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1 Delft University of Technology On parametric transversal vibrations of axially moving strings Ali, Rajab DOI 1.433/uuid:39e166c-9a f-ce18cfb4b14 Publication date 16 Document Version Final published version Citation (APA Ali, R. (16. On parametric transversal vibrations of axially moving strings DOI: 1.433/uuid:39e166c- 9a f-ce18cfb4b14 Important note To cite this publication, please use the final published version (if applicable. Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s and/or copyright holder(s, unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 1.

2 ON PARAMETRIC TRANSVERSAL VIBRATIONS OF AXIALLY MOVING STRINGS

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4 On parametric transversal vibrations of axially moving strings Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op donderdag 7 October 16 om 1:3 uur door Rajab ALI Master of Science in Mathematics, The University of Sindh, Jamshoro. geboren te Sanghar, Sindh-Pakistan.

5 This dissertation has been approved by the promotor: Prof. dr. ir. A.W. Heemink copromotor: Dr. ir. W. T. van Horssen Composition of the doctoral committee: Rector Magnificus, Prof. dr. ir. A. W. Heemink, Dr. ir. W. T. van Horssen, Independent members: Prof. dr. ir. C. W. Oosterlee, Prof.dr.ir. Peter Steeneken, Prof.dr.J.Molenaar, Prof. dr. A. K. Abramian, Prof.dr.Igor V. Andrianov, Chairman Promotor, Delft University of Technology Copromotor, Delft University of Technology Delft University of Technology Delft University of Technology Wageningen University, The Netherlands Russian Academy of Sciences, Russia RWTH Aachen University, Aachen This thesis has been completed in fulfillment of the requirements of the Delft University of Technology, for the award of the PhD degree. The research described in this thesis was carried out in the section of Mathematical Physics at the Delft Institute of Applied Mathematics (DIAM Delft University of Technology, The Netherlands. This research was supported by Quaid-e-Awam University Nawabshah Sindh Pakistan under the faculty development program of Higher Education Commission (HEC of Pakistan and the Delft University of Technology, the Netherlands. ISBN Copyright c 16 by Rajab Ali R.Ali@tudelft.nl All rights reserved. No part of this publication may be reproduced in any form or by any means of electronic, mechanical, including photocopying, recording or otherwise, without the prior written permission from the author. Printed in The Netherlands by: Sieca repro services.

6 Educating the mind without educating the heart is not education at all. Aristotle Dedicated to my Parents and Teachers with gratitude and love

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8 Contents List of Figures vi List of Tables viii 1 Introduction Background Aim of the Research The Mathematical Model Applied Mathematical Methods Outline of the Thesis On resonances and the applicability of Galerkin s truncation method for an axially moving string with time-varying velocity Introduction Equations of motion Harmonically varying velocity about a low mean speed Application of the two timescales perturbation method A general resonance case: Ω = mπ (m is positive odd integer Application of the truncation method Analysis of the infinite dimensional system ( Near resonance case : Ω = mπ + εδ Analysis of the infinite dimensional system( Harmonically varying velocity about a relatively high constant mean speed Application of the two timescales perturbation method Ω = k π(1 V, a general resonance case Application of the truncation method Analysis of the infinite dimensional system ( Conclusions and remarks On the asymptotic approximation of the solution of an equation for a nonconstant axially moving string Introduction This chapter is slightly revised version of published article 1- On resonances and the applicability of Galerkin s truncation method for an axially moving string with time-varying velocity, J. Sound Vib., vol. 344, p: 1-17, February 15 This chapter is slightly revised version of published article - On the asymptotic approximation of the solution of an equation for a non-constant axially moving string, J. Sound Vibr., vol. 367, p:3-18, January 16. v

9 vi CONTENTS 3. Equations of motion The construction of asymptotic approximations The non-resonant case (up to O(ε on timescales of 1 ε Ω = (m 1π, a pure resonance case Specific initial conditions Energy of the system Results and discussion Conclusion On parametric stability of a non-constant axially moving string near-resonances Introduction Equations of motion A perturbation approach to construct approximations Ω = (m 1π + εδ, a detuned resonance case Case 1: δ < α Case : δ = α Case 3: δ > α Stability analysis Conclusion Conclusions and Future Work 65 A Infinite dimensional system of coupled ODEs (Eq.(.8 69 B Infinite dimensional system (Eq.( C Proof of Integral (Eq.( D The infinite dimensional system (Eq.( E The infinite dimensional system (Eq. ( Bibliography 85 Summary 91 Samenvatting 93 Acknowledgments 95 List of publications and presentations 97 About the author 99 3 This chapter is slightly revised version of accepted article 3- On parametric stability of a non-constant axially moving string near-resonances, J. Vibr. Acoust., vol.139, p: , October 16. vi

10 List of Figures 1.1 A moving horizontal conveyor belt system A moving cable car system A schematic model of moving belt system between two fixed points Integration in the characteristic ξ (or σ direction m = 1, V =.8, α =.5, ɛ =.1, x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u m = 1, V =, α = 1, ɛ =.1, x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u m = 1, V =.8, α =.5, ɛ =., x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u m = 1, V =, α = 1, ɛ =., x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u Logarithmic scale of energy with ɛ =.1, V =.8, α =.5. (a Approximated energy of system. (b Energy of the system Logarithmic scale of energy with ɛ =.1, V =, α = 1. (a Approximated energy of system. (b Energy of the system Logarithmic scale of energy with ɛ =.,V =.8, α =.5. (a Approximated energy of system. (b Energy of the system Logarithmic scale of energy with ɛ =., V =, α = 1. (a Approximated energy of system. (b Energy of the system m = 1, ɛ =.1, V =.8, α =.5, δ =.1, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u m = 1, ɛ =.1, V =.8, α =.5, δ =.1. (a Approximated energy. (b Energy of the system m = 1, ɛ =.1, V =.8, α =.5, δ =α, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u m = 1, ɛ =.1, V =.8, α =.5, δ =α. (a Approximated energy. (b Energy of the system m = 1, ɛ =.1, V =.8, α =.5, δ =-α, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u m = 1, ɛ =.1, V =.8, α =.5, δ =-α. (a Approximated energy. (b Energy of the system vii

11 viii LIST OF FIGURES 4.7 m = 1, ɛ =.1, V =.8, α =.5, δ =15, x =.5. (a The stable first order asymptotic approximation (v. (b The stable numerical solution (u m = 1, ɛ =.1, V =.8, α =.5, δ =15. (a Approximated energy. (b Energy of the system Approximate instability region in the (α, Ω-plane for ε =.1 : the amplitude α of the velocity fluctuation of the belt versus the frequency Ω of the velocity fluctuation of the belt. The boundaries of the instability region are given by Ω = (m 1π+εδ with δ = α and m = 1,, viii

12 List of Tables.1 Approximations of the eigenvalues of the truncated system (.8 for m = 3 and m = Approximations of the eigenvalues of the truncated system (.67 for k = Approximations of the eigenvalues of the truncated system (.67 for k = Approximations of the eigenvalues of the truncated system (.67 for k = 3 and k = ix

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14 Chapter 1 Introduction Imagination is more important than knowledge. Albert Einstein 1.1 Background Mechanical structures such as belts, cables and tapes are often referred to as axially moving materials or axially moving continua, due to their larger dimension in axial direction in comparison to the other two directions. Axially moving continua have wide range of applications in many branches of engineering, such as aerospace, architectural, civil, chemical, automotive and mechanical engineering. Despite the numerous engineering applications of such systems, vibrations have limited their stability and applications because of excessive amplitudes resulting mainly from resonances. Vibrations, most commonly, transverse vibrations are often caused by the eccentricity of a support roller, irregular speed of the driving motor, material non-uniformity; and/or vibrations developed by natural phenomena such as an earthquake and wind-forces. Severe vibrations are, in general, undesirable attribute in mechanical structures as it reduces efficiency, creating unwanted sound vibrations and greatly affecting human comfort. In addition, it is evident that the severity of the vibrations in the systems can also lead to operational and maintenance problems, including an increase of the energy consumption of the system and even damage to structures 4. The Tacoma Narrow suspension bridge in USA collapsed on 7 November 194 due to 4 mile-per-hour wind, is a classic example of a structural failure. The main objective of the mechanical or civil engineers, physicists and applied mathematicians is to understand and mitigate the structural and mechanical vibrations through testing (experimentally and analytically and predicting the dynamic responses of structural or mechanical systems at multiple operational conditions. Axially moving continua can be modeled as flexible strings and beams, depending on the bending stiffness of the belt. If the bending stiffness is negligible, the systems are in the class of string-like 5, 6, 7, 8, 9; otherwise, they are in the class of beam-like 1, 11, 1, 13, 14. In addition to this, these systems behave linearly 11, 15, 16, 17 at very small amplitudes, while nonlinearities in these systems occur at large amplitudes 18, 19,, 1. Several studies have been devoted to dynamic investigations of axially moving systems. The relevant research on the vibration analysis of axially moving materials undertook in the last decade of nineteenths century, when Skutch published his study in German language on the transverse vibrations of a traveling elastic string moving through two pinholes and determined its fundamental 1

15 CHAPTER 1. INTRODUCTION Figure 1.1: A moving horizontal conveyor belt system. resonance frequency by superposition of two waves propagating in opposite directions. In the early 195s, one of the first papers in the English-language on this subject was published by Sack 3 to investigate the vertical vibrations of an undamped string subject to a harmonic excitation which travels under constant tension at a constant speed over fixed supports. Archibald and Emslie 4 derived the same equations of motion for the traveling strings by using a variational approach and examined the linear transverse oscillations in the string with a constant speed along the longitudinal direction where one or both of its ends are sinusoidally excited. Mahalingam 5 studied the parametric transverse displacement of the power transmission chains under tension fluctuations. Transversal vibrations of a moving strip subjected to constant speed and tension are examined by Thurman and Mote 6. Swope and Ames 7 determined the response of the transverse vibrations of a translating string with a constant speed and examined the properties of wave propagation in the string. Recently, Gaiko and van Horssen 8 analyzed the transverse oscillations of a traveling string under a constant speed and examined the effects of damping at boundaries. While, the transverse vibrations in an axially moving string under internal damping is examined in Refs. 9, 3. In addition, Akaya and van Horssen 31 studied the various damping properties of waves in a semi-infinite string by means of D Almebert s method. All above-mentioned researchers considered the axial speed of the moving system to be constant. Axially moving system moving with a constant axial speed, however, have limited practical applications. In reality, there are various components of belt systems which can vary the axial speed and cause the accelerations and decelerations in motion, e.g. the eccentricity of pulleys and other belt imperfections. The systems, therefore, often exhibit transverse vibrations due to axial variation speed termed as parametric vibrations. The time-dependent velocity of axially moving continua consists of a constant mean speed along with some (small harmonic variations. These harmonic variations in the time-dependent velocity may cause parametric resonances (instability which adversely effect the response of the system and may lead to failure of the system. In other words, parametric resonance is generally excited by the time-varying coefficients (i.e. speed variation in the governing equations of motion. Resonant vibrations in mechanical structures occur when the excitation frequency (frequency of velocity variations coincides with one of the natural frequencies of the system. In order to achieve an optimal design of the systems, the major task is to evaluate and assess all possible resonance frequencies and to keep them away from the natural frequencies of the system. The most fundamental and crucial step in analyzing the oscillatory motion in structural or mechanical systems is the development of mathematical models. The mathematical models that govern the transverse vibrations of axially moving continua can

16 1.1. BACKGROUND 3 Figure 1.: A moving cable car system. be described in terms of the second-order (string-like and fourth-order (beam-like (nonlinear hyperbolic partial differential equations with variable-coefficients. Such partial differential equations include Coriolis acceleration component, because the axially moving continua belongs to the class of distributed gyroscopic systems 3. Since closed-form (analytic solutions of partial differential equations with Coriolis acceleration term are in general not available; several other numerical and approximate techniques have been employed to facilitate the vibration analysis, and to investigate the (instability of an axially moving continua. There are comprehensive studies on the mathematical analysis of parametric transverse vibrations of axially moving systems subject to the speed variation. Mote 33 used the Laplace transform technique to investigate the parametric vibrations of an axially moving string with respect to the time-varying speed driven harmonically at one end by replacing the varying speed by its time-averaged values. Wickert and Mote 34 developed a modal analysis solution to the linear transverse vibrations of axially moving strings and beams under arbitrary excitations and initial conditions. They obtained the natural frequencies using the Green s function method. Zhu and Guo 35 determined the free and forced responses of a translating string with an arbitrarily velocity profile by using the method of characteristic transformations. Van Horssen 5 employed the method of Laplace transforms to determine the exact responses of an axially moving string due to arbitrary lateral vibration of supports. Zhu and Zheng 36 investigated parametric instability of a translating string in terms of exact response with a sinusoidally varying length by means of characteristic transformations. Pellicano et al. 37 used the Galerkin s truncation method to investigate the parametric instability in power transmission belts. Zhu et al. 38 used the characteristic transformation without discretizing the governing partial differential equations to investigate the parametric instability behavior in terms of bounded displacement and unbounded vibratory energy of a translating string under sinusoidally varying speed. The method of multiple scales is a powerful tool to investigate the parametric vibrations of axially moving strings under speed variation. The development of an approximate analytic method, i.e. the method of multiple scales in investigating the axially moving system, started in the early 197s (see for instance 39. Pakdemirli and co-investigators 4, 41, 4 used the method of multiple scales based on the truncation method to construct the approximations of the governing partial differential equation of an axially moving string subject to the time-dependent velocity. Ghayesh et al. 8, 43 approximated the nonlinear transversal vibrations of an axially moving string by means of the perturbation method in association with the Galerkin truncation method. Chen et al. 9, 44, 45, 46 examined the parametric instability in the axially moving string by using the method of multiple scales based on the truncation method. Parker and Lin 3

17 4 CHAPTER 1. INTRODUCTION Figure 1.3: A schematic model of moving belt system between two fixed points. 47 studied the parametric vibrations of a translating string using the perturbation method. In most of the references mentioned above, the method of multiple scales based on the truncation method has been used without any valid justification of mode-truncation to approximate the transverse vibrations of an axially moving string. However, one should be careful in using the truncation method for string-like problems. The non-applicability of truncation method for stringlike problems, for instance, has been proven with mathematical justification in Refs. 48, 49, 5. The detailed comments on this mathematical justification for the truncation method are presented in Chapter and Chapter 3 of this thesis. 1. Aim of the Research Most of the previous studies on the subject of stability analysis of an axially moving string focused only on determination of the instability for the lower resonances and the (nonapplicability of the truncation method. The dynamic (instability of an axially moving string with timedependent velocities at lower resonance frequencies have been extensively studied (see for instance in Refs., 3, 48, 49, 5, 51; however, their stability analysis for higher resonances has developed slowly. In understanding the dynamic (instability in the axially moving system, all resonance frequencies must be taken into account. The main goal of the present thesis is to gain a better insight into the linear dynamics of axially moving strings which are fixed between two supports. The transverse vibrations of conveyor belt systems, for instance, can be modeled by these string-like problems. The dynamic (instability in the parametric transverse vibrations of the axially moving string under speed variations will mainly be studied at higher resonances and the (nonapplicability of truncation method will also be discussed in detail. 1.3 The Mathematical Model In this thesis, a belt moving with an axial velocity V between two fixed points that are a distance L apart, as shown in Figure.1.3. The transverse displacement of the belt can be modeled as an axially moving string. The mathematical model of an axially moving string is based on the following assumptions: The density ρ (mass m per unit length of the string is assumed to be constant. The string is perfectly elastic and offers no resistance to bending. 4

