Mote [56]. It was found that inertia of pulleys, pulley support rigidity, band tension, as well as transport speed were parameters that exerted an influence on the degree of the motion coupling between the spans. In [55] by the same authors, an analysis of vibrations of the band – pulleys system under pulse excitation of the band edge motion was conducted together with an experimental verification of the results obtained. The fact that a coupling of the motion between spans which were formed by the band moving between pulleys was proven exerted a direct influence on the way the system vibrations were controlled. A decrease in band transverse vibrations in the primary span can be achieved by an application of an active vibration damper that acts on the secondary span, pulleys or the pulley support system [16,60]. Recently, numerous studies devoted to the nonlinear analysis of the axially moving beam, where an effect of internal damping of the beam material has been analyzed, have been published. To solve the problems imposed, the following has been used: an increment method of harmonics balance [51], a method for analysis of nonstationary, complex modes of beam deflections [6], and an integration method of the system of differential ordinary equations obtained as a result of the discretization of motion equations with partial derivatives [8,32,63]. The method of numerical integration of the discretized equations of motion was used in the paper by Xiao-Dong Yang and Li-Qun-Chen [63]. Dynamics of the viscous beam performing an unsteady motion was analyzed. The rheological Kelvin-Voigt model was used to describe the beam material. In the mathematical model of the axial strain, the geometrical nonlinearity in the form of the Lagrange strain was taken into account. An axially moving thin steel plate performing an unsteady motion, described with the beam model, was subjected to analysis. In Figs. 2.8 and 2.9 two bifurcation diagrams presenting the results obtained in [63] are shown. These figures show bifurcation diagrams of the dimensionless transverse displacement of the beam as a function of the amplitude of a harmonic change in the transport speed  1, for two different frequencies of these changes. In both cases for low values of  1, a rectilinear static equilibrium position of the beam occurs as a state of the dynamic equilibrium. For a low value of frequency (!= 0.55 – Fig. 2.8), the equilibrium position losses stability and a periodic motion with narrow chaotic bursts can be observed along with an increase in the

24

2 State of Knowledge on Dynamics of Axially Moving Systems

Fig. 2.8 Effects of the dimensionless amplitude of axial speed fluctuation (! = 0.55); (a) displacement, and (b) velocity [63]

amplitude  1. At high values of the amplitude  1, a reverse situation is observed, namely: a chaotic motion is broken by narrow windows of a periodic motion. For a high value of frequency in the transport speed changes (!=12 – Fig. 2.9), a periodic motion is observed in the initial range of the amplitude value  1, then it is locally broken by narrow chaotic bursts at medium values of the amplitude, to transform into a quasi-periodic motion at high values of the amplitude. The bifurcation diagrams in Figs. 2.8 and 2.9, as well as the remaining results presented in [63] show that the occurrence of a chaotic or quasi-periodic motion in the investigated beam system moving with a harmonically variable speed depends on unsteady motion parameters. For the steady frequency of changes in speed, this motion occurs along with an increase in the speed of change amplitude, as well as at an increase in the mean value of the transport speed around which these changes occur.

Fig. 2.9 Effects of the dimensionless amplitude of axial speed fluctuation (! = 12.0); (a) displacement, and (b) velocity [63]

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25

2.3 Plate Systems As follows from the description presented in the previous sections, to avoid too complex a mathematical description, the string or beam theory was used in numerous earlier studies to model axially moving two-dimensional systems. Although this simplification leads in many cases to satisfactory results, however it cannot be applied in numerous instances. Such situations occur when the plate width and its orthotropic properties, a change in the distribution of load along the plate width, a lack of free ends of the plate, composite plates or intermediate linear or point supports of the plate are taken into consideration. In all these instances, it is necessary to employ the two-dimensional plate theory in dynamic investigations.

2.3.1 Numerical Investigations The first paper in which a dynamic analysis of the plate model of a wide blade of the band saw was analyzed was authored by Ulsoy and Mote [52]. The model of the plate moving with a constant axial speed was analyzed on the assumption of typical membrane stresses effect. The differential equation of motion with partial derivatives was discretized with the Ritz method. Approximate boundary conditions on free edges of the plate were taken into consideration, which – as pointed out in further studies – affects fundamentally an accuracy of the solution, especially in the overcritical range of the plate motion. Ulsoy and Mote investigated coupled transverse and torsional vibrations of the axially moving plate. Frequencies of both transverse and torsional vibrations are strongly dependent on the state of stresses existing in the plate plane. This dependence is illustrated in Fig. 2.10, where a part of the results published in [52] for a linear distribution of stresses along the plate width, whose plot is shown in the bottom of Fig. 2.10. The figure depicts a diagram of changes in dimensionless values of lowest frequencies of transverse vibrations !11 and torsional vibrations !12, as a function of changes in the ratio of parameters describing the membrane stress distribution along the plate width. The obtained results point out the fact that an increase in the ratio of the parameters results in an increase in the frequency of torsional vibrations and a decrease in the frequency of transverse vibrations. The distribution of stresses shown in Fig. 2.11 is typical for the actual prestressed blades. The investigation results show that increasing of the stress variation significantly increases the torsional natural frequency while slightly reducing the transverse frequency. Later studies devoted to investigations of the dynamics of moving plate systems were issued in the 1990s. Owing to the regions of industrial applications, these investigations were focused mainly on analyses of the dynamics of the moving web, that is to say, a thin plate characterized by low flexural stiffness.

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Fig. 2.10 Lowest transverse and torsional natural frequencies for Nx linear in y[52]

In 1995 Lin and Mote [24] published their paper, in which – starting with the Karman theory [20] – the equations of motion were determined, and then the results of investigations of transverse displacements and a distribution of stresses under equilibrium for an axially moving isotropic web under the transverse loading were presented. In this publication, an influence of boundary regions of the web on displacements and a stress distribution in the equilibrium state was also investigated. A year later, the same authors issued [25,26], where regions of instability of wrinkling and the corresponding wrinkling modes of stationary material webs were determined on the basis of the analysis of eigenvalues of the system under consideration. The fundamental study showing vibration characteristics and results of investigations of stability of axially moving plates within the linear theory was published by Lin in 1997 [23]. The vibrations of two-dimensional, axially moving isotropic plate with freely supported ends and two free ends were analyzed. A constant tension force along the transport speed acted on the freely supported ends in the plate plane. The boundaries of instability regions of the plate motion were determined in two ways: through the determination of existence of the nontrivial equilibrium position using a static analysis and through an examination of eigenvalues of

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27

Fig. 2.11 Lowest transverse and torsional natural frequencies for Nx quadratic in y [52]

the discretized system using a dynamic method. The plate transport speed corresponding to the stability boundary (critical speed) was determined as the lowest transport speed at which there existed a nontrivial equilibrium position of the plate or the lowest speed at which the real part of one of eigenvalues became nonzero. The results of the investigations conducted by Lin prove that both the velocities coincide. Lin proved that the critical value of the plate transport speed, similarly as in the case of the string, was equal to the speed of transverse wave propagation in the plate. The value of the critical speed increases if the flexural stiffness of the plate increases and the length to width ratio (slenderness) of the plate decreases. The results of the conducted investigations show as well that the one-dimensional theory of beam always overestimates the value of the plate critical speed, whereas the string theory always underestimates this speed. This regularity demonstrated in [23] is illustrated in Fig. 2.12. For various slendernesses of the plate ( = l/b), a relationship between the dimensionless critical speed C* and the dimensionless plate stiffness e has been plotted in this figure.

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Fig. 2.12 The speed at the onset of instability C*, with different slenderness ratios [23]

On the basis of the Mindlin-Reissner plate theory, X.Wang in [54] presented a description of the finite element for an axially moving thin plate. It was found that it was possible to determine not only the critical transport speed of the web but a distribution of normal stresses and shear stresses in the moving web with such finite elements. With the developed finite elements, numerical calculations of eigenfrequencies and critical speeds of the axially moving paper web were carried out. Among works published in the 1990s and 2000s which are devoted to axially moving web control is worth to note the studies of Young et al. [64,65], a practical study of web position in wallpaper machine [53] and the study of transverse vibration control of an axially accelerating web [43]. In these works, beside dynamic analysis, methods to control both lateral and longitudinal motion of the web are included. In addition, modeling and control of multiple web spans is studied in [65]. Using the date from the real wallpaper machine, mathematical model is obtained and validated against the real system response in [53]. Facilitate the Lyapunow analysis for the vibration regulation in [43] a control strategy for an axially moving web system is presented. In 2002 Luo and Hutton in [29] presented a formulation of the moving triangular isotropic plate finite element loaded with membrane and gyroscopic forces. The mathematical model of this element was derived through a modification of the Kirhchoff discrete theory. Some exemplary results of dynamic calculations of the axially moving plate under loading with a variable distribution presented in this study were verified by a comparison to the results of calculations obtained with the Releigh-Ritz method. Another formulation of the finite element was presented in [21], where equations of the modal finite element within the frequency domain were formulated on the basis of the Kantorowitz’ method. This element was then applied to

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29

dynamic calculations of the axially moving plate. The results of these calculations were compared in [21] to the analytical solution and to the FEM solution. In 2004 an extensive study by A.C. Luo and H.R. Hamidzadech [28] was issued, where equilibrium, membrane forces and buckling stability of thin plates were analyzed analytically. The plate model analyzed in this study is presented in Fig. 2.13. Within the Karman linear theory of plates and the nonlinear membrane theory, in [28] motion equations of the axially moving orthotropic plate were derived. Solutions to the motion equations were obtained with a perturbation method. Numerical computations were conducted for simple supports of all plate ends and for constant longitudinal and transverse vibrations. For the sinusoidal mode of the plate deflection: w ¼ " fmn ðÞ sin ðmpXrÞ sin ðnpYÞ;

(2:13)

where " ¼ h=b;  ¼ c1 t þ x; X ¼ =b; r ¼ b=a; Y ¼ y=b; fmn ð Þ – amplitude of the mode (m,n), under constant longitudinal loading and at a lack of transverse loading, an analytical relationship defining the critical transport speed of the plate was expressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 h i2 P Ehp2 h ðmrÞ2 þn2 þ ckr ¼ 2 0 h 12 0 ð1  2 Þ ðmrÞ b

(2:14)

On the basis of the solution presented in [28] for a steel plate of the following parameters: E= 21011 N/m2; 0 = 7.8103 kg/m3; = 0.3; a = 2 m; b = 1 m that moves with a speed c1 = 31.4 m/s, at a lack of longitudinal and transverse loadings (P = Q= 0), a distribution of buckling deformations along individual directions is shown in Fig. 2.14. The membrane forces presented in Fig. 2.15 correspond to the deformations from Fig. 2.14. The authors of [28] have formulated an important conclusion that while analyzing high modes of the plate deflection, the nonlinear plate theory should

Fig. 2.13 An axially moving plate [28]

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Fig. 2.14 Equilibrium displacement solutions in (a) the z-direction, (b) the x-direction and (c) the y-direction [28]

Fig. 2.15 Membrane forces in (a) the x-direction, (b) the y-direction and (c) shear membrane force for equilibrium [28]

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31

be the basis for analysis even if amplitudes of the deflection are very low compared to the plate thickness. Recently a study by S. Hatami, M. Azhari and M. M. Saadatpour [16] devoted to free vibrations of axially moving multi-span composite plates has been published. On the basis of the linear theory for thin plates, two investigations methods have been developed in this paper, namely: an exact analytical method and an approximated finite strip method (FSM). In the exact method, the plate stiffness matrix was determined for each span. Its elements are functions of eigenfrequencies, transport speeds and membrane forces the plate is loaded with. Employing the analytical method, eigenfrequencies of the whole multi-span systems are obtained after gathering these components in one matrix. The exact solutions thus obtained, although subject to numerous limitations, are of great value indeed. They provide much information that allow us to check validity, convergence and exactness of numerical methods used in dynamic analyses of axially moving plates. Some computational results of eigenfrequencies for the 4-span orthotropic plate that travels axially, obtained with the exact method, are presented in Fig. 2.16. Figure 2.16 depicts a change in the value of the generalized matrix determinant of stiffness for the whole 4-span system expressed in the logarithmic scale as a function

Fig. 2.16 Absolute of overall stiffness matrix determinant respect to frequency parameter in the exact method [16]

of the dimensionless eigenfrequency .This diagram shows that the first and second eigenvalue are related to the vibration mode in which one half of the sinusoid occurs on the plate width (n = 1), whereas the third eigenvalue is related to the vibration mode in which two halves of the sinusoid occur on the plate width (n = 2).

2.3.2 Experimental Investigations of Axially Moving Plate Systems The earliest experimental investigation results of axially moving band saw blades have been presented in the already quoted publications by C. D. Mote Jr, S. Naguleswaran [40] and A. G. Ulsoy, C. D. Mote Jr [52]. The blades are axially

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Table 2.2 Band saw blade parameters [52] (License No. 1870791412125) Description Parameter The paper [40]

The paper [52]

Length Width Thickness Density Initial static tension Wheel support constant

0.5953 27.5 1.4732 7800 25900 0.21

L [m] B [cm] H [mm]  [kg/m3] R0 [N]

[-]

0.3681 0.9525 0.5461 7754 76.22 0

moving plates with membrane stresses dependent on edge loading, thermal effects, and purposely induced initial stresses. Their motion is constrained by moveable hydrodynamic pressure guides as shown in Fig. 2.17. The parameters data of the blades investigated in [40] and [52] are presented in Table 2.2. Figure 2.18 shows the comparison of the received experimental data and the analytical prediction of the first two flexural frequencies of the table top band used in [52] Experimental investigations of the fundamental natural frequency of the blade at four axial tension values were also conducted on a full scale production band mill. The investigation results have been published in [52] and presented in Fig. 2.19. Among many references published in the late 1990s and the early 2000s which are devoted to the dynamic investigations of axially moving systems, special attention should be drawn to the publication by J. Moon and J. A. Wickert [34] and the publication by F. Pellicano, A. Fregolent, A. Bertuzzi and F. Vestroni [46], both devoted to analysis of nonlinear forced vibrations of power transmission belts. These are the only publication to the knowledge of the author of the

Fig. 2.17 Band coordinates and geometry [52]

2.3 Plate Systems Fig. 2.18 Comparison of analytical curves to experimental data [40] for the two lowest transverse natural frequencies [52]

Fig. 2.19 Comparison of analytical (solid) and experimental results for the fundamental natural frequency at four axial tension levels [52]

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present monograph in which the results of theoretical non-linear analyses of axially moving system have been compared to the experimental measurement results. The same theoretical non-linear model proposed in the publication by J. A. Wickert [58] has been used in both investigations. The experimental investigations included in [34] were carried out on the laboratory test stand, whose general view is presented in Fig. 2.20. A seamless timing belt with a span of L= 860 mm was driven by a variable speed rotor over drive and idle pulleys of a common pitch radius equal to r = 63 mm. Non-linear vibrations of the laboratory belt system were excited by pulleys having slight eccentricity. An idle pulley was mounted on rotation and translation edges in order to minimize those excitation sources that were associated with the pulley misalignment. A Michelson-type laser interferometer was used to measure transverse vibrations of the belt. Reference paths and target light beams were established with fiber optic leads. The interferometer measured alternations in lengths of those two light paths through the interference fringes generated by superposition of coherent beams that reflected from a stationary reference surface and a moving target, that is to say, the belt itself. The belt was coated with a thin layer of retroreflective paint to ensure that sufficient light was scattered from the belt and into the optical head. Particles within the paint matrix ensured that a portion of the incident light was returned into the source optical fiber regardless of the belt finite amplitude or slope. Owing to this technique, vibration measurements were made with a strong signal-to-noise ratio. A displacement resolution (sub-micron) and bandwidth (DC to 100 kHz) exceeded the test requirements. The eccentricity of each pulley with respect to its shaft rotation axis was recorded with an eddy current probe, whose linear range was of approx. 2 mm. Measurements were made on top lands of the pulley teeth. Such eccentricity may follow from manufacturing tolerances (especially in the case of cast components) and a clearance between the pulley hub and shaft necessary for assembly. The measured profiles indicated peak-to-peak run-out levels for the drive and idle pulleys of 0.71 and 0.42, correspondingly. The maxima were shifted in phase by approx. 240° of rotation. The relative phase ’ can be nulled

Fig. 2.20 Schematic of the test used to measure the vibration of power transmission belts [34]

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35

Fig. 2.21 Measured speed-dependent belt response amplitudes over resonance of the first two vibration modes. Data in the non-resonance range 10–14 m/s are omitted for clarity [34]

or adjusted to an arbitrary value by proper synchronization of the pulleys, but under the test conditions, ’ will generally be non-zero. Figure 2.21 presents measurements of the belt peak-to-peak response amplitude versus the running speed over 0 < V < 17 m/s. This range encompasses resonance of the first two modes. The speed was increased monotonically and it was then held constant. Each amplitude measurement was taken about 1 cm from the driven pulley. The time records of transient response shown in Fig. 2.22 obtained as the speed either increased or decreased through resonance of the fundamental mode near 9 m/s., are associated with the amplitude in Fig. 2.21. When the running speed was increased from 8.4 to 9.2 m/s in Fig. 2.22a, the amplitude decreased sharply from the maximal value of 1.98 mm to 0.57 near V= 8.9 m/s, as in Fig. 2.21. In the alternative case of Fig. 2.22b, when the speed was instead decreased through the same range, the amplitude grew from 0.52 to 1.51 during passage through V= 8.5 m/s.

Fig. 2.22 Measured time records depict jumps in amplitude over the first resonance region for monotonically (a) increasing, and (b) decreasing running speeds [34]

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Similar non-reversible behavior, in which the vibration amplitude was dual values and path dependent, was observed at speeds 15.7 m/s and 16.1 m/s, and corresponds to resonance of the belt second mode. Such classic ‘‘jump’’ behavior is characteristic of structural nonlinearity, and it is attributed here to the longitudinal stretching resulting from a finite amplitude motion in the nearresonance and resonance region. The second experimental laboratory test stand is presented in the publication by F. Pellicano, A. Fregolent, A. Bertuzzi and F. Vestroni [46]. The general view of this stand is shown in Fig. 2.23. In this stand a rubber belt with joint moves on two pulleys. The radius of each pulley is 0.1 m and the distance between axes is 1.01 m. A fixed tensioner there is on the lower branch of the belt. To excite the transverse vibrations of the laboratory belt system the left driven pulley has possessed varied eccentricity. The aim of the experimental tests was identification of the primary and parametric non-linear resonances of the transmission belt. A laser telemeter with the cut-off frequency 1000 Hz and the range 0.04 m was used to measure directly the transverse displacement of the belt. Because in resonance conditions large oscillations appear, the transducer was located in proximity of the right pulley, where the amplitude of oscillation was within measurement limits. The behavior of the system in primary resonance conditions with different pulley eccentricities was investigated. The introductory qualitative comparison between the actual vibration shape in the primary resonance condition and the predicted first analytical mode shows a good agreement. Figure 2.24 shows the behavior of the stationary amplitude of vibration in correspondence with the measurement points versus the excitation frequency for different eccentricities. Dots mean the actual amplitudes and balls represent the average values. The continuous lines in Fig. 2.24 present the analytical backbone and the amplitude – frequency curves obtained through the theoretical perturbation approach to the experimental data reported in [46]. In the next part of experimental investigations the parametric resonances were analyzed. Because of the parametric excitation furnished by the axial

Fig. 2.23 Experimental set-up [46]

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37

Fig. 2.24 Frequency amplitude curves. Primary resonance, eccentricity: (a) 0.004 m; (b) 0.0032 m; (c) 0.0013 m; (d) 0.0006 m; (. . .) measured amplitude; (ooo) mean amplitude; (__) analytical solution [46]

tension fluctuation the actual system responds with a one-half subharmonic, i.e. with half excitation frequency and with a spatial shape close to the first mode. Authors have noted that the effect of the direct resonance on the second mode is suppressed by the presence of the parametric instability. It was experimentally observed, that when the excitation frequency approaches twice that of the linear mode, this mode vibrates with very large amplitude and most of the energy transfers from high to low frequency. Figure 2.25a shows the response amplitude of the first mode in correspondence with the measurement points, versus the excitation frequency for the pulley eccentricity 0.004 m. The ratio /2!1 means that the excitation frequency is normalized with respect to the first linear natural frequency !1 at the velocity v such that  (v)/2!1(v) = 2, which corresponds to the dimensional frequency !1 = 34.4 p rad/s. Like previously balls represent measured amplitudes and continuous lines represent the analytical curves. The next series of tests was performed for the pulley eccentricities: 0.0023, 0.0013 and 0.0006 m, using the values of parameters identified in direct resonance. The dimensionless tension fluctuation assumed the values: 0.35, 0.2 and 0.1 for the eccentricities: 0.0023, 0.0013 and 0.0006 m, respectively.

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Fig. 2.25 Amplitude frequency response curves. Parametric resonance, eccentricity: (a) 0.004 m; (b) 0.0032 m; (c) 0.0013 m; (d) 0.0006 m; (ooo) measured amplitude; (__) analytical solution [46]

Comparisons between analytical response and measured amplitudes of the belt vibrations are shown in Fig. 2.25b–d.

2.4 Final Remarks The source of a lively interest in investigations of the dynamic behavior of axially moving systems lies in the fact that the object under consideration has become more and more complex over the years. Starting from axially moving one-dimensional systems (strings, threads), through beam systems, the researchers’ attention is focused on axially moving plate systems at present. Such a chronological order in the development of the object under investigations is clearly visible while one studies the literature devoted to this topic, although some works that supplement the state of knowledge specially on nonlinear dynamics of both string and beam systems are still issued now. It is visible that both geometrical and physical nonlinearity play vital roles in dynamic behaviour of axially moving system. The survey shows also the dynamic behavior of the orthotropic plate systems with viscoelastic properties, which include paper webs, has been the least recognized so far. In the literature already published, only isotropic systems have been analyzed in practice. The Author of this book has aimed at filling this gap to some extent at least.

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48. Pelicano F, Vestroni F (2002) Complex dynamics of high-speed axially moving systems. Journal of Sound and Vibration 258(1): 31–44 49. Sack R A (1954) Transverse oscillations in traveling strings. British Journal of Applied Physics 5: 224–226 50. Simpson A (1973) Transverse modes and frequencies of beams translating between fixed end supports. Journal of Engineering Science 15(3): 159–164 51. Sze K Y, Chen S H, Huang J L (2005) The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration 281: 611–626 52. Ulsoy A G, Mote C D Jr (1982) Vibrations of wide band saw blades. Journal of Engineering for Industry ASME 104(1): 71–78 53. Wang H, Logghe D, Miskin D (2005) Physical modelling and control of lateral web position for wallpaper making processes. Control Engineering Practice 13: 401–412 54. Wang X (1999) Numerical analysis of moving orthotropic thin plates. Computers & Structures 70: 467–486 55. Wang K W, Mote C D Jr (1987) Band/wheel system vibration under impulsive boundary excitation. Journal of Sound and Vibration 115(2): 203–216 56. Wang K W, Mote C D Jr (1986) Vibration coupling analysis of band/wheel mechanical systems. Journal of Sound and Vibration 109(2): 237–258 57. Wickert J A (1993) Analysis of self-excited longitudinal vibration of a moving tape. Journal of Sound and Vibration 160(3): 455–463 58. Wickert J A (1992) Non-linear vibration of a traveling tensioned beam. International Journal of Non-Linear Mechanics 27(3): 503–517 59. Wickert J A, Mote C D Jr (1990) Classical vibration analysis of axially-moving continua. Journal of Applied Mechanics ASME 57: 738–744 60. Wickert J A, Mote C D Jr (1988) Current research on the vibration and stability of axially moving materials. Shock and Vibration Digest 20: 3–13 61. Wickert J A, Mote C D Jr (1989) On the energetics of axially moving continuum. Journal of Acoustical Society of America ASME 85(3): 1365–1368 62. Wu W Z, Mote C D Jr (1986) Parametric excitation of an axially moving band by periodic edge loading. Journal of Sound and Vibration 110(1): 27–39 63. Yang X-D, Chen L-Q (2005) Bifurcation and chaos of an axially accelerating viscoelastic beam. Chaos, Solitions and Fractals 23(1): 249–258 64. Young G E, Reid K N (1993) Lateral and longitudinal dynamic behaviour and control of moving webs. Journal of Dynamic Systems, Measurement, and Control ASME 115: 309–317 65. Young G E, Shelton J J, Kardamilas C (1989) Modeling and control of multiple web spans using state estimation. Journal of Dynamic Systems, Measurement, and Control ASME 111: 505–510 66. Zhang L, Zu J W (1999) Nonlinear vibration of parametrically excited moving belts, Part I, II. Journal of Applied Mechanics 66(2): 396–409 67. Zhang N-H, Chen L-Q (2005) Non-linear dynamics of axially moving viscoelastic strings. Chaos, Solitions and Fractals 24(4): 1065–1074

Chapter 3

Dynamical Analysis of the Undamped Axially Moving Web System

3.1 One-Layered Orthotropic Web Dynamics of the thin web moving axially with a constant velocity is considered in this chapter. The geometrical dimensions of the system under analysis along with the assumed system of coordinates are presented in Fig. 3.1. In order to describe the dynamical behavior of the web, the following initial assumptions have been considered: y h

c

b x

l

Fig. 3.1 Physical model of the moving web system

z

1) the web is characterized by orthotropic properties, 2) the phenomenon of elastic wave propagation in the web plane has not been taken into account, 3) the Karman’s nonlinear theory of thin plates has been adopted in the analysis [2].