18 1.4. APPLIED MATHEMATICAL METHODS 5 The string is tightly stretched, which implies that the tension T in the string is regarded as a constant. In addition, the tension T is assumed as large compared to the weight of the string so that the effect of gravity is negligible. Effect due to internal viscosity of the string is also neglected. Only the motion of portion of the string between the pulleys are determined. It is assumed that no external forces are acting on the string. The displacement slope is small, u x 1, which ensures that the transverse displacement is small compared to the length of the string. Using these assumptions the governing equations of motion for an axially moving string can be written in the form 48, 5: where u tt + V u xt + V t u x + (V c u xx =, t >, < x < L, (1.1 u(x, t x t L c V ρ T : the displacement in the transversal (lateral direction : the axial position along the string : the time : the distance between the pulleys : wave speed : the time-varying velocity of a belt : the mass density of a belt : the tension of a belt and where c = T ρ. The time-dependent velocity V is assumed to be a harmonically varying function about a constant mean speed, that is, V (t = V + εα sin(ωt where the mean speed V = O(ε or V = O(1 with ε (i.e. < ε 1 and V, α, Ω are all positive constants. The terms u tt, u xt, u xx on the left hand side of Eq. (1.1 represent, respectively, the accelerations of local inertia, Coriolis acceleration associated with the rotation, and the centrifugal acceleration associated with the curvature. The velocity fluctuation frequency Ω may give rise to all kinds of resonances in the system. Furthermore, the belt is fixed at both ends, boundary constraints for the belt system are u(, t =, u(l, t =, t, (1. and the initial conditions are given by u(x, = φ(x, and, u t (x, = ψ(x, < x < L, (1.3 where φ(x and ψ(x represent, respectively, the initial displacement and velocity of the belt system. 1.4 Applied Mathematical Methods In this thesis, the two timescales perturbation method, the Fourier-mode expansion method, the Laplace transform method and the method of characteristic coordinates will be explored. The 5

19 6 CHAPTER 1. INTRODUCTION motion of an axially moving string is described by a second order homogeneous linear partial differential equations with variable coefficients. To obtain the solution in closed (analytic form is usually not possible, an analytic-approximate method, such as, the method of multiple scales, is often used to construct approximations for the solutions of these differential equations. For a complete overview of the method of multiple scales, the interested reader is referred to standard texts such as 39, 53, 54. In order to apply the multiple timescales perturbation method, the following steps are used: The original initial-boundary value problem is first converted into a perturbation problem by introducing a small dimensionless parameter ε (where < ε 1. In this thesis we will convert the original problem into a perturbation problem by the inclusion of the velocity function V (t into the governing equations of motion. The solution of the perturbation problem is then presumed in the form: u(x, t = u(x, t; ε. (1.4 The assumed solution close to ε = is expanded into a power series in ε as follows u(x, t; ε = u (x, t + εu 1 (x, t + ε u (x, t +..., (1.5 where all u is for i =, 1,,... are O(1 on time and space scales. The straight forward expansion (1.5 may break down when the O(ε i+1 term dominates the O(ε i, for some i =, 1,,... as the time t becomes large because of resonances leading to unbounded (so-called secular terms. These secular terms should be avoided, because these terms may cause non-uniformities and inconsistencies in the asymptotic solutions. To develop an asymptotic solution of the perturbation problem valid for all time t up to O(ε 1, additional different timescales t = t, t 1 = εt, t = ε t,..., should be introduced. The solution u(x, t; ε is then assumed to be a function of these timescales: u(x, t; ε = w(x, t, t 1,... This new function can be expanded in power series in ε: w(x, t, t 1 ; ε = w (x, t, t 1,... + εw 1 (x, t, t 1, , (1.6 where w i = O(1 for i =, 1,,..., and explicit computations of all w i guarantee the elimination of secular terms. Based on the expansion (1.6, the following transformations for the time derivatives are introduced: d(. dt = (. t + ε (. t 1 + ε (. t +..., (1.7 d (. dt = (. t + ε (. + ε ( (. t t 1 t + (. +..., (1.8 1 t 1 t and so on. Substitution of (1.6-(1.8 into a perturbation problem and equating the coefficients of like powers of small parameter ε yields the sequence of differential equations, which can be solved sequentially (if possible. Such ordinary and/or partial differential equations can also be solved by using the method of Laplace transform. The interested reader can view the method of Laplace transform in Refs. 55, 56, 57. The method of characteristic coordinates was discovered by the French mathematician Jean Le Rond d Alembert in 18th century to study the wave equation. In solving the equations of motion governing transverse vibrations of axially moving strings, the method of characteristic coordinates is of great interest, in the sense that it avoids the problem of convergence and computational difficulties of infinite series (as has been noticed in 1, 48, 49, 5 in the Fourier series approach. A crucial requirement for the applicability of the method of characteristic coordinates to the governing equations of motion for the axially moving strings over the finite domain is that the 6

20 1.5. OUTLINE OF THE THESIS 7 initial-boundary value problem be first replaced into an initial value problem. This replacement requires for small V (t an odd and -periodic extension of the dependent variable u(x, t as well as the initial values φ(x and ψ(x on the infinite interval < x < (method of reflection, so that the Dirichlet type boundary conditions are satisfied. Having transformed the initial-boundary value problem to an initial-value problem, the solution of the governing equations of motion is assumed to be a function of the characteristic coordinates: u(x, t = v(σ, ξ, where σ = x t, ξ = x + t. The expansion of the assumed solution close to ε = into a power series in the small parameter ε yields v(σ, ξ = v (σ, ξ + εv 1 (σ, ξ +... This straightforward expansion causes secular terms. To obtain secular-free approximations valid on long timescales, the two timescales perturbation method is used. By introducing σ = x t, ξ = x + t and τ = εt ( t a fast-scale over which oscillations occur and τ a slow-scale over which amplitudes evolve ; the solution is then assumed in the form: u(x, t = w(σ, ξ, τ w (σ, ξ, τ + εw 1 (σ, ξ, τ +... (1.9 The introduction of σ, ξ, and τ leads to the following transformations: (. t = (. σ + (. ξ + ε (. τ, (1.1 (. t = (. σ + (. ξ (. ( σ ξ + ε (. ξ τ (. + ε (. σ τ τ, (1.11 (. x = (. σ + (. ξ, (1.1 (. x = (. σ + (. ξ + (. σ ξ, (1.13 (. x t = (. σ + (. ( ξ + ε (. σ τ + (.. (1.14 ξ τ Substitution of Eqs. (1.9-(1.14 into the perturbation problem and equating coefficients of like powers of ε produces the sequence of differential equations, which can be solved one by one. 1.5 Outline of the Thesis The thesis is organized as follows. In Chapter an initial-boundary value problem is considered for the linear transverse vibrations of an axially moving string. Accurate asymptotic approximations valid on long timescales of O(ε 1, are constructed by means of the two timescales perturbation method in combination with either a Fourier-mode expansion method or the method of Laplace transforms. All explicit approximations of the energy of the system are computed up to O(ε and the (nonapplicability of the truncation method is discussed. In Chapter 3 the linear transverse parametric vibrations of an axially moving string under speed variations are studied. The displacement-response subject to the specific harmonic initial conditions is determined analytically at higher resonances by using the two timescales perturbation method in combination with the method of characteristic coordinates. Also explicit approximations for the energy of the system up to O(ε are computed. The results obtained for the displacement-response and the energy of the system on long timescales are verified with a 7

21 8 CHAPTER 1. INTRODUCTION numerical finite difference method. Chapter 4 deals with the analysis of the parametric (instability of an axially moving string in the neighborhood of resonance frequencies. Explicit approximations for the amplitude-response and the energy of the system up to O(ε are computed near resonances by means of the two timescales perturbation method in combination with the method of characteristic coordinates. It has been shown for what values of the detuning parameter the amplitude-response and the energy of the system are (unstable. All approximations obtained for the amplitude-response and the energy of the system are verified using the numerical finite difference method. In Chapter 5 conclusions about the parametric (instability of an axially moving string with time-dependent speed are presented, and some possibilities for future research related to this work are discussed. 8

22 Chapter On resonances and the applicability of Galerkin s truncation method for an axially moving string with time-varying velocity 1 Nature laughs at the difficulties of integration. Pierre-Simon Laplace As was described in the introduction, an axially moving string system may experience (instabilities due to harmonic variations in the axial speed. This time-varying velocity of the axially moving system consists of a constant mean value along with some small harmonic variations. These harmonic variations in the time-dependent velocity may cause a resonance (instability in the system. In this chapter the (instability of an axially moving string subject to the time-dependent velocity at all resonant frequencies is examined. The two timescales perturbation method in conjunction with the Fourier-mode expansion method and the Laplace transform method is employed to the equations of motion in search of infinite mode approximate solutions. All explicit approximations for the energy of the system are computed. In addition, it is shown that mode-truncation does not yield accurate approximations on long timescales, that is, on a timescale of order ε 1..1 Introduction Axially moving strings can represent many engineering devices such as serpentine belts, aerial cables, power transmission belts, plastic films, magnetic tapes, paper sheets and textile fibers. Roll eccentricity of pulleys, small variations in the driving force of the pulley and other belt imperfections can lead to varying belt speed. In other words, (unknown sources can lead to varying belt speed, and so leading to undesirable oscillations in such systems. In engineering, understanding the transverse vibrations of axially moving strings is important for the safe design, construction and operation of a variety of machines and structures. Analysis of transverse vibrations of axially moving strings is a challenging subject, which has been studied for many years by several researchers and still of interest today. Much research has been done in this area to study the linear vibrations of axially moving strings, which was reviewed in Refs. 3, 4, 5. In all of these in- 1 This chapter is slightly revised version of published article 1- On resonances and the applicability of Galerkin s truncation method for an axially moving string with time-varying velocity, J. Sound Vib., vol. 344, p: 1-17, February 15 9

23 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 1 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY vestigations, it is found that linear analysis is applicable to small amplitude vibrations. The axial speed plays a significant role on the dynamics of axially moving strings. Both the constant and time-varying axial speed cases have been examined in the literature 6, 7, 1, 1, 48, 49, 58. While most of the studies deal with constant axial velocity, some paper addressed the influence of speed variation on the vibrations, see in Refs. 16, 41, 59, 6. Miranker 5 was the first to derive the equations of motion for a string traveling with time-dependent axial velocity. Several techniques have been employed to analyze the governing equations of motion for axially moving strings with constant and time-varying speed. Thurman and Mote 6 used the Linsted perturbation method and the averaging method to study linear and non-linear models of an axially moving string, and found that the influence of non-linearities increases with the increase of transport speed. Mote 33 studied the problem of an axially accelerating string driven harmonically at one end, obtained approximate solutions and investigated the stability of resulting constant coefficient equation by means of the Laplace transform method. The transverse vibrations of traveling strings and the moving beam were studied by Wickert and Mote 34 by using the Green s function method; they obtained modal functions of string and found that orthonormal modal function become singular when the string transports at critical speed. Pakdemirli and Boyaci 61 used the Lindstedt- Poincar technique and the two timescales perturbation method, respectively, to approximate the transverse oscillations of the beam-like and string-like problems subject to constant axial speed. However, in that paper the approximation is truncated to a single mode of vibration for stringlike problem and it is shown that the frequencies and amplitudes of vibration may grow or decay in time. While, the same problem was studied by van Horssen 5 using the Laplace transform techniques and found that the truncation to a single mode of vibration leads to inaccurate results on long timescales. A two timescales perturbation method was also used in Refs. 41, 4 to investigate the stability in transverse vibrations of an axially moving string with a time varying speed. The truncation method is then used to approximate the solutions. In these papers, the approximation is truncated to a single mode of vibration. However, the solution of the partial differential equation can consist of infinitely many interactions between vibration modes. So this truncation can cause inaccurate results on long timescales (see for instance Refs.11, 6, 63. Recently, Chen9 used the method of multiple scale to investigate the approximate analytical solution of the non-linear transverse vibration of axially accelerating strings with a harmonically varying velocity about a constant mean speed and with longitudinally varying tension and found that the exact internal resonances among the first three frequencies exists, while the effects of infinite exact internal resonances may be neglectable. On the other hand, Sandilo and van Horssen 5 studied the axially moving string with harmonically varying length about constant mean length, and found that the Galerkin s truncation method can not be applied to obtain asymptotic approximations on long time-scales. In this chapter, the transverse vibrations of axially moving strings with time dependent velocity varying harmonically about low constant mean speed and about relatively high constant mean speed will be studied. In case of low speed, the governing equation of motion is first discretized by using the Fourier sine series, and then the two timescales perturbation method is employed, while in the case of relatively high constant speed, the Laplace transform method is used followed by the two timescales perturbation method to obtain the analytical approximate solution. The Laplace transform method is, however, well-described in elementary text books on partial differential equations see for instance Ref. 56. Additionally, It will be shown that the truncation method cannot be applied to string-like problems in all axial speed cases. This chapter is organized as follows. Section. presents the mathematical formulation of the axially moving string system. The governing equations of motion with the time-dependent velocity at a low mean speed is studied in section.3. In section.4, the governing equations of motion 1

24 .. EQUATIONS OF MOTION 11 is examined with the harmonically varying velocity about a relatively high constant mean speed. Finally, the conclusions of this chapter are presented in section.5.. Equations of motion The schematic representation of an axially moving string with length L, moving with velocity V is shown in Figure 1.3. We will assume that the string is fixed at x = and x = L, where L is the distance between the two supports (that is, it is assumed that there is no displacement of the string in vertical direction at the supports. The equations of motion describing an axially moving string is obtained either through the application of Hamilton s principle see Ref. 6 or Newton s second law of motion see Ref.7. Let t be the time, x be the spatial coordinate along the longitude of motion, V be the timevarying axial speed of the string and u(x, t be the transversal displacement of the string at spatial coordinate x and time t. Then the equation describing the transversal displacement of the string in vertical direction is given by u tt + V u xt + V t u x + (V c u xx =, t >, < x < L, (.1 where c is the wave speed due to a pretension of the string, and c = T ρ, in which T is assumed to be non-zero constant tension in the string, and ρ is the (constant mass of the string per unit length. The boundary and initial conditions are given by and u(, t =, u(l, t =, t, (. u(x, = φ(x, and, u t (x, = ψ(x, < x < L, (.3 where φ(x and ψ(x represents the initial displacement and initial velocity of the string respectively. To put the equation in non-dimensional form, we introduce the following dimensionless quantities: x = x L, V = V c, t = ct L, u(x, t u (x, t = L, Ω = LΩ c, φ (x = φ(x L, ψ (x = ψ(x. c Substitution of Eq. (.4 into the initial-boundary value problem (.1-(.3 yields the dimensionless form of equation of motion: (.4 u tt u xx = V t u x V u xt V u xx, t, < x < 1, (.5 with non-dimensional boundary conditions and initial conditions u(, t =, u(1, t =, t, (.6 u(x, = φ(x, and u t (x, = ψ(x, < x < 1. (.7 The asterisks indicating the dimensionless quantities are dropped in Eqs.(.5-(.7 and henceforth. 11

25 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 1 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 3 In the following sections we present the methods to obtain an accurate approximations of the solutions of the governing equations of motion (.5-(.7 for an axially moving string at low and relatively high mean speed of the system..3 Harmonically varying velocity about a low mean speed In this section, the methods for solving the initial-boundary value problems (.5-(.7 with harmonically varying velocity about a low mean speed are discussed. We assume that the velocity is a harmonically varying function about a constant mean speed V (t = ε(v + α sin(ωt, (.8 where V and α are, respectively, the mean speed and the amplitude of order ε, and Ω is the velocity fluctuation frequency of O(1, and ε is a dimensionless small parameter with < ε 1. It is also assumed that V > α, which guarantees that the belt will always move forward in one direction. In addition, belt velocity V is assumed to be small compared to the wave speed c of the belt in this case; consequently, the term V u xx will have no contribution to the solutions up to O(ε on time-scales of order 1 ε. If the velocity function (.8 is substituted into Eq. (.5, one obtains the following perturbation problem: u tt u xx = ε αω cos(ωtu x ( V + α sin(ωt u xt ε uxx V + α sin(ωt, (.9 with boundary conditions and initial conditions u(, t; ε =, u(1, t; ε =, t, (.1 u(x, ; ε = φ(x, and u t (x, ; ε = ψ(x, < x < 1. (.11 The following Fourier series expansion (that is, an eigenfunction expansion 48 u(x, t = u n (t; ε sin(nπx, (.1 for u(x, t is assumed, so that the Dirichlet type boundary conditions (.1 are satisfied. Substitution of Eq. (.1 into the initial-boundary value problem (.9-(.11 leads to ü n +(nπ u n sin(nπx = ε nπ αω cos(ωtu n + ( V +α sin(ωt u n cos(nπx+o(ε. (.13 The following orthogonality properties for the set of functions sin(nπx on < x < 1 should be observed: 1 sin(nπx sin(kπxdx = { for n k, 1/ for n = k, (.14 1

26 .3. HARMONICALLY VARYING VELOCITY ABOUT A LOW MEAN SPEED 13 1 { for n ± k is even, cos(nπx sin(kπxdx = k for n ± k is odd. (n k π (.15 Multiplying Eq. (.13 by sin(kπx on both sides and then integrating w.r.t. x from x = to x = 1, and by using Eqs. (.14and (.15 we obtain: ü k + (kπ u k = ε n±k is odd where u k must satisfy the following initial conditions: u k ( = nk n k 4αΩ cos(ωtu n + 8 ( V + α sin(ωt u n + O(ε, 1 φ(x sin(kπxdx, u k ( = 1 (.16 ψ(x sin(kπxdx. (.17 Equation (.16 is an infinite dimensional system of ordinary differential equations, which cannot be solved exactly. In what follows, we will apply the two timescales perturbation method to obtain O(ε accurate approximations of the solutions of (.16 for different values of Ω on the timescales of O(ε Application of the two timescales perturbation method This section presents the application of a two timescales perturbation method for constructing formal approximations of the solution of the infinite dimensional system of ordinary differential equations (.16. For a more complete overview of this perturbation method (i.e. the method of multiple scales the reader is referred to Refs. 53, 64, 65. A straight-forward expansion in ε may have secular terms which signals the nonuniform validity of approximations for large values of t. In order to get rid of these unbounded (so-called secular terms, we will use the two timescales perturbation method. These secular terms may occur on the right-hand side of Eq. (.16. To avoid these terms we introduce the fast time-scale t = t and slow time-scale t 1 = εt, and assume that u k (t can be expanded in a formal power series in ε, that is, u k (t; ε = w k (t, t 1 ; ε. (.18 In terms of the new variables, the following transformations are needed for the time derivatives: du k dt = w k t + ε w k t 1, (.19 d u k dt = w k t Substitution of Eqs. (.19 and (. into Eq. (.16 yields: + ε w k + ε w k. (. t t 1 t 1 w k t +ε w k t t 1 +(kπ w k = ε n±k is odd nk n k 4αΩ cos(ωtw n +8 ( V +α sin(ωt w n +O(ε. t (.1 13

27 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 14 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 4 An approximation of w k (t, t 1 ; ε is sought in the form w k (t, t 1 ; ε = w k (t, t 1 + εw k1 (t, t (. Substitution of Eq. (. into Eq. (.1 and then equalization of the coefficients of ε and ε in the resulting equation leads to the O(1-problem and O(ε-problem for w k and w k1 : O(1 : w k (, = for k = 1,,3, w k t φ(x sin(kπxdx, + (kπ w k =, (.3 t w k (, = 1 ψ(x sin(kπxdx, (.4 O(ε : w k1 t + (kπ w k1 = w k t t 1 + n±k is odd nk n k 4αΩ cos(ωt w n + 8(V + α sin(ωt w n, t (.5 w k1 (, =, w k1 (, = w k (,, (.6 t t 1 for k = 1,,3,.... The solution of the O(1-problem can be written as follows: w k (t, t 1 = A k (t 1 cos(kπt + B k (t 1 sin(kπt, (.7 where the amplitudes A k and B k are functions of slow time t 1, which are so determined as to make the solution of the O(ε-problem for w k1 (t, t 1 free of secular terms. As stated above, it is assumed that w k (t, t 1 and w k1 (t, t 1,... are bounded on timescales of O(ε 1 ; these secular (unbounded terms may destroy the accuracy of the approximations on long timescales, so they should be avoided. By plugging Eq.(.7 into Eq.(.5, the resulting equation (see appendix.a will give rise to resonances when Ω = (k + nπ, Ω = (k nπ or Ω = (n kπ with k ± n is odd. In other words, secular terms in the solution of the O(ε-problem will occur when Ω is an odd multiple of π. In what follows the following two resonance cases will be considered to prevent these secular (unbounded terms. case 1: Ω = mπ (m is positive odd integer case : Ω mπ (as opposed to Ω = mπ is referred to as near-resonance. We will consider these cases for m > 1, where m is a positive odd integer. The solution of the problem for m = 1 is studied in A general resonance case: Ω = mπ (m is positive odd integer In this case, it is assumed that the velocity fluctuation frequency (Ω of the axially moving string is equal to m th times the natural frequency of the string, where m is a positive odd integer, that is, Ω = mπ. In order to preclude the appearance of secular terms in w k1 (t, t 1, we substitute 14

28 .3. HARMONICALLY VARYING VELOCITY ABOUT A LOW MEAN SPEED 15 Ω = mπ into O(ε-problem (.5; the following conditions must be applied to A k (t 1 and B k (t 1 (see appendix.a: da k = (k + mb d t (k+m + (k mb (k m (m kb (m k, 1 db k = (k + ma d t (k+m + (k ma (k m + (m ka (m k, 1 (.8 where t 1 = αt 1 m and k = 1,, 3,... And for non positive indices k, the functions A k and B k are defined to be zero. For convenience, we will drop the bar from t 1. System (.8 is an infinite dimensional system of coupled ordinary differential equations (ODEs. The solution of the system (.8 for A k and B k can yield the amplitude-response and the energy of the system. However, it can clearly be seen from the system that there are infinitely many interactions between the vibration modes, and cannot be easily solved. In the subsequent sections, the system (.8 will be analyzed by using the truncation method and also the energy of the belt system will be computed in terms of infinite dimensional system..3.3 Application of the truncation method In this section the infinite dimensional system of coupled ordinary differential equations (.8 will be studied by using the truncation method, that is, the infinite dimensional system will be truncated to a finite dimensional one (that is, only a finite number of vibration modes are considered. For the case m = 1, it is shown in Ref. 48 that the truncated system (.8 has only purely imaginary eigenvalues and/or zero eigenvalues. In this study, we will consider the case m > 1, where m is a positive odd integer. We will truncate the system for m = 3 and m = 5 and use just some first few modes and neglect the higher order modes. For example, truncating the infinite dimensional system (.8 for m = 3 to the first four modes, we obtain: where Ẋ = AX (.9 A 1 4 B 1 4 A 1 X = B A 3, and A = 1. B 3 A 4 1 B 4 1 This system has eigenvalues ± i, ± i and,,,. Using the computer software package Maple, the eigenvalues of system (.8 have been computed up to 1 modes for m = 3 and m = 5 and are listed in Table.1. From Table.1, it can be seen that the eigenvalues of the truncated system are always either reals or purely imaginary or complex with both the positive and negative real part. It is well known in mathematics that in this case no conclusion can be drawn for the infinite dimensional system. 15

29 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 16 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 5 No.of modes m = 3 eigenvalues of matrix A (all multiplicity 1 ± 4 3, ± 6 4,, ± i 8 5, ± i, ±.4.48i 1 6 ± i, ±.4.48i, ±4.4i 1 7, ±4.4i, ±.89i, ±5.64i 14 8,, ±4.4i, ±.89i, ±5.64i 16 9,,, ±.89i, ±5.64i, ±8.48i 18 1,, ±.8i, ±.55i, ±8.48i, ±9.96i m = 5 1, 4 3, ± ±, ± 6 8 5, ±, ± 6 1 6,, ± 6,± i 1 7,,, ± i, ± i 14 8,, ± i, ± i, ± i 16 9, ± i, ± i, ±.64i, ±5.57i 18 1 ± i, ± i, ±.64i, ±5.57i, ±7.7i Dimension eigenspace of A Table.1: Approximations of the eigenvalues of the truncated system (.8 for m = 3 and m = Analysis of the infinite dimensional system (.8 In the preceding section, the infinite dimensional system (.8 was truncated to finite dimensional one and was shown that the mode approximation leads to the stable and/or unstable solution. In the following, we shall compute the energy of the belt system and demonstrate that the results obtained by applying the truncation method are not valid on long timescales, that is, on a timescales of O(ε 1 in all cases. By introducing X k (t 1 = ka k (t 1 and Y k (t 1 = kb k (t 1, system (.8 yields: { dxk dt 1 = k Y (m k + Y (k+m + Y (k m, dy k (.3 dt 1 = kx (m k + X (k+m + X (k m, for k = 1,, 3,..., and the functions X k and Y k are zero for non positive indices k. Then it can be deduced that: { X k Ẋ k = k X k Y (m k + X k Y (k+m + X k Y (k m, (.31 Y k Ẏ k = ky k X (m k + Y k X (k+m + Y k X (k m. 16

30 .3. HARMONICALLY VARYING VELOCITY ABOUT A LOW MEAN SPEED 17 By adding both the equations in (.31, and by taking the sum from k = 1 to, it follows that: 1 d (Xk dt + Y k =m (X (k+m Y k Y (k+m X k 1 k=1 k=1 + (1( X (m 1 Y 1 Y (m 1 X 1 + (( X (m Y Y (m X (.3. + (m ( X Y (m Y X (m + (m 1( X 1 Y (m 1 Y 1 X (m 1. By differentiating Eq. (.3 with respect to t 1 on both sides, we obtain (see appendix B: 1 d dt k=1 1 (X k + Y k = m and then by putting k=1 (X k + Y k = W (t 1 into Eq. (.33 yields: The solution of (.34 is: d W (t 1 dt 1 (Xk + Y, (.33 k=1 k 4m W (t 1 =. (.34 W (t 1 = C 1 e mt 1 + C e mt 1, (.35 where C 1 and C are arbitrary constants and can be determined by using the initial conditions. The energy of belt system can be approximated using the function W (t 1 (see in Ref. 48. For C 1, W (t 1 (so the energy increases exponentially if t 1 increases. Thus W (t 1 is unbounded in t 1 and increases as t 1 increases. This behavior is different from the behavior of A k and B k as obtained by applying the truncation method. If we apply the truncation method, we obtain the mixture behavior to the system (.8 due to combination of real, complex and purely imaginary eigenvalues. In other words, truncation method yields both the stable and unstable approximations for the (unstable solution. This implies that the approximations obtained by applying the truncation method to system (.8 are not accurate on long time scales, that is, on time-scales of order ε 1. In what follows, we shall investigate the (instability of an axially moving system in the neighborhood of resonances..3.5 Near resonance case : Ω = mπ + εδ In the previous section, the instability of the axially moving string was examined at resonances. In this subsection, we shall investigate the stability of the system near the resonances, that is, Ω mπ, where m is a positive odd integer. To do so, we express the nearness of Ω by employing the relation: Ω = mπ + εδ, (.36 where δ is a detuning parameter of O(1 and ε is a small dimensionless parameter, that is, < ε 1. Plugging Eq. (.36 into Eq. (.5 yields for the O(ε for w k1 : 17

31 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 18 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 6 w k1 t + (kπ w k1 = w k t t 1 + n±k is odd nk n k 4α ( mπ + εδ cos ( (mπ + εδt wn ( + 8 V + α sin ( w n (mπ + εδt. (.37 t In order to prevent the secular terms (by carrying out the same operations as described in section.3., A k and B k have to satisfy: da k = (k + m A d t (k+m sin(δ t 1 + B (k+m cos(δ t 1 1 (k m A (k m sin(δ t 1 B (k m cos(δ t 1 (m k A (m k sin(δ t 1 + B (m k cos(δ t 1, db k = (k + m A d t (k+m cos(δ t 1 + B (k+m sin(δ t 1 1 (k m A (k m cos(δ t 1 + B (k m sin(δ t 1 (m k A (m k cos(δ t 1 B (m k sin(δ t 1, (.38 where t 1 = αt 1 m and k = 1,, 3,..., and the functions A k and B k are defined to be zero for k. For convenience, we will again drop the bar from t 1. It should be noticed that for δ =, we can again obtain system (.8. The system (.38 is an infinite dimensional system of coupled ODEs, which is in fact not easy to solve for the functions A k and B k. In the following, we shall compute the energy of the system near the resonances and will demonstrate the behavior of the system for different values of the detuning parameter δ..3.6 Analysis of the infinite dimensional system(.38 By introducing X k (t 1 = ka k (t 1 and Y k (t 1 = kb k (t 1, system (.38 yields: dx k dt 1 = (k X (k+m sin(δt 1 + Y (k+m cos(δt 1 X (k m sin(δt 1 +Y (k m cos(δt 1 X (m k sin(δt 1 Y (m k cos(δt 1, dy k dt 1 = (k X (k+m cos(δt 1 + Y (k+m sin(δt 1 X (k m cos(δt 1 Y (k m sin(δt 1 X (m k cos(δt 1 + Y (m k sin(δt 1, (.39 for k = 1,, 3,..., and the functions X k, Y k are zero for k. Then it can be deduced from (.39 that: X k Ẋ k = (k X k X (k+m sin(δt 1 + X k Y (k+m cos(δt 1 X k X (k m sin(δt 1 + X k Y (k m cos(δt 1 X k X (m k sin(δt 1 X k Y (m k cos(δt 1, (.4 Y k Ẏ k = (k Y k X (k+m cos(δt 1 + Y k Y (k+m sin(δt 1 Y k X (k m cos(δt 1 Y k Y (k m sin(δt 1 Y k X (m k cos(δt 1 + Y k Y (m k sin(δt 1. 18