3.1.1 Formulation of Nonlinear Equations of the Web Motion We consider a case of the web motion in a time interval (t0,t1) according to the Lagrange’s description. In the time interval between the initial and final position, various trajectories of motion of the system under analysis can be considered. The actual trajectories differ from the other ones in that aspect that they satisfy the Hamilton’s principle: Zt1 

ðU  V þ W Þdt ¼ 0

(3:1)

t0

K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_3, Ó Springer-Verlag Berlin Heidelberg 2008

43

44

3 Dynamical Analysis of the Undamped Axially Moving Web System

where U – kinetic energy of the system; V = Vb+Vm – potential energy of the system; Vb – energy of the elastic strain of bending; Vm – energy of elastic strains of the membrane state; W – work performed by external forces. The following nonlinear geometrical relationships between strains and displacements have been assumed: 1 2 w ; 2 ;x 1 "y ¼ v; y þ w2; y ; 2

"x ¼ u; x þ

(3:2)

xy ¼ 2"xy ¼ u; y þ v; x þ w; x w; y ; x ¼ w; xx ; y ¼ w; yy ; x ¼ w; xy where a comma in the index denotes a partial derivative. A relation between the Young moduli and Poisson numbers for main orthotropy directions of the web takes the following form: Ex yx ¼ Ey xy

(3:3)

Relations between cross-sectional forces and strains are expressed by:

Nx ¼

Eh Eh ð"x þ "y Þ ; Ny ¼ ð"y þ "x Þ 1   2 1   2

Nxy ¼ Ghxy ¼ 2Gh"xy ; My ¼ Dðw;yy þ w;xx Þ ;

Mx ¼ Dðw;xx þ w;yy Þ

(3:4)

Mxy ¼ D1 w;xy

where E ¼ Ex ; yx ¼ ;

 ¼ xy ;  ¼

Ey yx ¼ ; E 

Eh3 Gh3 ; D1 ¼ D¼ 2 12ð1   Þ 6

(3:5)

The individual components of energy that occur in the Hamilton’s principle (3.1) can be expressed as follows:

3.1 One-Layered Orthotropic Web

Z

1 U ¼ h 2

½ðc þ u;t þ cu;x Þ2 þ ðv;t þ cv;x Þ2 þ ðw;t þ cw;x Þ2  dS;

S

Z

1 Vm ¼ 2

45

½Nx "x þ Ny "y þ Nxy xy  dS;

S

Vb ¼

1 2

Z

½Mx x þ My y þ 2Mxy xy  dS;

S

W ¼

Zb

Nx0 ðyÞjx¼0 ðujx¼l ujx¼0 Þ dy þ

0

þ

(3:6) Ny0 ðxÞjy¼0 ðvjy¼b vjy¼0 Þ dx þ

0

Zb

Nxy0 ðyÞjx¼0 ðujx¼l ujx¼0 Þ dy þ

0

þ

Zl

Zl

Nxy0 ðxÞjy¼0 ðvjy¼b vjy¼0 Þ dx

0

Z qz w dSi ; S

where S – web area, qz – web transverse loading, Nx0 ; Ny0 ; Nxy0 – external loading forces of the web. After determining variation increments on the basis of 3.1 and grouping the individual terms of variations, the following variational system of equilibrium equations is obtained: Zt1 Z t0

½ h ðw;tt  2cw;xt  c2 w;xx Þ þ qz þ ðNx w;x Þ;x þ ðNy w;y Þ;y þ ðNxy w;x Þ;y þ

S

þ ðNxy w;y Þ;x þ Mx;xx þ 2Mxy;xy þ My;yy  w dS dt ¼ 0 Zt1 Z t0

S

Zt1

Z

t0

S

½ hðu;tt  2cu;xt  c2 u;xx Þ þ Nx;x þ Nxy;y  u dS dt ¼ 0

½ hðv;tt  2cv;xt  c2 v;xx Þ þ Ny;y þ Nxy;x  v dS dt ¼ 0

Variational boundary conditions:

(3:7)

46

3 Dynamical Analysis of the Undamped Axially Moving Web System

Z l Zt1 0

t0

Zbi

Zt1

0

t0

Z l Zt1

½ hðc2 þ cu;t þ c2 u;x Þ  Nx þ Nx0 ðyÞ u dy dtjx¼l x¼0 ¼ 0

i ¼bi ½  Ny þ Ny0 ðxÞ v dx dtjyy¼0 ¼0

t0

0

Zb Zt1

½ hðcv;t þ c2 v;x Þ  Nxy þ Nxy0 ðyÞ v dy dtjx¼l x¼0 ¼ 0

t0

0

Z l Zt1 0

t0

Zb

Zt1

0

t0

Z l Zt1 0

½Nxy þ Nxy0 ðxÞ u dx dtjy¼b y¼0 ¼ 0

(3:8) i ½  Ny w;y  Nxy w;x  My;y  2Mxy;x  w dx dtjy¼b y¼0 ¼ 0

½ hðcw;t þ c2 w;x Þ  Nx w;x  Nxy w;y  Mx;x  2Mxy;y  w dy dtjx¼l x¼0 ¼ 0

My  w;y dx dtjy¼b y¼0 ¼ 0

t0

Zb Zt1 Mx  w;x dy 0

dtjx¼l x¼0

¼ 0;

t0

Zt1

x¼l y¼b 2 Mxy x¼0  w dt ¼ 0 y¼0

t0

Variational initial conditions: Zb Z l 0

0

Zb

Zl

0

0

Zb

Zl

0

0

1 ½ hðw;t þ c w;x Þ w dy dxjt¼t t¼t0 ¼ w ð0Þ

1 ½ hðc þ u;t þ c u;x Þ u dy dxjt¼t t¼t0 ¼ uð0Þ

1 ½ hðv;t þ c v;x Þ v dy dxjt¼t t¼t0 ¼ vð0Þ

(3:9)

3.1 One-Layered Orthotropic Web

47

Equations of motion (3.7) have the following form of differential equations of equilibrium:  hðw;tt  2cw;xt  c2 w;xx Þ þ Mx;xx þ 2Mxy;xy þ My;yy þ þ qz þ ðNx w;x Þ;x þ ðNy w;y Þ;y þ ðNxy w;x Þ;y þ ðNxy w;y Þ;x ¼ 0

(3:10)

 hðu;tt  2cu;xt  c2 u;xx Þ þ Nx;x þ Nxy;y ¼ 0

(3:11)

 hðv;tt  2cv;xt  c2 v;xx Þ þ Nxy;x þ Ny;y ¼ 0

(3:12)

If the propagation of elastic waves in the web plane x  y is not taken into account and if kinetostatic effects are neglected, then equilibrium equations (3.11) and (3.12) for the stationary web (c = 0) can be reduced to the strain inseparability equation in the following form:     1 E F; xxxx þ F; xxyy  2   þ  F; yyyy ¼ w2;xy  w;xx w;yy Eh G

(3:13)

where F – the Airy function of cross-sectional forces that fulfils the relationships: Nx ¼ F;yy ; Ny ¼ F;xx ; Nxy ¼ F;xy The mathematical model that describes the transverse motion and the field of cross-sectional forces of the axially moving orthotropic web on the assumptions taken has a form of the system of two coupled nonlinear differential equations with partial derivatives:  hðw;tt  2cw;xt  c2 w;xx Þ þ Mx;xx þ 2Mxy;xy þ My;yy þ þ qz þ F;yy w;xx  2 F;xy w;xy þ F;xx w;yy ¼ 0     1 Ei  F;xxxx þ F;xxyy  2   þ  F;yyyy ¼ w2;xy  w;xx w;yy Eh G

(3:14)

(3:15)

Equations (3.14) and (3.15) yield a nonlinear mathematical model of the axially moving orthotropic material web. This model is the basis for dynamic analyses presented in further parts of the book.

3.1.2 Solution to the Mathematical Model In a further part of considerations devoted to a solution to the mathematical model of the orthotropic axially moving web, it has been assumed that the web is loaded along the longitudinal direction only:

48

3 Dynamical Analysis of the Undamped Axially Moving Web System

Nx0 ðyÞ 6¼ 0 ;

Ny0 ¼ Nxy0 ¼ 0 ;

(3:16)

The following boundary conditions that correspond to the simple support at both ends of the web have been assumed (Fig. 3.1): w ðx ¼ 0; yÞ ¼ w ðx ¼ l; yÞ ¼ 0 ; Mx ðx ¼ 0; yÞ ¼ Mx ðx ¼ l; yÞ ¼ 0

(3:17)

To determine transverse displacements of the web and the function F, two systems of orthogonal functions as the conditions along longitudinal unloaded edges have been introduced: Z l Z l Z l Z l Ny vdx ¼ 0 ; Nxy udx ¼ 0 ; My w;y dx ¼ 0 ; Qy wdx ¼ 0 (3:18) 0

0

0

0

where Qy– shear force. The introduced orthogonal functions have the form as follows: f1 ¼ w ; f2 ¼ w; ¼ f3 ¼ w;

w;y ; b

(3:19)

þ w; ;

f4 ¼ E2 ðw;

þ w; Þ; þ 4w; ;

and a1 ¼ F ; a2 ¼ F ; ; a3 ¼ F ; a4 ¼ ðF;

(3:20)

  F; ;   F; Þ; þ

E F; ; G

where ¼

x ; b

¼

y ; b

E1 ¼

E ; Gð1    2 Þ

E2 ¼ E1 

(3:21)

After introducing the system of orthogonal functions (3.20), cross-sectional forces (3.4) can be expressed as: 1 1 1 1 F; ¼ 2 a2; ; Ny ¼ 2 F; ¼ 2 a1; b2 b b b (3:22) 1 1 Nxy ¼  2 F;  ¼  2 a2; b b Because it is impossible to find the exact solution to the nonlinear differential equation (3.14), an approximate solution to the mathematical model of the Nx ¼

3.1 One-Layered Orthotropic Web

49

system has been sought in further investigations. It has been assumed that orthogonal functions (3.19) that express a web deflection and its derivatives are approximated with a function series of the linearized system that is described by the linear part of equation (3.14): f1 ¼

M X N X

Tmn ðtÞ F1mn ð Þ sin

mp b l

Tmn ðtÞ F2mn ð Þ sin

mp b l

m¼1 n¼1

f2 ¼

M X N X m¼1 n¼1

f3 ¼

M X N X m¼1 n¼1

f4 ¼

M X N X

(3:23) mp b Tmn ðtÞ F3mn ð Þ sin l Tmn ðtÞ F4mn ð Þ sin

m¼1 n¼1

mp b l

where F1mn ðy Þ; F2mn ðy Þ; F3mn ðy Þ; F4mn ðy Þ – unknown functions of the dimensionless transverse coordinate y , Tmn(t) – unknown function of time. While determining the function F1mn ðy Þ=F4mn ðy Þ, it has been assumed that they correspond to the modal frequencies of free vibrations of the stationary plate on assumption that there is no transverse load, i.e., qz = 0. A harmonic mode of the function Tmn(t) has been assumed: Tmn ðtÞ ¼ ejw mn t

(3:24)

where j is an imaginary unit. If we take into account (3.23) and (3.24) in (3.19), and put them into linearized equation (3.14), we obtain a coupled, homogenous system of differential equations of the first order with respect to the transverse direction of the web y , on the basis of which the unknown functions Fmn can be determined: d F1mn ¼ F2mn ; d   dF2mn mpb 2 ¼ F3mn þ  F1mn ; l d " #   dF3mn 1 mpb 2 ¼ F4mn þ 4 F2mn ; E2 l d     dF4mn mpb 4 d F2mn m p b 2 ¼ E1 F1mn þ E2 þ l l d d   2 b2  h 12 b2 mpb 2 þ F1mn b2 w 2mn  2 F1mn E1 ð1    2 Þ D D1 l h

(3:25)

50

3 Dynamical Analysis of the Undamped Axially Moving Web System

where w mn – modal frequency of free vibrations of the linearized system, D – dimensionless longitudinal displacement of the web. Having introduced the system of orthogonal functions (3.20) into the differential equation of strain inseparability (3.15), we obtain: a2 ¼ a1; a2; ¼ a3 þ  a1; a3; ¼ a4 

E a2; G

(3:26)

 1  2 a1; þ  a4;    a2; ¼ f2;  f1; f2; Eh The following solutions to the set of equations (3.26) have been foreseen:  ðm  iÞpb þ mi A1mnij ð Tmn Tij ð1  mi ÞA1mnij ð Þ cos l m n j i  2 ðm þ iÞpb þ A2mnij ð Þ cos þ Nx0 b2 l 2  XXXX ðm  iÞpb þ mi B1mnij ð a2 ¼ Tmn Tij ð1  mi ÞB1mnij ð Þ cos l m n j i  ðm þ iÞpb þ B2mnij ð Þ cos þ Nx0 b2 l  XXXX ðm  iÞpb þ mi C1mnij ð Tmn Tij ð1  mi ÞC1mnij ð Þ cos a3 ¼ l m n j i  ðm þ iÞpb þ C2mnij ð Þ cos þ Nx0 b2 l  XXXX ðm  iÞpb þ mi D1mnij ð a4 ¼ Tmn Tij ð1  mi ÞD1mnij ð Þ cos l m n j i  ðm þ iÞpb þ D2mnij ð Þ cos l a1 ¼

XXXX

Þþ

Þþ

Þþ

Þþ

(3:27) whereA1mnij . . . D2mnij – unknown functions y , mi – Kronecker’s delta. After introducing (3.27) into (3.26), we obtain a non-homogenous, coupled system of differential equations of the first order with respect to y :

3.1 One-Layered Orthotropic Web

51

dA1mnij ¼ B1mnij ; d dA2mnij ¼ B2mnij ; d   dB1mnij ðm  iÞp b 2 ¼ C1mnij  mi  A1mnij ; l d   dB2mnij ðm þ iÞp b 2 ¼ C2mnij   A2mnij ; l d   dC1mnij E ðm  iÞp b 2 ¼ D1mnij þ mi B1mnij ; G l d   dC2mnij E ðm þ iÞp b 2 ¼ D2mnij þ B2mnij ; G l d (     ) dD1mnij A1mnij ðm  iÞp b 4 d B1mnij ðm  iÞp b 2 ¼ mi   þ l l d  d "      # Eh mp b ip b dF2ij mp b 2 F2mn F2ij þ F1mn ; þ 2 l l l d (     ) dD2mnij A2mnij ðm þ iÞp b 4 d B2mnij ðm þ iÞp b 2 ¼   þ l l d  d "     # Eh mp b ip b dF2ij mp b 2 F2mn F2ij  F1mn þ 2 l l l d

(3:28)

With the numerical method of transition matrix on the basis of the system of equations (3.28), the unknown functions A1mnij, A2mnij, B1mnij, B2mnij, C1mnij, C2mnij, D1mnij, D2mnij can be determined analogically as it has been done in the case of the function F1mn(y ) . . . F4mn(y ). Taking into account relations (3.22) and (3.27), components of the crosssectional forces can be expressed by the following formulas:

1 XXXX dB1mnij ðm  iÞpb dB1mnij þ mi cos þ Nx ¼ 2 Tmn Tij ð1  mi Þ b m n i j l d d

dB2mnij ðm þ iÞpb þ cos þ Nx0 l d

52

3 Dynamical Analysis of the Undamped Axially Moving Web System

(   1 XXXX ðm  iÞ pb 2 ðm  iÞpb þ Ny ¼ 2 Tmn Tij ð1  mi ÞA1mnij cos b m n i j l l 



ðm þ iÞ pb 2 ðm þ iÞpb cos l l   1 XXXX ðm  iÞ pb ðm  iÞpb þ ¼ 2 Tmn Tij ð1  mi ÞB1mnij sin b m n i j l l  

ðm þ iÞ pb ðm þ iÞpb þ B2mnij sin l l  A2mnij

Nxy

(3:29)

Having determined orthogonal functions (3.19), (3.20) and taken into account the first equation (3.7), nonlinear equation of equilibrium (3.14) takes the following form: Z



12b2 2 ðb f1;tt þ 2cb f1;t þ c2 f1;&& Þ  E1 f1;   E2 f2;  f4; þ Gh2

Si 4

þ

12 qz b 12 þ ½a2; f1; þ a1; f2; G h3 G h3

 2 a2; f2;  f1 dS ¼ 0

(3:30)

After taking into account foreseen solutions (3.23) and (3.27) in equilibrium equation (3.30), this equation has been discretized with the Galerkin orthogonization procedure. Thus, the (M  N) system of ordinary differential equations whose solution defines still unknown time functions Tkr has been obtained. XX d 2 Tkr dTmn þ þ ðz1kr c2  z2kr Þ Tkr þ d0mnkr c 2 dt dt m n XXXXXX þ dmnijpqkr Tmn Tij Tpq ¼ z3kr qz

z0kr

m

n

i

j

p

(3:31)

q

where k = 1, 2, . . . M, r = 1, 2, . . . N. A variability in the following coefficients has been assumed: n, j, q, r = 1, 2, . . . N; m, i, p, k = 1, 2, . . . M. The coefficients of equations (3.31) are defined by the following relationships:

3.1 One-Layered Orthotropic Web

ZZ

12b4 2 kp b d d ; F sin2 l Gh2 1kr   ZZ 12b2 2 kp b 2 kp b d d ; ¼ F sin2 l l Gh2 1kr

z0kr ¼  z1kr

53

z1kr b2 w 2 z2kr ¼  2 rl ; kp b

dmnijpqkr

l

12b4 kp b d d ; F1kr sin 3 l Gh   ZZ 24b3 mp b mp b kp b sin d d ; ¼ F1mn F1kr cos l l l Gh2 (   ZZ dB1ijpq 12 mp b 2 ði  pÞp b mp b sin þ F ¼  cos 1mn 3 Gh l l l d (3:22)   dB2ijpq mp b 2 ði þ pÞp b mp b  sin þ F1mn cos l l l d   ði  pÞp b 2 dF2mn ði  pÞp b mp b  ð1  ip Þ A1ijpq sin þ cos l l l d   ði þ pÞp b 2 dF2mn ði þ pÞp b mp b  A2ijpq sin þ cos l l l d   ði  pÞp b mp b ði  pÞp b sin þ 2 ð1  ip Þ B1ijpq F2mn l l l   mp b ði þ pÞp b mp b ði þ pÞp b þ 2 B2ijpq sin  cos F2mn l l l l

mp b kp b d d  cos F1kr sin l l

z3kr ¼ d0mnkr

ZZ

The discretized mathematical model in the form of ordinary differential equations (3.31) was the basis for dynamical investigations in which flexural vibrations and flexural-torsional vibrations of the web were analysed. Flexural vibration modes are characterized by the fact that the web longitudinal axis y = b/2 is the symmetry axis for them. For flexural-torsional vibration modes, the longitudinal axis of the web y = b/2 is the antisymmetry axis. The modes of vibrations of the axially moving web are discussed in detail in Chap. 4 of this book. In the case flexural vibrations and flexural-torsional vibrations are analysed independently, it can be assumed that N = 1, and in relationships (3.23)/(3.32) n = r = j = q = 1. Thus, the way the relations have been formulated in this chapter can be simplified considerably.

54

3 Dynamical Analysis of the Undamped Axially Moving Web System

For the liner paper web the discretized mathematical model is shown in Appendix A (A.1). For the fluting paper web the discretized mathematical model is shown in Appendix A (A.2).

3.1.3 Results of Comparative Studies During the first stage of dynamic investigations of the moving web, a discretized mathematical model in the form of the (MN) system of ordinary differential equations (3.31) was applied. Before proper investigations started, some calculations to test the numerical computation code that had been built in on the basis on the mathematical model employed were performed. In those test calculations, we used data from the literature that characterize axially moving real objects, i.e. blades. The first real object subjected to the test computations was a band-saw presented in Ulsoy and Mote [5]. The parameters that characterize the band-saw blade mentioned in the study are as follows: l = 1.5 m, b = 05 m, h = 3 mm, E = 0.2 106 MPa. The results of calculations of lowest flexural (w 11) and flexuraltorsional (w 12) free vibrations that were performed at various transport speeds with the linear model have been published. The formulated differential equation of motion with partial derivatives has been discretized by Ulsoy and Mote with the Ritz method. Employing the linearized form of equations (3.31) for M = 5 and N = 5, calculations of the first frequency of flexural free vibrations w 11 and flexuraltorsional free vibrations w 12 of the band-saw blade for various transport speeds and different stiffness of the guide roller support, presented in [5], for the values of the dimensionless coefficient:  = 1 (rigid support) and  = 0 (free support), were conducted. The model of free support in (3.31) is obtained by neglecting the second term that includes c2. The comparative studies were carried out in the undercritical range of transport speeds. Figure 3.2 shows a comparison of the 140 [s–1]

ω ω12

120 × 100

ω11

× × ×

× × × ×

80

×

κ=0

κ=0 × κ=1 × ×

×

× × κ=1

60 ×

40

Fig. 3.2 Results of comparatives studies (band-saw blade, – test calculations,  [5])

20 0

×× ×× ×

×

0

10

20

30

40

×

c × 50 60 [m/s]

3.1 One-Layered Orthotropic Web

55

obtained computational results and the results published in [5]. The differences between the results obtained and the results quoted in [5] did not exceed 1%. The second real object subjected to tests was a wide axially moving paper web that was analysed dynamically by Wang [6]. The parameters that characterize the web are as follows: l = 2 m, b = 8 m, h = 0.7 mm, Ex = 7.44 GPa, Ey = 3.47 GPa, Gxy = 2.04 GPa. In the study quoted, the attention was focused mainly on identification of the stress field in the moving wide web. The results of calculations of lowest frequencies of flexural free vibrations (w 11) of the moving paper web in the subcritical range of transport speed for various values of the axial tension force were published as well. The dynamic calculations in [6] were performed with the finite element method, employing an element whose description was generated on the basis of the Mindlin - Reissner plate theory. In Fig. 3.3, results of the test calculations for M = 20 and N = 1 and the results published in [6] are shown. The differences in the results did not exceed –4% in this case. The difference between values of frequencies of flexural vibrations that remains nearly constant within the whole range of transport speed is probably caused by the fact that the web was not sufficiently densely divided into finite elements by Wang. As has been mentioned earlier, Wang’s attention was focused mainly on identification of stress fields and modes of web deflection in the subcritical range in the numerical calculations. On the other hand, in the literature known to the author of the present book, information on the dynamic behaviour of such objects that could be employed in comparative studies is lacking. The third stage of investigations was a comparison of the numerical results obtained with linearized equations to the results published by Lin [4]. This comparison was performed for dimensionless quantities and was aimed at determination of the critical speed skr versus the plate stiffness y and the web isotropic slenderness l/b. Such a relationship, comprising also ranges of stiffness ω [1/s]

30

N = 150 N/m

25

N = 100 N/m

20 N = 50 N/m 15 10 5

c

0 0

2

4

6

8

10

12

14

16

18 [m/s]

Fig. 3.3 Results of comparative studies (paper web, – test calculations, - - [6])

56

3 Dynamical Analysis of the Undamped Axially Moving Web System 1,5

skr

beam model 1,4

l / b = 0,25 l / b = 0,5 l / b=1 l / b = 10

1,3 1,2 1,1

ψ 1 0

0,02

0,04

0,06

0,08

0,1

0,12

Fig. 3.4 Dimensionless critical speeds of the web

of thick plates, is presented in Fig. 2.12, Chap. 2. The results of analogous calculations with the generated mathematical model are presented in Fig. 3.4. The range of changes in values of the dimensionless flexural stiffness of the isotropic web, due to real stiffnesses of paper, was limited in the investigations whose results are shown in Fig. 3.4 to the maximum value of y = 0.1. The same plot presents also the results of calculations of the dimensionless critical speed obtained for the beam model of the web. The beam model of the web is analysed in Chap. 6 of this book. A comparison of the obtained results with the results for the analogous range of changes in the plate stiffness in [4] shows that the differences are insignificant and do not exceed 1%. The plots in Fig. 3.4 indicate that an increase in the web width increases the value of the critical transport speed for the given plate stiffness. Although the beam model overestimates the value of the critical speed of the wide web, but in the range of paper stiffness, a degree of this overestimation introduced by the beam model is very slight. Owing to this fact, the beam model can be useful as less complex than the plate model in some dynamic computations of a wide paper web.

3.1.4 Results of Dynamic Investigations of the Moving Paper Web The literature survey presented in Chap. 2 points out to the fact that parameters which affect the dynamic behavior of the moving web are numerous. Among them, the most significant are: transport speed, plate stiffness, state of stresses in the web, and web geometry. During the investigations of dynamic stability of the paper web, an influence of orthotropic properties of the web in relation to a majority of the above-mentioned factors was analyzed.

3.1 One-Layered Orthotropic Web

57

In the initial stage, the dynamic behavior of the paper webs under consideration was subject to numerical investigations with a linearized mode. The linearized mathematical model presented in the form of system of (A.1) was employed to investigate the dynamic behavior of the axially moving paper web in the undercritical and overcritical range of transport speed. The scope of numerical investigations comprised the investigations of an effect of transport speed and longitudinal load of the web on its dynamic behavior. The results of calculations of the two lowest eigenfrequencies of flexural vibrations (w 11 and w 21) in the undercritical and overcritical range of transport speed are shown in Fig. 3.5 for the liner paper web, and in Fig. 3.6 for the fluting paper web. On the basis of the obtained results, one can state that all eigenfrequencies decrease their absolute values with an increase in transport speed. A stability loss for the divergent-type motion occurs at the critical speed ccr, at which the first eigenfrequency of flexural vibrations (w 11) vanishes. The web dynamic behavior in the overcritical range is mostly affected by time histories of 150

ω

125 100 75 50

Fig. 3.5a Three lowest flexural eigenvalues of the liner paper web (r = 1, Nx0= 50 N/m)

25

c

0 0

3

6

9

12

15

Im (ωi)

ω21 Divegence inst. region

ω11

c Re (ωi)

STABILITY REGION Flatter inst.region ccr = 15.11 m/s

Fig. 3.5b Two lowest eigenvalues of the liner paper web in neighborhood of the critical transport speed (r = 1, Nx0= 50 N/m)

58

3 Dynamical Analysis of the Undamped Axially Moving Web System Im (ω i)

ω11

ω21

Re (ω i) STABILITY REGION Divergence inst. region ccr = 20.178 m/s cf = 20.215 m/s ccr1 = 20.2 m/s

c Flatter inst. region

Fig. 3.6 Lowest flexural eigenvalues of the fluting paper web in neighborhood of the critical speed (r = 1, Nx0= 50 N/m)

lowest frequencies of free flexural vibrations. Frequencies of flexural-torsional vibrations of the paper web at an alternation in transport speed change in a similar way as flexural vibrations do, but attain higher values. (Fig. 3.7). With an increase in transport speed at the speed ckr1, a narrow region of web motion stability occurs. Further on, the two lowest flexural frequencies join and the flatter-type motion instability takes place (a pair of complex coupled eigenvalues of the system occurs) at the transport cf.

(a)

ω 3 2,5 2 1,5 1 0,5

c 0 20,1 20,12 20,14 20,16 20,18 20,2 20,22 20,24 (b)

ω 0,3

Fig. 3.7 Eigenvalues in the circum-critical range, (a) imaginary part, (b) real part (flutting paper, – w 11, - - w 12, r = 1, Nx0= 50 N/m)

0,25 0,2 0,15 0,1 0,05 c 0 20,1 20,12 20,14 20,16 20,18 20,2 20,22 20,24

3.1 One-Layered Orthotropic Web

59

The presented results of calculations allowed us to draw a conclusion that while investigating the dynamic behavior of the axially moving uniformly loaded web, first of all we should focus our attention on flexural vibrations of the systems under analysis. An effect of the web longitudinal load on critical speeds was investigated in the range of load changes from 50 N/m up to 2000 N/m. In the case of the liner paper, these values are equal to 0.3% and 12.5%, respectively, whereas for the flutting paper, they equal 0.75% and 30.3% , correspondingly, of the web breaking load. The investigation results of critical speeds of the liner paper and fluting paper webs, for different longitudinal loads, are presented in Fig. 3.8. The dynamic behavior of the nonlinear model of the axially moving liner paper web in individual ranges of transport speed was determined through a direct integration of system of (A.1) with the Runge-Kutta method. The dynamic behavior of the web was presented in the form of phase portraits and time histories of free vibrations of the first generalized coordinate q1 of the solution to system of (A.1). In the undercritical range of transport speed of the linerized model of the liner paper web, the following initial conditions were assumed: q1(0) = 0.1; q_ 1 ð0Þ ¼ q2 ð0Þ ¼ ::::: ¼ q_ 4 ð0Þ ¼ 0. In the undercritical range of transport speed (Figs. 3.9 and 3.10), free vibrations occur around the trivial equilibrium position. Phase trajectories of the undamped motion have a character of centers, and frequencies of free vibrations decrease with an increase in the web transport speed. In the nonlinear system, described with model (A.1), in the divergent instability region of the linearized system, one can observe a different form of dynamic behavior. Instead of an unlimited increase in the web deflection, oscillations of transverse displacements of the web around a new equilibrium position occur. A transition of the nonlinear system to this region causes that the amplitude of

ccr

140 120 fluting paper

100

liner paper

80 60 40 20 0 0

200

400

600

Nx0 800 1000 1200 1400 1600 1800 2000

Fig. 3.8 Critical speeds of paper webs (r = 1)

60

3 Dynamical Analysis of the Undamped Axially Moving Web System

(a) 4

(b)

0.1 q1

q1,t

2

0

0 –2 –4 –0.12 –0.06

–0.1

q1

0

0.06

t 0

0.12

1

2

3

4

Fig. 3.9 Phase portrait (a) and time history (b); c=10 m/s; a = 0

(a)

(b) 0.2

0.1

q1,t

q1

0.1 0

0

–0.1 –0.2 –0.12

q1 –0.06

0

0.06

t

–0.1 0

0.12

5

10

15

Fig. 3.10 Phase portrait (a) and time history (b); c=15 m/s; a = 0

(a)

(b) 3

20

q1

q1,t 1.5

10

0

0

–1.5

–10

–3 –20

q1 –10

0

10

20

–20

t 0

50

100

150

200

Fig. 3.11 Phase portrait (a) and time history (b); c=15.12m/s

free vibrations increases several times at the initial time instant if compared to the undercritical range (Fig. 3.11). It proves that a bifurcation of the node-saddle type solution occurs at the critical speed ccr= 15.11 m/s. An increase in transport speed in the undamped system is followed by an increase in the amplitude of nonlinear vibrations (Fig. 3.12). In the flatter-type instability region of the

3.1 One-Layered Orthotropic Web

61

(a)

(b) 50

q1,t

50 q1

25

25

0

0

–25

–25

–50 –50

q1 –25

0

25

–50

50

0

t 100

50

Fig. 3.12 Phase portrait (a) and time history (b); c=15.23m/s, q1(0) = 4, q1,t(0) = –3

linearized system beside the new equilibriums, free vibrations of the nonlinear system under initial conditions close to zero take place around the trivial equilibrium position (Fig. 3.13). Nonlinear vibrations forced by the harmonically variable transverse load of the liner paper web were analyzed by direct simulation of the web motion. The simulation was performed through an integration of discretized mathematical model (A.1) during adequately long time to eliminate initial unsteady vibrations. To achieve this, first 2000 periods of vibrations were rejected, and then the next 100 were considered. A value of the vibration period was determined on the basis of the excitation frequency. Thus, for each value of the parameter under analysis, Poincare maps on the phase plane defined by the first generalized coordinate q1 and its derivate with respect to time q_ 1 were determined. A comparison of Poincare maps corresponding to variable values of the parameter under investigation has allowed for plotting bifurcation diagrams. While preparing a bifurcation diagram, the final state of the system corresponding to the current value of the bifurcation parameter defines the initial conditions for the web motion simulation for the subsequent value of the analyzed bifurcation parameter. (a)