32 .3. HARMONICALLY VARYING VELOCITY ABOUT A LOW MEAN SPEED 19 By adding both equations in (.4, and then by taking the sum from k = 1 to, we obtain: 1 k=1 d dt 1 (X k + Y k = m (Y k X (k+m X k Y (k+m cos(δt 1 (X k X (k+m + Y k Y (k+m sin(δt 1 k=1 + (1(Y 1 Y (m 1 X 1 X (m 1 sin(δt 1 (X 1 Y (m 1 + Y 1 X (m 1 cos(δt 1 + ((Y Y (m X X (m sin(δt 1 (X Y (m + Y X (m cos(δt 1. + (m (Y Y (m X X (m sin(δt 1 (Y X (m + X Y (m cos(δt 1 + (m 1(Y 1 Y (m 1 X 1 X (m 1 sin(δt 1 (Y 1 X (m 1 + X 1 Y (m 1 cos(δt 1. By differentiating Eq. (.41 twice with respect to t 1, we obtain: 1 d 3 dt 3 k=1 1 (X k + Y k = δ k=1 d dt 1 (X k + Y k + m and then by putting k=1 (X k + Y k = W (t 1 into Eq. (.4 yields: d 3 W (t 1 dt 3 1 Integration of Eq. (.43 w.r.t. t 1, gives: d W (t 1 dt 1 k=1 (.41 d (Xk dt + Y k, ( (δ 4m dw (t 1 dt 1 =. (.43 + (δ 4m W (t 1 = D 1, (.44 where D 1 is a constant of integration. Solution of Eq. (.44 leads to the following results: for δ < m : W (t 1 = D 1 4m δ + D cosh( 4m δ t 1 + D 3 sinh( 4m δ t 1, for δ = m : W (t 1 = D 1 + D t D 3t 1, (.45 for δ > m : W (t 1 = D 1 δ 4m + D cos( δ 4m t 1 + D 3 sin( δ 4m t 1, where D 1, D, and D 3 are constants of integration. The important conclusions of these solutions are: for δ < m, W (t 1 increases exponentially, for δ = m, W (t 1 increases polynomially; so W (t 1 (the energy of the infinite dimensional system is unbounded for δ m and finally for δ > m, W (t 1 (so the energy is bounded due to the trigonometric solutions. 19

33 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 7.4 Harmonically varying velocity about a relatively high constant mean speed The preceding sections considered the transverse vibrations of an axially moving string subject to the harmonically varying velocity about a low constant mean speed. This section discusses the analysis of the governing equations of motion (.5-(.7 with a time-varying velocity about a relatively high constant mean velocity. The lowest resonance case for the problem is studied in 49. In this study, we generalize the work of 49 to all higher resonance cases. Assuming that the velocity is harmonically varying about a constant mean velocity V, one writes V (t = V + εα sin(ωt, (.46 where V, α and Ω are some positive constants, and ε is a small dimensionless parameter with < ε 1. Unlike the low mean speed case, the mean speed V is assumed to be O(1 in this case. In other words, the mean speed V is considered to be of same order as the wave speed c. Substitution of Eq.(.46 into an initial-boundary value problem (.5-(.7 yields the following perturbation problem: ( u tt + V u xt + (V 1u xx = ε α sin(ωtu xt V α sin(ωtu xx αω cos(ωtu x + O(ε, (.47 with the boundary and initial conditions (.6 and (.7 respectively. In the following subsection, the system (.47 will be analyzed using the two timescales perturbation method in conjunction with the Laplace transform method to find approximations valid on long timescales, that is, on a timescales of O(ε Application of the two timescales perturbation method In this section the application of two timescales perturbation method to the perturbation problem (.47 is considered. Assuming an expansion for u(x, t of the form u(x, t; ε = v(x, t, t 1 v (x, t, t 1 + εv 1 (x, t, t , (.48 in which t = t and t 1 = εt are the usual fast and slow timescales. In terms of the new variables, the following transformations are needed for the time derivatives: u t = v t + ε v t 1, (.49 u t = v t + ε v + ε v t t 1 t (.5 1 Substituting Eqs. (.48-(.5 into Eq. (.47, separating terms at each order of ε, we obtain the O(1 and O(ε-problem as follows: O(1 : v t v + V t x (1 V v =, (.51 x

34 .4. HARMONICALLY VARYING VELOCITY ABOUT A RELATIVELY HIGH CONSTANT MEAN SPEED 1 O(ε : v 1 t + V v 1 t x + (V 1 v 1 x = v t t 1 V v t 1 x α sin(ωt v t x V α sin(ωt v x αω cos(ωt v x (.5 The solution of the O(1 problem can be found by means of the Laplace transform method 49. v (x, t, t 1 = F 1n (x ( A n (t 1 cos(ω n t B n (t 1 sin(ω n t +F n (x ( A n (t 1 sin(ω n t + B n (t 1 cos(ω n t, (.53 where F 1n (x = cos(nπ(v + 1x cos(nπ(v 1x, F n (x = sin(nπ(v + 1x sin(nπ(v 1x, (.54 and ω n = nπ(1 V, n Z+ are the natural frequencies of the belt system. In Eq. (.53, A n (t 1 and B n (t 1 are still arbitrary functions and can be used to eliminate secular terms in the solution of the O(ε-problem. Substituting Eqs. (.53 and (.54 into Eq. (.5, the O(ε-problem becomes where v 1 t v 1 + V t x + (V 1 v 1 x An ( Θ n (x, t 1 = t 1 + = sin(ω n t Θ n + cos(ω n t Θ n F n F 1n ω n V x An ( F 1n Θ n (x, t 1 = F t n ω n V 1 x ( F1n ϕ n (x, t 1 = α A n x ω F n n V x ( ϕ n (x, t 1 = α A n sin(ωt sin(ω n t ϕ n + sin(ωt cos(ω n t ϕ n +cos(ωt sin(ω n t ζ n + cos(ωt cos(ω n t ζ n, F n x ω F 1n n V x and + B ( n t 1 + B ( n t 1 F 1n F n ω n + V x,, F n F 1n ω n V x ( Fn + B n x ω F 1n n + V x, ( F1n + B n x ζ n (x, t 1 = αω ω n V F n x, F n A n x + B n ζ F 1n n (x, t 1 = αω A n x B n F 1n x F n x,. (.55 (.56 According to Eq. (.55, if the axial speed variation frequency Ω is equal to (or close to an integer times a natural frequency ω n of the linear generating system (.51; resonances may occur. The first resonance case, that is Ω = π(1 V has been studied in Ref.49. In this work, the k th resonance case, that is, Ω = k π(1 V where k Z + will be investigated. 1

35 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 8.4. Ω = k π(1 V, a general resonance case In this case, we assume that Ω = ω k, that is, Ω = k π(1 V, where k Z + and fixed. Plugging Ω = k π(1 V into Eq. (.55 yields v 1 t v 1 + V t x + (V 1 v 1 x + 1 = sin(ω n t Θ n + cos(ω n t Θ n cos(ω n k t ( ϕ n + ζ n + cos(ωn+k t ( ϕ n + ζ n +sin(ω n k t ( ϕ n + ζ n + sin(ωn+k t ( ϕ n + ζ n, (.57 where functions Θ, Θ, ϕ, ϕ, ζ and ζ are given by Eq. (.56. In Eq. (.57, it should be observed that ω n = ω n and ω =. In order to find v 1 at all higher resonances, we will apply the Laplace transform method to Eq. (.57. After taking the Laplace transform on both sides of Eq. (.57, we will calculate the poles, and then the residues, and then use the convolution integral theorem to find the inverse Laplace transform, we obtain k { 1 v 1 (x, t, t 1 = 4 (Ψ 1n + Ψ n (Ψ1 n + Ψ3 1n }(t sin(ω n t { 1 1( 1 + ( Ψ1n Ψ 4 n + Ψ 8 1n n } Ψ3 (t cos(ω n t { 1 ( + Ψ1n + 4 Ψ 1( 1 n + Ψ 8 n + Ψ3 1n + Ψ n + 1n } Ψ4 (t sin(ω n t n=k +1 { 1 1( 1 + ( Ψ1n Ψ 4 n + Ψ 8 1n Ψ3 n + Ψ 1n n } Ψ4 (t cos(ω n t k 1 1 } + Ψ 8{ 5 1n + Ψ6 n (t sin(ω n t + 1 { } Ψ 6 8 1n + Ψ5 n (t cos(ω n t + terms with non-secular behavior, (.58 where Ψ 1n (x, t 1 = w n F 1n + p n F n, Ψ n (x, t 1 = w n F n p n F 1n, Ψ 1n (x, t 1 = w n F 1n + p n F n, Ψn (x, t 1 = w n F n p n F 1n, Ψ l 1n (x, t 1 = wnf l 1n + p l nf n, Ψ l n (x, t 1 = wnf l n p l nf 1n, (.59

36 .4. HARMONICALLY VARYING VELOCITY ABOUT A RELATIVELY HIGH CONSTANT MEAN SPEED 3 with l = 1,, 3, 4, 5, 6, and w 1 n = 1 w n = 1 w 3 n = 1 w n = 1 w 5 n = 1 1 w 6 n = 1 1 ϕ n+k + ζ n+k ( F πn n dx, p 1 n = 1 ϕ n k + ζ n k ( F πn n dx, p n = 1 ϕ n+k + ζ n+k πn 1 1 ( F n dx, p 3 n = ϕ n k + ζ n k ( F πn n dx, p 4 n = 1 ϕ k n + ζ k n πn w n (x, t 1 = 1 w n (x, t 1 = 1 ϕ k n + ζ k n πn 1 1 ( F n dx, p 5 n ( F n dx, p 6 n Θ n (x, t 1 ( F πn n dx, Θ n (x, t 1 ( F πn n dx, = 1 = ϕ n+k + ζ n+k ( F πn 1n dx, ϕ n k + ζ n k ( F πn 1n dx, ϕ n+k + ζ n+k ( F πn 1n dx, 1 p n (x, t 1 = 1 p n (x, t 1 = 1 ϕ n k + ζ n k ( F πn 1n dx, ϕ k n + ζ k n ( F πn 1n dx ϕ k n + ζ k n ( F πn 1n dx. 1 1 Θ n (x, t 1 ( F πn 1n dx, Θ n (x, t 1 ( F πn 1n dx. (.6 It follows from Eq. (.58 that the solution of the O(ε-problem does not contain secular terms if and only if Ψ 1n (x, t 1 + Ψ ( n (x, t Ψ 1 n (x, t 1 + Ψ 3 1n (x, t 1 Ψ 5 1n (x, t 1 + Ψ 6 n (x, t 1 =, ( Ψ 1n (x, t 1 Ψ n (x, t Ψ 1 1n (x, t 1 Ψ 3 n (x, t 1 + Ψ 6 1n (x, t 1 + Ψ 5 n (x, t 1 =, (.61 for n = 1,,... k 1, and Ψ 1n (x, t 1 + Ψ ( n (x, t Ψ 1 n (x, t 1 + Ψ 3 1n (x, t 1 + Ψ n (x, t 1 + Ψ 4 1n (x, t 1 =, ( Ψ 1n (x, t 1 Ψ n (x, t Ψ 1 1n (x, t 1 Ψ 3 n (x, t 1 + Ψ 1n (x, t 1 Ψ 4 n (x, t 1 =, (.6 for n = k, k + 1,.... By defining A and B, it follows from Eqs. (.61 and (.6 that for all n = 1,,..., Ψ 1n (x, t 1 + Ψ ( n (x, t Ψ 1 n (x, t 1 + Ψ 3 1n (x, t 1 + Ψ n (x, t 1 + Ψ 4 1n (x, t 1 Ψ 5 1n (x, t 1 + Ψ 6 n (x, t 1 =, ( Ψ 1n (x, t 1 Ψ n (x, t Ψ 1 1n (x, t 1 Ψ 3 n (x, t 1 + Ψ 1n (x, t 1 Ψ 4 n (x, t ( Ψ 5 n (x, t 1 + Ψ 6 1n (x, t 1 =. We rewrite Eq. (.63 in a more simple form as { F 1n (xφ 1n (t 1 + F n (xφ n (t 1 =, F 1n (xφ n (t 1 F n (xφ 1n (t 1 =, (.64 3

37 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 4 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 9 where ( Φ 1n (t 1 = w n (t 1 p n (t w n 3 (t 1 p 1 n (t 1 + w n 4 (t 1 p n (t 1 w n 5 (t 1 p 6 n (t 1, ( Φ n (t 1 = w n (t 1 + p n (t w n 1 (t 1 + p 3 n (t 1 + w n (t 1 + p 4 n (t 1 + w n 6 (t 1 p 5 n (t 1. (.65 System (.64 can be viewed as a system for two unknowns Φ 1n (t 1 and Φ n (t 1. It should be observed from Eq. (.64 that the determinant of this system is equal to (F 1n + (F n, for all x (, 1. It then follows that Φ 1n (t 1 = and Φ n (t 1 =, or equivalently, ( w n (t 1 p n (t w n 3 (t 1 p 1 n (t 1 + w n 4 (t 1 p n (t 1 w n 5 (t 1 p 6 n (t 1 =, ( w n (t 1 + p n (t w n 1 (t 1 + p 3 n (t 1 + w n (t 1 + p 4 n (t 1 + w n 6 (t 1 p 5 n (t 1 =. (.66 It can be seen that the system (.66 involves the functions da n dt 1, db n dt 1, A n (t 1 and B n (t 1. After some lengthy, but elementary calculations, the following system of ODEs for A n (t 1 and B n (t 1 can be obtained from (.66: = (n + k d t µ k A (n+k + η k B (n+k 1 + (k n µ k A (k n + η k B (k n, = (n + k d t η k A (n+k + µ k B (n+k 1 + (k n η k A (k n µ k B (k n, da n db n (n k µ k A (n k η k B (n k (n k η k A (n k + µ k B (n k for n = 1,, 3,..., and the functions A n and B n are zero for n, where (.67 t 1 = αt 1 k, µ k = ( 1k +1 sin(k πv, η k = ( 1 k cos(k πv 1. (.68 For convenience, we will drop the bar from t 1. System (.67 is an infinite dimensional system of coupled ordinary differential equations (ODEs, which exhibits infinitely many interactions between the vibration modes for Ω = ω k. The solution of this coupled ODEs for the functions A n (t 1 and B n (t 1 will yield the amplitude-response and the energy of the system. As described, there are infinitely many interactions between the vibration modes, the closed-form solution of the coupled ODEs is not easily attainable. In following subsections we deal with investigating the system (.67 by truncating the infinite dimensional system to finite dimension one and by computing the energy of the belt system in terms of the infinite dimensional system..4.3 Application of the truncation method In this subsection, the infinite dimensional system (.67 will be truncated to finite one. To do so, we will use some first few modes and neglect the higher order modes. We truncate the system for k =, k = 3, k = 4, k = 5 upto 1 modes by using the computer software package Maple. The truncation of the system (.67 for k = 1 up to 1 modes is given in Ref.49. It is shown that the system yields either the zero or purely imaginary eigenvalues in the first resonance case. From Table., Table.3 and Table.4, it can be seen that the eigenvalues of the truncated system are always either real, complex or purely imaginary. It is well known in mathematics that in this case no conclusion can be drawn for the infinite dimensional system. Moreover, it can 4

38 .4. HARMONICALLY VARYING VELOCITY ABOUT A RELATIVELY HIGH CONSTANT MEAN SPEED 5 No.of modes eigenvalues of matrix A k = 1 ± µ + η,, ± µ + η 4 3,, (± i µ + η, (± i µ + η, 6 4 (± i µ + η, (± i µ + η, ± 8 (µ + η i, ± (µ + η i 5 (± i µ + η, (±.8 4.3i µ + η, ± 1 (µ + η i, ± (µ + η i, ±.84 µ + η 6,, (± i µ + η, (±.8 4.3i µ + η, ±4 1 (µ + η i, ± (µ + η i, ±.84 µ + η 7,, (± i µ + η, (± i µ + η, (± i 14 µ + η, (± i µ + η, ±4 (µ + η i, ±4 (µ + η i 8 (± i µ + η, (± i µ + η, (± i 16 µ + η, (± i µ + η, ±8.65 µ + η i, ±8.65 µ + η i, ±.6 µ + η i, ±.6 µ + η i 9 (± i µ + η, (± i µ + η, (± i 18 µ + η, (±.1 1.i µ + η, ±8.65 µ + η i, ±8.65 µ + η i, ±.6 µ + η i, ±.6 µ + η i, ±.77, 1,, (± i µ + η, (± i µ + η, (± i µ + η, (±.1 1.i µ + η, ±11.78 µ + η i, ±11.78 µ + η i, ±4.61 µ + η i, ±4.61 µ + η i, ±.77 µ + η Dimension eigenspace of A Table.: Approximations of the eigenvalues of the truncated system (.67 for k =. be seen from these tables that by increasing the number of modes the eigenvalues do not seem to converge. In what follows, we will examine the infinite dimensional system of coupled ODEs (.67 by computing the energy of the axially moving system..4.4 Analysis of the infinite dimensional system (.67 By introducing X n (t 1 = na n (t 1 and Y n (t 1 = nb n (t 1 in the system (.67, we obtain dx n dt 1 = n µ k X (n+k + η k Y (n+k µ k X (n k + η k Y (n k + µ k X (k n +η k Y (k n, (.69 dy n dt 1 = n η k X (n+k + µ k Y (n+k η k X (n k µ k Y (n k + η k X (k n µ k Y (k n, 5

39 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 6 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 1 No.of modes k = 4 eigenvalues of matrix A 1,,, ± µ 4 + η ± µ 4 + η 4, ± 3(µ 4 + η 4, ± 3(µ 4 + η 4 6 4,, ± µ 4 + η 4, ± 3(µ 4 + η 4, ± 8 3(µ 4 + η 4 5,,,, ± (µ 4 + η 4, ± (µ 4 + η 4 i, ± 1 (µ 4 + η 4 i 6,,,, (± i µ 4 + η 4, (±1. 3.3i 1 µ 4 + η 4, ± (µ 4 + η 4 i, ± (µ 4 + η 4 i 7,, (± i µ 4 + η 4, (±1. 3.3i 14 µ 4 + η 4, ±.51 µ 4 + η 4 i, ±.51 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i 8 (± i µ 4 + η 4, (±1 3.3i µ 4 + η 4, ± µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±.51 µ 4 + η 4 i, ±.51 µ 4 + η 4 i, ±5.66 µ 4 + η 4 i, ±5.66 µ 4 + η 4 i 9,, (± i µ 4 + η 4, (±1 3.3i 18 µ 4 + η 4, ±.51 µ 4 + η 4 i, ±.51 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±7.4 µ 4 + η 4 i, ±7.4 µ 4 + η 4 i 1,, (± i µ 4 + η 4, (± i µ 4 + η 4, ±.51 µ 4 + η 4 i, ±.51 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±4.9 µ 4 + η 4 i, ±7.4 µ 4 + η 4 i, ±7.4 µ 4 + η 4 i, ±1.68 µ 4 + η 4 Dimension eigenspace of A Table.3: Approximations of the eigenvalues of the truncated system (.67 for k = 4. for n = 1,, 3,..., and the functions X n and Y n are defined to be zero for all n. Then it can be deduced that: X n Ẋ n = n µ k X (n+k X n + η k Y (n+k X n µ k X (n k X n +η k Y (n k X n + µ k X (k nx n + η k Y (k nx n, (.7 Y n Ẏ n = n η k X (n+k Y n + µ k Y (n+k Y n η k X (n k Y n µ k Y (n k Y n + η k X (k ny n µ k Y (k ny n. By adding both the equations in (.7, and by taking the sum from n = 1 to, it follows 6

40 .4. HARMONICALLY VARYING VELOCITY ABOUT A RELATIVELY HIGH CONSTANT MEAN SPEED 7 No.of modes k = 3 eigenvalues of matrix A (all multiplicity 1 ± (µ 3 + η 3 4 3, ± (µ 3 + η 3 6 4,, ± (µ 3 + η 3 i 8 5, (± i µ 3 + η 3, (±.4.48i µ 3 + η (± i µ 3 + η 3, (±.4.48i µ 3 + η 3, ±4.4 1 µ 3 + η 3 i 7, ±4.4 µ 3 + η 3 i, ±.89 µ 3 + η 3 i, ±5.64 µ 3 + η 3 i 14 8,, ±4.4 µ 3 + η 3 i, ±.89 µ 3 + η 3 i, ± µ 3 + η 3 i, 9,,, ±.89 µ 3 + η 3 i,±5.64 µ 3 + η 3 i, ± µ 3 + η 3 i 1,, ±.8 µ 3 + η 3 i, ±.55 µ 3 + η 3 i, ±8.48 µ 3 + η 3 i, ±9.96 µ 3 + η 3 i k = 5 1, 4 3, ± 6(µ 5 + η ± µ 5 + η 5, ± 6(µ 5 + η 5 8 5, ± µ 5 + η 5, ± 6(µ 5 + η 5 1 6,, ± 6(µ 5 + η 5, ± (µ 5 + η 5 i 1 7,,, ± (µ 5 + η 5 i, ± (µ 5 + η 5 i 14 8,, ± (µ 5 + η 5 i, (± i µ 5 + η 5, (± i 16 µ 5 + η 5 9, (± i µ 5 + η 5, (± i µ 5 + η 5, ± µ 5 + η 5 i, ±5.57 µ 5 + η 5 i 1 (± i µ 5 + η 5, (± i µ 5 + η 5, ±.64 µ 5 + η 5 i, ±5.57 µ 5 + η 5 i, ±7.7 µ 5 + η 5 i Dimension eigenspace of A Table.4: Approximations of the eigenvalues of the truncated system (.67 for k = 3 and k = 5. that: 7

41 CHAPTER. ON RESONANCES AND THE APPLICABILITY OF GALERKIN S TRUNCATION METHOD FOR AN 8 AXIALLY MOVING STRING WITH TIME-VARYING VELOCITY 11 1 d dt 1 (X n + Y n = k η k (Y n X (n+k X n Y (n+k µ k (X n X (n+k + Y n Y (n+k + (1 µ k (X 1 X (k 1 Y 1 Y (k 1 + η k (X 1 Y (k 1 + Y 1 X (k 1 + ( µ k (X X (k Y Y (k + η k (X Y (k + Y X (k. + (k µ k (X (k X Y (k Y + η k (X (k Y + Y (k X + (k 1 µ k (X (k 1X 1 Y (k 1Y 1 + η k (X (k 1Y 1 + Y (k 1X 1. (.71 Differentiating Eq. (.71 with respect to t 1 on both sides (by following a similar calculation as in appendix B, we obtain: 1 d dt 1 (Xn + Yn = k (ηk + µ k (Xn + Yn, (.7 and then by putting (X n + Y n = W (t 1 into Eq. (.7 yields: where d W (t 1 dt 1 4k (η k + µ k W (t 1 =, (.73 µ k = ( 1 k +1 sin(k πv, η k = ( 1 k cos(k πv 1, (.74 with k Z +. The solution of Eq. (.73 is W (t 1 = K 1 e k η k +µ k t 1 + K e k η k +µ k t 1, (.75 where K 1 and K are both constants of integration and k Z +. From Eq. (.75 it is clear that for K 1, W (t 1 (so the energy increases if t 1 increases. In a similar way as discussed in section.3, we conclude that the truncation method leads to erroneous results on long timescales, that is, on time scales of O(ε 1..5 Conclusions and remarks This chapter is devoted to investigate the linear transverse vibrations of an axially moving string traveling with a harmonically varying velocity. In the analysis, it has been assumed that the string moves in one direction with a time-varying speed V (t, that is, V (t = ε(v + α sin(ωt and/or V (t = V + εα sin(ωt, where < ε 1 and V, α, Ω are some positive constants. A two timescales perturbation method has been employed in search of infinite mode approximate solutions. In both of these velocity cases, it turned out that there are infinitely many values of Ω 8

42 .5. CONCLUSIONS AND REMARKS 9 giving rise to resonances in the axially moving string. For V = O(ε, in fact that happens when the velocity fluctuation frequency Ω is equal (or close to an odd multiple of the natural frequency of the constant velocity system, that is, when Ω = mπ, where m is positive odd integer. However, the velocity fluctuation frequency Ω gives rise to resonance in the system for relatively high mean speed, that is, for V = O(1 if Ω = k π(1 V, where k Z +. The infinite dimensional system of coupled ordinary differential equation has been analyzed using the truncation method and by computing the energy of the belt system. In order to analyze the infinite dimensional system by using the truncation method, we considered the first few modes and neglected the higher order modes. For the low mean speed case, it is shown in Table.1 that the 4 th, 7 th, 8 th, 9 th and 1 th vibration modes lead to the stable approximations, while 1 st, nd, 3 rd, 5 th and 6 th mode lead to the unstable approximations due to the real and complex eigenvalues with both the positive and negative real parts. In Tables.,.3 and.4, analogous situation of stable and unstable approximations for different vibration modes can be witnessed. In other words, the eigenvalues obtained through the truncation method do not converge for large times. It is, therefore, observed that the truncation (in case of instability leads to the stable and/or unstable approximations of the (unstable solution. However, the analysis of energy in terms of the infinite dimensional system for both the velocity cases yields exponential behavior that grows without bound. In addition, the energy analysis of the system for the low speed cases in the neighborhood of resonances lead to the following results: δ < m: the energy of the belt system increases exponentially and hence is unbounded, δ = m: the energy of the belt system increases polynomially and also unbounded, and δ > m: the energy of the belt system varies trigonometrically and hence is bounded (δ is a detuning parameter. It turns out that all approximations obtained for energy are valid on long timescales, that is, on timescales of order ε 1. This is due to the fact that all energy approximations confirm the instability in the system, but this instability is not always obtained when the truncation method is applied. It, is therefore, concluded that the truncation method is no more applicable on long timescales. Moreover, it is also to be expected that for other boundary conditions on the bounded domain, similar difficulties and inaccuracies may occur. 9

43

44 Chapter 3 On the asymptotic approximation of the solution of an equation for a non-constant axially moving string If you want to find the secrets of the universe, think in terms of energy, frequency and vibration. Nikola Tesla The parametric (instability of an axially moving string under harmonically varying velocity about low and relatively high constant mean speed cases in terms of the response of individual vibrational modes and the energy was discussed in Chapter. However, the dynamic response analysis of the system also plays a very important role to examine the (instability in axially moving system. In this chapter, the energy and the response, such as the amplitude of the system subject to a (low constant mean speed case are computed by means of an analytic-approximate technique, i.e. the two timescales perturbation method in combination with the method of characteristic coordinates. It is found that infinitely many resonances in the system can arise, when the velocity fluctuation frequency is equal (or close to an odd multiple of the lowest natural frequency of the system. The results obtained for the amplitude-response and the energy of the axially moving system from the analytic-approximate technique are seen in full agreement with those computed using the finite-difference numerical technique. It is shown that both the energy and the amplituderesponse of the system grow on long timescales, that is, on a timescale of order ε Introduction Axially moving strings have applications in many mechanical systems such as serpentine belts, aerial cables, power transmission belts, plastic films, magnetic tapes, paper sheets and textile fibers. To meet the design challenges in these so varied technical applications, it is essential to develop better understanding of the dynamics of such systems. The transverse vibrations and stability are of substantial interest in the study of dynamics of axially moving systems. The transverse vibrations of axially moving strings have been treated at length in early works such as Refs. 3, 4, 5. Furthermore, the study of dynamics of axially moving strings with time This chapter is slightly revised version of published article - On the asymptotic approximation of the solution of an equation for a non-constant axially moving string, J. Sound Vibr., vol. 367, p:3-18, January

45 3 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING dependent velocity has been gaining much attention for the last two decades see for instance Refs. 6, 7, 9, 1, 1, 4, 48, 49, 58. The equation of motion for transverse vibrations of moving string with time-varying velocity was first derived by Miranker 5, but no solution was presented in the paper. However, several techniques have been employed to analyze the governing equations of motion for axially moving strings with time-varying speed over the past 1 years. The authors in Refs. 11, 5 have used the two timescales perturbation method in combination with the Fourier series and the Laplace transform method to obtain asymptotic approximations of the solutions for the equation of axially moving strings. It is observed in these papers that the elimination of secular terms in fact leads to an infinite dimensional system of coupled ordinary differential equations. It is moreover shown that the infinite dimensional system is mathematically difficult to solve due to infinitely many interactions between the vibration modes, and that for string like problems truncating the infinite dimensional system to a finite one is usually not allowed. Although this truncation method is a very popular method 16, 6, one should be careful in using this method for string-like problems. However, it was possible to study the energy of the system by using of the infinite dimensional system in order to prove the (instabilities of vibrations of axially moving strings 1. In this chapter, an initial boundary value problem for an axially moving string with harmonically varying velocity about a low constant mean speed will be studied. The asymptotic approximations of the analytic solution are constructed on long timescales by using the method of characteristic coordinates (in combination with the two timescales perturbation method without using the Fourier series. The technique of characteristic coordinates outlined in this chapter turns out to be suitable to develop asymptotic solutions of initial-boundary value problems without computational difficulties induced by truncating infinite Fourier series. For more details concerning this kind of method the interested reader is referred to 66, 67, 68, 69. In addition, Wu and Zhu 7 used the method of characteristics to investigate the parametric instability in a taut string subject to the time-varying boundary conditions. Meanwhile, it will also be proved in this chapter that the elimination of secular terms indeed leads to the same infinite dimensional system of coupled ordinary differential equations obtained in chapter to study the same problem. We therefore, alternatively, find the solution of the infinite dimensional system in this study. Furthermore, we will use the particular harmonic initial values in order to find the first-term asymptotic approximation in terms of the amplitude-response and compare it with the numerical results obtained using the finite difference numerical technique directly from the governing partial differential equation. The energy of the belt system will also be computed from the asymptotic approximations and will be compared with the energy obtained in 1, 48. In addition, the numerical simulation of the energy will also be presented. This chapter is organized as follows. Section 3. presents the governing equations of motion for an axially moving string. Formal approximations are constructed using the two timescales perturbation method in combination with the method of characteristic coordinates in section 3.3. Numerical verifications of all approximations are presented in section 3.4. The conclusions of this chapter are presented in section Equations of motion An axially moving string with length L is shown in Figure 1.3. The string moves with a time dependent velocity V between two fixed end points. The linear equation of motion that governs the transverse vibrations of the axially moving string is derived from Hamilton s principle (see Ref. 6 and can be written as follows: 3