(b) 4

5

q1,t

q1

2.5

2

0

0

–2.5

–2

–5 –4

q1 –2

0

2

4

–4

t 0

20

40

Fig. 3.13 Phase portrait (a) and time history (b); =15.23m/s, q1(0) = 1

62

3 Dynamical Analysis of the Undamped Axially Moving Web System

Fig. 3.14 Bifurcation diagram, a = 2; w = 10 rd/s

q1

100

0

–100 c

12

12.5

13

13.5

14

14.5

15

Figure 3.14 presents a bifurcation diagram as a function of transport speed of the liner paper web at constant values of the excitation amplitude and frequency. Within the transport speed range c < 12.5 m/s, the system is asymptotically stable, its dynamic response tends to the static equilibrium position. In the transport speed range 12.5 m/s < c < 15 m/s, a motion of the web takes place, which one can call as quasi-regular. The Poincare maps in Fig. 3.15 a, b, c show the course of this motion. A concentration of points which forms regular areas one can observe in these maps. In this range of transport speed, the region occupied by the attractor on Poincare maps increases with an increase in transport velocity. To investigate this motion the frequency analysis of the time history of the first generalized coordinate q1 by using the FFT procedure has been carried out. The investigation results for the transport speed c = 14.1 m/s are shown in Fig. 3.16. Regular distribution of dominant peaks with the initial peak at the excitation frequency w = 10 rd/s = 1.59 Hz is visible in Fig. 3.16. At the transport speed c = 15 m/s, a chaotic motion develops suddenly. The Poincare map in Fig. 3.15d. and the frequency analysis of the time history of the first generalized coordinate in Fig. 3.17 characterize the motion at this transport velocity. Figure 3.18 shows a resonance plot of the nonlinear system of the liner paper web, at the constant transport speedc = 2 m/s and the constant dimensionless excitation amplitude a = 2. It is worth noticing that in this

200

(a)

(b)

q1 150

(c)

600

1000

q1

qq11

2⋅104

qq11

400

0

100

500

200 50 –5

0

q1

5

0 –40

(d)

–20

0

20

40

q1

–50

0

q1

50

–2⋅104 –200 –100

0

100

q1

200

Fig. 3.15 Poincare maps, (a) c = 13.0 m/s, (b) c = 13.6 m/s, (c) c = 14.1 m/s, (d) c = 15.0 m/s

3.1 One-Layered Orthotropic Web

63

Fig. 3.16 FFT analysis (a = 2, w = 10 rd/s, c = 14.1 m/s) Amplitude

10

1

0.1

0.01

0

5

10 15 Frequency (Hz)

5

10 15 Frequency (Hz)

20

25

Amplitude

10

1

0.1

Fig. 3.17 FFT analysis (a = 2, w = 10 rd/s, c = 15 m/s)

0.01 0

20

25

250

q1

200 150 100 50

Fig. 3.18 Resonance plot, c = 2 m/s, a = 2

ω [rd /s]

0 0

10 20 30 40 50 60 70 80 90 100

case the two lowest resonance frequencies of flexural vibrations, defined by the linearized model, are equal to w 1r = 46.63 rd/s and w 2r = 98.85 rd/s, correspondingly. Figure 3.19 depicts a bifurcation diagram as a function of dimensionless excitation amplitude of the liner paper web at constant values of transport

64

3 Dynamical Analysis of the Undamped Axially Moving Web System

Fig. 3.19 Bifurcation diagram, c = 10 m/s, w = 10 rd/s

20 q1 10

0

–10

–20

0

10

20

30

a

speed and excitation frequency. Within the amplitude values a < 3, the system is asymptotically stable, and its dynamic response tends to the static equilibrium position. For the range of amplitude values 32 > a  3, a quasi-periodic motion of the web, which is characterized by Poincare maps in Fig. 3.20a,b,c, occurs. At a = 32, a bifurcation that results in an irregular motion, which is represented by the strange attractor resembling a human face in the Poincare map in Fig. 3.20d, occurs. (a)

(b)

q1,t

q1,t

250

250

200 200 q1 –1

0

q1

1

(c)

–1

0

1

–50

0

50

(d)

q1,t 250

q1,t 2000

200 0

150 q1 –1

0

1

q1

Fig. 3.20 (a) a = 18.8; (b) a = 23.4; (c) a = 25.8; (d) a = 33.

3.2 Multi-Layered Composite Web

65

3.2 Multi-Layered Composite Web It is well-known that since their appearance in the 1960s, the use of modern composite materials has been growing rapidly around the world. Composites have been commonly used in aircraft industry, army, automotive industry and marine structures, as well as in packaging. With progress in the technology of composite materials, composite plates made of solid laminates, sandwich laminates, and laminates reinforced with stiffeners are widely used in packages, cars, wagons, ships, airplanes and marine structures. The positive factors promoting the use of fibrous composites in the construction of load carrying elements include good wear (e.g. friction) resistance, fatigue life (under time-variable loads), sound insulation properties or electromagnetic transparency. Corrugated board is a highly efficient composite material. It is used extensively as a packaging material because of its high strength and stiffness properties. These properties allow it to be manufactured into boxes and trays for transporting, storing and distributing a wide range of consumer products. Paper and corrugated board properties derive from raw materials and papermaking processes. Relatively recent technical developments allow high speed formation of the web into a simultaneous or sequential multi-layered structure like corrugated board. Paper is generally considered to be an anisotropic fibrous composite material. Theoretical models describe mechanical properties of paperboard including those based on a thin-walled orthotropic plate structure. As was said excessive vibrations of moving board webs in the industry increase defects and can lead to failure of the web. In the paper industry involving motion of thin materials, stress analysis in the moving web is essential for the control of wrinkle, flutter and sheet break. Although the mechanical behavior of axially moving materials has been studied for many years, little information is available on the dynamic behavior and stress distribution in the axially moving multi-layered paper and board materials.

3.2.1 Mathematical Model of the Axially Moving Multi-Layered Web To formulate the problem, a three-layered composite consisting of two thin facings sandwiching a core is shown in Fig. 3.21. This model refers to the three-layered corrugated board structure. The facings are made of high strength materials having good properties under tension, whereas the siding core in the form of a corrugated trapezoidal plate is made of lightweight materials. Sandwich composites combine lightness and flexural stiffness. Figure 3.21 also depicts the coordinate system. The x-axis refers to the machine direction, the y-axis refers to the cross or transverse direction. The machine and cross

66

3 Dynamical Analysis of the Undamped Axially Moving Web System

Fig. 3.21 Schematic view of the sandwich composite macrostructure

x

facings core y

z

directions form the plane of the structure, and z-axis refers to the out-of-plane (or through-thickness) direction. Let us consider a thin-walled web built of plate elements (panels). The web under consideration is a multi-layer plate made of orthotropic materials. The composite has been modeled as a thin-walled laminar construction with a rigid core. The classical laminated panel theory [1] is used in the theoretical analysis, the effects of shear deformation through the thickness of the laminate are neglected and the results given are those for thin laminated panels. The materials they are made of are subjected to the Hooke’s law. For each panel component, precise geometrical relationships are assumed in order to enable the consideration of both out-of-plane and in-plane bending of the plate:  "1 ¼ u1;1 þ 0:5 u21;1 þ u22;1 þ u23;1 ;  "2 ¼ u2;2 þ 0:5 u21;2 þ u22;2 þ u23;2 ; "3 ¼ u1;2 þ u2;1 þ u1;1 u1;2 þ u2;1 u2;2 þ u3;1 u3;2 ;

(3:33)

"4 ¼ h u3;11 ; "5 ¼ h u3;22 ; "6 ¼ 2 h u3;12 ; where "1= "x, "2= "y, "3=2"xy= gxy, "4= h x, "5= h y, "6= h xy, u1  u; u2  ; u3  w – components of the displacement vector in the x, y, z axis direction, respectively, h – thickness of the plate. The reduced expressions ð::Þ;1 ¼ @ ð::Þ=@ u1 and ð::Þ;2 ¼ @ ð::Þ=@ u2 have been used in (3.33) Using the classical plate theory [1], the constitutive equation for the laminate is taken as follows:  fN g ¼

 ½A½B f"g ¼ ½Kf"g ½B½D

(3:34)

3.2 Multi-Layered Composite Web

67

where N 1X ij Þ ðzk  zk1 Þ ðQ k h k¼1

(3:35)

Bij ¼

N 1 X ij Þ ðz2  z2 Þ ðQ k1 k k 2h2 k¼1

(3:36)

Dij ¼

N 1 X ij Þ ðz3  z3 Þ ðQ k1 k k 3h3 k¼1

(3:37)

Aij ¼

A11 6A 6 21  6 6 A61 ½B ¼6 6B ½D 6 11 ¼6 4 B21

A12 A22

A16 A26

B11 B21

B12 B22

A62 B12

A66 B16

B61 D11

B62 D12

B22

B26

D21

D22

3 B16 B26 7 7 7 B66 7 7 D16 7 7 7 D26 5

B61

B62

B66

D61

D62

D66

K11

K12

K13

K14

K15

K16

6K 6 21 6 6 K31 6 6K 6 41 6 4 K51

K22

K23

K24

K25

K32 K42

K33 K43

K34 K44

K35 K45

K52 K62

K53 K63

K54 K64

K55 K65

K26 7 7 7 K36 7 7 K46 7 7 7 K56 5 K66

2

 ½K ¼

½A ½B

2

K61

(3:38)

3

ij is the transformed in which Aij = Aji, Bij = Bji, Dij = Dji, Kij = Kji and Q reduced stiffness matrix. On the left-hand side of (3.34), the elements N1, N2, N3 (dimensionless sectional forces) and N4, N5, N6 (dimensionless sectional moments) have the following forms: N1 ¼

Ny Nxy My Mxy Nx Mx ; N2 ¼ ; N3 ¼ ; N4 ¼ ; N5 ¼ ; N6 ¼ ; (3:39) E0 h E0 h E0 h E0 h2 E0 h2 E0 h2

where E0 is the elastic modulus of reference. The reverse relation with respect to (3.34) can be written as: f"g ¼ ½K1 fNg ¼ ½KfNg

(3:40)

68

3 Dynamical Analysis of the Undamped Axially Moving Web System

In the constitutive matrix of (3.34), the extensional stiffness submatrix [A], detailed in (3.35), is related to the in-plane response of the laminate. The bending stiffness submatrix [D], described by (3.37), is associated with the out-of-plane bending response of the laminate. The interaction submatrix [B], illustrated by (3.36), is a measure of a coupling between the membrane and the bending action. Thus, it is impossible to stretch a laminate that has the non-zero Bij terms without bending and/or twisting the laminate at the same time. Such a laminate cannot be subjected to moment without suffering simultaneously from extension of the middle surface. Let us suppose now that a multi-layered web of the length l is considered. The web moves at the constant velocity c in the x direction. The geometry of the considered model is shown in Fig. 3.22. Rotary inertia of the web has been neglected and a symmetrical transversal heterogeneity of the web is assumed in further considerations. The equations of dynamic stability of the moving composite structure have been derived using the Hamilton’s principle. Actual trajectories differ from the other ones that satisfy the Hamilton’s principle for the web and can be written as: ðt1 ð 1 ½ðc þ u1;t þ cu1;1 Þ2 þðu2;t þ cu2;1 Þ2 þ  Ldt ¼ ðU  V þ WÞdt ¼ 2 ðt1

ðt1

to

to

to

ð

þðu3;t þ cu3;1 Þ2 dW 

W

(3:41)   ð 1 "1 þ 2 "2 þ 3 "3 ÞdW þ hp0 ðx2 Þu1 dx2 xx11 ¼‘ ¼0 gdt ¼ 0 ðb

W

0

where U – kinetic energy, V– internal elastic strain energy, W – work of external forces, p0(x2)– external load in the plate middle surface. W ¼ ‘  b  h ¼ S  h: After grouping the components at respective variations, the following system of equations of motion has been obtained:

y c

Fig. 3.22 Axially moving multi-layered web

b l

z

x

3.2 Multi-Layered Composite Web

ðt‘ ð

69

f½N1 ð1 þ u1;1 Þ þ N3 u1;2 ;1 þ ½N2 u1;2 þ N3 ð1 þ u1;1 Þ;2

to S

þ ðu1;tt  2cu1;1t  c2 u1;11 Þgu1 dSdt ¼ 0 ; ðt‘ ð

f½N1 u2;1 þ N3 ð1 þ u2;2 Þ;1 þ f½N2 ð1 þ u2;2 Þ þ N3 u2;1 ;2

to S

(3:42)

þ ðu2;tt  2cu2;1t  c2 u2;11 Þgu2 dSdt ¼ 0 ; ðt‘ ð

½ðhN4;1 þ N1 u3;1 þ N3 u3;2 Þ;1 þ ðhN5;2 þ 2hN6;1 þ N2 u3;2 þ N3 u3;1 Þ;2 þ

to S

þ ðu3;tt  2cu3;1t  c2 u3;11 Þu3 dSdt ¼ 0 ;

where

 ¼

N 1X k ðzk  zk1 Þ: h k¼1

(3:43)

The boundary conditions for x1=const: ðt1 ðb

   2   c þ cu1;t þ c2 u1;1  N1 þ N1 u1;1 þ N3 u1;2  hp0 ðx2 Þ u1 dx2 dtjx1 ¼ 0;

t0 0

ðt1 ðb

     c u2;t þ c2 u2;1  ½N3 þ N1 u2;1 þ N3 u2;2  u2 dx2 dtjx1 ¼ 0 ;

t0 0

ðt1 ðb

(3:44) N4 u3;1 dx2 dtjx1 ¼ 0 ;

t0 0

ðt1 ðb

     c u3;t þ c2 u3;1  ðhN4;1 þ 2hN6;2 þ N1 u3;1 þ N3 u3;2 Þ u3 dx2 dtjx1 ¼ 0:

t0 0

The boundary conditions for x2=const:

70

3 Dynamical Analysis of the Undamped Axially Moving Web System

ðt1 ðl

½N2 þ N2 u2;2 þ N3 u2;1 u2 dx1 dtjx2 ¼ 0 ;

t0 0

ðt1 ðl

½N3 þ N2 u1;2 þ N3 u1;1 u1 dx1 dtjx2 ¼ 0 ;

t0 0

ðt1 ðl

(3:45) N5 u3;2 dx1 dtjx2 ¼ 0 ;

t0 0

ðt1 ð‘

ðhN5;2 þ 2hN6;1 þ N2 u3;2 þ N3 u3;1 Þu3 dx1 dtjx2 ¼ 0 :

t0 0

The boundary conditions for the plate corner (x1= const and x2= const): ðt1

2N6 jx1 jx2 u3 dt ¼ 0:

(3:46)

t0

Equations (3.42) form a system of equilibrium equations. Relations (3.44), (3.45) and (3.46) correspond to the boundary conditions on the edges and in the plate corner, respectively.

3.2.2 Solution of the Mathematical Model The problem of dynamic stability has been solved with an asymptotic perturbation method. In the solution of the problem, the Koiter’s asymptotic expansion  and the fields of sectional [3] has been employed. The fields of displacements U  have been expanded into power series with respect to the dimensionless forces N amplitude of the web deflection n (the amplitude of the n-th free vibration frequency of the extension system divided by the thickness h1 of the web assumed to be the first one):  ¼ U ð0Þ þ 1 U ð1Þ þ    þ n U ðnÞ U  ¼ N ð0Þ þ 1 N ð1Þ þ    þ n N ðnÞ N where

(3:47)

3.2 Multi-Layered Composite Web

71

ð0Þ ; N ð0Þ – pre-critical static fields, U ðnÞ  ðnÞ  U ; N – the n – order fields of displacements and sectional forces, respectively. After substitution of expansions (3.47) into governing equations (3.42), continuity conditions and boundary conditions (3.44)/(3.46), the boundary value problems of the zero and first order can been obtained. The zero approximation describes the pre-critical static state, whereas the first order approximation refers to the linear problem of dynamic stability, and allows for determination of the eigenvalues, the eigenvectors and the critical speeds of the system. The pre-critical static state has been solved similarly like previously in the case of the homogenous plate, using the exact transition matrix method. The inertial forces from the in-plane displacements u and v are neglected. The pre-critical solution is assumed as: ð0Þ

ð0Þ

ð0Þ

u1 ¼ ðx  ‘=2ÞD; u2 ¼ y D K12 =K22 ; u3 ¼ 0

(3:48)

where D is the actual loading of the web. Including (3.47) in (3.34), the inner sectional forces of the pre-critical static state for the assumed homogeneous field of displacements (3.48) are expressed as follows: ð0Þ ¼ ðK11  K2 =K22 ÞD ; N 12 1 ð0Þ ¼ 0 ; N 2 ð0Þ ¼ ðK31  K32 K21 =K22 ÞD ; N 3 ð0Þ ¼ ðK41  K42 K21 =K22 ÞD ; N 4

(3:49)

ð0Þ ¼ ðK51  K52 K21 =K22 ÞD ; N 5 ð0Þ ¼ ðK61  K62 K21 =K22 ÞD: N 6 In the one before last dynamical component of the third equation of the set of equations (3.42) there is a derivative with respect to x and t. Because of the incompatibility of trigonometric functions in the x direction, the Galerkin orthogonalization procedure has been used to find an approximate solution to this equation. Numerical aspects of the problem being solved for the first order fields have resulted in an introduction of the following new orthogonal functions with the n-th harmonic for the composite web in the sense of the boundary conditions for two longitudinal edges:

72

3 Dynamical Analysis of the Undamped Axially Moving Web System ðnÞ ðoÞ ðnÞ aðnÞ ¼ N2 ð1 þ DK21 =K22 Þ þ  N3 d; =b; ðnÞ ð0Þ ðnÞ bðnÞ ¼ N3 ð1   DÞ þ  N3 c; =b; ðnÞ

cðnÞ ¼ u1 ; ðnÞ dðnÞ ¼ u2 ;

(3:50)

ðnÞ

eðnÞ ¼ u3 ; ðnÞ fðnÞ ¼ u3; =b ¼ eðnÞ ; =b; ðnÞ

gðnÞ ¼ N5 ; ðnÞ ðoÞ ðnÞ ; =b; hðnÞ ¼ h gðnÞ ; =b þ 2hN6; =b þ  N3 e

where  ¼ x=b and  ¼ y=b: The expansion into power series (3.47) and new functions (3.50) have been taken into account in the third equation of the set of equations (3.42). The solutions of the third equation of the set of equations (3.42) for the first order problem can be written in the following form: a ¼

c ¼

N X

npb ; Tn ðtÞAðnÞ ðÞ sin ‘ n¼1

b ¼

N X

npb ; Tn ðtÞCðnÞ ðÞ cos ‘ n¼1

d ¼

N X

npb ; Tn ðtÞBðnÞ ðÞ cos ‘ n¼1

N X

ðnÞ ðÞ sin npb; Tn ðtÞD ‘ n¼1 N X

npb e ¼ ; Tn ðtÞEðnÞ ðÞ sin ‘ n¼1 g ¼

N X

N X

npb ; Tn ðtÞGðnÞ ðÞ sin ‘ n¼1

h ¼

N X

 ðnÞ ðÞ sin npb: Tn ðtÞH ‘ n¼1

where Tn ðtÞ ¼ ejw n t in the first order problem solution, j ¼ ðnÞ

ðnÞ

(3:51)

npb f¼ ; Tn ðtÞFðnÞ ðÞ sin ‘ n¼1

ðnÞ

ðnÞ

ðnÞ

ðnÞ

ðnÞ

pffiffiffiffiffiffiffi 1. ðnÞ

The eigenfunctions A ; B ; C ; D ; E ; F ; G ; H (with the n-th harmonic) are initially unknown functions that refer to eigenvalues of the non-moving tensioned web (for c = 0) that will be determined by the numerical method of transition matrices. In the modal analysis of the non-moving tensioned web, the obtained system of homogeneous ordinary differential equations has been solved by the transition matrix method, having integrated numerically the equilibrium equations in order to obtain the relationships between the state vectors on two longitudinal edges. In the system of the third equation of the set of equations (3.42) for the non-moving tensioned web, there are two components ð0Þ and N ð0Þ . The component N ð0Þ has an insignificant of the pre-critical loading N 1

3

3

ð0Þ . effect on the value of critical load in comparison with N 1

3.2 Multi-Layered Composite Web

73

In the general case of the moving composite web (i.e. for c 6¼ 0 and Tn ðtÞ 6¼ e jw n t ), because of the incompatibility of trigonometric functions in the x-direction, after substituting (3.50) into the third equation of the set of equations (3.42), the Galerkin orthogonalization method has been used. In this way the system of N ordinary differential equations with respect to the function Tn ðtÞ can be determined in the following form: N X d 2 Tm dTn a1nm þ Tm a0m ¼ 0 a þ 2m 2 dt dt n¼1

m ¼ 1; 2; . . . N

(3:52)

On the basis of (3.52), one can determine eigenvalues, eigenvectors and critical speeds of the moving composite web system. The system of ordinary differential equations for investigated corrugated board is shown in Appendix A (A.4).

3.2.3 Results of the Comparative Studies The numerical investigations were carried out on the basis of the mathematical model, which has been presented in the previous subsection. First, the calculation results were compared with the available solutions. A comparison of dimensionless fundamental natural frequencies (w *=w b2(/(Eyh2))1/2) of the simply supported anti-symmetric angle-ply laminate, received in the present study and by Jones [1], is shown in Fig. 3.23. The following dimensionless numerical data were used: Ex/Ey= 40, Gxy/Ey= 0.5, nxy = 0.25. One can observe that the majority of discrepancies of the compared results occurs for a two-layered laminate. When the number of layers increases, there are fewer and fewer discrepancies in the compared values. ω∗

26

6 layers 24 4 layers

22 20 18 16

2 layers

14 12

θ [°]

10 0

10

20

30

40

Fig. 3.23 Comparison of the numerical results, – ply angle, (_) – present method, (- -) – Jones [1]

74 (a)

3 Dynamical Analysis of the Undamped Axially Moving Web System (b) r = 0.81

1

r

(c)

r

1.625

1.625

0.5 3.25 mm

3.25 mm

3.25 mm

Fig. 3.24 Steel core profiles: trapezoidal profile (a), semicircular profile (b), sinusoidal profile (c)

To verify the proposed model, comparative investigations were carried out for various steel core profiles: trapezoidal, semicircular and sinusoidal. The crosssections and geometric dimensions of these profiles are shown in Fig. 3.24. The symmetry conditions at the ends of these profiles were assumed. First, the transverse natural frequency calculation results of simply supported corrugated steel profiles were compared with the analogous values received from the finite element analysis. The ANSYS 5.7 software package was used in those investigations. The investigation results for transverse natural frequencies for the trapezoidal, semicircular and sinusoidal profile are shown in Tables 3.1, 3.2 and 3.3, respectively. The numerical data were as follows: length l = 1000 mm, thickness h = 0.12 mm, E = 2105 MPa, Poisson’s ratio = 0.3, mass density  = 7850 kg/m3. The effect of two thin facings on transverse natural vibrations was studied in different investigations. The cross-sections and geometric dimensions of the two three-layered steel profiles E and C, which were investigated, are shown in Fig. 3.25. To find parameters of the three-layered profile, the middle layer was homogenized (see Chap. 1). The rigid internal core was replaced by a substitute orthotropic plate. The mass and stiffness of the substitute plate were the same Table 3.1 Trapezoidal wave – transverse natural frequencies comparison [Hz] Nat. freqen. 1 2 3 4 5 6 7 8 9

10

Comp.result 4.895 19.597 44.052 78.309 122.347 176.159 239.734 313.079 396.17 489.80 ANSYS 4.901 19.585 44.057 78.312 122.24 176.15 239.71 313.02 396.06 488.82

Table 3.2 Semicircular wave – transverse natural frequencies comparison [Hz] Nat. freqen. 1 2 3 4 5 6 7 8 9

10

Comp.result 4.456 17.827 40.108 71.300 111.397 160.396 218.290 285.078 360.748 445.293 ANSYS 4.505 18.019 40.543 72.074 112.610 162.160 220.700 288.250 364.800 450.340 Table 3.3 Sinusoidal wave – transverse natural frequencies comparison [Hz] Nat. freqen. 1 2 3 4 5 6 7 8 9

10

Comp.result 4.2468 16.986 38.219 67.941 106.151 152.844 208.017 271.664 343.778 424.354 ANSYS 4.1512 16.605 37.360 66.418 103.780 149.430 203.390 265.650 336.200 415.050

3.2 Multi-Layered Composite Web

75

(a)

(b) 0.12

0.12

1.625

3.225

0.12 3.25 mm

0.12 7.15 mm

Fig. 3.25 Three-layered steel profiles: E-profile (a), C-profile (b)

as the original corrugated profile. The thickness hCand the width bC of the substitute plates were as follows: – hC  bC = 1.625 mm  3.25 mm – for the E-profile, – hC  bC = 3.225 mm  7.15 mm – for the C-profile. The transverse natural frequency calculations of the simply supported steel E-profile and C-profile were carried out during the numerical investigations. The results were compared with the analogous values received from the finite element analysis. A comparison of the investigation results for the E- and C-steel profile is shown in Tables 3.4 and 3.5, respectively. The investigation results in Tables 3.4 and 3.5 show that the greatest discrepancy does not exceed 7%. This comparison confirms that one can take into consideration the classical laminated plate theory with the Kirchhoff-Love’s hypothesis in this analysis. Table 3.4 Steel E-profile – transverse natural frequencies comparison [Hz] Nat. freqen. 1 2 3 4 5 6 7 8 9

10

Comp.result 6.1513 24.605 55.364 98.426 153.789 221.457 301.429 393.704 498.250 615.162 ANSYS 5.8203 23.272 52.350 93.038 145.320 209.160 284.530 371.400 469.710 579.430

Table 3.5 Steel C-profile – transverse natural frequencies comparison [Hz] Nat. freqen. 1 2 3 4 5 6 7 8 9

10

Comp.result 11.532 46.131 103.796 184.526 288.322 415.184 565.111 738.104 934.165 1152.334 ANSYS 11.702 46.777 105.140 186.640 291.090 418.190 567.630 738.990 931.780 1145.400

3.2.4 Results of the Dynamic Investigations of the Axially Moving Corrugated Board Web The dynamic behavior of the axially moving three-layered corrugated board web was investigated analogously as for paper webs the corrugated board was made of. The results of the investigations of paper webs have been presented in Sect. 3.1.4.

76

3 Dynamical Analysis of the Undamped Axially Moving Web System

In the initial stage, the dynamic behavior of the corrugated paper web was investigated numerically with a linearized model. The linearized model was formulated after the rejections of nonlinear terms in system of equations (A.4). The linearized model dynamic analysis comprised investigations of the effect of transport speed and longitudinal load of the corrugated paper web on its dynamic behavior. The calculation results of the lowest eigenfrequencies of flexural vibrations of the corrugated board web are presented in Fig. 3.26. Similarly as for component papers, all eigenfrequencies decrease their absolute values with an increase in transport speed in the corrugated board. Divergence instability of the web motion appears at the critical speed ccr = 20.4 m/s, at which the first eigenmode of flexural vibrations w 11 equals zero (Fig. 3.26b). On should notice that the value of the critical speed of the corrugated board web 550

ω31

ω 500 450 400 350 300 250 200 150 100 50 0

ω21 ω11 C

0

5

10

15

20

25

30

35

40

Fig. 3.26a Lowest eigenvalues of the corrugated board web; r = 1, Nx0= 50 N/m; . . .. real part (b) Im (ωi)

ω21 ω11

Re (ωi) STABILITY REGION Divergence inst. region cf = 36.6 m/s Flatter inst. cf = 37.7 m/s region

ccr = 20.4 m/s

c

Fig. 3.26b Two lowest eigenvalues in the neighborhood of the critical transport speed (corrugated board, r = 1, Nx0= 50 N/m)

3.2 Multi-Layered Composite Web

ccr

77

140 120 fluting paper 100

liner paper

80 corrugated board 60 40 20 Nx0

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Fig. 3.27 Critical speeds of the investigated webs (r = 1)

under the investigated longitudinal tension Nx0 = 50 N/m is higher than critical speeds of analogous webs of the liner and fluting paper. Such a relationship can be seen only for low longitudinal tensions Nx0 < 100 N/m. The investigation results of the linearized longitudinal model shown in Fig. 3.27 point out to the fact that the value of the critical speed increases with an increase in the longitudinal tension, but in the load range Nx0 > 100 N/m it is situated beyond the critical speeds of both analogous component paper webs. The dynamic behavior of the corrugated board web in the range of supercritical transport speeds of the linearized system is characterized by a considerably wider range of transport speeds at which divergence instability occurs if compared to the analogous behaviors of component paper webs. A higher flexural stiffness of the corrugated board web than the paper web results in separation of frequencies of free vibrations in the case of the former one. This is a direct cause of an increase in the divergence instability region of the corrugated board linearized model. The dynamic behavior of the nonlinear model of the corrugated board web has been presented in the form of phase portraits and time histories of free vibrations of the first generalized coordinate q1 of the solution to system of equations (A.4). Like in the case of the paper web, free vibrations occur around the trivial equilibrium position in the subcritical range of transport speed (Fig. 3.28). Phase portraits of the undamped motion have a character of centers, and the frequencies of free vibrations decrease with an increase in the web transport speed. In the nonlinear system in the divergence instability region of the linearized system, oscillations of transverse displacements around a new equilibrium position occur instead of an unlimited increase in the web deflection. A transition of the nonlinear system into this region takes place in a different way than in the case of the paper web. It can be clearly seen in the form of evolution of

78

3 Dynamical Analysis of the Undamped Axially Moving Web System

(a) 10 q1,t 5

(b) 1.2 q1 0.6

0

0

–5

–0.6 q1

–10 –2

–1

0

1

–1.2

2

0

2.5

5

t 10

7.5

Fig. 3.28 Phase portrait (a) and time history (b); c=20.2 m/s, a = 0, r = 1, Nx0= 50 N/m

phase portraits shown in Figs. 3.28 and 3.29. These trajectories depict the eigenmotion of the corrugated board web at two transport speeds that do not differ much. A bifurcation of the trivial solution that occurs at the critical speed of the linearized system ccr = 20.4 m/s corresponds to a pitchwork-type bifurcation in the damped system. An increase in transport speed in the undamped system is followed by an increase in the amplitude of nonlinear vibrations (Fig. 3.30). In the flatter-type (a) 6 q1,t

(b) 2.5 q1

3

1.25

0

0

–3

–1.25 q1

–6 –2.5

–1.25

0

1.25

–2.5

2.5

t 0

5

10

20

15

Fig. 3.29 Phase portrait (a) and time history (b); c=20.457 m/s, a = 0, r = 1, Nx0= 50 N/m

(a)

(b) 2000

50

q1,t

q1 1000

25

0

0

–1000

–25

–2000 –50

q1 –25

0

25

50

–50

t 0

2.5

5

7.5

10

Fig. 3.30 Phase portrait (a) and time history (b); c=38.0 m/s, a = 0, r = 1, Nx0= 50 N/m

3.2 Multi-Layered Composite Web Fig. 3.31 Bifurcation diagram, a = 2, w = 20 rd/s

79 30

q1 15 0 –15 –30 12

cb114.5

17

cb219.5

22

instability region of the linearized system (c > 37.7 m/s), free vibrations of the linearized system occur around the trivial equilibrium position under the initial condition close to zero, apart from new equilibrium positions. The nonlinear vibrations forced with harmonically variable transverse loading of the corrugated board web were analyzed similarly as for the liner paper web. First 2000 vibration periods were rejected and then the next 100 were analyzed. The vibration period values were determined on the basis of the excitation frequency. Thus, for each value of the parameter under analysis, Poincare maps on the phase plane determined by the first generalized coordinate q1 and its first derivative with respect to time q_ 1 were defined. The Poincare maps corresponding to variable values of the parameter under investigation allowed for plotting bifurcation diagrams. While drawing a bifurcation diagram, Figure 3.31 shows a bifurcation diagram as a function of transport speed of the corrugated board web at constant values of the dimensionless amplitude a = 2 and the excitation frequency w = 20 rad/s. Within transport speeds c < 14 m/s, the system is asymptotically stable, its dynamic response tends to the stable equilibrium position. At the speed cb1 = 14 m/s, a Hopf bifurcation appears and a quasi-periodic motion of the web that is represented by the Poincare map in Fig. 3.32 and the phase portrait and time histories in Fig. 3.33 can be observed in the transport speed range of 14 m/s < c < 19 m/s. 44

q1,t 40.5 37 33.5

Fig. 3.32 Poincare map, c = 17.1 m/s

q1 30 –0.25

–0.13

0

0.13

0.25

80

3 Dynamical Analysis of the Undamped Axially Moving Web System

(a)

(b)

80

3

q1,t

q1

40

1.5

0

0

–40

–1.5

–80 –3

q1

–1.5

0

1.5

3

–3

t

0

1.25

2.5

3.75

5

Fig. 3.33 Phase portrait (a) and time history (b); c=17.1 m/s; a = 2, r = 1, Nx0 = 50 N/m

For the transport speed cb2 = 19 m/s, a web motion, which has been referred to as quasi-regular in the description of the dynamic behavior of the moving liner paper web, occurs. Its characteristics are illustrated by the time history in Fig. 3.34 and by the Poincare map in Fig. 3.35. On the map, one can see a concentration points that forms a regular region. With an increase in transport speed, the region occupied by the fields of attraction on the Poincare maps grows as well. For the transport speed of the corrugated board web c = 21.5 m/s, an irregular motion of the system under consideration rapidly develops (Fig. 3.36). In this range of the web transport speed that covers the divergence instability region of the linearized system the chaotic motion coexists with the quasi-periodic one (Fig. 3.37). Figure 3.38 presents a resonance plot of the nonlinear system of the corrugated board web, for the constant transport speed c = 2 m/s and the constant dimensionless excitation amplitude a = 2. It is worth noticing that

(a) 250

(b) 10

q1,t

q1 125

5

0

0

–125

–5

–250 –10

q1 –5

0

5

10

–10

t 0

1.25

2.5

3.75

5

Fig. 3.34 Phase portrait (a) and time history (b); c=19.6 m/s, a = 2, r = 1, Nx0= 50 N/m

3.2 Multi-Layered Composite Web

81

(a) 130

(b) 180

q1,t

q1,t

122.5

150

115

120

107.5

90

100 –0.5

q1 –0.25

0

0.25

60 –1

0.5

q1 –0.5

0

0.5

1

Fig. 3.35 Poincare maps, a = 2, r = 1, Nx0= 50 N/m, c = 19.6 m/s (a), c = 20.8 m/s (b)

(b) 40

(a) 1500

q1

q1,t 750

20

0

0

–750

–20

–1500 –40

q1 –20

0

20

40

–40

t 0

0.5

1

1.5

2

Fig. 3.36 Phase portrait (a) and time history (b) c=30.0 m/s, a = 2, r = 1, Nx0= 50 N/m

2000

q1,t 999.5 –1 –1001.5

Fig. 3.37 Poincare map, c = 21.5 m/s

–2002 –25

q1 –12.5

0

12.5

25

the resonance plot of the liner paper web shown in Fig. 3.18 has been determined for the same transport speed and the same excitation amplitude. In the case of the corrugated board under analysis, the two lowest resonance frequencies of flexural vibrations, determined by the linearized model, are equal to w 1r = 63.67 rd/s and w 2r = 229.56 rd/s, respectively.

82

3 Dynamical Analysis of the Undamped Axially Moving Web System

Fig. 3.38 Resonance plot, c = 2 m/s, a = 2

20

q1 15 10 5

ω

0 0

40

80

120

160

200

240

3.3 Final Remarks Chapter 3 is devoted to an analysis of the dynamic behavior of axially moving one-layered and multi-layered webs. Mathematical models of the moving web systems are derived on the basis of the asymptotic perturbation method for the classical thin-walled plate theory. In the one-layered web case, the non-linear mathematical model has the form of two coupled partial differential equations. Geometrical non-linearity of the moving web is taken into account. In the multi-layered web case, the moving web is treated as a thin-walled composite structure in the elastic range, being under axial tension. The equations of dynamic stability of the moving composite structure have been derived using the Hamilton’s principle. In both cases, the mathematical model solution is based on the numerical method of the transition matrix, using Godunov’s orthogonalization. To verify the proposed mathematical models, the comparative studies were carried out using the data of real moving objects from the literature and the results from the finite element analysis. A comparison of the obtained results shows a good accuracy. The numerical calculations were carried out for the liner paper and the fluting paper used in corrugated board and for three-layered corrugated board web structures. The numerical data of the investigated paper structures have been received from the experimental investigations, and are presented in Chap. 1. For the constant axial tension of the web, the calculation results of the linearized systems show that the lowest flexural and flexural-torsional natural frequencies decrease during an axial velocity increase in the subcritical region of transport speed. At the critical transport speed, the fundamental flexural eigenfrequency vanishes, thus indicating divergence instability. In the supercritical region, a narrow region of web motion stability occurs with an increase in transport speed. Further on, the two lowest flexural frequencies join and the flatter-type motion instability takes place. In the undercritical range of transport speed of the nonlinear system, free vibrations occur around the trivial equilibrium position. Phase portraits of the undamped motion have a character of centers, and frequencies of free

References

83

vibrations decrease with an increase in the web transport speed. Instead of an unlimited increase in the web deflection in the divergence instability region of the linearized system, one can observe oscillations of transverse displacements of the web around a new equilibrium position in the nonlinear system. Then, an increase in transport speed is followed by an increase in the amplitude of nonlinear vibrations. The investigations of nonlinear vibrations forced by the harmonically variable transverse load of the web show a quasi-regular motion of the web, which characterizes a regular distribution of dominant peaks in the frequency analysis plot in the supercritical region.

References 1. Jones A R (1968) An experimental investigation of the in-plane elastic moduli of paper. Tappi Journal 51(5): 203–209 2. von Karman T (1910) Festigkeitsprobleme in Mashinenbau. Encyklopedie der Matheatischen Wissenschaften Vol. IV: 349 3. Koiter W T (1976) General theory of mode interaction in stiffened plate and shell structures. WTHD Report 590, Delft 4. Lin C C (1997) Stability and vibration characteristics of axially moving plates. International Journal of Solid and Structures 34(24): 3179–3190 5. Ulsoy A G, Mote C D Jr (1982) Vibrations of wide band saw blades. Journal of Engineering for Industry ASME 104(1): 71–78 6. Wang X (1999) Numerical analysis of moving orthotropic thin plates. Computers & Structures 70: 467–486

Chapter 4

Displacements of the Web in Equilibrium States of the Linearized System

A mathematical model of the axially moving orthotropic plate, derived in Sect. 4.2, describing a transverse motion of the web and a field of sectional forces is applied in this section. A static analysis involving determination of the non-trivial equilibrium positions existence is used in the investigations of stability of the web motion. Equations of equilibrium positions of the axially moving web performing uniform motion have been derived. Transverse displacements and wrinkling of the webs made of two kinds of papers and the three-layered corrugated board composed of these papers have been investigated.

4.1 Mathematical Models of the Axially Moving Orthotropic Web The equilibrium static state of the stationary web and the dynamic state of the axially moving web with uniform motion, when time-dependent forces do not interact, is generally referred to as the equilibrium state of the web. A nonuniform loading distribution at the edges and out-of-parallel location of the tension-guide rolls can lead to wrinkling in the case of the stationary web and the moving web as well. When the web is uniformly loaded, deflections arise in the region of the axially moving web with uniform motion. The forms of deflections are dependent on the transport speed of the web. A mathematical model of the axially moving orthotropic web with uniform motion which describes the equilibrium states was analytically solved to determine the displacements of the web. The linearized mathematical model of the system (Fig. 4.1) considered in Chapter 3 has been determined on the basis of (3.14), (3.15). Having neglected the non-linear components, these equations have the following form:  hðw; tt  2cw; xt  c; t w; x  c2 w; xx Þ þ Mx; xx þ þ 2Mxy; xy þ My; yy þ qz þ F; yy w; xx  2 F; xy w; xy þ F; xx w; yy ¼ 0  F; xxxx þ F; xxyy

 E  2   þ  F; yyyy ¼ 0; G

K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_4, Ó Springer-Verlag Berlin Heidelberg 2008

(4:1)

(4:2)

85

86

4 Displacements of the Web in Equilibrium States of the Linearized System

Fig. 4.1 Model of the web

y h

c

b

x

l z where qz – external loading. Under normal conditions the web moves between rolls loaded nearly uniformly with the longitudinal tension force. In this case, the Airy stress function F in (4.2) has the form: F ¼ A y2 ;

(4:3)

where A is the constant which depends on the longitudinal load. After double differentiation of (4.3) with respect to y and x on the basis of (3.13), one can write: F; yy ¼ 2A ¼ Nx0 ; F; xx ¼ Ny ¼ 0;

F; xy ¼ Nxy ¼ 0:

(4:4)

Substituting (4.3) and (4.4) into (4.1) and (4.2) and assuming a shortage of the external load qz = 0, a homogenous, linearized equation of the equilibrium state of the axially moving web with uniform motion which is tensioned with a constant longitudinal load Nx0 is obtained:  hc2 w;xx  Nx0 w;xx þ D w;xxxx þ 2ð  D þ D1 Þw;xxyy þ  D w;yyyy ¼ 0;

(4:5)

where D¼

E h3 ; 12ð1    2 Þ

D1 ¼

G h3 : 6

In the case of the non-uniform longitudinal loading, (4.1) can be expressed in the following form in an analogical way:  hc2 w;xx  F;yy w;xx þ 2F;xy w;xy  F;xx w;yy þ Dw;xxxx þ 2 ð  D þ D1 Þ w;xxyy þ  Dw;yyyy ¼ 0:

(4:6)

Equations (4.5) and (4.6) can be transposed to dimensionless forms by using the following terms: w x y b z ¼ ;  ¼ ; ’ ¼ ; r ¼ ; cw ¼ h l l l D D1 "¼ ; "1 ¼ ; Nx0 l2 Nx0 l2

sffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi Nx0 c h ;s¼ ¼c ; cw Nx0 h

F F¼ : Nx0 l2

(4:7)

4.2 Solution to the Mathematical Model

87

The dimensionless equation of the equilibrium state of the axially moving orthotropic web which is non-uniformly loaded along the width has the form: s2 z;  F;’’ z; þ 2F;’ z;’  F; z;’’ þ þ " ½z; þ 2ð  þ "1 ="Þz;’’ þ  z;’’’’  ¼ 0

(4:8)

The equation of the equilibrium state of the axially moving orthotropic web which is uniformly loaded is: ðs2  1Þz; þ "½z; þ 2ð  þ "1 ="Þz;’’ þ  z;’’’’  ¼ 0

(4:9)

Simple boundary conditions have been assumed at the ends  ¼ 0 and  ¼ 1:     (4:10) z¼0 ¼ 0; z; ¼0 ¼ 0; ¼1

¼1

and the free longitudinal ends ’ ¼ 0 and ’ ¼ b=l ¼ r     z;’’’ þ ð2  Þz;’ ’¼0 ¼ 0; z;’’ þ  z; ’¼0 ¼ 0: ’¼r

’¼r

(4:11)

Homogenous, partial differential equations (4.8) and (4.9) together with boundary conditions (4.10) and (4.11) constitute two mathematical models to determine displacements of the web in equilibrium states.

4.2 Solution to the Mathematical Model of the Web Loaded with Constant Longitudinal Force Under normal working conditions of winding machines, the web loaded with a nearly constant longitudinal tension force displaces between rolls. In this case, the linearized mathematical model of the system is expressed by (4.9) together with boundary conditions (4.10) and (4.11). To determine displacements of the web in equilibrium states, a solution to differential equation (4.9) has been sought and the dimensionless coefficient S has been introduced in the form: S¼

s2  1 : "

(4:12)

Then, (4.9) can be expressed in the following form: S z; þ z; þ 2 ð þ "1 ="Þz;’’ þ  z;’’’’ ¼ 0:

(4:13)

Comparing (4.13) with the equation of the stationary plate subjected to tension load along the longitudinal direction [5], one can infer that non-trivial equilibrium positions do not appear for S < 0. Thus, a condition for the existence of multiple equilibrium positions of the web is expressed as:

88

4 Displacements of the Web in Equilibrium States of the Linearized System

S0

or

s2  1:

(4:14)

The solution to (4.13) enables us to determine non-trivial equilibrium positions of the web zmk(m, k = 1, 2, . . .) complying with multiple values Smk. The solution to (4.13) is forecast in the following form: zmk ¼ Ck e’ sinðmp Þ:

(4:15)

Substituting (4.15) into (4.13), one receives the characteristic equation: 4  2ð þ

h i "1 1 ÞðmpÞ2 2 þ ðmpÞ2 ðmpÞ2  S ¼ 0:  "

(4:16)

Having introduced the following notations: "2 ¼  þ

"1 ; "

"3 ¼

  "1 2 1 1 ¼ "22  ;  þ    "

(4:17)

the solutions of characteristic equation (4.16) have the forms: 1 ¼ 2 ¼ 3 ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

"2 ðmpÞ2 þ mp "3 ðmpÞ2 þ S= ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

"2 ðmpÞ2  mp "3 ðmpÞ2 þ S= ; rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "2 ðmpÞ2 þ mp

(4:18)

"3 ðmpÞ2 þ S= :

Solutions (4.18) comply with three sets of non-trivial equilibrium positions: – for 0  S5ðmpÞ2 : zm1 ¼ ½C11 sinhð1 ’Þ þ C12 coshð1 ’Þ þ C13 sinhð2 ’Þ þ þ C14 coshð2 ’Þ sinðmpÞ;

(4:19)

– for 4ðmpÞ2 : zm2 ¼ ½C21 sinhð1 ’Þ þ C22 coshð1 ’Þ þ C23 sinð3 ’Þ þ þ C24 cosð3 ’Þ sinðmp Þ ;

(4:20)

– for S ¼ ðmpÞ2 : zm3 ¼ ½C31 sinhð1 ’Þ þ C32 coshð1 ’Þ þ C33 þ C34 ’ sinðmpÞ;

(4:21)

where C11, C12, . . . C34 – constants, which are determined from boundary conditions (4.11).

4.2 Solution to the Mathematical Model

89

To determine the coefficients Sm1, which describe non-trivial equilibrium positions in the range 05S5ðmpÞ2 , the expression in the bracket of (4.19) is substituted into boundary conditions (4.11). Having introduced the notations: Q1 ¼ 31m  ð2  Þ1 m2 p2 ;

Q2 ¼ 32m  ð2  Þ2 m2 p2

Q3 ¼ 21m   m2 p 2 ;

Q4 ¼ 22m   m2 p2

Q5 ¼ 33m  ð2  Þ 3 m2 p2 ;

Q6 ¼ 23m   m2 p2 ;

(4:22)

boundary conditions (4.11) can be expressed as: Q1 C12 þ Q2 C14 ¼ 0; Q1 C11 sinhð1m rÞ þ Q1 C12 coshð1m rÞ þ Q2 C13 sinhð2m rÞþ Q2 C14 coshð2m rÞ ¼ 0; Q3 C12 þ Q4 C14 ¼ 0

(4:23)

Q3 C11 sinhð1m rÞ þ Q3 C12 coshð1m rÞ þ Q4 C13 sinhð2m rÞþ Q4 C14 coshð2m rÞ ¼ 0: One receives a condition for the existence of multiple equilibrium positions (4.19) in the range of transport speeds 05S5ðmpÞ2 from the condition of the non-zero solution to (4.23) with respect to the constants C11. . .C14.   2Q1 Q2 Q3 Q4 ½coshð1m rÞ coshð2m rÞ  1  Q21 Q24 þ Q22 Q23 (4:24) sinhð1m rÞ sinhð2m rÞ ¼ 0 To determine the coefficients Smk, which describe non-trivial equilibrium positions in the range S4ðmpÞ2 , the expression in the bracket of (4.20) is substituted into boundary conditions (4.11). Then, boundary conditions (4.11) can be expressed as follows: Q1 C22 þ Q5 C24 ¼ 0; Q1 C21 sinhð1m rÞ þ Q1 C22 coshð1m rÞ þ Q5 C23 sinhð2m rÞþ Q5 C24 coshð2m rÞ ¼ 0; Q3 C22 þ Q6 C24 ¼ 0

(4:25)

Q3 C21 sinhð1m rÞ þ Q3 C22 coshð1m rÞ þ Q6 C23 sinhð2m rÞþ Q6 C24 coshð2m rÞ ¼ 0: One receives a condition for the existence of multiple equilibrium positions (4.20) in the range of transport speeds S4ðmpÞ2 from the condition of the nonzero solution of (4.25) with respect to the constants C21. . .C24.

90

4 Displacements of the Web in Equilibrium States of the Linearized System

  2Q1 Q5 Q3 Q6 ½coshð1m rÞ coshð2m rÞ  1  Q21 Q26 þ Q25 Q23 sinhð1m rÞ sinhð2m rÞ ¼ 0:

(4:26)

Equations (4.24) and (4.26) show that non-trivial equilibrium positions depend on the slenderness ratio of the web r = b/l, the longitudinal load and the transport speed through the coefficient S described by (4.12). The critical speed of the axially moving web corresponding to the speed at the onset of instability is the lowest transport speed at which the non-trivial equilibrium positions exist.

4.3 Displacements of the Web Loaded with Constant Longitudinal Force The values of transport speeds at which the non-trivial equilibrium positions appear depend on the slenderness ratio and the longitudinal load of the web. The range Nx0 = (502000) N/m of the longitudinal load of the papers and the three-layered corrugated board has been applied in the numerical investigations. In the case of the liner paper the border values mean 0.3% and 12.5%, and for the fluting paper: 0.75% and 30.3% of the breaking load, respectively. The wave speeds of the investigated webs under the border loads and their dimensionless plate stiffnesses related to the unit length are shown in Table 4.1. Figure 4.2 shows deflection forms of the orthotropic liner paper, which are connected with the non-trivial equilibrium positions z11, z21 and z31 (the lowest flexural forms) and z12, z13, z22, z23, z32 and z33 (the lowest flexural-torsional forms). Calculation results of transport speeds of the liner paper at which the nontrivial equilibrium positions occur under the load Nx0 = 50 N/m for various slenderness ratios are shown in Fig. 4.3. Figure 4.3 shows that z11 is the deflection form corresponding to the critical transport speed of the axially moving web. For the specified slenderness of the web r= b/l, the dimensionless transport speed s11(r), at which the form z11 appears, is the lowest speed, when the non-trivial equilibrium position exists. Thus, on the basis of (4.7) and (4.14), the critical speed can be determined as: ccr ¼ cw

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ " s11

(4:27)

Table 4.1 Wave speeds and the plate stiffnesses " and "1 Parameters Liner paper Fluting paper N0=2000 [N/m] N0=50 [N/m] N0=2000 [N/m] N0=50 [N/m] cw [m/s] [-] [-]

15.075 3.8310–4 1.6510–4

95.346 9.5710–6 0.4110–5

20.162 8.1410–5 4.2410–5

127.515 2.0310–6 1.0610–6

4.3 Displacements of the Web Loaded with Constant Longitudinal Force z

z

91

z

ϕ μ

μ

μ

(a) z11

(b) z12

z

(c) z13 z

z

μ

ϕ

μ

μ

(e) z22

(d) z21 z

(f) z23

z

z

ϕ μ

μ

μ (g) z31

(h) z32

(i) z33

Fig. 4.2 Deflection forms of the web

15,5 c [m/s] 15,45

z32 z22

z12

15,4

z13

z33 z23

z31

15,35 15,3 15,25

z21

15,2 15,15

z11

15,1

r

15,05 0

2

4

6

8

10

Fig. 4.3 Equlibrium. positions of the liner paper (Nx0=50 N/m)

To compare the investigation results, the dynamic mathematical model derived in Chapter 3 has been used. Fig. 4.4 shows the plot of two smallest eigenvalues of the liner paper web versus the transport speed. The eigenvalues are received using the Galerkin method with comparison functions zm1 in (4.19). At the critical transport speed the real part of the smallest natural frequency !11 impends to be non-zero. Figures 4.3 and 4.4 show the same values of the critical transport speeds predicted by static and dynamic analysis.

92

4 Displacements of the Web in Equilibrium States of the Linearized System

Fig. 4.4 Eigenvalues of the liner paper (r=0.5; Nx0=50 N/m)

Im (ωi) ω21 ω11

Re (ωi)

c [m/s]

Stability region

Divergence instability region

ccr

2-nd stability region

c [m/s] 20,3 20,28

z13 z22 z12

20,26 20,24

z33 z32

z23

z31

20,22 20,2

z21

20,18

z11

20,16 20,14 20,12

Fig. 4.5 Transport speeds of the fluting paper (Nx0= 50 N/m)

20,1 0

2

4

6

8

10

r [-] c [m/s]

100

z12

z32 z33

z13 z31

75

z22

z23

z21

50

z11

25

Fig. 4.6 Transport speeds of the corrugated board (Nx0= 50 N/m)

0 0

2

4

6

8

10 r [-]

The results of analogical investigations of transport speeds, at which the non-trivial equilibrium positions appear, are presented in Fig. 4.5 for the fluting paper, and in Fig. 4.6 for the corrugated board composed of the papers under investigation.

4.3 Displacements of the Web Loaded with Constant Longitudinal Force Fig. 4.7 Transport speeds of the liner paper (Nx0= 100 N/m)

21,6

93

c [m/s] z32 z33

z12 z13

21,55

21,5 z31

z22

21,45 21,4

z21

21,35

z11

z23

21,3 0

2

4

6

8

10

r [-] c [m/s]

100

z32 z 33

z31

75

z22

z12 z21

50

z23 z13

z11 25 Fig. 4.8 Transport speeds of the corrugated board (Nx0= 100 N/m)

0 0

2

4

6

8

10

r [-]

Figures 4.3, 4.5 and 4.6 show that the critical speed increases when the slenderness of the web enlarges. The three-layered corrugated board web of the same geometrical dimensions as the component papers is characterized by greater value of the critical speed than the web of the liner paper and the web of the fluting paper. In the range of slenderness ratio r< 6 of both the investigated papers, many different equilibrium positions occur in a small range of transport speeds. In this case, an occurrence of jumps between deflection modes may be observed. The investigation results of transport speeds for the two-times greater longitudinal load Nx0= 100 N/m and for different slenderness of the liner paper web are presented in Fig. 4.7. The analogical results for the three-layered corrugated board composed of the investigated papers are presented in Fig. 4.8. A comparison of Figs. 4.3 and 4.7 shows that two-times greater longitudinal loading of the liner paper causes an increase in the critical speed by 1/3 of the initial value. In the case of the corrugated board, the same increase in longitudinal loading causes an increase in the critical speed by 12% of the initial value. The investigation results of critical speeds of the liner paper, fluting paper

94

4 Displacements of the Web in Equilibrium States of the Linearized System

Fig. 4.9 Critical speeds of different webs (r = 1)

ccr 140 [m/s] 120 100 80 60 40 20 0

fluting paper corrugated board-static analysis liner paper corrugated board-dynamic analysis

0

500

1000

1500

Nx0 [N/m] 2000

and three-layered corrugated board webs in the whole investigated range of the longitudinal load for r = 1 are presented in Fig. 4.9. Dotted line in Fig. 4.9 shows the results of dynamic investigations from the previous section (compare Fig. 3.27). These plots show in the case of corrugated board the static analysis overestimates the value of the critical speed of the axially moving web by more than 1/3 of the dynamic value.