46 3.. EQUATIONS OF MOTION 33 u tt + V u xt + V t u x + (V c u xx =, t >, < x < L, (3.1 where u(x, t is the displacement in the transverse direction, x denotes the axial position along the string, t is the time, c > is the wave speed, and L is the distance between the pulleys. Furthermore, it is assumed that the velocity V is small with respect to the wave speed, that is, V < c. This is a reasonable assumption in already many applications. The boundary and initial conditions for u(x, t are respectively given by and u(, t = u(l, t =, t, (3. u(x, = φ(x, and, u t (x, = ψ(x, < x < L. (3.3 Transforming the Eqs.(3.1-(3.3 into non-dimensional form, one obtains u tt u xx = V t u x V u xt V u xx, t, < x < 1, (3.4 with boundary conditions and initial conditions u(, t = u(1, t =, t, (3.5 u(x, = φ(x, and u t (x, = ψ(x, < x < 1, (3.6 where the following dimensionless quantities are used: x = x L, V = V c, t = ct L, u(x, t u (x, t = L, φ (x = φ(x L, ψ (x = ψ(x c The asterisks indicating the dimensionless quantities are dropped in Eqs.(3.4-(3.6 and henceforth. In this paper, the velocity of the string is assumed to be a small harmonic variation about a constant low mean speed V (t = ε(v + α sin(ωt, (3.8 where α is a amplitude, Ω is a velocity fluctuation frequency and ε is a dimensionless small parameter with < ε 1. Additionally, it is also assumed that V > α, which guarantees that the belt will always move forward in one direction. When the velocity function (3.8 is substituted into Eq. (3.4, one obtains the following perturbation problem u tt u xx = ε αω cos(ωtu x (V + α sin(ωtu xt + O(ε, (3.9 (3.7 with boundary conditions and initial conditions u(, t; ε = u(1, t; ε =, t, (3.1 u(x, ; ε = φ(x, and u t (x, ; ε = ψ(x, < x < 1. (3.11 In the following sections, the mathematical techniques for obtaining the solutions of the governing equations of motion (3.9-(3.11 will be presented. 33

47 34 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING 3.3 The construction of asymptotic approximations In this section a two timescales perturbation method in conjunction with the method of characteristic coordinates will be used to construct asymptotic approximations of the solutions of the initial-boundary value problem (3.9-(3.11, which are valid on long timescales (that is, on timescales of order 1 ε. The straightforward expansion turns out to be not applicable to solve (3.9-(3.11 due to the presence of secular (unbounded terms. For a more complete overview of the perturbation method the reader is referred to Refs. 53, 64, 65. To avoid computational difficulties with and errors due to truncation of the Fourier series representation for the solution of the initial-boundary value problem, it is convenient to use characteristic coordinates σ = x t, ξ = x + t. In this approach the initial-boundary value problem is transformed into an initial value problem by extending the dependent variable u(x, t as well as the initial values φ(x and ψ(x to odd and -periodic functions in x 71. This is accomplished by multiplying each term in Eq.(3.9 which is not already odd in x, (i.e., terms like u x and u xt with H(x (see also 1, 48, where H(x = 4 sin((n 1πx. (3.1 (n 1π The function H is an odd and -periodic in x, and on < x < 1, H is equal to 1. So, Eq.(3.9 then becomes ( u tt u xx = ε αω cos(ωth(xu x V + α sin(ωt H(xu xt + O(ε. (3.13 Eq.(3.13 is solved by using the two timescales perturbation method together with the method of characteristic coordinates. Let us assume that the solution u(x, t of (3.13 is a function depending on the characteristic variables σ, ξ and the slow timescale τ, that is, u(x, t = w(σ, ξ, τ, (3.14 where σ = x t, ξ = x + t, τ = εt. The introduction of σ, ξ, and τ leads to the following transformations: u t = w σ + w ξ + ε w τ, (3.15 u t = w σ + w ξ w σ ξ + ε( w ξ τ w + ε w σ τ τ, (3.16 Substitution of(3.14-(3.19 into (3.13 yields: 4w σξ = ε u x = w σ + w ξ, (3.17 u x = w σ + w ξ + w σ ξ, (3.18 u x t = w σ + w ξ + ε( w σ τ + w (3.19 ξ τ ( Ω(ξ σ (w ξτ w στ αω cos ( V + α sin ( Ω(ξ σ ξ + σ H( 34 H ( ξ + σ (w σ + w ξ + O(ε. ( w σσ + w ξξ (3.

48 3.3. THE CONSTRUCTION OF ASYMPTOTIC APPROXIMATIONS 35 Furthermore, the function w can be expanded in power series in ε: w(σ, ξ, τ = w (σ, ξ, τ + εw 1 (σ, ξ, τ + O(ε, (3.1 where w i (σ, ξ, τ and all of its derivatives are assumed to be of O(1. Inserting Eq.(3.1 into Eq.(3. and equating to zero the coefficients of like powers of ε in the resulting equation, we have the following equations: O(1: 4w σξ =, < σ < ξ <, τ >, (3. O(ε: ( + w (σ, σ, = φ (σ, < σ = ξ <, τ =, (3.3 w σ (σ, σ, + w ξ (σ, σ, = ψ (σ, < σ = ξ <, τ =, (3.4 ( Ω(ξ σ 4w 1σξ = w ξτ + w στ αω cos V + α sin ( Ω(ξ σ ξ + σ H( H ( ξ + σ (w σ + w ξ (w σσ w ξξ, < σ < ξ <, τ >, (3.5 w 1 (σ, σ, = φ 1 (σ, < σ = ξ <, τ =, (3.6 w 1σ (σ, σ, + w 1ξ (σ, σ, = w τ (σ, σ, + ψ 1 (σ, < σ = ξ <, τ =, (3.7 in which the function H is defined as in (3.1, and where φ i (σ is the O(ε i -part of φ(σ, and ψ i (σ is the O(ε i -part of ψ(σ. The general solution of (3.-(3.4 is obtained by direct integration. Hence, the solution of Eq.(3. can be expressed as w (σ, ξ, τ = f (σ, τ + g (ξ, τ, (3.8 where f and g are arbitrary functions of the characteristic variables σ and ξ and the slow timescale τ, which in turn are functions of x and t. The functions f and g can be obtained by demanding that w 1 does not contain secular terms and that w satisfies the initial conditions (3.3 and (3.4, implying that f and g have to satisfy f (σ, + g (σ, = φ (σ and f (σ, + g (σ, = ψ (σ, where the prime indicates a derivative with respect to the first argument. By eliminating g (σ, from the previous two equations, it follows that f (σ, = 1 (φ (σ ψ (σ. From the odd and -periodic extension of the dependent variable of the problem (3.9-(3.11, it also follows that f and g have to satisfy g (σ, τ = f ( σ, τ and f (σ, τ = f (σ +, τ for - < σ < and τ. Equation (3.5 can now be expressed in terms of the characteristic variables in the form ( Ω(ξ σ ( ξ + σ 4w 1σξ = g ξτ + f στ αω cos H (f σ + g ξ ( ( Ω(ξ σ + V + α sin H ( ξ + σ (f σσ g ξξ. (3.9 In order to obtain an expansion for w 1 valid for times as large as O(ε 1 demands that w 1σ and w 1ξ must be of O(1 for t = O(ε 1. To obtain w 1σ and w 1ξ from Eq.(3.9, we integrate Eq.(3.9 with respect to ξ (or σ on the interval σ ξ ξ in the characteristic ξ direction (see Figure 3.1, leading to the following expression for w 1σ ( a similar expression is obtained for w 1ξ : 35

49 36 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING Figure 3.1: Integration in the characteristic ξ (or σ direction. ξ 4w 1σ (σ, ξ, τ + 4w 1σ (σ, σ, τ = ξ ( ξ + σ Ωαf σ H σ ξ ( ξ + σ Ωα g ξh σ σ cos cos ξ ξ g ξτ d ξ + f στ d ξ + V σ ( Ω( ξ σ ξ ( ξ + σ d ξ + αf σσ H σ ( Ω( ξ σ ξ ( ξ + σ d ξ α g ξ ξh σ σ ( f σσ g ξ ξ H sin sin ( ξ + σ ( Ω( ξ σ ( Ω( ξ σ d ξ d ξ d ξ. (3.3 It should be observed that the boundedness condition on w and w 1 (that is, w and w 1 are of O(1 demands that secular (unbounded terms in w 1 (that is, terms linear in ξ σ = t must be removed. It can be seen from the fourth and fifth terms in the right-hand side of Eq.(3.3 that the velocity fluctuation frequency (Ω gives rise to secular terms only when Ω = (m 1π, where m Z + and fixed, that is, ξ σ ξ σ ( ξ + σ H cos ( ξ + σ H sin ( Ω( ξ σ ( Ω( ξ σ d ξ = d ξ = (ξ σ sin((m 1πσ + NST, (3.31 (m 1π (ξ σ cos((m 1πσ + NST, (3.3 (m 1π where NST stands for nonsecular terms. In a similar way, one can find secular terms in the last two integrals of Eq.(3.3 only when Ω = (m 1π. However, no resonances occur at the O(ε- level, when Ω is not equal to (or in a neighborhood of (m 1π, that is, one cannot find terms like ξ σ = t. It is interesting to point out that the higher order resonances indeed occur in the axially moving system, but in this study we are only interested in O(ε accurate approximations of the solutions on timescales of order 1 ε. The higher order resonances only contribute to O(ε and do not contribute to O(1 on these timescales of O(ε 1. 36

50 3.3. THE CONSTRUCTION OF ASYMPTOTIC APPROXIMATIONS The non-resonant case (up to O(ε on timescales of 1 ε When Ω (m 1π (or not within an order ε-neighborhood of (m 1π, with m Z + and fixed only resonances occur due to the first and second term in the right hand side of Eq.(3.3. It can be shown elementarily from Eq.(3.3 that secular terms in w 1 can be avoided if and f στ =, (3.33 g ξτ =. (3.34 Integration of Eq.(3.33 and Eq.(3.34 with respect to τ, keeping σ and ξ respectively fixed, yields f σ = F (σ, (3.35 g ξ = G (ξ, (3.36 where F and G are arbitrary functions of σ and ξ respectively. Next, integration of Eq.(3.35 with respect to σ and Eq.(3.36 with respect to ξ, regarding τ as fixed gives f = P (σ + M (τ, (3.37 g = Q (ξ + N (τ, (3.38 where P = F, Q = G, and the functions P, Q, M, N are arbitrary functions. The first order approximation w in non-resonant case then becomes, w (σ, ξ, τ = P (σ + Q (ξ + R (τ, (3.39 where R (τ = M (τ + N (τ. We can rewrite Eq.(3.39 in terms of the variables x and t as w (x, t, τ = P (x t + Q (x + t + R (τ, (3.4 where P, Q and R are determined by the initial conditions and periodicity conditions. In what follows, we will investigate the motion of the axially moving strings in pure resonance in which Ω is equal to (m 1π Ω = (m 1π, a pure resonance case In this section, it is assumed that the velocity fluctuation frequency Ω is equal to an odd multiple of the natural frequency of an axially moving string, that is, Ω = (m 1π for some fixed m Z +. Plugging Ω = (m 1π into Eq.(3.3 and then performing the integration by parts leads to ξ 4w 1σ (σ, ξ, τ = 4w 1σ (σ, σ, τ σ ( +(ξ σ f στ α sin (m 1πσ ξ +4α g ξ σ g ξτ d ξ + V ξ f σ + ( (n 1π( ξ + σ cos σ ( f σσ g ξ ξ H 4α ( (m 1π cos (m 1πσ sin ( (m 1π( ξ σ ( ξ + σ d ξ f σσ d ξ + NST, (

51 38 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING where NST stands for nonsecular terms. The last integral in (3.41 can become unbounded if the integrals over a period are non-zero. It turns out that the integral can be written in a part which is O(1 and in a part which is linear in t = ξ σ, that is, we have 1 + ξ σ ξ g ξ σ { ξ = g ξ σ g ϕ cos g ϕ cos ( (n 1π( ξ + σ cos sin ( (n 1π( ξ + σ cos ( (n 1π(ϕ + σ sin ( (n 1π(ϕ + σ sin ( (m 1π( ξ σ d ξ ( (m 1π( ξ σ ( (m 1π(ϕ σ sin ( (m 1π(ϕ σ } dϕ d ξ dϕ. (3.4 Substitution of Eq.(3.4 into Eq.(3.41 gives ξ ξ ( ξ + σ 4w 1σ (σ, ξ, τ = 4w 1σ (σ, σ, τ g ξτ d ξ + V (f σσ g ξ ξh d ξ σ σ ( { ξ ( (n 1π( ξ + σ ( (m 1π( ξ σ +4α g ξ cos sin σ 1 } ( (n 1π(ϕ + σ ( (m 1π(ϕ σ g ϕ cos sin dϕ d ξ 4α +(ξ σ f στ α sin((m 1πσf σ + (m 1π cos((m 1πσf σσ +α g ϕ ( (n 1π(ϕ + σ cos sin ( (m 1π(ϕ σ dϕ + NST, (3.43 where NST stands for nonsecular terms. It should be observed that the linear term ξ σ = t, which follows that ξ σ = O(ε 1 on a timescale of order ε 1. So, w 1σ will be of O(1 on timescales of O(ε 1 if the expression inside the brackets in the right-hand side of (3.43 has to be zero since it generates the secular terms (that is, the terms linear in ξ σ. In order to avoid the secular terms, it follows from Eq.(3.43 that the function f has to satisfy +α 4α f στ α sin((m 1πσf σ + (m 1π cos((m 1πσf σσ ( (n 1π(ϕ + σ ( (m 1π(ϕ σ g ϕ cos sin dϕ =. (3.44 By following similar calculations as above, the elimination of secular terms in the function w 1ξ (σ, ξ, τ yields 4α g ξτ α sin((m 1πξg ξ + (m 1π cos((m 1πξg ξξ ( (n 1π(ξ + ϕ ( (m 1π(ξ ϕ +α f ϕ cos sin d ϕ =. 38 (3.45

52 3.3. THE CONSTRUCTION OF ASYMPTOTIC APPROXIMATIONS 39 The condition g (ϕ, τ = f ( ϕ, τ implies that Eq.(3.44 and Eq.(3.45 are equivalent. The last integral in (3.44 (see appendix C can be written as g ϕ ( (n 1π(ϕ + σ cos By means of Eq.(3.46, we can rewrite Eq.(3.44 as sin ( (m 1π(ϕ σ dϕ = sin((m 1πσf σ. (3.46 f στ α sin((m 1πσf σ + α (m 1π cos((m 1πσf σσ =. (3.47 It is worthwhile to note that the Eq.(3.47 leads to the same infinite dimensional system of coupled ordinary differential equations when a Fourier series approach is used to approximate the solution (see appendix D. This infinite dimensional system was obtained in 1 to study the same problem. However, it is hard or impossible to solve the infinite dimensional system analytically, and only limited results were obtained for the infinite dimensional system. The general solution of the partial differential equation (3.47 subject to the initial condition f σ (σ, = 1 (φ (σ ψ (σ which after some handling leads to f σ (σ, τ = e ατ (φ (σ ψ (σ(1 + sin((m 1πσ cos ((m 1πσ + e 4ατ ( 1 + sin((m 1πσ (3.48 The equation (3.48 is an explicit solution of Eq.(3.47 for f σ. In the following subsection, we will use some specific initial values in order to solve Eq.(3.48 for f. We will eventually be able to find the first order approximation w by using the odd and periodicity properties of the resulting function f Specific initial conditions In this section, we will study the problem (3.9-(3.11 for some particular initial values, that is, and u(x, = φ(x = sin((m 1πx, (3.49 u t (x, = ψ(x =. (3.5 With these initial values, the solution of Eq.(3.47 (and Eq. (3.48 is given by e 4ατ sin((m 1πσ f (σ, τ = (e 4ατ + 1, ( λ(τ sin((m 1πσ where λ(τ = e 4ατ 1 e 4ατ + 1 (3.5 By using the oddness and periodicity properties, the Eq.(3.8 can be rewritten as w (σ, ξ, τ = f (σ, τ f ( ξ, τ. (3.53 Therefore, the first order approximation w is of the form 39