4.4 Web Loaded with a Non-Uniform Longitudinal Force In installations used to rewind wide material webs, there are guide rolls of a curvilinear surface (spread rolls). These rolls are used to eliminate wrinkling and sacking of the web along the transverse direction and to avoid slating (fluttering) of longitudinal edges during rewinding [1]. On the other hand, an excessive curvature of the spread rolls surface can cause wrinkling of the web. Thus, analyzing the behavior of the web in equilibrium states, an influence of the spread rolls curvature on wrinkling of the web surface is investigated. An axially moving orthotropic web with a constant dimensionless speed s, loaded with an in-plane tension force of non-linear distribution along the web width, is considered. This loading has been presented as a sum of the constant, dimensionless unit load N0 and the small non-linear component Nx of sinusoidal distribution (Fig. 4.10). The Nx component has been described by using the  parameter which determines the amplitude of loading. p p y ¼  sin ’ (4:28) Nx ¼  sin b r The Gorman and Singhal’s method [2] is applied to determine the Airy stress function under the loading Nx along edges  ¼ 0 and  ¼ 1. The stress function y

Fig. 4.10 Non-linear distribution of the web loading

l

1

λ

b x

4.4 Web Loaded with a Non-Uniform Longitudinal Force

95

is determined in the form of a sum of products of harmonic and hyperbolic functions representing the solution to (4.2). For the edge loading symmetric with respect to the axis y = b/2 or ’ ¼ r=2, only odd components appear in the series. p ’ F ¼ F1 þ F2 þ F3 ¼  sin r

        p 1 1 p 1 C1 cosh   þ C2   sinh þ r 2 2 r 2 þ

Rp X

Bp sinðpp Þ

p¼1;3;:::

h

     r r r i D1p cosh pp ’  þ D2p ’  sinh pp ’  þ 2 2 2 Rq qp X ’ Bq sin þ r q¼1;3;:::

        qp 1 1 qp 1 D3q cosh   þ D4q   sinh ; r 2 2 r 2

(4:29)

where Rp ; Rq ! 1. To determine the constants C1 and C2, a condition for vanishing of normal and shear stresses of the F1 component along the edges  ¼ 0 and  ¼ 1 for  ¼ 1 is formulated:  x h þ Nx j¼0 ¼ 0; xy ¼0 ¼ 0 (4:30) ¼1

¼1

Substituting (4.28) and (4.29) into boundary conditions (4.30), one obtains: p p  p  p þ 2r cosh 2r 2 r2 sinh 2r 2 r sinh 2r  

  C1 ¼  ; C2 ¼ p p2 pr þ sinh pr p r þ sinh pr

(4:31)

To determine the constants D1p and D2p, a condition for vanishing of normal and shear stresses of the F2 component along the edges ’ ¼ 0 and ’ ¼ r is formulated:   y ’¼0 ¼ 0 ;  xy h þ Nxy ’¼0 ¼ 0 ’¼r

(4:32)

’¼r

From boundary conditions (4.32), we have: D1p

    r sinh pp2 r 2 cosh ppr 2 ; D2p ¼  ¼ pp½pp r þ sinhðpp rÞ pp½pp r þ sinhðpp rÞ

(4:33)

96

4 Displacements of the Web in Equilibrium States of the Linearized System

To determine the constants D3q and D4q, a condition for vanishing of normal and shear stresses of the F3 component along the edges  ¼ 0 and  ¼ 1 is formulated: x j¼0 ¼ 0 ; ¼1

  xy h þ Nxy ¼0 ¼ 0 ¼1

From boundary conditions (4.34), one gets:

D3q

    r sinh qp 2 r cosh qp 2r 2r  qp ; D4q ¼  qp ¼ qp qp qp qp þ sinh þ sinh r r r r

The constants Bp and Bq, which there are in the F2 and F3 components of the stress function (4.29), are determined to ensure vanishing of shear stresses along the edges of the web. The state of non-dimensional stresses inside the web arising from the load Nx, determined on the basis of function (4.29) and dependences (4.31), (4.33) and (4.35), is presented by the following formulas:     p

p2 p 1 1 p 1 ’ C1 cosh ð  Þ þ C2 ð  Þ sinh ð  Þ x ¼  2 sin r r 2 2 r 2 r h   r Bp sinðpp Þ p2 p2 D1p þ 2ppD2p cosh pp ð’  Þ 2 p¼1;3;...  i r r þp2 p2 D2p ð’  Þ sinh pp ð’  Þ 2 2

  R q  X q2 p2 qp qp 1  ’ D3q cosh ð  Þ Bq 2 sin r r 2 r q¼1;3;...   1 qp 1 þD4q ð  Þ sinh ð  Þ ; 2 r 2    p p2 p p 1 y ¼  sin ’ ð  Þ C1 þ 2 C2 cosh r r r 2 r2   p2 1 p 1 þC2 2 ð  Þ sinh ð  Þ 2 r 2 r þ



Rp X

h  r p2 p2 Bp sinðpp Þ D1p cosh pp ð’  Þ 2 p¼1;3;...  i r r þD2p ð’  Þ sinh pp ð’  Þ 2 2 Rp X

(4:36)

4.4 Web Loaded with a Non-Uniform Longitudinal Force

97

   qp q2 p2 qp qp 1 ’ ð  Þ D þ 2D cosh 3q 4q r r r 2 r2 q¼1;3;...   q2 p 2 1 qp 1 þD4q 2 ð  Þ sinh ð  Þ ; 2 r 2 r

    p p p p 1 ’ C1 þ C2 sinh ð  Þ

xy ¼  cos r r r r 2   p 1 p 1 þC2 ð  Þ cosh ð  Þ r 2 r 2 þ

Rq X

Bq sin

h   r Bp pp cosðpp Þ pp D1p þ D2p sinh pp ð’  Þ 2 p¼1;3;...  i r r þD2p pp ð’  Þ cosh pp ð’  Þ 2 2

  R q   X qp qp qp qp 1 þ ’ D3q þ D4q sinh ð  Þ Bq cos r r r r 2 q¼1;3;...   qp 1 qp 1 þD4q ð  Þ cosh ð  Þ r 2 r 2 Dependencies (4.36) for the convexity of the roll  = 1, the slenderness of the web r = 0.5 and the number of the series components Rp, Rq = 9 are presented in Figs. 4.11, 4.12, and 4.13. The distribution of the normal stresses x in Fig. 4.11 is nearly uniform away from the edges of the web. The normal stresses y in Fig. 4.12 become the compressive ones in the internal part of the web. The shear stresses xy in Fig. 4.13 are antisymmetric with respect to  ¼ 1=2 and j ¼ r=2. The lack of vanishing of the shear stresses along the transverse edges visible in Fig. 4.13 follows from small numbers of the stress function components Rp, Rq = 9. The results of the similar investigations of the stationary, isotropic web by Lin’s and Mote’s [3,4] show that increasing the numbers up to Rp, Rq = 41 slightly changed the results received. þ

Rp X

σx

Fig. 4.11 Dimensionless normal stresses x ¼ F;yy on the web surface (Rp, Rq = 9)

μ

ϕ

98

4 Displacements of the Web in Equilibrium States of the Linearized System

Fig. 4.12 Dimensionless normal stresses y ¼ F;xx on the web surface (Rp, Rq = 9)

σy

ϕ

μ

τxy

Fig. 4.13 Dimensionless shear stresses xy ¼ F;xy on the web surface (Rp, Rq = 9)

μ

ϕ

In the considered case, the non-uniform longitudinal load of the web from the spread rolls is modeled by superposition of the unit tension force N0 and the Nx force nonlinearly distributed on the width of the web (Fig. 4.11): p (4:37) N ¼ N0 þ Nx ¼ N0 ð1 þ  sin ð ’ÞÞ r The dimensionless linearized equilibrium equation of the web (4.8) in this case has the form: ðs2  1Þ z;  F;’’ z; þ 2 F;’ z;’  F; z;’’ þ " ½z; þ þ 2ð  þ "1 ="Þz;’’ þ  z;’’’’  ¼ 0 :

(4:38)

where (4.29) determines the Airy stress function from the force Nx. A magnitude of the variable component Nx in (4.37) which causes wrinkling is obtained from the solution to an eigenvalue problem derived from (4.38), which satisfies the appropriate boundary conditions.

4.5 Wrinkling of the Web Loaded with a Non-Uniformly Distributed Longitudinal Force The web loaded with the longitudinal force N, whose distribution is shown in (4.37), is numerically investigated. The eigenvalue problem formed by (4.38) together with simple support boundary conditions is solved using the Galerkin method with the following comparison functions:

4.5 Non-Uniformly Distributed Longitudinal Force Fig. 4.14 Amplitude of the load causing wrinkling (liner paper, r = 0.5; – N0= 50 N/ m; . . . N0= 100 N/m; - N0= 2000 N/m)

99

λ cr 0.15 0.1 0.05 0

s 0

0.2

0.4

0

0.2

0.4

0.6

0.8

1

λcr 0.15 0.1

Fig. 4.15 Amplitude of the load causing wrinkling (liner paper r = 1; – N0= 50 N/m; - - N0= 2000 N/m)

0.05 0

s

zij ¼ sinðipÞ sin

  jp ’ ; r

0.6

0.8

1

(4:39)

where i, j =1, 2, 3, . . .. The amplitude values cr of the load component Nx causing wrinkling of the liner paper are shown in Figs. 4.14 and 4.15. These values correspond to the smallest positive eigenvalues of (4.38) for various transport speeds sand various longitudinal loads of the liner paper web of r= 0.5. The web wrinkles when   cr and is smooth when 5cr . When the transport speed increases, the amplitude of load of the liner paper web, which causes wrinkling, diminishes. Enlarging the N0 value results in a decrease in the cr value. A more considerable influence of the latter factor on the change of cr can be observed in the range of slight loads of the web. A double decrease in the slenderness ratio of the liner paper web from r= 0.5 to r= 1 results in an increase in the cr by about 5% (Fig. 4.15). The web of a higher slenderness ratio has lower sensibility of the amplitude cr to a change in the longitudinal load. An increase in the load up to 2000 N/m causes a decrease in the cr value by about 2%. The dimensionless stresses in the web at the beginning of wrinkling along particular directions are described by the following dependencies: xcr ¼ 1 þ cr x1

xcr ¼ cr xy1 ycr ¼ 1 þ cr y1 ;

100

4 Displacements of the Web in Equilibrium States of the Linearized System

Fig. 4.16 Longitudinal stresses at the beginning of wrinkling (liner paper, r = s= 0.5; N0= 2000 N/m)

σxcr [N/m]

x

y

σycr [N/m] y

Fig. 4.17 Transverse stresses at the beginning of wrinkling (liner paper, r = s = 0.5; N0 = 2000 N/m)

x

τxcr [N/m]

Fig. 4.18 Shear stresses at the beginning of wrinkling (liner paper, r = 0.5; N0 = 2000 N/m)

x

y

where x1 ; xy1 y1 stresses along particular directions under  ¼ 1 (Figs. 4.11, 4.12 and 4.13). A distribution of stresses at the beginning of wrinkling of the liner paper web of the slenderness ratio r= 0.5 under the longitudinal load N0=2000 N/m and the transport speed s = 0.5 is presented in Figs. 4.16, 4.17, and 4.18. Decreasing the paper flexural stiffness causes also diminishing the cr value. A dependence of the cr on the dimensionless transport speed s of the liner paper web and the fluting paper web of the same slenderness ratio r = 0.5 under the same longitudinal load N0 = 50 N/m is shown in Fig. 4.19. In the

4.6 Final Remarks Fig. 4.19 Amplitude of the load causing wrinkling of the liner and fluting paper web (r = s = 0.5; N0 = 50 N/m)

101 λcr 0.15

Liner paper web

0.1

Fluting paper web

0.05 s

0

0

0.2

0.4

0.6

0.8

1

case of the fluting paper web, nearly a quintuple decrease in the plate stiffness in comparison to the liner paper web causes a decrease in the cr value by about 4%.

4.6 Final Remarks On the base of static analysis of the linearized system the instability of the orthotropic web transported with constant speed is predicted by determining the existence of non-trivial equilibrium positions. The derived equations show that non-trivial equilibrium positions depend on the slenderness ratio of the web, the longitudinal load and the transport speed. The critical speed of the axially moving web corresponding to the speed at the onset of instability is the lowest transport speed at which the non-trivial equilibrium positions exist. It is worth noting that the static analysis is less complicated than dynamic analysis and comparative investigations of both paper webs show the same values of the critical transport speeds predicted by both methods. In the case of corrugated board the investigation results show the static analysis overestimates the value of the critical speed of the multilayer web. In the second part an influence of the spread rolls curvature on wrinkling of the web surface is investigated. The Gorman and Singhal’s method is applied to determine the stress function under the longitudinal loading of the web. The stress function is determined in the form of a sum of products of harmonic and hyperbolic functions representing the solution to the strain inseparability equation. A magnitude of the variable component of the longitudinal load which causes wrinkling is obtained from the solution to an eigenvalue problem derived from the dimensionless linearized equilibrium equation of the web. Investigation results show when the transport speed increases, the amplitude of load of the paper web, which causes wrinkling, diminishes. Enlarging the initial load value results in a decrease of the critical amplitude of loading.

102

4 Displacements of the Web in Equilibrium States of the Linearized System

References 1. Adams R J (1992) The influence of rolls and reels on flutter and windage. Tappi Journal 75(11): 215–222 2. Gorman D J, Singhal R K (1993) A superposition Rayleigh-Ritz method for free vibration analysis of non-uniformly tensioned membranes. Journal of Sound and Vibration 162(3): 489–501 3. Lin C C, Mote C D Jr (1996) The wrinkling of rectangular webs under nonlinearly distributed edge loading. Journal of Applied Mechanics ASME 63: 655–659 4. Lin C C, Mote, C D Jr (1996) The wrinkling of thin, flat, rectangular webs. Journal of Applied Mechanics ASME 63: 774–779 5. Timoshenko S (1936) Theory of Elastic Stability, McGraw-Hill, New York

Chapter 5

Dynamics of the Axially Moving Viscoelastic Web

In the considerations devoted so far to dynamics of the axially moving web, it has been assumed that the web material is characterized by elastic properties. This assumption, which leads to a simplification in the mathematical model, is sufficient for many materials met in engineering applications. However, numerous materials of high practical significance, whose physical properties are not fully rendered by the elastic model, appear nowadays. One can find among them webs made of plastics or composites. Also, in the case of paper webs, although their elastic model is used in dynamic analyses, it does not describe fully paper properties. Paper is a very specific material, whose physical characteristics depend on its structure, raw materials composition, production technology, finishing, processing and hydrothermal state. Different properties of paper that result from its heterogeneity are shown in many works (e.g. [1,6,7]). On one hand, investigation results prove that two main orthotropy directions can be distinguished in the anisotropic paper web: parallel and perpendicular to its longitudinal axis. On the other hand, these investigations show that the viscoelastic rheological model should be used in the dynamic analysis. In the case of a stationary orthotropic plate, its two-dimensional rheological model was described fairly long ago by Sobotka [4,5]. Recently, two-dimensional an elastic-viscoelastic rheological element has been proposed [3]. In literature, one can find also works in which one-dimensional rheological models have been used to describe properties of the axially moving material. Some of these works have been presented in Chap. 2. One-dimensional rheological elements have been employed as well in the analyses of beam models of the web discussed in Chap. 6. In modeling the axially moving viscoelastic web, a two-dimensional rheological element has been assumed in this chapter. Using this rheological model and the plate theory, a differential equation of motion of the axially moving web has been derived. On the basis of the partial differential equation that governs the transverse vibrations of the system, a second-order Galerkin truncated system has been determined. The effects of transport speed and internal damping on the dynamic behavior of the axially moving web have been presented. K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_5,  Springer-Verlag Berlin Heidelberg 2008

103

104

5 Dynamics of the Axially Moving Viscoelastic Web

5.1 Two-Dimensional Rheological Model for Viscoelastic Materials A two-dimensional rheological model for orthotropic viscoelastic material is presented in Fig. 5.1. The material is stretched along its longitudinal direction with a constant stress x0 . In the plane-stress state, the strain components are given by the generalized Hooke’s law: "x ¼ a11 ðx0 þ x Þ þ a12 y "y ¼ a21 x þ a22 y

(5:1)

"xy ¼ a33 xy where "x ; "y ; "xy – strain tensor components, x ; y ; xy – stress tensor components, a11 . . .. a33 – coefficients of the elastic compliance matrix.

On the other hand, one can determine the strains "x and "y taking into account creep functions in both the directions: "x ðts Þ ¼ ðx0 þ x Þ x ðts Þ þ

Zts

x ðts  Þ

@ x d @

o

"y ðts Þ ¼ y y ðts Þ þ

Zts

y ðts  Þ

(5:2)

@ y d @

o

y

σy σx

σx0

σx0

σx

x

Fig. 5.1 Two-dimensional rheological model

σy

5.1 Two-Dimensional Rheological Model for Viscoelastic Materials

105

where x ; y – creep functions in the x and y directions, respectively, ts – transport time. The strain rates are expressed in the following form: 2 t 3 Zs dy d"x dx dx dx dx @ 4 @x 5 x ðts  Þ ¼ a11 þ a12 ¼ x0 þ x þ x þ d dt dt dt dt dt dt @ t @ 0

2 t 3 Zs d"y dy dy dy @y 5 dx @ 4 y ðts  Þ ¼ a21 þ a22 ¼ y þ y þ d dt dt dt dt dt @ t @ 0

(5:3)

In further considerations, the creep functions of a Zener-type viscoelastic element (Fig. 5.2) have been taken into account [8]: x ðtÞ ¼

1 E2x  EE1xx t  e 0x ; E1x E0x E1x

y ðtÞ ¼

E2y  EE0y1yy t 1  e E1y E0y E1y

(5:4)

where E0x = E1x+E2x, E0y = E1y+E2y,   x ¼ x ; y ¼ y – creep time constants, E2x E2y x ; y – internal viscous damping coefficients. To determine the shear stress, the relaxation function of the Zener-type viscoelastic element has been taken into consideration: xy

ðtÞ ¼ G1 þ G2 e

 Tt

G

(5:5)

;

where G = G1 + G2 – shear modulus of the material, TG – relaxation time constant. For the symmetric shear, the shear strain "xy is the same throughout the model, thus:

xy ¼

Zt

xy ðt

 Þ

@"xy d @

(5:6)

0

η E1

Fig. 5.2 Zener element

E2

106

5 Dynamics of the Axially Moving Viscoelastic Web

Substituting (5.4) into (5.3) and solving finally (5.3) with respect to x and y , we obtain: x ¼ M1 K1 "x

(5:7)

y ¼ M1 K2 "y

In (5.7) the differential operator M of the sixth order is given as follows: M¼

a211

a222



 d6 d5 d4 d3 d2 q6 6 þ q5 ðtÞ 5 þ q4 ðtÞ 4 þ q3 ðtÞ 3 þ q2 ðtÞ 2 ; dt dt dt dt dt

(5:8)

where    dy ðtÞ 1 1 dx ðtÞ ; þ þ x ðtÞ þ y ðtÞ E1x E1y dt dt    E2x x dy ðtÞ E2y y dx ðtÞ 1 1 q4 ðtÞ ¼ 2 þ þ 2 þ x ðtÞ þ y ðtÞ ; dt E1x E1y dt E1x E1y     E2y y 1 E2x E2y x y E2x x 1 q5 ðtÞ ¼ 2 þ y ðtÞ þ 2 þ x ðtÞ ; q6 ¼ ; E1y E1x E1x E1y E21x E21y (5:9)

q2 ðtÞ ¼

dx ðtÞ dy ðtÞ ; q3 ðtÞ ¼ dt dt



In (5.7) the differential operators K1 and K2 of the fifth order are given by: "

#  2 dy d 1 d K1 ¼ þ y ðtÞ þ ; E1y d t2 dt dt   2   5 1 d dx d 2 2 E2x x d K2 ¼ a11 a22 þ þ x ðtÞ þ : E1x d t2 dt dt E21x d t5 a211

a222

E2y y d 5 þ E21y d t5



(5:10)

5.2 Mathematical Model of the Moving Viscoelastic Web A viscoelastic moving web of the length l, the width b and the thickness h is considered. The web moves at a constant velocity c. The co-ordinates system and geometry are shown in Fig. 5.3.

y, v c

Fig. 5.3 Axially moving web

z, w

h l

b

x, u

5.2 Mathematical Model of the Moving Viscoelastic Web

107

The orthotropic material of the moving web obeys the generalized Hooke’s law. Taking into account both the out-of-plane and in-plane deformations, geometrical relationships in the Lagrange description have been assumed [2]: "x ¼ u;x  zw;xx "y ¼ v;y  zw;yy

(5:11)

2"xy ¼ u;y þ v;x  2zw;xy where "x ; "y ; "xy – strain tensor components of the middle surface of the web, u, v, w – displacement components of the web middle surface. The dependence between the Young modulus and the Poisson ratios is such that: Ex yx ¼ Ey xy

(5:12)

The relationships between stresses and loads in the middle surface of the web have the forms: Nx ¼

Z

h=2

x dz; Ny ¼

Z

h=2

Mx ¼

Z

y dz; Nxy ¼ Nyx ¼

h=2

h=2

x z dz; My ¼

h=2

h=2

Z

Z

h=2

xy dz h=2

h=2

y z dz; Mxy ¼ Myx ¼

h=2

Z

(5:13)

h=2

xy z dz h=2

where Nx, Ny, Nxy – in-plane stresses resultants, Mx, My, Mxy – bending moment resultants. The partial differential equilibrium equation resulting from the Hamilton’s principle and the corresponding expressions appearing in (5.11) for a transverse motion of the two-dimensional axially moving web have been derived in Chap. 3 and have the following form:  h ðw;tt  2cw;xt  c2 w;xx Þ þ Mx;xx þ 2Mxy;xy þ My;yy þ         þ Nx w;x ;x þ Ny w;y ;y þ Nxy w;x ;y þ Nxy w;y ;x þq ¼ 0

(5:14)

where  – mass density of the web, q – transverse loading of the web. To introduce a two-dimensional rheological model of the web material, stress-strain relationships (5.7) have been substituted into (5.13), taking into account the out-of-plane and in-plane deformations, respectively:

108

5 Dynamics of the Axially Moving Viscoelastic Web

Nx ¼ M1 K1 u;x h ; Ny ¼ M1 K2 v;y h ;

(5:15)

Nxy ¼ Lðu;y þ v;x Þ h ; Mx ¼ M1 K1 w;xx

h3 ; 12

My ¼ M1 K2 w;yy

h3 ; 12

(5:16)

@ð Þ d: @

(5:17)

Mxy ¼ 2 L w;xy

h3 ; 12

where the operator L takes the following form:

L¼

Zt

xy ðt

 Þ

0

In further considerations one assumes that the web is stretched only along its longitudinal direction, hence: Nx ¼ Nx0 ¼ x0 h ¼ const;

Ny ¼ Nxy ¼ 0;

q ¼ 0:

(5:18)

Substituting (3.15), (3.16) and (3.18) into governing (3.14), one receives:   hM w;tt  2 hMcw;xt   hMc2 w;xx  K1  K2

h3 w;xxxx  12

h3 h3 w;yyyy  2ML w;xxyy þ MNx0 w;xx ¼ 0 12 12

(5:19)

The boundary conditions referring to simple supports at transverse ends of the web are as follows: w jx¼0 ¼ 0; x¼l

 w;xx x¼0 ¼ 0

(5:20)

x¼l

The boundary conditions at longitudinal free ends of the web are:

 

 w;yyy þ 2  xy w;xxy y¼0 ¼ 0; y¼b



 w;yy þ xy w;xx y¼0 ¼ 0

(5:21)

y¼b

Partial differential equilibrium equation (5.19) and boundary conditions at the transverse and longitudinal ends of the web (5.20) and (5.21) constitute a mathematical model of the moving viscoelastic two-dimensional web.

5.3 Solution to the Problem

109

Then, introducing the dimensionless parameters: w x y c z¼ ; ¼ ; ¼ ; s¼ ¼c h l l cw sffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffi cw t P 0 P0  ¼t ¼ ; cw ¼ ; l A l A

rffiffiffiffiffiffiffi A ; P0 (5:22)

where A = b  h – cross-section area of the web, Po = Nx0  b – resultant longitudinal force, one receives governing equation (B.1), which is presented in Appendix B. Boundary conditions (5.20) and (5.21) in the non-dimensional form are following:  z j ¼0 ¼ 0; z;  ¼0 ¼ 0; ¼1 ¼1  

 z;

þ 2  xy z;  ¼0 ¼ 0;

¼b=l



 z;

þ xy z;  ¼0 ¼ 0:

(5:23)

¼b=l

Partial differential equation (B.1) with boundary conditions (5.23) constitute a mathematical model of the system in the dimensionless form that has been investigated.

5.3 Solution to the Problem Due to complexity of the problem, the solution to (B.1) cannot be obtained analytically. The problem has been solved using the Galerkin method. The following finite series representation of the dimensionless transverse displacement has been assumed: zð ; ; Þ ¼

n X

zi;1 ð ; Þ qgi ðÞ;

(5:24)

i¼1

where zi,1ð ; Þ – flexural modes of the web, qgi – i-th generalized displacement. To denote the flexural modes of the web, its non-trivial equilibrium positions have been determined (see Chap. 4). The web deflections of the fundamental flexural modes zi1 are described by sine functions along the axial direction and a function including hyperbolic sines and hyperbolic cosines along the width direction. For the considered orthotropic web, the modes zi1 have the form: zi1 ð ; Þ ¼ sinðip Þ½sinhði Þ þ A2i coshði Þ þ A3i sinhð i Þ þ A4i coshð i Þ

(5:25)

110

5 Dynamics of the Axially Moving Viscoelastic Web

where: A2i, A3i, A4i – constants, which are determined from boundary conditions (5.23), i = 1, 2 . . . n. The 2-term finite series representation of the dimensionless transverse displacement of the web has been employed in the numerical investigations. Substituting (5.24) into (B.1) and using orthogonality conditions, one determines a set of ordinary differential equations. For the liner paper web the equations (B.3) are presented in Appendix B. The fourth-order Runge-Kutta method was used to integrate ordinary differential equations and to analyze the dynamic behavior of the system.

5.4 Results of the Numerical Investigations The mathematical model of the axially moving web in which a two-dimensional rheological model had been considered was subject to dynamic investigations. The numerical calculations of dynamic stability were conducted for both the webs under analysis, namely: liner paper and fluting paper. Influences of internal damping of the web material, original tension and web slenderness on motion stability were investigated. The rheological parameters of the papers employed in the calculations have been presented in Table 1.1 in Chap. 1. Owing to the identification method of viscoelastic parameters E2 and Z; which is an approximated method, it has been assumed that the identified strain relaxation time Z=E attains the maximum value. Changing a value of this parameter, we investigated an effect of internal damping on the web motion stability. The calculations were conducted for the constant value of relaxation of shear stresses TG = 1 s. In the numerical calculations, it was also assumed that elements of the elastic compliance matrix of the web were constant: a11 ¼

1 :; Ex

a22 ¼

1 ; Ey

a12 ¼

xy ; Ey

a21 ¼

yx ; Ex

a33 ¼

1 : G

(5:26)

In the initial stage of the investigations, a comparison of the dynamic behavior of the undamped system and damped system of the moving paper web was made. In the case of the liner paper, the dynamic behavior of the undamped system is illustrated in Fig. 3.5 in Chap. 3. A divergent-type stability loss of motion appears then at the transport speed ccru = 15.11 m/s, when the lowest eigenvalue of flexural vibrations in the imaginary form changes into a real positive number. This is illustrated in the first column of Table 5.1 and in Fig. 5.4, which is a part of Fig. 3.5 situated in the vicinity of the critical speed. In the case of the damped system with a two-dimensional rheological element, the lowest eigenvalue of flexural vibrations has a complex form. In the range of undercritical transport speeds, the real part has a negative value that grows with an increase in transport speed. At the critical speed, the real part attains a positive value. This process is illustrated in the second column of

5.4 Results of the Numerical Investigations

111

Table 5.1 Comparison of eigenvalues (liner paper, r = 1, Nx0 = 50 N/m) 2-D undamped model 2-D damped model c [m/s]

Re() + i Im ()

c [m/s]

Re() + i Im ()

15.091 15.098 15.105 15.109

–i 0.0205 –i 0.0148 –i 0.0064 – 0.0004

14.85 14.865 14.879 14.895

–0.20  10–2–i 0.014 –0.14  10–2–i 0.0135 –0.6  10–2–i 0.0127 0.014  10–2–i 0.011

Im (ωi) ω11 ω1 Re (ωi)

c [m/s]

ccrd = 14.89 m/s ccru = 15.11 m/s

Fig. 5.4 Lowest eigenvalues of the damped and undamped system (liner paper, !1 – damped eigenvalue, !11 – undamped eigenvalue, r = 1; Nx0 = 50 N/m)

Table 5.1 and in Fig. 5.4, where the two lowest eigenvalues of damped vibrations are drawn with a thick line. The dynamic behavior of the fluting paper web has been presented in an analogous way in Table 5.2 and Fig. 5.5. The part of Fig. 5.5 drawn with a thin line is a part of the plot from Fig. 3.6 for the near-critical range Thick lines, similarly as in the previous case, denote the transport speed history of lowest eigenvalues of the damped system. In both the cases compared, the fact that internal damping was taken into consideration has been followed by a decrease in the critical speed if compared to the undamped system. In the case of the liner paper web, the critical value decreased by 1.45%, whereas in the case of the fluting paper – by 0.45%.

Table 5.2 Comparison of eigenvalues (fluting paper, r = 1, Nx0 =50 N/m) 2-D undamped model 2-D damped model c [m/s]

Re() + i Im ()

c [m/s]

Re() + i Im ()

20.160 20.170 20.177 20.178

–i 0.190 –i 0.109 –i 0.017 – 0.030

20.085 20.087 20.089 20.091

–1.9  10–4–i 0.0078 –1.2  10–4–i 0.0077 –4.9  10–5–i 0.0076 +1.9  10–5–i 0.0075

112

5 Dynamics of the Axially Moving Viscoelastic Web

Im (ωi) ω1

ω11

Re (ωi)

c [m/s]

ccrd= 20.09 m/s ccru= 20.178 m/s

Fig. 5.5 Lowest eigenvalues of the damped and undamped system (fluting paper, !1 – damped eigenvalue, !11 – undamped eigenvalue, r = 1; Nx0 = 50 N/m)

a)

scr

b) 1,01

scr

2000 N/m

1

500 N/m 0,99 100 N/m

0,98 0,97

λy

50 N/m

0,96 1

1,5

2

2,5

3

3,5

4

4,5

5

5,5

1,002 1 0,998 0,996 0,994 0,992 0,99 0,988 0,986

2000 N/m 100 N/m 500 N/m 50 N/m

λx 1

2

3

4

5

6

7

8

Fig. 5.6 Critical speeds of the liner paper web, r = 1, (a) x = 7.92 s = const., (b) y = 5.53 s = const.

The results of investigations of the influence of internal damping and initial tension on the critical speed of the liner paper web are presented in Fig. 5.6. As has been already mentioned, the value of internal damping was changed by decreasing values of creep constants along both directions, i.e.: the machine direction and the transverse one. This process was repeated for various values of initial tension within the range from 50 N/m to 2000 N/m. Figure 5.6a shows results of the investigations at the constant creep value along the machine direction x = 7.92 s. This value was obtained in the identification procedure for the machine direction. Under such conditions, a decrease in the relaxation constant of strains causes a decrease in the critical speed of the liner paper web. The larger the decrease, the smaller the web initial tension. For the lowest initial tension tested Nx0 = 50 N/m, a five-fold decrease in the creep constant along the transverse direction up to y = 1 s is followed by a decrease in the critical speed by 3%. The results of analogous investigations, at the fixed value of the creep constant along the transverse direction y = 5.53 s and a change in the creep constant value along the machine direction x, are presented in Fig. 6.5b. In this

5.4 Results of the Numerical Investigations a)

b) 1,01

scr

113

1,05

2000 N/m

1 0,99

scr

500 N/m

0,98

50 N/m

0,96 0,95

λy

0,94 1

1,5

2

2,5

3

2000 N/m

0,9 0,85 100 N/m 0,8 50 N/m 0,75

100 N/m

0,97

1

0,95 500 N/m

3,5

4

4,5

5

5,5

0,7 1

2

3

4

5

6

7

8

λx

Fig. 5.7 Critical speeds of the liner paper web, r = 0.5, (a) x = 7.92 s = const., (b) y = 5.53 s = const.

case, one can observe a similar character in changes in the critical speed, however the system sensitivity is lower. An eight-fold decrease in the creep constant along the machine direction was followed by a decrease in the critical speed by 0.5% only. In the next stage, an effect of web slenderness on its critical speed for various values of initial tension was analyzed. Figure 5.7 shows results of the conducted investigations, analogous to those made in the previous stage, but this time for the liner paper web of the slenderness r = 0.5. Figure 5.7a presents a plot of the dimensionless web critical speed as a function of the creep constant along the transverse direction for the fixed creep constant value along the machine direction. Figure 5.7b shows analogous results of investigations for the unaltered value of the creep constant along the transverse direction. Comparing the results of investigations depicted in Figs. 5.6 and 5.7, one can state that the narrower web of the slenderness r = 0.5 is characterized by higher sensitivity of the critical speed to changes in internal damping. The highest sensitivity occurred at changes in the creep constant along the machine direction. At the lowest initial tension Nx0 = 50 N/m, an eight-fold decrease in the strain relaxation constant was followed by a decrease in the critical speed of the liner paper web by as much as 22.4%. In the final stage, an effect of orhotropic properties of the web damped model on critical speeds was investigated. These investigations were conducted within the whole range of the initial tension analyzed for the liner paper web and for the fluting paper web we as well. In both the cases, it was assumed that the isotropic web was characterized along both the orthotropy directions by the same parameters as the orthotropic web along the machine direction. The results of these investigations are to be seen in Fig. 5.8. In both the cases, the fact that orthotropic properties of the web were considered has been followed by an increase in the web critical speed. This increase is higher for lower values of initial tension. Its highest values were attained at the tension 50 N/m and were equal to 0.11% for the liner paper web and 0.03% for the fluting paper web, respectively. In the case of the fluting

114

5 Dynamics of the Axially Moving Viscoelastic Web

a)

scr

b) 1

scr

1

0,998

0,999

0,996

0,998

0,994

0,997

Nx0

0,992 0

500

1000

1500

2000

Nx0

0,996 0

500

1000

1500

2000

Fig. 5.8 Critical speeds of the webs (r = 1, --- orthotropy, - - -  - isotropy), (a) liner paper, (b) fluting paper

paper web, the fact that its orthotropic properties were taken into account within high initial stresses does not have any practical effect on its critical speed.

5.5 Final Remarks A survey of investigations on dynamics of axially moving plate systems in Chap. 2 shows that there is a lack of studies devoted to dynamic investigations of two-dimensional systems in which the effect of rheological properties of the moving material has been analysed. On the other hand, the works in which a two-dimensional rheological model of stationary plates was considered in the dynamic analysis have been known for a long time. In this chapter, a two-dimensional rheological model has been employed to model the axially moving orthotropic web made of the material characterized by viscoelastic properties. To formulate the constitutive equation of this model, the generalized Hook’s law and the creep functions of the web material along the main directions of orthotropy have been employed. Rheological properties of the web both along the machine direction and the transverse direction have been described with the three-parameter Zener model. On the basis of the plate theory, a linear mathematical model of the moving web, whose approximated solution was sought with the Galerkin method, has been developed. The numerical investigations conducted for both the webs indicate that if the material internal damping is taken into consideration, then there is a decrease in the web critical speed. A level of this decrease depends first of all on the magnitude of initial tension. The higher the initial tension, the less significant the effect of internal damping on the critical speed. The investigations of the influence of rheological properties of the moving web and its slenderness on the critical speed have shown that the narrow web is characterized by the highest sensitivity to alternations in rheological parameters along the machine direction. It has been also found that in both the cases of paper webs, the fact that orhotropic properties were taken into consideration resulted in an increase in the critical speed.

References

115

References 1. Jones A R (1968) An experimental investigation of the in-plane elastic moduli of paper. Tappi Journal 51(5): 203–209 ´ 2. Kołakowski Z, Krolak M (1995) Interactive elastic buckling of thin-walled closed orthotropic beam-columns. Engineering Transactions 43 (4): 571–590 3. Marynowski K (2006) Two-dimensional rheological element in modeling of axially moving viscoelastic web. European Journal of Mechanics A/Solids 25: 729–744 4. Sobotka Z (1984) Rheology of Materials and Engineering Structures, Academia, Prague 5. Sobotka Z (1989) Theory of Plasticity and Limit Design of Plates, Elsevier 6. Szewczyk W, Marynowski K, Tarnawski W (2006) The analysis of Young’s modulus distribution in paper plane. Fibers & Textiles in Eastern Europe 58: 91–94 7. Uesaka T, Marakami K, Imamura R (1979) Biaxial tensile behaviour of paper. Tappi Journal 62(8): 111–114 8. Zener C (1937) Internal friction in solids. Physics Review 52: 230–240

Chapter 6

Beam Model of the Moving Viscoelastic Web

The results of analysis of the axially moving web dynamics obtained with the nonlinear mathematical model developed in Chap. 3 have been presented in the previous chapters. The complexity of this model as regards a complicated nature of the mathematical description causes that it is not convenient in engineering applications. The level of complexity of the mathematical description and its size are especially important in calculations of this kind, where computation time plays a significant role. On the other hand, the results of analyses obtained so far indicate that an application of the beam model of the moving web in dynamical calculations can sometimes result in satisfactory results (e.g. [1,3,4]). One can suppose that in some dynamical calculations aiding designing and building devices that rewind a broad web, its beam model can be useful as less complex in comparison to the plate model. An additional argument supporting the application of the beam model is a possibility to consider various rheological models of the web material in a much simpler way than in it has been presented in Chap. 5. In the considerations included in Chaps. 3 and 4, the web material has been treated as a continuum with elastic properties. An elastic model of the web as regards paper, despite the fact that it is applied, can turn out to be too little accurate. The results of dynamic calculations that take into account the two-dimensional rheological model, presented in Chap. 5, confirm this thesis. The results of comparative investigations of beam models, where various rheological models of the axially moving material were analyzed, can be interesting as well. This chapter presents a development of the nonlinear beam model of the axially moving web made of the material characterized by elastic-damping properties. The model has been used in the dynamic investigations whose results are presented in the further part of this chapter. Two rheological models, namely: the twoparameter Kelvin-Voigt model that has been most often used in the investigations so far and the three-parameter Poynting-Thompson model already discussed in Chap. 1, have been used to describe material properties in the comparative investigations. The results of dynamic analyses of free vibrations and parametric vibrations exited by the harmonically variable longitudinal tension force have been shown. The results of analyses of the beam model have been compared to the analogous results of the investigations conducted for the plate model. K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_6, Ó Springer-Verlag Berlin Heidelberg 2008

117

118

6 Beam Model of the Moving Viscoelastic Web

6.1 Nonlinear Beam Model of the Viscoelastic Web A viscoelastic axially moving beam model of the web with a length l is considered. The beam moves at an axial velocity c. The geometry of the system and its co-ordinates are shown in Fig. 6.1. Dynamics of the axially moving plate model of the web in a state of uniform initial stress has been investigated in Chap. 3. The mathematical model that describes the motion and the field of cross-sectional forces has the form of a system of three coupled differential equations of equilibrium (3.10), (3.11) and (3.12): hðw;tt  2cw;xt  c2 w;xx Þ þ Mx;xx þ 2Mxy;xy þ My;yy þ

(6:1)

þ qz þ ðNx w;x Þ;x þ ðNy w;y Þ;y þ ðNxy w;x Þ;y þ ðNxy w;y Þ;x ¼ 0 hðu;tt  2cu;xt  c2 u;xx Þ þ Nx;x þ Nxy;y ¼ 0

(6:2)

hðv;tt  2cvxt  c2 v;xx Þ þ Nxy;x þ Ny;y ¼ 0

(6:3)

In the case of transverse oscillations of the beam model, one should take into account only the first equation. The application of this model gives the following equation of motion in the y direction:  hz ðw;tt  2 c w;xt  c2 w;xx Þ þ Mx;xx þ ðNx w;x Þ;x ¼ 0;

(6:4)

where hz – equivalent thickness of the beam, Mx – bending moment, Nx – perturbated axial stress,  – mass density of the beam. The nonlinear strain component in the x direction is related to the displacement w by: " ðx; tÞ ¼

1 2 w ðx; tÞ: 2 ;x

(6:5)

Dependence (6.5) shows that the geometrical type of non-linearity has been taken into consideration in the beam model of the web.

y E,J,A,ρ,γ

w(x,t) P

Fig. 6.1 Axially moving beam model of the web

l

c

P

x

6.1 Nonlinear Beam Model of the Viscoelastic Web

119

6.1.1 Kelvin-Voigt Model of Material The model of internal damping introduced by Kelvin-Voigt (K-V) is shown in Fig. 6.2. For this two-parameter viscoelastic model of material, the differential constitutive equation is as follows:  ¼ E" þ  ";t ;

(6:6)

Displacement of the web w is a function of the x and t variables, and substituting (6.5) into (6.6) we obtain: ¼

1 E w2;x þ  w;x w;xt þ  c w;x w;xx 2

(6:7)

Let us assume that the axial tension is characterized as a periodic perturbation on the steady-state tension: Nx ¼ Nx0 þ N1 cos ðtÞ

(6:8)

where Nx0 – initial axial tension, N1 – amplitude of axial tension. Then, (6.4) can be written in the following form:  w;tt  2cw;xt  c2 w;xx þ

1 1 Mx;xx þ ;x w;x þ  hz 

þ ðNx0 þ N1 cosðtÞÞw;xx þ

(6:9)

1  w;xx ¼ 0 

The bending moment Mx is related to the displacement w by: Mx ¼ E Jz w;xx  Jz  w;xxt :

(6:10)

where Jz – equivalent inertial moment of the cross-section. Using (6.7), (6.9) and (6.10), one receives the nonlinear equation of the viscoelastic beam model with the K-V element: w;tt þ 2cw;xt þ c2 w;xx þ 

E Jz Jz  P0 þ P1 cosðtÞ w;xxxx þ wxxxxt  w;xx þ  Az  Az  Az

3E 2   w w;xx  2 ðw;x w;xt w;xx þ c w;x w2;xx Þ  ðw2;x w;xxt þ c w2;x w;xxx Þ ¼ 0: 2  ;x   (6.11)

γ

E

Fig. 6.2 Kelvin-Voigt model

x

120

6 Beam Model of the Moving Viscoelastic Web

The boundary conditions are as follows: w ð0; tÞ ¼ wðl; tÞ ¼ 0 ;

w;xx ð0; tÞ ¼ w;xx ðl; tÞ ¼ 0:

(6:12)

Let the dimensionless parameters be: w z¼ ; hz

sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi Az  cw t P0 ¼ ; ; t¼t P0 l l Az  sffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi Az  P0 : ! ¼ l ; cw ¼ P0 Az 

x c ¼ ; s¼ ¼c l cw

(6:13)

The substitution of (6.13) into (6.11) and (6.12) gives the dimensionless nonlinear equation of the viscoelastic beam model motion: z;tt þ 2 s z;t þ ð s2  1   cosð! tÞÞ z; þ " z; þ  z;t þ 

3  z2; z;  s ð2 z2; z; þ z2; z; Þ  ð2 z; z;t z; þ z2; z;t Þ ¼ 0 2

(6:14)

where ¼

l3

Jz  E Jz pffiffiffiffiffiffiffiffiffiffiffiffi ; " ¼ Pl 2 P Az

¼

E h2z Az  h2z Az P1 ; ¼ 3 pffiffiffiffiffiffiffiffiffiffiffiffi : ; ¼ 2 Pl P0 l P Az

(6:15)

The boundary conditions in the dimensionless form: z ð0; tÞ ¼ zð1; tÞ ¼ 0 ; z; ð0; tÞ ¼ z; ð1; tÞ ¼ 0

(6:16)

Equation (6.14) together with boundary conditions (6.16) constitute the mathematical model of the beam with the K-V element.

6.1.2 Poynting-Thompson Model of Material Generally, the one-dimensional constitutive equation of a differential-type material obeys the relation: s ¼  e;

(6:17)

where  and  are differential operators defined as: ¼

R X j¼0

aj

P X dj dj ;  ¼ bj j : j dt dt j¼0

(6:18)

6.1 Nonlinear Beam Model of the Viscoelastic Web

121

The Poynting-Thompson (P-T) rheological model is shown in Fig. 1.1b. For this three-parameter model, the differential constitutive equation is: a1 ;t þ a0  ¼ b1 ";t þ b0 ";

(6:19)

a1 ¼ 1 ; a0 ¼ ðE1 þ E2 Þ=  b1 ¼ E1 ; b0 ¼ E1 E2 =;

(6:20)

where

In this case, the bending moment Mx is given as: Mx ¼ E0 Jz w;xx  Jz  w;xxt :

(6:21)

where E0 ¼

E1 E2 : E1 þ E2

Taking into account (6.21), the governing equation (6.4) has the following form: w;tt þ 2 c w;xt þ c2 w;xx þ

Jz E0 Jz  1 w;xxxx þ w;xxxxt  ðN w;x Þ;x ¼ 0:  hz  hz  hz

(6:22)

To obtain a mathematical description of the viscoelastic beam model, one should multiply (6.22) by the operator . Using (6.5) and taking into account the dimensionless parameters (6.13), one receives: z;ttt þ 3 s z;tt þ ð3 s2  1   cosð!tÞÞ z;t þ ð s2  1   cosð!tÞÞs z; þ   þ g1 z;tt þ 2 g1 s z;t þ g1 s2  1   cosð!tÞ z; þ g2 z; þ g3 z;t þ 3 þ g4 z; þ g5 z;tt þ g5 s z;t  g6 z2; z;  g7 s ð2 z2; z; þ z2; z; Þ 2 2  g7 ð2 z; z;t z; þ z; z;t Þ ¼ 0; (6.23) where g1 ¼

E2 l ðE1 þ E2 ÞJz E2 ðE1 þ E2 ÞJz E2 Jz P1 ; g2 ¼ ; g3 ¼ þ ; ¼ ; 3 2 2 2 2 cw  A l cw  Al cw  A l cw P0

g4 ¼

ðE1 þ E2 ÞJz Jz E1 E2 h2z ðE1 þ E2 Þ h2z ; g5 ¼ ; g6 ¼ ; g7 ¼ ; 2 2 3 3  A z l cw  A z l cw  l cw  l 2 c2w

Az ¼ hz b:

(6.24)

Equation (6.23) together with boundary conditions (6.16) constitute the mathematical model of the beam with the P-T element.

122

6 Beam Model of the Moving Viscoelastic Web

6.1.3 Solution to the Problems The problems represented by (6.14) in the case of the beam model with the K-V element and by (6.23) in the case of the beam model with the P-T element, together with boundary conditions (6.16), have been solved using the Galerkin method. The following finite series representation of the dimensionless transverse displacement has been assumed: zð; tÞ ¼

n X

sinðip Þ qi ðtÞ;

(6:25)

i¼1

where qi(t) is the generalized displacement. The 4-term finite series representation of the dimensionless transverse displacement of the beam has been taken in the numerical investigations. The even order truncations are recommended because the gyroscopic coupling in the mathematical model is taken into consideration. Substituting (6.25) into the governing equations and using the orthogonality condition, one determines a set of ordinary differential equations. For n= 4, ordinary differential equations of the viscoelastic beam model with the Kelvin-Voigt rheological element are presented in Appendix C (C.1). A set of ordinary differential equations of the viscoelastic beam model with the PoyntingThompson rheological element is shown in the same Appendix C (C.2). To analyze the dynamic behavior of the considered system, the set of ordinary differential equations has been integrated. Poincare maps and bifurcation diagrams are modern techniques used to analyze non-linear systems. These maps are convenient tools to identify the dynamic behavior, especially chaos. In bifurcations diagrams, the dynamic behavior may be viewed globally over a range of parameters values and compared simultaneously with various types of motions. The Poincare maps and bifurcation diagrams have been determined for the non-dimensional displacement of the center of the moving beam in the following form:   1 v ; iT ¼ q1 ðiTÞ  q3 ðiTÞ (6:26) 2 where T = 2p/o; i=1, 2, 3 . . . The numerical investigations have been carried out for the beam model of the liner paper web. The fourth-order Runge-Kutta method has been used to integrate ordinary differential equations and to analyze the dynamic behavior of the system. The bifurcation diagrams are presented by varying the dimensionless parameters: the transport speed s, the amplitude of the tension periodic perturbation , and the internal damping coefficient  or g5, while the dimensionless frequency of the periodic perturbation o = p is kept constant. For each set of parameters, the first 2000 points of the Poincare map have been discarded in order to exclude the transient vibration and the displacement of the next 100 points have been plotted on the bifurcation diagrams.

6.2 Investigations Results of the Model with the K-V Element

123

6.2 Investigations Results of the Model with the K-V Element 6.2.1 Linearized System At first, the linearized beam model of the system was investigated. To show the dynamic behavior of the web natural damped oscillations of the dimensionless displacement v given by (6.26) for different values of axial speed s of the beam model were investigated. In subcritical region of transport speeds (s< scr) one can observe free flexural damped vibrations around trivial equilibrium position (Fig. 6.3). In supercritical transport speeds (s > scr) for small internal damping the web experiences divergent instability and next flutter instability (Fig. 6.4). The location of instability regions of the linearized system (C.1) with the K-V model of axially moving material is shown in Fig. 6.5.

(a)

(b)

0.15

0.5

0.075

v 0.25

0

0

– 0.075

– 0.25

v,t

t – 0.15 – 0.5

– 0.25

0

0.25

v 0.5

– 0.5

0

75

150

225

300

Fig. 6.3 The phase portrait (a) and time history (b) of the solution of the linearized system (C.1), (s=0.95, b = 510–4)

(b)

(a) 12

50

v,t 6

v 25

0

0

–6

–25

–12 –50

v –25

0

25

50

–50

t 0

125

250

375

500

Fig. 6.4 The phase portrait (a) and time history (b) of the solution of the linearized system (C.1), (s=1.04, b = 510–5)

124 Fig. 6.5 Instability regions of the linearized beam model with the K-V element

6 Beam Model of the Moving Viscoelastic Web 1,00E–02

β

Divergence instability region

1,00E–03 Flutter instability region

Stability region 1,00E–04

s

scr = 1.002

1,00E–05 0,6

0,8

1

1,2

1,4

6.2.2 Non-Linear System At first, a parametrically unexcited system was investigated ( = 0). To show the dynamic behavior of the web, a bifurcation diagram of the dimensionless displacement v given by (6.26) is presented in Fig. 6.6. The dimensionless transport speed s has been used as the bifurcation parameter. One can observe a supercritical pitchwork-type bifurcation at the transport speed s = sb1 = 1.01. For s < scr , only one attractor exists (v = 0) and for s > scr this critical point becomes a repeller and one can observe two attractors (non-zero critical points). It is worth to note that dimensional bifurcation speed of the beam model (cb1 = 15.22 m/s) is greater than the dimensional bifurcation speed of the liner paper web predicted by the non- linear plate model (ccr = 15.11 m/s). Though the analysis of the linearized system predicts exponentially growing oscillations in the parametrically unexcited system, non-linear damped oscillations which tend to the stable critical point occur (Fig. 6.7). If the transport speed is further increased at sb2 = 1.08 then a Hopf-type bifurcation occurs. This bifurcation leads to the appearance a limit cycle motion (Fig. 6.8). It is worth to note that this region of transport speed covers the flatter instability regions of both the linearized beam model and the linearized plate model. The Poincare map of the dimensionless displacement v in this region has a regular form, which is shown in Fig. 6.9. If the transport speed is further increased at sb3 = 1.18 then the third bifurcation occurs. At the transport speeds above the bifurcation point, the parametrically unexcited non-linear system exhibits a global irregular motion.

10

v 5 0 –5

Fig. 6.6 Bifurcation diagram ( =5  10–4;  = 0)

–10 0.5

sb1 sb2

0.68

0.85

1.03

s 1.21

6.2 Investigations Results of the Model with the K-V Element (a)

(b)

v,t

v

125

0.8 0.08

0.55

0.04

0.3

0

0.05

v

–0.04 –0.2

0.05

0.3

0.55

–0.2

0.8

t 0

150

300

450

600

Fig. 6.7 The phase portrait (a) and time history (b) of the solution of the nonlinear system (C.1), (s=1.05,  = 510–4) (a)

(b)

2

2.5

v,t

v

1

1.75

0

1

–1

0.25

–2 –0.5

t

v 0.25

1

1.75

–0.5

2.5

0

25

50

75

100

Fig. 6.8 The phase portrait (a) and time history (b) of the solution of the linearized system (C.1), (s=1.15, =510–4) 2

v,t 1 0 –1

Fig. 6.9 Poincare map (s= 1.15;  =510–4;  = 0)

v

–2 0

0.56

1.13

1.69

2.25

Next, the non-linear parametrically excited system was investigated. A bifurcation diagram of the dimensionless displacement v versus the dimensionless transport speed s for the specific values of the parameters  and  is shown in Fig. 6.10. Only the stable region of transport speed (s < sb1) of the parametrically unexcited system has been considered in these investigations.

126

6 Beam Model of the Moving Viscoelastic Web

Fig. 6.10 Bifurcation diagram ( =510–4;  = 0.5)

8

v 4 0 –4

s

–8 0.53

(a)

0.65

0.77

0.9

1.02

(b)

0.25

0.5

v,t

v,t

0.25

0.13

0

0

–0.25

–0.13

–0.5 –0.25

v –0.13

0

0.13

–0.25

0.25

v 0

100

200

300

400

Fig. 6.11 The phase portrait (a) and time history of the solution of the nonlinear system (C.1), (s = 0.798;  = 0.5;  =510–4)

At the transport speed s = 0.795, much lower than in the unexcited case, a Hopf-type bifurcation of the zero critical point occurs. This bifurcation leads to the appearance of the limit cycle motion. Figures 6.11 and 6.12 show a phase portrait and Poincare map of the system motion in this region of the transport speed. This limit cycle coexists with the zero critical point in the range 0.795 < s < 0.92. 0.5

v,t 0.25 0 –0.25

Fig. 6.12 Poincare map (s = 0.798;  = 0.5;  =510–4)

–0.5 –0.4

v –0.2

0

0.2

0.4

6.2 Investigations Results of the Model with the K-V Element Fig. 6.13 Poincare map (s=0.99; =0.5;  =510–4)

127

20

v,t 10 0 –10 –20 –8

v –4

0

4

8

If the transport speed is further increased (s > 0.92), the limit cycle motion coexists with a chaotic motion. The Poincare map in Fig. 6.13 shows the dynamic behavior of the investigated system in this region of the transport speed. The bifurcation of the dimensionless displacement v given by (6.26) against the non-dimensional internal damping coefficient b for fixed s = 0.9,  = 0.5 is shown in Fig. 6.14. When the damping coefficient has been taken as the bifurcation parameters one can observe dynamic behavior characteristic for non-linear systems which are determined in multi-dimensional phase space. In this case, for small values of internal damping (b < 10–4), irregular motion occurs. The Poincare map in Fig. 6.15 shows the dynamic behavior of the investigated model in this region of internal damping. With an increase in b, one can observe a direct transition from a chaotic to quasi-periodic attractor (Fig. 6.16). In the region of damping b > 3104, the quasi-periodic attractor coexist with periodic attractor. Figure 6.17 shows the Poincare map of a period 6-motion. Six points represent six periodic orbits in the bifurcation diagram in Fig. 6.14. At b = 310–3 the inverse Hopf bifurcation occurs and the system is asymptotically stable with its response tending to zero.

10

v 5

0

–5

Fig. 6.14 Bifurcation diagram (s = 0.9;  = 0.5)

–10 1.10–4

β 1.10–3

0.01

128 Fig. 6.15 Poincare map (s = 0.9;  = 0.5;  =1.810–4)

6 Beam Model of the Moving Viscoelastic Web 35

v,t 17.5

0

–17.5

–35

v –5

0

5

–0.5

0

0.5

–0.5

0

0.5

1.5

v,t 0.75

0

–0.75

Fig. 6.16 Poincare map (s=0.9; =0.5; =3.910–4)

–1.5

v

1.5

v,t 0.75

0

–0.75

Fig. 6.17 Poincare map (s = 0.9;  = 0.5;  =7.810–4)

–1.5

v

The bifurcation diagram in Fig. 6.18a shows Poincare maps of the dimensionless displacement v given by (6.26) against the perturbation amplitude  for fixed s= 0.95 and b = 510–4. In this case, the system is asymptotically stable with its response tending to zero for  < 0.2. At the perturbation amplitude b1 = 0.2, the zero critical point looses its stability and a Hopf-type bifurcation

6.3 Investigations Results of the Model with a P-T Element (a)

(b)

3

2 v,t 1

0

0

6

v

–6

0

αb2

0.13 0.25 0.38

(c)

2

v,t

1 0

–1

–1

α –2

v –2

–3

αb1

129

0.5

–0.5 –0.25

0

0.25

0.5

–4

v –2

0

2

4

Fig. 6.18 Bifurcation vs. the amplitude of the periodic perturbation (s = 0.95; b =510–4), (a) bifurcation plot, (b) Poincare map ( = 0.38), (c) Poincare map ( = 0.39)

occurs. With an increase in , a quasi-periodic motion appears (Fig. 6.18b). At the perturbation amplitude b2 = 0.39, an explosive bifurcation occurs and an irregular motion appears (Fig. 6.18c).