53 4 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING e 4ατ sin((m 1πσ w (σ, ξ, τ = (e 4ατ λ(τ sin((m 1πσ + sin((m 1πξ (3.54 (1 λ(τ sin((m 1πξ We can rewrite Eq.(3.54 in terms of the variables x and t as e 4ατ sin((m 1π(x t w (x, t, τ = (e 4ατ λ(τ sin((m 1π(x t sin((m 1π(x + t +, 1 λ(τ sin((m 1π(x + t (3.55 where λ(τ is given in (3.5. Since w is -periodic in t, we can expand w in its Fourier series representation where and w (x, t, τ = a k (τ = b k (τ = a n (τ cos(nπt + b n (τ sin(nπt sin(nπx, ( w (x, t, τ sin(kπx cos(kπt dxdt 1, (3.57 sin (kπx cos (kπt dxdt 1 w (x, t, τ sin(kπx sin(kπt dxdt 1 (3.58 sin (kπx sin (kπt dxdt The integrals in Eq.(3.57 and Eq.(3.58 are not easy to solve due to complicated structure of w (x, t, τ in Eq.(3.55. It also makes clear that the applicability of the truncation method is problematic for string-like problems Energy of the system In this subsection, we will compute the energy of the conveyor belt system from Eq.(3.55 and compare this energy with the energy found in terms of infinite dimensional system in Ref.48. The dimensionless total energy of the conveyor belt system is E(t = 1 1 w t + w xdx + O(ε. (3.59 The substitution of Eq.(3.55 for w into Eq.(3.59, after some elementary calculations yields E(t = π e ατ + e ατ + O(ε. (3.6 4 It should be observed that the energy found in 1, 48 is identical with the energy equation ( Results and discussion In this section, the approximate solution w to the unstable transverse vibrations of an axially moving string system for the lowest resonance case (that is, at m = 1 will be discussed. The first order asymptotic approximation w (as given by (3.55 of the solution of initial-boundary 4

54 3.5. CONCLUSION 41 value problem will be compared with the numeric solutions calculated using the numerical finite difference method. In addition, the approximated energy obtained from the first order approximation w and the numerical energy obtained using the finite difference numerical method of the belt system will also be discussed. The unstable displacement-response of the system with ε=.1 for V =.8, α =.5, and for V =, α = 1 is depicted in the Figures 3. (a and 3. (b, respectively. The displacement-response obtained from the asymptotic approximations w (shown in Figures 3.(a and 3.3 (a increases as the time t increases. Similarly, Figures 3.(b and 3.3(b demonstrate the displacement-response u obtained numerically from the original partial differential equation (Eqs.(3.9-(3.11, which also grows as the time t progresses. From these figures, it can clearly be seen that the asymptotic approximations w and the numerical solution u are in good agreement on long timescales, that is, on timescales of order ε 1. On the other hand, the unstable displacement-response of the system with ε=. for V =.8, α =.5 and for V =, α = 1, is shown in figures 3.4 and 3.5, respectively. In this case, Figures 3.4 and 3.5 show that the amplitude-response of the system increases as the time t increases. Moreover, as can be seen from these figures, the agreement between the asymptotic solutions w and the numerical solutions u looks reasonably good on the timescales of order ε 1. However, a little bit worse results can be observed when the time t gets larger than the time interval of order 1 ε. The effects of the system parameters on the energy of an axially moving string with harmonically varying speed in terms of the logarithmic scale are illustrated in the Figures Figures 3.6 and 3.7 exhibit the logarithmic scale of approximated energy and the energy of the system with ε=.1 for V =.8, α =.5 and for V =, α = 1, respectively. The numerical simulation of the dimensionless total energy of the belt system E(t = 1 1 (u t + V u x + u xdx, where V is the velocity function of an axially moving string is obtained using the same initial conditions (3.49 and (3.5 with the finite difference numerical method. It can be seen in the Figures 3.6 and 3.7 that the approximated energy (Figures 3.6(a and 3.7(a and the numerical energy (Figures 3.6(b and 3.7(b grow linearly with a same slope of angle α. Moreover, it can easily be seen from these energy graphs that there is an excellent agreement between the approximated energy and the numerical energy graphs on the timescales of order ε 1. Unlike the energy graphs (Figures 3.6 and 3.7, the logarithmic scale of approximated energy and the energy of the system with ε=. (shown in the Figures 3.8 and 3.9 clearly illustrate the linear growth in the energy of the system with a same slope of angle α only on timescales of order ε 1. However, the linear growth in these energy graphs cannot be seen with a same slope of angle α when the time t gets larger than the time interval of O(ε 1. Meanwhile, the energy of the system found in terms of the infinite dimensional system (see for instance in Ref.1, 48 looks reasonably the same as the energy found from the Eq.(3.55 in this paper. 3.5 Conclusion In this chapter, the unstable transverse vibrations of an axially moving string are investigated. The axial velocity is assumed to be a harmonically varying function about a (low constant mean speed. Approximate solutions are constructed by using the two timescales perturbation method in combination with the method of characteristic coordinates. It is found that the instabilities occur in the system only when the frequency of the velocity fluctuations is equal (or close to an odd multiple of the lowest natural frequency of the constant velocity system. In order to examine instability of an axially moving string system in terms of amplitude and energy, only the first (lowest resonance frequency is taken into account. The displacement-response of the system obtained from the asymptotic approximations w and the numerical solutions u is found to be growing on long timescales, that is, on timescales of order ε 1. Moreover, the logarithmic scale of 41

55 4 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING (a (b Figure 3.: m = 1, V =.8, α =.5, ɛ =.1, x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u. (a (b Figure 3.3: m = 1, V =, α = 1, ɛ =.1, x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u. 4

56 3.5. CONCLUSION 43 (a (b Figure 3.4: m = 1, V =.8, α =.5, ɛ =., x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u. (a (b Figure 3.5: m = 1, V =, α = 1, ɛ =., x =.5. (a The unstable first order approximation w. (b The unstable numerical solution u. 43

57 44 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING log(e(t (a t (b Figure 3.6: Logarithmic scale of energy with ɛ =.1, V =.8, α =.5. (a Approximated energy of system. (b Energy of the system..5 log(e(t (a t (b Figure 3.7: Logarithmic scale of energy with ɛ =.1, V =, α = 1. (a Approximated energy of system. (b Energy of the system. 44

58 3.5. CONCLUSION 45.5 log(e(t (a t (b Figure 3.8: Logarithmic scale of energy with ɛ =.,V =.8, α =.5. (a Approximated energy of system. (b Energy of the system log(e(t (a t (b Figure 3.9: Logarithmic scale of energy with ɛ =., V =, α = 1. (a Approximated energy of system. (b Energy of the system. 45

59 46 CHAPTER 3. ON THE ASYMPTOTIC APPROXIMATION OF THE SOLUTION OF AN EQUATION FOR A NON-CONSTANT AXIALLY MOVING STRING the approximated and the numerical energy demonstrates the linear growth in the energy of the system with the same slope of angle α on timescales of order ε 1. However, a little bit worse results are observed in the displacement-response and the energy of the system when the time t gets larger than the time interval of order ε 1. The analytical-approximate solution for the lowest resonance displacement-response and the energy of the system are in good agreement with that obtained from the finite difference numerical method on long timescales. In addition, the energy of the system found by the first order approximation w also shows good agreement with the energy found in terms of the infinite dimensional system in Refs. 1, 48. Furthermore, it is also verified that the elimination of secular terms gives rise to the same infinite dimensional system of coupled ordinary differential equations (.8 obtained in Chapter using the Fourier series approach. It should be understood, however, that the solution of the infinite dimensional system is difficult to attain due to infinitely many interactions between the vibration modes. Therefore, the method of characteristic coordinates has been used to obtain the first order approximation of the solution. Alternatively, the solutions of infinite dimensional system of coupled ordinary differential equations have been found in this chapter. Meanwhile, the method of characteristic coordinates can also be used to the problems related with the transverse vibrations of axially moving string with time-dependent velocity subject to the O(1-mean speed, but special attention has to be paid to extending all functions in x outside the interval, 1. This will be a possible extension of the method, and can be a subject for future research. 46

60 Chapter 4 On parametric stability of a non-constant axially moving string near-resonances 3 All stable processes we shall predict. All unstable processes we shall control. John von Neumann The parametric instability in the axially moving string at resonances in terms of the amplituderesponse and the energy of the system was discussed in chapter 3. However, the (instability of the system in the neighborhood of resonances is yet to be addressed. The aim of this chapter is to investigate the (instability of an axially moving string in the neighborhood of resonances. Both the amplitude-response and the energy of the axially moving system near the resonances are computed by means of the two timescales perturbation method in combination with the method of characteristic coordinates. The effects of the detuning parameter on the amplitudes of vibrations and on the energy of the system are also presented through numerical simulations. 4.1 Introduction Axially moving strings can be found in many engineering systems and devices such as serpentine belts, aerial cables, power transmission belts, plastic films, magnetic tapes, paper sheets and textile fibers. These systems, however, are prone to noise and vibrations. The vibrations, particularly, the transverse vibrations occur in such systems due to forced and parametric (external excitations. Parametric vibrations of an axially moving string results from two major factors, that is, periodical variations in the tension 5, 7, 73 and in the axial transport velocity 6, 1, 1, 4, 48, 49, 58. In designs of belt systems, these periodic fluctuations in tension and in velocity may lead to parametric resonances and/or instabilities in the systems. This behavior can result in failure of belt structures during its motion. Several techniques have been employed to analyze the parametric vibrations for axially moving strings subject to periodically fluctuating speed and tension. Mote 73 first studied the parametric vibrations of an axially moving string due to the variation in tension using a modal analysis. Naguleswaran and Williams 74 developed a numerical solution by employing Galerkin s truncation method up to first four terms to investigate the instability of band saw blades and belt drives 3 This chapter is slightly revised version of accepted article 3- On parametric stability of a non-constant axially moving string near-resonances, J. Vibr. Acoust., vol.139, p: , October

61 48 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES 3 subject to the periodic variation in the band tension. They found that the system can give rise to parametric instabilities only when the natural frequency of the system is at twice the tension fluctuations. Ariartnam and Asokanthan 75 analyzed the dynamic stability of a chain drive, which was modeled as an axially moving string with periodically fluctuating tension by means of Galerkin s truncation method and the averaging method. Mockensturm et al. 76 examined the primary, sum type and difference type parametric resonances for an axially moving string with tension fluctuations using Galerkin s method. The governing equations of motion for parametric vibrations of an axially moving string under periodically fluctuating speed was first derived by Miranker 5 by using a variational procedure. The stability of an accelerating transporting string with harmonic excitation at one end is examined in 33 by using the Laplace transform technique. Zhu et al. 38 investigated the parametric instability in a translating string in terms of bounded displacement-response and unbounded vibratory energy under sinusoidally varying velocity by using the method of characteristic transformations. The dynamic stability in transverse vibrations of an axially accelerating string was investigated by Pakedemirli et al. 41 using Galerkin s truncation method. Chen et al. 77 used the Galerkin truncation method to study the bifurcation and chaos of parametrically excited viscoelastic axially moving strings with geometric nonlinearities. However, the (nonapplicability of Galerkin s truncation method to problems for the parametric vibration of axially moving strings with respect to time-dependent speed is investigated in Refs. 1, 11, 48, 5. Recently, it is proven in by using of the method of characteristic coordinates that the truncation method is not always applicable to approximate the transversal vibrations of axially moving strings. In addition, Gaiko and van Horssen 78 also discussed the (nonapplicability of Galerkin s truncation method for the transverse vibrations of a vertically moving string subject to the harmonically time-varying length. Zhu and Guo 35 found exact solutions in terms of free and forced responses for the translating string with an arbitrary velocity profile using the method of characteristic transformations. In this chapter, we will examine the parametric vibrations of an axially moving string under a harmonically varying speed around a (low constant mean speed near the resonances. We construct the approximations of the solutions by using the two timescales perturbation method together with the method of characteristic coordinates. Moreover, in this study, it is also proved that the elimination of secular terms leads to a similar infinite dimensional system of coupled ordinary differential equations, which was studied in 1 to study the same problem for axially moving string near the resonances. Furthermore, we will use particular (harmonic initial values in order to find the displacement-response of the system and compare it with the numerical results obtained directly from the original partial differential equation in the neighborhood of the first resonance by using the numerical finite difference method. The energy of the belt system will also be computed from the asymptotic approximations and will be compared with the energy obtained in Ref.48, 1. In addition, the numerical simulation of the energy computed through the finite difference method will also be presented. The chapter is organized as follows. In section 4., the governing equations of motion for an axially moving string are presented. In section 4.3, the method of multiple scales together with the method of characteristic coordinates are applied to the governing equations of motion to construct approximations of the solution. In section 4.4, stability conditions are discussed in terms of the displacement-response and the energy of the system. Finally, in section 4.5 of this chapter some conclusions are presented. 48

62 4.. EQUATIONS OF MOTION Equations of motion Consider an axially moving string, with a time dependent velocity V traveling between two fixed supports at a distance L as shown in Fig.1.3. The governing equation of motion describing the transverse vibrations of the axially moving string can be derived from Hamilton s principle (see Ref. 6 and can be written as follows: u tt + V u xt + V t u x + (V c u xx =, t >, < x < L, (4.1 where u(x, t is the displacement of the string in the transverse direction, x denotes the coordinate in horizontal direction, t is the time, c > is the wave speed, and L is the distance between the pulleys. Furthermore, it is assumed that the velocity V is small with respect to the wave speed, that is, V < c. The boundary and initial conditions for u(x, t are, respectively, given by and u(, t = u(l, t =, t, (4. u(x, = φ(x, and, u t (x, = ψ(x, < x < L, (4.3 where φ(x represents the initial displacement, and ψ(x the initial velocity of the string. We introduce the dimensionless quantities x = x L, V = V c, t = ct L, u(x, t u (x, t = L, φ (x = φ(x L, ψ (x = ψ(x c The substitution of Eq.(4.4 into Eqs.(4.1-(4.3 gives the dimensionless linear equations of motion from axially moving string model with boundary conditions and initial conditions (4.4 u tt u xx = V t u x V u xt V u xx, t, < x < 1, (4.5 u(, t = u(1, t =, t, (4.6 u(x, = φ(x, and u t (x, = ψ(x, < x < 1. (4.7 The asterisks indicating the dimensionless quantities are dropped in Eqs.(4.5-(4.7 and henceforth. The velocity of the string is assumed to be a harmonically varying function about a constant low mean velocity of order ε: V (t = ε(v + α sin(ωt, (4.8 where ε, V, α and Ω are some positive constants with V >, V > α, and ε is a dimensionless small parameter with < ε 1. The condition V > α guarantees that the belt will always move forward in one direction. The substitution of the velocity function (4.8 into Eq. (4.5 yields the following perturbation problem u tt u xx = ε αω cos(ωtu x ( V + α sin(ωt u xt + O(ε, t, < x < 1, (4.9 49