6.3 Investigations Results of the Model with a P-T Element A three-parameter Poynting-Thompson rheological model was used to identify rheological parameters of the liner paper under consideration. The identification method has been presented in Chap. 1. The determined values of rheological parameters of the analysed paper along the machine direction have been employed in dynamic investigations of its beam model.

6.3.1 Linearized System The stability and instability regions of the linearized system (C.2) in the form of a stability boundary map in the internal damping – transport speed area are shown in Fig. 6.19. The boundaries have been calculated for three amplitude values of the tension periodic perturbation:  = 0,  = 0.25,  = 0.5. The analysis of the linearized system predicts exponentially growing oscillations in 1,00E-01

g5 1,00E-02

Stability region

1,00E-03

α=0

1,00E-04 0

0,1

0,2

0,3

0,4

s 0,5

0,6

0,7

Fig. 6.19 Stability boundaries of the linearized system (C.2); . . .  = 0.25, – – –  = 0.5

130

6 Beam Model of the Moving Viscoelastic Web

the supercritical region of transport speed. The maximum value of the dimensionless damping coefficient g5 that corresponds to the maximum value of the rheological parameter g that fulfills identification condition (see Sect. 1.1.1) has been denoted by the horizontal dash-dot line. Above this value of damping, the error made during the identification procedure exceeds the assumed accuracy limit.

6.3.2 Nonlinear System At first, a parametrically unexcited nonlinear system was investigated ( = 0). To show the dynamic behaviour of the beam, a bifurcation diagram of the dimensionless displacement v given by (6.26), Poincare maps, a phase portrait and a time history for g5 = 1.0  10–2 are presented in Figs. 6.20, 6.21 and 6.22, respectively. The dimensionless transport speed s has been used as the bifurcation parameter. To obtain Poincare maps in this parametrically unexcited case, the dimensionless fundamental natural frequency o0 = 0.63 has been taken into consideration. In Fig. 6.20 one can observe a supercritical Hopf-type bifurcation at the transport speed s = sb1 = 0.53. Though the analysis of the linearized 7.5

v 3.75 0 –3.75

Fig. 6.20 Bifurcation diagram: g5 = 10–2,  = 0

–7.5 0.5

sb1

sb2

s

0.53

0.56

0.6

0.63

–0.5

0

0.5

1

5

v,t 2.5 0 –2.5

Fig. 6.21 Poincare map: g5 = 10–2,  = 0, s= 0.54

–5

v –1

6.3 Investigations Results of the Model with a P-T Element (a) 4 v,t

131

(b) 0.6

2

v 0.3

0

0 –0.3

–2

v –4 –0.6

–0.3

0

0.3

t –0.6

0.6

0

25

50

75

100

Fig. 6.22 Phase portrait (a) and time history (b) of the solution of (C.2); s= 0.54, g5 = 0.01,  =0

system predicts exponentially growing oscillations for s > scr, nonlinear damped oscillations that tend to the stable limit cycle motion occur. The Poincare map in Fig. 6.21 and the phase portrait and time history in Fig. 6.22 show the dynamic behaviour of the nonlinear system in this region of transport speed for the initial conditions close to zero. If the transport speed is increased further at sb2 = 0.58, the second bifurcation occurs. The small quasi-periodic attractor presented in Fig. 6.23a transforms into a larger attractor shown in Fig. 6.23b. At transport speeds above the bifurcation point, the parametrically unexcited non-linear system exhibits a global motion between two center points. The Poincare maps in Fig.6.23b and the phase portrait in Fig. 6.24 show the dynamic behaviour of the nonlinear system in this region of transport speed. With an increase in s one can observe a direct transition from quasi-periodic attractor to chaotic attractor. Next, a nonlinear parametrically excited system was investigated. A bifurcation diagram of the dimensionless displacement v versus the dimensionless transport speed s for the specific amplitude value of the tension periodic perturbation  = 0.25 and the internal damping coefficient g5 = 10–2 is shown in Fig. 6.25. In this case, the system is asymptotically stable with its

(b)

(a) 10

15

v,t

v,t 5

7.5

0

0

–5

–7.5

–10 –1

v –0.5

0

0.5

1

–15

–5

v –2.5

Fig. 6.23 Poincare map: g5 = 102,  = 0, (a) s= 0.56, (b) s= 0.585

0

2.5

5

132

6 Beam Model of the Moving Viscoelastic Web

(a)

(b) 4

15

v,t

v 7.5

2

0

0

–7.5

–2

–15 –5

v –2.5

0

2.5

–4

5

t

0

100

200

300

400

Fig. 6.24 Phase portrait (a) and time history (b) of the solution of (B.2); s= 0.585 g5 = 0.01,  = 0

50

v 25

0

–25 sb1

Fig. 6.25 Bifurcation diagram: g5 = 102,  = 0.25

–50

0.4

0.41

0.42

Sb2 0.43

0.44

s 0.45

response tending to zero for s < 0.425. Figure 6.26 shows a Poincare map of the system behavior in the region of transport speed above the bifurcation point. If the transport speed is increased further (sb2 = 0.445), the second Hopf bifurcation occurs and a chaotic motion appears. 4 v,t

2 0 –2

Fig. 6.26 Poincare map: g5= 10–2,  = 0.25, s= 0.444

–4 –2.5

–1.25

0

1.25

v 2.5

6.3 Investigations Results of the Model with a P-T Element Fig. 6.27 Bifurcation diagram:  = 0.1, s= 0.53

133

180 v

90 0 –90 g5

–180 1.10–4

1.10–3

0.01

A bifurcation diagram of the dimensionless displacement v given by (6.26) against the non-dimensional internal damping coefficient g5 for  = 0.1 and the axial speed s = 0.53 is shown in Fig. 6.27. For small values of internal damping (g5 < 2.810–4), a quasi-periodic motion occurs. Figure 6.28 shows a Poincare map of the system behavior in this region. With an increase in internal damping at g5 = 2.810–4, a Hopf-type bifurcation occurs and a quasi-periodic and periodic motion occurs (Fig. 6.29). Finally, an inverse Hopf-type bifurcation 5 v,t

2.5 0 –2.5

Fig. 6.28 Poincare map: g5=1.3310–4,  = 0.1, s= 0.53

v

–5 –100

(a)

50

0

50

100

(b)

10

10

v,t

v,t

5

5

0

0

–5

–5 v

–10 –50

–25

0

25

50

v

–10 –5

–2.5

0

2.5

Fig. 6.29 Poincare map: s= 0.53,  = 0.1, (a), g5 ¼ ~ 337  103 (b) g5 ¼ ~ 445  103

5

134

6 Beam Model of the Moving Viscoelastic Web

occurs at g5 = 6.010–3 and the system is asymptotically stable with its response tending to zero. Finally, dimensionless amplitude of the tension periodic perturbation  has been used as the bifurcation parameter. For small values of this bifurcation parameter and small values of transport speed, the system is stable with its response is tending to zero. The bifurcation diagram in Fig. 6.30 shows Poincare maps of the dimensionless displacement v against the perturbation amplitude  for the transport speed value s= 0.4 and the dimensionless damping g5 = 10–2. One can observe a supercritical pitchwork-type bifurcation of the zero critical point at the tension periodic perturbation b1 = 0.37. This bifurcation leads to the appearance of a limit cycle motion (Fig. 6.31). If the tension perturbation is increased further at b2 = 0.46, the second Hopf-type bifurcation occurs. At the perturbations above the bifurcation point, at first the system exhibits a local motion and, next, a global motion between two center points. The Poincare maps in Fig. 6.32 show the dynamic behaviour of the non-linear system in this region of transport speed. At b3 = 0.513, an explosive bifurcation occurs and a chaotic motion appears.

50

v 25

0

–25

Fig. 6.30 Bifurcation diagram: g5 =10–2, s= 0.4

–50 0.3

v,t

αb1

αb2

αb3

α

0.36

0.41

0.47

0.53

–1

0

1

2

3 1.5 0

–1.5 Fig. 6.31 Phase portrait: g5 =10–2,  = 0.4, s= 0.4

–3 –2

v

6.4 Final Remarks

135

(a) 3

(b)

v,t

v,t

4

1.5

2

0

0

–1.5

–2

v –3 –5

–2.5

0

2.5

5

v –4 –5

–2.5

0

2.5

5

Fig. 6.32 Poincare maps: s= 0.4, g5 =10–2, (a)  = 0.498, (b)  = 0.512

6.4 Final Remarks The present chapter discusses the results of dynamic investigations of the axially moving web of the liner paper obtained by means of the beam model. For comparison reasons, two rheological models, namely: the two-parameter Kelvin-Voigt model, which has been most often used in the investigations so far, and the three-parameter Poynting-Thompson model, have been employed to describe the web material properties. The generated mathematical models in the form of differential equations with partial derivatives are of different orders. The model with a K-V element is described with the second order equation, whereas the model with a P-T element is described with the third order equation. In both cases, the equations have been discretized to find approximate solutions by means of the Galerkin method. The dynamic behavior of the systems has been analyzed through direct integration of a set of ordinary differential equations. While comparing the dynamic behavior of the nonlinear beam model with a K-V element and the nonlinear plate model presented in Chap. 3, we can see that the transport speed at which a pitchwork-type bifurcation of the zero critical point occurs is slightly higher than predicted by the plate model with respect to the bifurcation of the static equilibrium position. A similar dependence has been also found in the investigations of linear models [2]. The second difference follows from the fact that the plate model is an undamped model and here a node-saddle type bifurcation has occurred. Both nonlinear models describe a similar dynamic behavior of the web within the range of overcritical transport speeds. In the case of parametric excitation of the beam model with a K-V element generated by an alternation in the tension force, a bifurcation of the zero solution occurs much earlier than the linear model predicts. First, a quasiperiodic motion appears, and then with an increase in the transport speed, an irregular chaotic motion takes place.

136

6 Beam Model of the Moving Viscoelastic Web

Another scenario of the dynamic behavior of the liner paper web has been obtained as a result of an application of the nonlinear beam model of the web with a P-T element. The basic difference lies in the fact that this model predicts an appearance of the bifurcation at the zero equilibrium position for the transport speed slightly higher that 50% of the speed of transverse wave propagation in the web. As a result of the Hopf bifurcation, a quasiperiodic motion of the system occurs then. With an increase in transport speed, this motion changes into a chaotic one at the transport speed equal to approximately 65% of the web wave speed. The scenario of the dynamic behavior of the nonlinear beam model with a P-T element at the parametric excitation generated by changes in tension force is similar to that one predicted by the previously discussed model with a K-V element. The basic difference lies in the fact that the above described phenomena appear at values of transport speeds that are half lower. Depending on the parameter under analysis, first a Hopf bifurcation or a pitchwork-type bifurcation leads to a quasi-periodic motion and then, with an increase in transport speed, to a chaotic motion.

References 1. Hwang S J, Perkins N C (1994) High speed stability of coupled band/wheel systems: Theory and experiment. Journal of Sound and Vibration 169(4): 459–483 2. Lin C C (1997) Stability and vibration characteristics of axially moving plates. International Journal of Solid and Structures 34(24): 3179–3190 3. Moon J, Wickert J A (1997) Non-linear vibration of power transmission belts. Journal of Sound and Vibration 200(4): 419–431 4. Pelicano F, Fregolent A, Bertizzi A, Vestroni F (2001) Primary parametric resonances of a power transmission belt: theoretical and experimental analysis. Journal of Sound and Vibration 224(4): 669–684

Chapter 7

Concluding Remarks

The results of analyses and investigations discussed in the book allow for formulating numerous conclusions and final remarks that are presented below.

 The up-to-date survey of the knowledge on dynamic analyses of axially moving systems in Chap. 2 allows for stating that this subject belongs to one of the most currently developed in scientific centers leading in the field of dynamic investigations. It is despite the fact that dynamic investigations of axially moving systems have been already conducted for more than half a century. In the early stage, mainly one- and two-dimensional systems (strings, beams) were the object of interest for scientists. The dynamic behavior of plate systems with viscoelastic properties, which include paper webs, has been the least recognized so far. In the literature already published, only isotropic systems have been analyzed in practice. The Author of this book has aimed at filling this gap to some extent at least.  The mathematical model derived in Chap. 3, which describes transverse vibrations of the axially moving orthotropic web, includes the most important factors that exert an influence on the system dynamic behavior. The following can be mentioned among these factors: transport speed, web geometry, magnitude and distribution of axial stresses, and nonlinear geometrical relationships between web strains and displacements. This last factor has caused that the derived mathematical model is a nonlinear one. It describes the transverse motion and the field of web cross-sectional forces, and takes a form of a system of two coupled nonlinear differential equations with partial derivatives. The coupling of the equations takes place through a variable that determines the web transverse displacement and the Airy stress function. Because it is impossible to find an exact solution to this system of equations, it has been decided to determine its approximate solution. In seeking this solution, an introduction of two systems of orthogonal functions rendering the conditions along longitudinal, unloaded edges of the web has been of principal importance. It has been assumed that these functions are approximated by a series of functions of the linearized system, which is described by the linear part of the mathematical model. An application of the numerical method of matrix transition, the Godunov orthogonalization K. Marynowski, Dynamics of the Axially Moving Orthotropic Web, DOI: 10.1007/978-3-540-78989-5_7, Ó Springer-Verlag Berlin Heidelberg 2008

137

138









7 Concluding Remarks

method and the Galerkin orthogonalization procedure has resulted finally in the formulation of a system of ordinary differential equations, whose solution allows for determination of the displacements sought. The dynamic investigations of the undamped linearized system have shown that for the paper web under constant initial axial tension in the whole transport speed range under consideration, flexural vibrations decide about dynamic behaviors. The dynamic behavior of the paper web in the subcritical range is characterized by the fact that lowest eigenvalues of flexural vibrations are situated close to one another. With an increase in transport speed, these values decrease to vanish eventually at the critical speed. The range of critical transport speeds corresponding to the divergenttype instability of the five lowest modes of flexural vibrations does not exceed the value of 3 m/s. In the nonlinear system, instead of an unlimited increase in the web deflection in the divergence instability region of the linearized system, one can observe oscillations of transverse displacements around a new equilibrium position. Then, an increase in transport speed is followed by an increase in the amplitude of nonlinear vibrations. From the point of view of engineering applications, the calculation model presented in Chap. 4 is the most practical in dynamic calculations of the axially moving paper web. In this model, a static analysis based on the determination of existence of nontrivial equilibrium positions of the web was employed in the stability investigations of the web motion. It is important that orthotropic properties of the web can be considered in this model. For the one-layered paper webs under investigation, the results of static analyses complied with the results of dynamic analyses conducted on the basis of the much more complex dynamic model presented in Chap. 3. In the case of the three-layered corrugated board, the values of the critical speed obtained from the static model were approx. by 1/3 higher than from the dynamic model. It might not result from the imperfection of the calculation method itself. Comparative studies with the available solutions show good accuracy. The identification accuracy of parameters of the corrugated board treated as a composite structure exerts a considerable influence on the results. Experimental investigations should verify these theoretical results. The results of investigations of steady motion of the web linearized system equipped with spread rolls point out to the fact that an increase in transport speed is followed by a decrease in the amplitude value of axial loading that causes wrinkling of the web. It is synonymous with stating that an increase in transport speed of the web, the spread roller with lower and lower curvilinearity of its surface area may result in the web surface wrinkling. The linear mathematical model presented in Chap. 5 takes into account viscoelastic properties of the web material along both the main directions of orthotropy. Rheological properties have been described in this model with a Zener-type three-parameter element. The results of the numerical calculations show that if internal damping is considered in the axially moving liner paper web, then its critical speed decreases. A level of this decrease depends

7 Concluding Remarks

139

mainly on the magnitude of the initial longitudinal tension load of the web. The higher the tensioning load, the lower the degree the critical speed is decreased due to internal damping.  The nonlinear beam model of the axially moving web has been presented in this monograph mainly because various rheological models of the web material can be taken easily into account in it. The results of investigations including two rheological models, namely: the two-parameter Kelvin-Voigt model and the three-parameter Poynting-Thompson model, have been discussed. The results show that if viscoelastic properties of the web material are considered, then they exert a considerable effect on the dynamic behavior of the system under investigation. The results of dynamic investigations of the paper web with the three-parameter model differ much from the results obtained with the two-parameter Kelvin-Voigt model. It confirms the thesis that the latter model, despite the fact that it is employed eagerly by various researchers due to its simple structure, does not render the viscoelastic properties of paper sufficiently. Using the least complex mathematical procedures, the three-parameter rheological model should be used at least to describe the dynamic behavior of the web with viscoelastic properties, according to the Author’s opinion.  The Author is conscious that the presented results do not deplete the description of a very wide and complex issue of the dynamic behavior of the axially moving orthotropic web. The main attention has been focused on the presentation of the up-to-date state of investigations conducted in this field, which draws unceasing interest of researchers in numerous scientific centers located in distant parts of the world. Attention has also been focused on the exact presentation of various mathematical models that can be employed in dynamic investigations of such objects. The models differ considerably with respect to the complexity of the mathematical description and the aim of investigations the model is to be used for has to be analyzed thoroughly before their application.  The results of analysis of the dynamic behavior of the axially moving orthotropic material web allows for determination of further directions of dynamic investigations in this field. As far as the orthotropic web is concerned, in the Author’s opinion, analyses should be extended by investigations comprising nonlinearity in the two-parameter model. It concerns both the geometrical nonlinearity, as well as that one which follows from the physical properties of the moving material. The results obtained so far show that the effect of nonlinearity on the behavior of the web obtained during the investigations is very strong indeed. Also, the consideration of internal damping of the support system of web guide rollers can affect significantly the system dynamic behavior. The future experimental investigations of the system for rewinding a wide material orthotropic web may determine more accurately the practical applicability range of the analyses conducted than the results of the comparative studies presented in this book.

Appendix A

For the liner paper web (l = b = 1 m, Nx0 = 50 N/m, M = 4, N = 1), the discretized mathematical model of flexural vibrations has the following form:

q€1 ¼ 9:8696 c2 q1  2251:4873 q1 þ 5:1678cq_ 2 þ 2:0025cq_ 4  0:0345q31 þ 0:0565q21 q3  0:3726q1 q22 þ 0:2807q1 q2 q4  0:6462q1 q23  1:1737q1 q24  0:7125q22 q3  2:561q2 q3 q4 þ 1784:2645a sinð!tÞ; q€2 ¼ 39:4784c2 q2  9107:2604q2  5:5037cq_ 1 þ 9:4312cq_ 3  0:7813q32  0:3725q21 q2 þ 0:1154q21 q4  1:3513q1 q2 q3  2:0785q1 q3 q4  4:2305q2 q23  6:1783q2 q24  5:2049q23 q4 ; q€3 ¼ 88:8265c2 q3  20871:9018q3  9:7716cq_ 2 þ 13:5241cq_ 4 þ 0:0188q31

ðA:1Þ

 4:5119q33  0:5983q21 q3  0:6767q1 q22  2:0421q1 q2 q4  4:0767q22 q3  9:6046q2 q3 q4  19:672q3 q24 þ 6245:7313a sinð!tÞ; q€4 ¼ 157:9137c2 q4  38053:4899q4  2:2722cq_ 1  13:907cq_ 3 þ 0:106q21 q2  0:9228q21 q4  2:0118q1 q2 q3  5:3585q22 q4  4:6749q2 q23  18:5287q23 q4  16:5805q34 For the fluting paper web (l = b = 1 m, Nx0 = 50 N/m, M = 4, N = 1), the discretized mathematical model of flexural vibrations has the following form:

q€1 ¼ 9:8696 c2 q1  4018:1166 q1 þ 5:1534 c q_ 2 þ 1:9907 c q_ 4  0:0145q31 þ 0:0272 q21 q3  0:1656q1 q22 þ 0:1273q1 q2 q4  0:1552q1 q23  0:4947q1 q24  0:3248 q22 q3  1:1398q2 q3 q4 þ 46114:4762 a sinð! tÞ 141

142

Appendix A

q€2 ¼ 39:4784 c2 q2  16111:1523 q2  5:519 c q_ 1 þ 9:4151 c q_ 3  0:336q32  0:167q21 q2 þ 0:0557q21 q4  0:6294q1 q2 q3  0:9698q1 q3 q4  1:8935q2 q23  2:7021q2 q24  2:3713 q23 q4 q€3 ¼ 88:8265 c2 q3  36394:2888 q3  9:7883 c q_ 2 þ 13:5054 c q_ 4  0:0094q31  1:9519q33  0:2709 q21 q3  0:3176q1 q22  0:9668q1 q2 q4  1:8448 q22 q3  4:4656q2 q3 q4  8:7725q3 q24 þ 16213:9038 a sinð! tÞ

ðA:2Þ

q€4 ¼ 157:9137c2 q4  65061:3814 q4  2:2856 c q_ 1  13:9262 c q_ 3 þ 0:0543 q21 q2  0:4111 q21 q4  0:9607q1 q2 q3  2:4188 q22 q4  2:1968q2 q23  8:3687 q23 q4  7:1732q34 For the corrugated board web (l = b = 1 m, Nx0 = 50 N/m, M = 4, N = 1), the discretized mathematical model of flexural vibrations has the following form: q€1 ¼ 9:8696 c2 q1  4104:9611 q1 þ 5:1678 c q_ 2 þ 2:0025 c q_ 4  12:3983q31 þ 20:3247 q21 q3  133:9965q1 q22 þ 100:9449q1 q2 q4  232:3581q1 q23  422:0457q1 q24  256:1892 q22 q3  920:8636q2 q3 q4 þ 719:9215 a sinð! tÞ; q€2 ¼ 39:4784 c2 q2  52838:108 q2  5:5037 c q_ 1 þ 9:4312 c q_ 3  280:9734q32  133:9326 q21 q2 þ 41:5043 q21 q4  485:9227q1 q2 q3  747:4065 q1 q3 q4  1521:2984 q2 q23  2221:6875 q2 q24  1871:6558 q23 q4 ;

ðA:3Þ

q€3 ¼ 88:8265 c2 q3  255699:2956 q3  9:7716 c q_ 2 þ 13:5241 c q_ 4 þ 6:7669q31  1622:5785 q33  215:1327 q21 q3  243:3395 q1 q22  734:3217q1 q2 q4 1466:0261 q22 q3  3453:8061 q2 q3 q4  7074:2398q3 q24 þ 252:007 a sinð! tÞ;

q€4 ¼ 157:9137 c2 q4  795346:0946 q4  2:2722 c q_ 1  13:907 c q_ 3 þ 39:2033 q21 q2  331:827 q21 q4  723:4162 q1 q2 q3  1926:9887 q22 q4  1681:1175 q2 q23  6663:2357 q23 q4  5962:7478 q34 Notations in (A.1), (A.2) and (A.3): qk – k-th generalized coordinate, k = 1, 2, 3, 4, a – amplitude of the web transverse loading, ! – frequency of the web transverse loading.

Appendix B

The governing equation of the moving viscoelastic two-dimensional web in dimensionless form:

@8z @7z @6z @5z @4z @9z þ  ðÞ þ  ðÞ þ  ðÞ þ  ðÞ þ  1 2 3 4 12 @ 8 @ 7 @ 6 @ 5 @ 4 @ 7 @2   @8z   @7z þ s2  1  11 þ 22 ðÞ þ 1 ðÞ s2  2 ðÞ  21 ðÞ þ 32 ðÞ 6 2 @ @ @ 5 @2   @6z   þ 2 ðÞ s2  3 ðÞ  31 ðÞ þ 42 ðÞ þ 3 ðÞ s2  4 ðÞ  41 ðÞ þ 52 ðÞ 4 2 @ @



  @4z @5z @8z @7z 2 þ 2 þ  ðÞ s   ðÞ   ðÞ þ 2 s ðÞ s 4 5 51 1 @ 3 @2 @ 2 @2 @ 7 @ @ 6 @

þ 22 ðÞ s þ"x2 ðÞ

@6z @5z @4z @9z þ 23 ðÞ s 4 þ 24 ðÞ s 3 þ "x1 5 4 5 @ @ @ @ @ @ @ @

ðB:1Þ

@8z @7z @6z @9z @8z þ "x3 ðÞ 3 4 þ "x4 ðÞ 2 4 þ "y1 5 4 þ "y2 ðÞ 4 4 4 4 @ @ @ @ @ @ @ @ @ @

þ "y3 ðÞ

@7z @6z @ 11 z @ 10 z þ " ðÞ   þ ½    ðÞ  y4 11 1 21 @ 3 @4 @ 2 @4 @ 7 @2 @2 @ 6 @2 @2

þ ½2 ðÞ  31 ðÞ

@9z @8z þ ½  ðÞ   ðÞ  ¼ 0; 3 41 @ 5 @2 @2 @ 4 @2 @2

where

1 ðÞ ¼

11 ¼

q5 ðtÞl q4 ðtÞl2 q3 ðtÞl3 q2 ðtÞl 4 Nx0 ; 2 ðÞ ¼ ; 3 ðÞ ¼ ; 4 ðÞ ¼ ; 1 ¼ ; 2 3 4 q6 cw q6 cw q6 cw q6 cw  hc2w

h2 E1x h2 E2x x Nx0 q5 ðtÞ l h2 q5 ðtÞE1x ; 12 ¼ ; 2 ðÞ ¼ ; 21 ðÞ ¼ ; 2 2 3 32  hq6 cw 12 l cw 12 l cw 12 l q6 c3w

143

144

Appendix B

22 ðÞ ¼

h2 q5 ðtÞE2x x Nx0 q4 ðtÞ l 2 h2 q4 ðtÞE1x ;  ðÞ ¼ ;  ðÞ ¼ ; 3 31 12 l 2 q6 c2w  hq6 c4w 12l q6 c4w

32 ðÞ ¼

h2 q4 ðtÞE2x x Nx0 q3 ðtÞ l3 h2 q3 ðtÞE1x l ;  ðÞ ¼ ;  ðÞ ¼ ; 4 41 12 l q6 c5w 12 lq6 c3w  hq6 c5w

42 ðÞ ¼

h2 q3 ðtÞE2x x Nx0 q2 ðtÞ l 4 h2 q2 ðtÞE1x l 2 ;  ðÞ ¼ ;  ðÞ ¼ ; 5 51 12 q6 c4w  hq6 c6w 12l q6 c6w

h2 p2y h2 p1y ðtÞ h2 q2 ðtÞE2x x l ; " ¼ ; " ðÞ ¼ ; x1 x2 12 q6 c5w 12 q6 c4w 12 l q6 c3w    2  d y =dt h2 l d y =dt2 h2 l h2 p2x "x3 ðÞ ¼ ; " ðÞ ¼ ; " ¼ ; x4 y1 6 q6 c5w 12 q6 c6w 12 l q6 c3w  2  d xy =dt2 h2 l h2 p1x ðtÞ ðd x =dtÞh2 l "y2 ðÞ ¼ ; "y3 ðÞ ¼ ; "y4 ðÞ ¼ ; 12 q6 c6w 12 q6 c4w 6 q6 c5w 52 ðÞ ¼

1 ¼

ðB:2Þ

h2 G1 h2 G 2 T G h2 q5 ðtÞG1 h2 q5 ðtÞG2 TG ;  ¼ ;  ðÞ ¼ ;  ðÞ ¼ ; 11 2 21 6 l 2 c2w 6 l 3 cw 6 lq6 c3w 6 l 2 q6 c2w

3 ðÞ ¼ 41 ðÞ ¼

h2 q4 ðtÞG1 h2 q4 ðtÞG2 TG h2 q3 ðtÞG1 l ;  ðÞ ¼ ;  ðÞ ¼ ; 31 4 6 q6 c5w 6 q6 c4w 6 q6 c3w h2 q3 ðtÞG2 TG h2 q2 ðtÞG1 l2 h2 q2 ðtÞG2 TG l ;  ðÞ ¼ ;  ðÞ ¼ : 5 51 6 q6 c5w 6 q6 c4w 6 q6 c6w

The set of ordinary differential equations of motion of the moving viscoelastic two-dimensional web (n = 2):     ð6Þ q1 ¼ p2 4 s2  5  51 þ 1xy 5  p2 "x4  "y4 1y q1     þ p2 3 s2  4  41 þ 52 þ 1xy ð4  51 Þ  p2 "x3  "y3 1y q_ 1     þ p2 2 s2  3  31 þ 42 þ 1xy ð3  41 Þ  p2 "2  "y2 1y  4 q€1     þ p2 1 s2  2  21 þ 32 þ 1xy ð2  31 Þ  p2 "x1  "y1 1y  3 q1    ð4Þ  þ p2 s2  1  11 þ 22 þ 1xy ð1  21 Þ  2 q1     ð5Þ þ p2 12  1  1xy 11  1 q1 þ 24 1c s q_ 2 þ 23 1c s q€2 þ 22 1c s q2 ð4Þ

ð5Þ

ðB:3Þ

þ 21 1c s q2 þ 21c s q2 ; ð6Þ

ð4Þ

ð5Þ

q2 ¼  24 2c s q_ 1  23 2c s q€1  22 2c sq2  21 2c s q1  22c s q1     þ 4p2 4 s2  5  51 þ 2xy 5  4p2 "x4  "y4 2y q2

Appendix B

145

    þ 4p2 3 s2  4  41 þ 52 þ 2xy ð4  51 Þ  4p2 "x3  "y3 2y q_ 2     þ 4p2 2 s2  3  31 þ 42 þ 2xy ð3  41 Þ  4p2 "2  "y2 2y  4 q€2     þ 4p2 1 s2  2  21 þ 32 þ 2xy ð2  31 Þ  4p2 "x1  "y1 2y  3 q€2     ð4Þ þ 4p2 s2  1  11 þ 22 þ 2xy ð1  21 Þ  2 q2     ð5Þ þ 4p2 12  1  2xy 11  1 q2 ; where

R1 R1 ixy ¼

R1 R1

zi1 ð; Þ zi1; ð; Þd d

0 0

R1 R1

; iy ¼ z2i1 ð;

R1 R1

Þ d d

0 0

R1 R1 1c ¼

0 0

0 0

i ¼ 1; 2:

; z2i1 ð;

Þ d d

0 0

R1 R1

z11 ð; Þ z21; ð; Þd d R1 R1

zi1 ð; Þ zi1; ð; Þd d

0 0

; 2c ¼ z211 ð;

Þ d d

ðB:4Þ

z21 ð; Þ z11; ð; Þd d

0 0

R1 R1 0 0

: z221 ð;

Þ d d

Appendix C

For n = 4 the ordinary differential equations of the viscoelastic beam model with Kelvin-Voigt rheological element are given: q€1 ¼ ð s2  1   cos !Þ p2 q1  " p4 q1 þ ð16=3Þs q_ 1 þ ð32=15Þ s q_ 4   p4 q_ 1  3  p4 ½ð3=8Þq21 q3 þ ð1=8Þq31 þ q1 q22 þ 2 q1 q2 q4 þ ð9=4Þq1 q23 þ 6 q2 q3 q4 þ 4 q1 q24 þ ð3=2Þ q22 q3  þ 2 p4 s ½ð704=105Þ q32 þ ð96256=2145Þ q34 þ ð56=15Þ q21 q2 þ ð1016=105Þ q21 q4 þ ð144=7Þ q1 q2 q3 þ ð7936=315Þ q22 q4  ð160=3Þ q1 q3 q4 þ ð1496=35Þ q2 q23 þ ð4432=55Þ q23 q4 þ ð20086528=266805Þ q2 q24   2  p4 ½ð3=4Þ q1 q3 q_ 1 þ 2 q1 q4 q_ 2 þ ð3=8Þ q21 q_ 1 þ 2 q1 q2 q_ 4 þ ð3=8Þq21 q_ 3 þ 2 q2 q4 q_ 1 þ 2 q1 q2 q_ 2 þ 3 q2 q3 q_ 2 þ ð3=2Þ q22 q_ 3 þ 4 q24 q_ 1 þ 6 q2 q4 q_ 3 þ 6 q2 q3 q_ 4 þ 6 q3 q4 q_ 2 þ ð9=2Þ q1 q3 q_ 3 þ 8 q1 q4 q_ 4 þ q22 q_ 1 þ ð9=4Þ q23 q_ 1  q€2 ¼ 4ð s2  1   cos !Þ p2 q2  16 " p4 q2  ð16=3Þs q_ 1 þ ð48=5Þ s q_ 3  16  p4 q_ 2  3  p4 ½q21 q2 þ 2 q32 þ 9 q2 q23 þ 3 q1 q2 q3 þ q21 q4 þ 6 q1 q3 q4 þ 16 q24 q2 þ 9 q23 q4  þ 2 p4 s ½ð4=15Þ q31 þ ð22356=385Þ q33 þ ð784=15Þ q22 q3 þ ð976=105Þ q1 q22 þ ð508=35Þ q1 q23 þ ð468=35Þ q21 q3 þ ð3776=63Þ q1 q2 q4 þ ð81728=3465Þ q1 q24 þ ð107712=605Þ q2 q3 q4 þ ð1219616=4095Þ q3 q24   2  p4 ½3 q1 q2 q_ 3 þ 3 q1 q3 q_ 2 þ 6 q22 q_ 2 þ 2 q1 q4 q_ 1 þ q21 q_ 4 þ 3 q2 q3 q_ 1 þ 2 q1 q2 q_ 1 þ 18 q2 q3 q_ 3 þ q21 q_ 2 þ 9 q23 q_ 2 þ 6 q1 q4 q_ 3 þ 6 q1 q3 q_ 4 þ 6 q3 q4 q_ 1 þ 18 q3 q4 q_ 3 þ 32 q2 q4 q_ 4 þ 16 q24 q_ 2 þ 9 q23 q_ 4  q€3 ¼ 9ð s2  1   cos !Þ p2 q3  81" p4 q3  ð48=5Þs q_ 2 þ ð97=7Þ s q_ 4  81 p4 q_ 3  3  p4 ½ð81=8Þq33 þ ð1=8Þq31 þ ð9=4Þ q21 q3 þ 6 q1 q2 q4 þ ð3=2Þq1 q22 þ 18 q2 q3 q4 þ 36 q3 q24 þ 9 q22 q3  þ 2 p4 s ½ð192=45Þ q32 þ ð79872=315Þ q34 þ ð72=35Þ q21 q2 þ ð1072=35Þ q21 q4 þ ð5328=105Þ q1 q2 q3 þ ð52992=385Þ q22 q4 þ ð2016=33Þ q1 q3 q4 þ ð19224=385Þ q2 q23 147

148

Appendix C

þð119664=455Þ q23 q4 þ ð18176=455Þ q2 q24   2  p4 ½ð9=2Þ q1 q3 q_ 1 þ 6 q1 q4 q_ 2

þ ð3=8Þ q21 q_ 1 þ 3 q1 q2 q_ 4 þ ð9=4Þ q21 q_ 3 þ 6 q2 q4 q_ 1 þ 6 q1 q2 q_ 4 þ 18 q2 q3 q_ 2 þ ð3=2Þ q22 q_ 1 þ 36 q24 q_ 3 þ 18 q2 q4 q_ 3 þ 18 q2 q3 q_ 4 þ 18 q3 q4 q_ 2 þ 72 q3 q4 q_ 4 þ 9 q22 q_ 3 þ ð243=8Þ q23 q_ 3  2

2

ðC:1Þ 4

q€4 ¼ 16 ð s  1   cos !Þ p q4  256 " p q2  ð32=15Þ s q_ 1  ð96=7Þ s q_ 3  256  p4 q_ 4  3  p4 ½q21 q2 þ 32 q34 þ 9 q2 q23 þ 6 q1 q2 q3 þ 4q21 q4 þ 16 q22 q4 þ 36 q23 q4  þ 2 p4 s ½ð104=105Þ q31  ð21384=455Þ q33  ð10656=385Þ q22 q3 þ ð3424=315Þ q1 q22 þ ð36792=495Þ q1 q23 þ ð88=105Þ q21 q3 þð24846976=266805Þ q1 q2 q4 þ ð178304=2145Þ q1 q24 þ ð70016=273Þ q2 q3 q4

þ ð36928=315Þ q3 q24   2  p4 ½6 q1 q2 q_ 3 þ 6 q1 q3 q_ 2 þ 9 q23 q_ 2 þ 8 q1 q4 q_ 1 þ 4q21 q_ 4 þ 6 q2 q3 q_ 1 þ 2 q1 q2 q_ 1 þ 18 q2 q3 q_ 3 þ q21 q_ 2 þ 96 q24 q_ 4 þ 32 q2 q4 q_ 2 þ 72 q3 q4 q_ 3 þ 16 q22 q_ 4 þ 36 q23 q_ 4  For n = 4 the ordinary differential equations of the viscoelastic beam model with Poynting-Thompson rheological element are given: q1 þ 8s€ q2 þ ðð 3s2  1   cos ð!ÞÞ p2  g3 p4 Þq_ 1 q1 ¼  ðg1 þ g5 p4 Þ€ þ ðð16=3Þg1 þ ð128=3Þ g5 p4 Þs q_ 2 þ ðg1 ð s2  1   cos ð!ÞÞ p2  g2 p4 Þq1 þ ðð32=3Þp2 sðs2  1   cos ð!ÞÞ þ ð128=3Þ g5 p4 sÞq2 þ ðð256=15Þp2 sðs2  1   cos ð!ÞÞ þ ð4096=15Þ g4 p4 sÞq4 þ ðð32=15Þg1 s þ ð4096=15Þ g5 p4 sÞq_ 4 þ ð16=5Þs€ q4  3 g6 p4 ðð1=8Þq31 þ q1 q22 þ ð9=4Þq1 q23 þ ð3=8Þq21 q3 þ ð3=2Þq22 q3 þ 2 q1 q2 q4 þ 4 q1 q24 þ 6 q2 q3 q4 Þ  2g7 p4 ðð3=8Þ q21 q_ 1 þ ð3=8Þ q21 q_ 3 þ ð3=4Þ q1 q3 q_ 1 þ ð3=2Þ q22 q_ 3 þ 3q2 q3 q_ 2 þ 2q1 q2 q_ 2 þ q22 q_ 1 þ ð9=4Þ q23 q_ 1 þ ð9=2Þ q1 q3 q_ 3 þ 2q1 q4 q_ 2 þ 2q1 q2 q_ 4 þ 2q2 q4 q_ 1 þ 6q2 q4 q_ 3 þ 6q2 q3 q_ 4 þ 6q3 q4 q_ 2 þ 8q1 q4 q_ 4 þ 4 q24 q_ 1 Þ þ 2g7 p4 ðð144=7Þ q1 q2 q3 þ ð7936=315Þ q22 q4 þ ð20086528=266805Þ q2 q24 þ ð4432=55Þ q23 q4 þ ð96256=2145Þ q34 Þ q1  ðg1 þ 16g5 p4 Þ€ q2 þ ð72=5Þs€ q3  ðð16=3Þg1 þ ð8=3Þg5 p4 Þsq_ 1 q2 ¼  8s€ þ ð4ð 3s2  1   cos ð!ÞÞ p2  16 g3 p4 Þq_ 2 þ ðð48=5Þ g1 þ ð1944=5Þg5 p4 Þs q_ 3 þ ðð8=3Þp2 ðs2  1   cos ð!ÞÞ  ð8=3Þg4 p4 Þsq1 þ ð4g1 p2 ðs2  1   cos ð!ÞÞ  16 g2 p4 Þq2 þ ðð1944=5Þg4 p4  ð216=5Þp2 ðs2  1   cos ð!ÞÞÞsq3  3 g6 p4 ðq21 q2 þ 2 q32 þ 9 q2 q23 þ 3 q1 q2 q3 þ q21 q4 þ 6 q1 q3 q4 þ 16 q24 q2 þ 9 q23 q4 Þ

Appendix C

149

 2 g7 p4 ð3 q1 q2 q_ 3 þ 3 q1 q3 q_ 2 þ 6 q22 q_ 2 þ 2 q1 q4 q_ 1 þ q21 q_ 4 þ 3 q2 q3 q_ 1 þ 2 q1 q2 q_ 1 þ 18 q2 q3 q_ 3 þ q21 q_ 2 þ 9 q23 q_ 2 þ 6 q1 q4 q_ 3 þ 6 q1 q3 q_ 4 þ 6 q3 q4 q_ 1 þ 18 q3 q4 q_ 3 þ 9 q23 q_ 4 þ 16 q24 q_ 2 þ 32 q2 q4 q_ 4 Þ þ 2g7 p4 sðð4=15Þ q31 þ ð22356=385Þ q33 þ ð784=15Þ q22 q3 þ ð976=105Þ q1 q22 þ ð508=35Þ q1 q23 þ ð468=35Þ q21 q3 þ ð3776=63Þ q1 q2 q4 þ ð81728=3465Þ q1 q24 þ ð107712=605Þ q2 q3 q4 þ ð1219616=4095Þ q3 q24 Þ q3  ð72=5Þs€ q2 þ ð9ð 3s2  1   cos ð!ÞÞ p2 q3 ¼  ðg1 þ 81g5 p4 Þ€  81g3 p4 Þq_ 3  ðð48=5Þg1 þ ð384=5Þ g5 p4 Þs q_ 2 þ ð9g1 ð s2  1   cos ð!ÞÞ p2  81g2 p4 Þq3 þ ðð96=5Þp2 sðs2  1   cos ð!ÞÞ  ð384=5Þ g5 p4 sÞq2 þ ðð768=7Þp2 sðs2  1   cos ð!ÞÞ q4 þ ð12288=7Þ g4 p4 sÞq4 þ ðð96=7Þg1 s þ ð12288=7Þ g5 p4 sÞq_ 4 þ ð144=7Þs€  3 g6 p4 ðð81=8Þq33 þ ð1=8Þq31 þ ð9=4Þ q21 q3 þ 6 q1 q2 q4 þ ð3=2Þq1 q22 þ 18 q2 q3 q4 þ 36 q3 q24 þ 9 q22 q3 Þ  2 g7 p4 ðð9=2Þ q1 q3 q_ 1 þ 6 q1 q4 q_ 2 þ ð3=8Þ q21 q_ 1 þ 3 q1 q2 q_ 2 þ ð9=4Þ q21 q_ 3 þ 6 q2 q4 q_ 1 þ 6 q1 q2 q_ 4 þ 18 q2 q3 q_ 2 þ ð3=2Þ q22 q_ 1 þ 36 q24 q_ 3 þ 18 q2 q4 q_ 3 þ 18 q2 q3 q_ 4 þ 18 q3 q4 q_ 2 þ 72 q3 q4 q_ 4 þ 9 q22 q_ 3 þ ð243=8Þ q23 q_ 3 Þ þ 2g7 p4 s ðð192=45Þ q32 þ ð72=35Þ q21 q2 þ ð79872=315Þ q34 þ ð1072=35Þ q21 q4 þ ð5328=105Þ q1 q2 q3 þ ð52992=385Þ q22 q4 þ ð2016=33Þ q1 q3 q4 þ ð19224=385Þ q2 q23 þ ð119664=455Þ q23 q4 þ ð18176=455Þ q2 q24 Þ

ðC:2Þ

q1  ðg1 þ 256g5 p4 Þ€ q4  ð144=7Þs€ q3  ðð32=15Þg1 q4 ¼  ð16=5Þs€ þ ð16=15Þg5 p4 Þsq_ 1 þ ð16ð 3s2  1   cos ð!ÞÞ p2  256 g3 p4 Þq_ 4  ðð96=7Þ g1 þ ð3888=7Þg5 p4 Þs q_ 3 þ ðð16=15Þp2 sðs2  1   cos ð!ÞÞ  ð16=15Þ g4 p4 sÞq1 þ ð16g1 p2 ðs2  1   cos ð!ÞÞ  256 g2 p4 Þq4 þ ðð3888=7Þg4 p4 þ ð432=7Þp2 ðs2  1   cos ð!ÞÞÞsq3  3 g6 p4 ðq21 q2 þ 32 q34 þ 9 q2 q23 þ 4q21 q4 þ 6 q1 q2 q3 þ 16 q22 q4 þ 9 q2 q23 þ 36 q23 q4 Þ  2 g7 p4 ð6 q1 q2 q_ 3 þ 6 q1 q3 q_ 2 þ 9 q23 q_ 2 þ 8 q1 q4 q_ 1 þ 4q21 q_ 4 þ 6 q2 q3 q_ 1 þ 2 q1 q2 q_ 1 þ 18 q2 q3 q_ 3 þ q21 q_ 2 þ 96 q24 q_ 4 þ 32 q2 q4 q_ 2 þ 72 q3 q4 q_ 3 þ 16 q22 q_ 4 þ 36 q23 q_ 4 Þ þ 2g7 p4 s ½ð104=105Þ q31  ð21384=455Þ q33  ð10656=385Þ q22 q3 þ ð3424=315Þ q1 q22 þ ð36792=495Þ q1 q23 þ ð88=105Þ q21 q3 þ ð24846976=266805Þ q1 q2 q4 þ ð178304=2145Þ q1 q24 þ ð70016=273Þ q2 q3 q4 þ ð36928=315Þ q3 q24 Þ

Index

Airy stress function, 86, 94, 98, 137 Analytical method, 31 Anisotropic web, 2, 103 Asymptotic perturbation method, 70, 82 Axially moving isotropic plate, 26, 55, 56, 93, 97, 113 Band buckling, 17 Band-saw blade, 54 Beam model linearized, 123, 124 nonlinear, 22, 118, 135, 136, 139 of the liner paper web, 122 of the web, 2, 56, 103, 117, 118 parameters, 20 viscoelastic, 120, 121, 122, 147, 148 Belt system, 23, 34–37 Bending stiffness submatrix, 68 Bernoulli beam, 17 Bifurcation cascade, 16 diagrams, 16, 23–24, 24, 60–61, 64, 79, 122, 124–128, 130–134 double-period, 16 explosive, 129, 134 Hopf, 20, 79, 124, 126–128, 132–133, 136 node-saddle type, 135 parameter, 61, 124, 127, 130, 134 pitchwork-type, 78, 124, 134, 135, 136 point, 19, 124, 131, 132 speed of beam, 16, 124 symmetrical saddle-node, 19 tangential, 16 trivial solution, 78 zero equilibrium position, 136 zero solution, 135 Bu¨rgers model, 2, 3, 6 CD, see Cross direction (CD) Classical multilayer plate theory, 8–9

Composites, 65, 103 materials, 10, 65 plates, 25, 31, 39, 65 structure, 10, 39, 82, 138 three-layered, 65 web, 65, 71, 73, 75 Corrugated board linearized model, 77 papers, 5, 9 properties, 65 stiffnesses, 10 structure, 6, 10 three-layered, 8, 10, 65, 75, 82, 85, 90, 93, 138 web, 2, 75–80 Corrugated trapezoidal plate, 65 Critical speed, 14, 17, 19, 20, 27, 28, 55, 57, 59, 60, 71, 73, 90, 101, 114, 138, 139 comparison, 15, 56, 59, 77, 93, 94, 112–114 neighborhood of, 21, 58, 110, 111 relationship, 21, 90 value of, 27, 28, 56, 76–78 Cross direction (CD), 6 C–type board, 10 Damping coefficient, 13, 105, 122, 127, 130, 131, 133 dimensionless, 134 force, 13 internal, 16, 23, 103, 110–114 negative, 13 Dynamic analysis, 1–3, 15, 22, 25, 28,, 76, 91, 94, 101, 103, 114 Poynting-Thompson model, 3 Zener model, 3 Eigenfrequency, 18, 31, 57, 82Eigenfunctions, 16, 22, 72 Eigenvalues in circum-critical range, 58

151

152 Elastic strains, 3 Equilibrium state of web, 2, 85–87 linearized system, 85, 94–102 Finite element, 1, 28, 55, 74, 82 Finite strip method (FSM), 31 Flexural-torsional vibrations, 53, 58 Flexural vibration modes, 53 Fluting paper web, 57, 59, 100, 113, 141 dynamic behavior of, 111 FSM, see Finite strip method (FSM) Galerkin method, 16, 19, 22, 52, 71, 73, 91, 98, 109, 114, 122, 135 Galerkin truncated system, second-order, 103 Godunov orthogonalization method, 82, 137–138 Geometrical nonlinearity, 44 Gorman and Singhal’s method, 94–95 Gyroscopic coupling, 122 Gyroscopic system, 18 Hamilton’s principle, 11, 20, 43, 44, 68, 82, 107 Homogenization of corrugated layer, 7 method, see Paper web Hooke’s law, 2, 66, 104, 107 Hopf-type bifurcation, 124, 126, 128, 130, 133–134 Instability, 1, 101 divergent-type of, 13, 17, 59, 76, 77, 80, 82, 83, 92, 123, 138 flatter-type of, 15, 58, 60, 78–79, 82, 123, 124 onset of, 28, 90, 101 parametric, 37 regions, 15, 21, 26, 123, 124, 129 Isotropic web, 26, 55, 56, 93, 97, 113 Kantorowitz method, 28 Karman theory, 26 linear, 28, 29 nonlinear, 43 Kelvin-Voigt rheological model, 4 Kirchhoff-Love hypothesis, 6, 75 Koiter’s asymptotic expansion, 70 Kronecker’s delta, 50 Krylov-Bogolubov method, 14, 21 Laboratory belt system, 34–37 Lagrange’s description, 43 Laminated panel theory, 66 Laminates, 65

Index Laser telemeter, 36 Liner paper, 5–10, 54, 57, 59–63, 77, 79–82, 90–94, 99–101, 110–114, 122, 129, 135, 136, 138 Longitudinal stresses, 100 Lyapunov method, 19, 28 Machine direction (MD), 2, 6 Michelson-type laser interferometer, 34 Mindlin-Reissner plate theory, 28, 55 Modal theory of perturbations, 13, 21 Moving triangular isotropic plate, 28 Moving viscoelastic web, 103–115 Multi-layered composite web, 65, 68, 82 comparative studies, results of, 73–75 dynamic investigations of axially moving corrugated board web, results of, 75–82 mathematical model, axially moving multi-layered web, 65–70 mathematical model, solution of, 70–73 Multiple equilibrium positions of web, 87–89 Negative damping, 13 Node-saddle type, bifurcation, 60, 135 Nonlinear equations of motion, 43–47 Nonlinear mathematical model, 47–54, 117 Nonlinear geometrical relationships strains and displacements, 44 Nonlinear plate theory, 29 Nonlinear stiffness of string, 15 Non-trivial equilibrium, 85–88 Non-uniform longitudinal loading, 86 One-dimensional constitutive equation, 120 One-dimensional rheological model, 103 One-layered orthotropic web, 43–47, 82, 138 comparative studies, results of, 54–56 moving paper web, results of dynamic investigations of, 56–64 nonlinear equations of web motion, formulation of, 43–47 solution to mathematical model, 47–54 Orthotropic plate 6, 8, 9, 29, 31, 38, 65, 74, 85, 103 Orthotropy, 2, 6, 44, 103, 113–114, 138 Paper properties of, 9 tension test of, 3–6 Paper web, 1–5, 28, 38, 54–63, 75–81, 91, 93, 99–101, 103, 110–114, 121–124, 136–139

Index identification of corrugated board as composite corrugated layer, 7 homogenization method, 7–9 identification results, 9–10 identification of rheological parameters identification method, 3–5 results of experimental identification, 5–6 rheological parameters, 6 machine direction (MD), 2 Perturbations amplitude, 128, 129, 134 modal theory of, 13, 21 Phase portraits, 59, 77–78, 82 Physical model of moving web system, 43 Plastic strain, 2 Plate, flexural stiffness, 1, 10, 25, 27, 56, 65, 77, 100 Poincare maps, 16, 61, 79, 122, 128, 130, 134 Poynting-Thompson (P-T) model of material, 3–6, 117, 120–121, 135, 139 Positive acceleration, 13 Poynting-Thompson (P-T) model, 3–6 of material, 120–121 parameters calculation, 4, 5 rheological, 121 three-parameter, 117 Quasi-periodic motion, 24, 64, 79, 129, 133, 136 Releigh-Ritz method, 25, 28, 54 Runge-Kutta method, 59, 110, 122 Sandwich composites, 65–66 Shear stresses, 28, 95–98, 100, 110 Sinusoidal distribution, 94 Solid laminates, 65 Solution, 1, 11, 19, 21, 23, 29, 72, 109, 122, 137 analytical, 29, 37, 38 approximated, 22, 48, 59, 71, 109, 114, 135, 137 asymptotic, 13, 15, 21 FEM, 29 linear, 14, 15, 19 to equation of motion, 12, 29, 30, 47, 48, 50, 52, 70, 72, 77, 82, 87, 88, 95, 98, 101, 123, 125, 126, 131, 132 trivial, 20, 78, 135 Stability, 17, 57, 138 boundaries, 27, 57, 58, 76, 92, 124, 129 buckling, 29 dynamic, 56, 68, 70, 71, 82, 110 linear analysis of, 22, 129

153 of equilibrium position, 23, 128 of system motion, 13, 15, 18, 58, 82, 85, 110 phenomena, 11 Steel core profiles, 74 Strain relaxation, 2, 110, 113 Stress function determination, 100 Stress relaxation, 2 String boundary control of, 19 dynamic investigations of, 15, 16, 17, 38 frequency of, 12, 15 linear system, 11, 13, 18, 19, 137 motion equation, 12 nonlinear stiffness, 15 nonlinear system, 13, 16, 27 wave propagation in, 12, 13 Telemeter, laser, 36 Tension test of paper, 3, 4 Theoretical perturbation approach, 36 Time history, 60–62, 78, 80, 81, 111, 123, 125, 126, 130, 131 Timoshenko beam, 17 Transition matrix method, 71–72 Transmission belt, identification, 1, 23, 32, 34–38 Transverse displacements and wrinkling of webs, 85, 98–101 Transverse-longitudinal beam vibrations, 13 Transverse stresses, 98, 100 Trapezoidal plate, corrugated, 65 Two-dimensional plate theory, 25 Two-dimensional rheological model for viscoelastic materials, 104–106 Undamped axially moving system, 43–83, 110–112, 135, 138 multi-layered composite web mathematical model, axially moving multi-layered web, 65–70 results of comparative studies, 73–75 results of dynamic investigations of axially moving corrugated board web, 75–82 solution of mathematical model, 70–73 one-layered orthotropic web, 43–47 formulation of nonlinear equations of web motion, 43–47 results of comparative studies, 54–56 results of dynamic investigations of the moving paper web, 56–64 solution to the mathematical model, 47–54 remarks, 82–83

154 Vibrations amplitude of, 36, 38, 60 analysis of, 23 control of, 23, 28 damped, 111, 123 flexural, 53–59, 63, 76, 81, 110, 138 flexural-torsional, 53, 54, 58 forced, 18, 32, 79 free, 54–61, 70, 77, 79, 82, 117 frequency of, 12, 13, 14, 37, 49, 50 investigation of, 26, 27, 28, 34–39, 61, 79 linear analysis of, 11, 12, 79 modes of, 15, 17, 31, 35, 53 nonlinear, 13, 14, 15, 19, 20, 34, 60, 61, 78, 138 parametric, 15, 117 torsional, 17, 25 transverse, 11–14, 17–21, 23, 25, 29, 34, 36, 74, 103, 137 Viscoelastic beam model, 117–136, 138, 139 Viscoelastic rheological models, 2–6, 15, 23, 103–115, 117, 121, 122, 129, 130, 135, 138, 139 Viscoelastic strains, 3–5 and stress relationship, calculation, 4–5 Viscoelastic web model, 38, 103–115, 137–139 dynamics of axially moving, 103–114 mathematical model of moving viscoelastic web, 106–109 relationships between stresses and loads, 107 remarks, 114

Index results of numerical investigations, 110–114 solution to problem, 109–110 two-dimensional rheological model for viscoelastic materials, 104–106 nonlinear beam model of Kelvin-Voigt model of material, 119–120 Poynting-Thompson model of material, 120–121 solution to the problems, 122 Wallpaper machine, 28 Web accelerating, 28 anisotropic, 103 boundary regions of, 26, 48, 96, 137 control of, 28 corrugated board, 65, 83 dynamic analysis of, 43–83, 85–102, 103–115, 138, 139 isotropic, 26, 55, 56, 93, 97, 113 paper, 1–6, 28, 38, 54–63, 91, 99, 100–103, 110–114, 137–139 properties, 1, 9, 55 stationary, 26, 47 stresses in, 28, 58 transport speed of, 28, 56, 59, 92–94, 113, 114 wrinkling of, 94, 98, 99 Winding machines, 87 Young moduli/modulus, 4, 44 Zener rheological model, 3, 6, 114