63 5 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES 3 with boundary conditions and initial conditions u(, t; ε = u(1, t; ε =, t, (4.1 u(x, ; ε = φ(x, and u t (x, ; ε = ψ(x, < x < 1. ( A perturbation approach to construct approximations In this section, we will construct an approximate-analytical solution of the initial boundary value problem (4.9-(4.11 by using the two timescales perturbation method in combination with the method of characteristic coordinates. In solving the equations of motion for axially moving strings, the method of characteristic coordinates is of great interest, in the sense that it avoids the problem of convergence and computational difficulties of infinite series (as has been noticed in 1, 48, 49, 5 in the Fourier series approach. The straightforward asymptotic expansion, however, gives a poor approximation for u on a long timescale due to the presence of secular (unbounded terms, a two timescales perturbation method in conjunction with the method of characteristic coordinates is used in constructing an asymptotic approximation of the solution of the governing equations of motion (4.9-(4.11 on a long timescale, that is, on a timescale of O(ε 1 (see for instance Refs., 66, 69. In this approach we first transform the initial-boundary value problem into an initial value problem. The boundary conditions (4.1 imply that the dependent variable u(x, t as well as the initial values φ(x and ψ(x should be extended to odd and -periodic functions in x 71. To do so, we multiply the terms in Eq.(4.9, which are not odd in x (i.e., u x and u xt with the Fourier series F (x of the function H(x (see also 1, F (x = 4 sin((n 1πx, (4.1 (n 1π where the function H, defined on R, is given by H(x = 1, for < x < 1, and H( = H(1 =, and H is odd and -periodic in x, we obtain, ( u tt u xx = ε αω cos(ωtf (xu x V + α sin(ωt F (xu xt + O(ε. (4.13 To solve Eq.(4.13 by using the two timescales perturbation method together with the method of characteristic coordinates, we seek the solution u(x, t of (4.13 as a function v depending on the characteristic variables σ, ξ and the slow timescale τ, that is, u(x, t = v(σ, ξ, τ, (4.14 where σ = x t, ξ = x + t, τ = εt. The introduction of σ, ξ, and τ leads to the following transformations: u t = v σ + v ξ + ε v τ, (4.15 u t = v σ + v ξ v ( σ ξ + ε v ξ τ v σ τ + O(ε, (4.16 u x = v σ + v ξ, (4.17 u x = v σ + v ξ + v σ ξ, (4.18 5

64 4.3. A PERTURBATION APPROACH TO CONSTRUCT APPROXIMATIONS 51 Substitution of(4.14-(4.19 into (4.13 yields: 4v σξ = ε ( V + α sin u x t = v σ + v ( ξ + ε v σ τ + v (4.19 ξ τ ( Ω(ξ σ (v ξτ v στ αω cos ( Ω(ξ σ ( ξ + σ F Following formal asymptotic expansion for v is assumed: F ( ξ + σ (v σ + v ξ + O(ε. ( v σσ + v ξξ (4. v(σ, ξ, τ = v (σ, ξ, τ + εv 1 (σ, ξ, τ + O(ε, (4.1 where v i (σ, ξ, τ and all of its derivatives are assumed to be bounded on the O( 1 ε timescale. By plugging Eq.(4.1 and its derivatives into Eq.(4. and equating to zero the coefficients of like powers of ε in the resulting equation, we have the following equations: O(1: O(ε: 4v σξ =, < σ < ξ <, τ >, (4. v (σ, σ, = φ (σ, < σ = ξ <, τ =, (4.3 v σ (σ, σ, + v ξ (σ, σ, = ψ (σ, < σ = ξ <, τ =, (4.4 ( + V + α sin ( Ω(ξ σ 4v 1σξ = v ξτ + v στ αω cos ( Ω(ξ σ ( ξ + σ F F ( ξ + σ (v σ + v ξ (v σσ v ξξ, < σ < ξ <, τ >, (4.5 v 1 (σ, σ, = φ 1 (σ, < σ = ξ <, τ =, (4.6 v 1σ (σ, σ, + v 1ξ (σ, σ, = v τ (σ, σ, + ψ 1 (σ, < σ = ξ <, τ =, (4.7 in which the function F is defined as in (4.1, and where φ i (σ is the O(ε i -part of φ(σ, and ψ i (σ is the O(ε i -part of ψ(σ. The general solution of (4.-(4.4 is obtained by direct integration with respect to σ and ξ yields v (σ, ξ, τ = f (σ, τ + g (ξ, τ, (4.8 f (σ, + g (σ, = φ (σ, f (σ, + g (σ, = ψ (σ, (4.9 where f and g are arbitrary functions of the characteristic variables σ and ξ and the slow timescale τ, and will be found by eliminating the secular terms from the equation of v 1, while the prime symbol denotes differentiation with respect to the first argument. Moreover, the elimination of g (σ, from the two equations in (4.9 yields f σ (σ, = 1 (φ (σ ψ (σ. From the odd and -periodic extension of the dependent variable of the problem (4.9-(4.11 it follows additionally that f and g have to satisfy g (σ, τ = f ( σ, τ and f (σ, τ = f (σ +, τ for - < σ < and τ. Eq.(4.5 can be written in terms of f and g as 51

65 5 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES 3 ( Ω(ξ σ ( ξ + σ 4v 1σξ = g ξτ + f στ αω cos F ( ( Ω(ξ σ ( ξ + σ + V + α sin F (f σ + g ξ (f σσ g ξξ. (4.3 In order to obtain an expression for v 1 valid for times as large as O(ε 1 demands that v 1σ and v 1ξ must be bounded for t = O(ε 1. To obtain these expressions for v 1, we integrate Eq.(4.3 w.r.t ξ (or σ from σ to ξ (see Fig.3.1, yielding ξ 4v 1σ (σ, ξ, τ + 4v 1σ (σ, σ, τ = ξ ( ξ + σ Ωαf σ F σ ξ ( ξ + σ Ωα g ξf σ σ cos cos ξ ξ g ξτ d ξ + f στ d ξ + V σ ( Ω( ξ σ ξ ( ξ + σ d ξ + αf σσ F σ ( Ω( ξ σ ξ ( ξ + σ d ξ α g ξ ξf σ σ ( f σσ g ξ ξ F sin sin ( ξ + σ ( Ω( ξ σ ( Ω( ξ σ d ξ d ξ d ξ. (4.31 It should be noticed that the boundedness conditions on v and v 1 (that is, v and v 1 are of O(1 on time-scales of O(ε 1 demands that secular terms in v 1 (that is, terms linear in ξ σ = t must be removed. It can be seen from the fourth and fifth terms in the right-hand side of Eq.(4.31 that the velocity fluctuation frequency (Ω gives rise to secular terms only when Ω = (m 1π, where m Z + and fixed, that is, ξ σ ξ σ ( ξ + σ F cos ( ξ + σ F sin ( Ω( ξ σ ( Ω( ξ σ d ξ = d ξ = (ξ σ sin((m 1πσ + NST, (4.3 (m 1π (ξ σ cos((m 1πσ + NST, (4.33 (m 1π where NST stands for nonsecular terms. In a similar way, one can find secular terms in the last two integrals of Eq.(4.31 only when Ω = (m 1π. However, no resonances occur, when Ω is not equal to (or in a neighborhood of (m 1π, that is, one cannot find terms like ξ σ = t in Eq.(4.31. The pure resonance case is recently studied in full detail in Ref.. In this study, the detuned resonance case will be studied for all m Ω = (m 1π + εδ, a detuned resonance case In this section, it is assumed that the velocity fluctuation frequency Ω is near to an odd multiple of the natural frequency of the axially moving string, that is, Ω (m 1π for some fixed m Z +. In order to show the nearness of Ω to the (m 1π, the fluctuation frequency can be written as Ω = (m 1π + εδ, where δ is a detuning parameter of O(1. Plugging Ω = (m 1π + εδ into Eq.(4.31, and then performing the integration by parts leads to 5

66 4.3. A PERTURBATION APPROACH TO CONSTRUCT APPROXIMATIONS 53 +(ξ σ ξ 4v 1σ (σ, ξ, τ + 4v 1σ (σ, σ, τ = σ f στ α sin ξ +4α g ξ σ ( (m 1πσ δτ f σ + ( (n 1π( ξ + σ cos g ξτ d ξ + V ξ sin σ ( f σσ g ξ ξ F ( ξ + σ 4α ( (m 1π cos (m 1πσ δτ ( (m 1π( ξ σ + δτ d ξ f σσ d ξ + NST, (4.34 where NST stands for nonsecular terms. The last integral in (4.34 can become unbounded if the integrals over a period are non-zero. It turns out that the integral can be written in a part which is O(1 and in a part which is linear in t = ξ σ, that is, we have 1 + ξ σ ξ g ξ σ { ξ = g ξ σ g ϕ cos g ϕ cos ( (n 1π( ξ + σ cos sin ( (n 1π( ξ + σ cos ( (n 1π(ϕ + σ sin ( (n 1π(ϕ + σ sin ( (m 1π( ξ σ + δτ d ξ ( (m 1π( ξ σ + δτ ( (m 1π(ϕ σ + δτ sin ( (m 1π(ϕ σ + δτ } dϕ d ξ dϕ. (4.35 Substitution of Eq.(4.35 into Eq.(4.34 gives ξ 4v 1σ (σ, ξ, τ = 4v 1σ (σ, σ, τ σ { ξ +4α g ξ cos σ g ϕ ξ ( ξ + σ g ξτ d ξ + V (f σσ g ξ ξf d ξ σ ( (n 1π( ξ + σ ( (m 1π( ξ σ + δτ sin } dϕ d ξ ( (n 1π(ϕ + σ ( (m 1π(ϕ σ + δτ α cos sin 4α +(ξ σ f στ α sin((m 1πσ δτf σ + (m 1π cos((m 1πσ δτf σσ +α g ϕ ( (n 1π(ϕ + σ cos sin ( (m 1π(ϕ σ + δτ dϕ + NST. (4.36 It should be observed that the linear term ξ σ = t, from which it follows that ξ σ = O(ε 1 on a timescale of order ε 1. So, v 1σ will be of O(1 on a timescales of O(ε 1 if the expression inside the brackets in the right-hand side of (4.36 is equal to zero since it generates secular terms (that is, terms linear in ξ σ. In order to avoid the secular terms, it follows from Eq.(4.36 that the function f has to satisfy 53

67 54 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES 3 4α f στ α sin((m 1πσ δτf σ + (m 1π cos((m 1πσ δτf σσ ( (n 1π(ϕ + σ ( (m 1π(ϕ σ + δτ +α g ϕ cos sin dϕ =. (4.37 By following similar calculations in order to eliminate secular terms in the function v 1ξ (σ, ξ, τ, it follows that g has to satisfy 4α g ξτ α sin((m 1πξ δτg ξ + (m 1π cos((m 1πξ δτg ξξ ( (n 1π(ξ + ϕ ( (m 1π(ξ ϕ + δτ +α f ϕ cos sin d ϕ =. (4.38 The condition g (ϕ, τ = f ( ϕ, τ implies that Eq.(4.37 and Eq.(4.38 are equivalent. The last integral in Eq.(4.37 (see Ref. can be written as g ϕ ( (n 1π(ϕ + σ cos Substitution of Eq.(4.39 into Eq.(4.37 gives sin ( (m 1π(ϕ σ + δτ dϕ = sin((m 1πσ δτf σ. (4.39 f στ + α (m 1π cos((m 1πσ δτf σσ α sin((m 1πσ δτf σ =. (4.4 It can be seen in Appendix E that Eq.(4.4 leads to the same infinite dimensional system of coupled ordinary differential equations which was obtained in 1 to study the same problem for a detuned case. However, it is hard or impossible to solve the infinite dimensional system analytically, and only limited results are obtained in 1 for the infinite dimensional system. The solution of Eq.(4.4 can be obtained by using the method of characteristics. The corresponding characteristic ODEs of the PDE (4.4 are written as dτ(r,s ds = 1, τ(r, =, dσ(r,s α cos((m 1πσ δτ ds = (m 1π, σ(r, = r, df σ (r,s ds = α sin((m 1πσ δτf σ, f σ (r, = 1 (φ (r ψ (r. The solution of the first ODE in Eq.(4.41 is given by By using Eq.(4.4, the first and second ODEs in Eq.(4.41 lead to τ ds = (4.41 τ(r, s = s. (4.4 z r dz α cos(z δ, (4.43 where z = (m 1πσ(r, s δs. (

68 4.3. A PERTURBATION APPROACH TO CONSTRUCT APPROXIMATIONS 55 The second and third ODEs in Eq.(4.41 give fσ (σ,τ (φ (r ψ (r df σ (r, s z f σ (r, s = α sin(zdz r α cos(z δ, (4.45 where z is given in Eq.(4.44. For the Eq.(4.43 and Eq.(4.45 it turns out (and similarly it turned out for the infinite dimensional system in 1 that three cases arise depending on the values of δ. Case 1: δ < α, Case : δ = α, and Case 3: δ > α. In the subsequent sections, the displacement-response and the energy of the system will be computed for each case and for particular (harmonic initial conditions Case 1: δ < α In this case, the general solution of equation (4.43 and Eq.(4.45 (or equivalently Eq.(4.4 subject to the general initial condition f σ (σ, = 1 (φ (σ ψ (σ is where f σ (σ, τ = (4α δ (φ (σ ψ (σe τ 4α δ, (4.46 p1 + q 1 cos((m 1πσ δτ + r 1 sin((m 1πσ δτ p 1 = α (e τ 4α δ δ e τ 4α δ, q 1 = αδ(e τ 4α δ 1, r 1 = α 4α δ (e τ 4α δ 1. (4.47 For the particular initial conditions, that is, for u(x, = φ(x = sin((m 1πx, (4.48 and u t (x, = ψ(x =, (4.49 the solution of Eq.(4.4 (and Eq. (4.46 is given by a 1 (1 + cos((m 1πσ δτ + c 1 sin((m 1πσ δτ f (σ, τ = R 1 {, p1 + q 1 cos((m 1πσ δτ + r 1 sin((m 1πσ δτ } (4.5 where a 1 = δ 4α δ (e τ 4α δ 1, c 1 = (α + δ{ δ(e τ 4α δ αe τ 4α δ }, (α δe τ 4α δ R 1 =, 4 (α(e τ 4α δ + 1 δe τ 4α δ (

69 56 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES amplitude.1 amplitude v o t (a.3 u t (b Figure 4.1: m = 1, ɛ =.1, V =.8, α =.5, δ =.1, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u. and where p 1, q 1 and r 1 are given in Eq.(4.47. And so, the first order approximation v is given by v (σ, ξ, τ = f (σ, τ f ( ξ, τ a 1 (1 + cos((m 1πσ δτ + c 1 sin((m 1πσ δτ R 1 {p 1 + q 1 cos((m 1πσ δτ + r 1 sin((m 1πσ δτ} (4.5 a 1(1 + cos((m 1πξ + δτ c 1 sin((m 1πξ + δτ {p 1 + q 1 cos((m 1πξ + δτ r 1 sin((m 1πξ + δτ} We can rewrite Eq.(4.5 in terms of the variables x and t as a 1 (1 + cos Φ + c 1 sin Φ v (x, t, τ = R 1 p 1 + q 1 cos Φ + r 1 sin Φ a 1(1 + cos Ψ c 1 sin Ψ, (4.53 p 1 + q 1 cos Ψ r 1 sin Ψ where Φ = { (m 1π(x t δτ } and Ψ = { (m 1π(x + t + δτ }. It should be noted that the first order approximations (Eqs.(4.53 leads to same approximations as given in, where the case δ = has been studied Case : δ = α In this case the two sub-cases arise: δ = α and δ = α. Case (a: δ = α In this case, the general solution of Eq. (4.43 and Eq.(4.45 (or equivalently Eq.(4.4 subject to the general initial condition f σ (σ, = 1 (φ (σ ψ (σ is given by f σ (σ, τ = (φ (σ ψ (σ, (4.54 p + q cos((m 1πσ ατ + r sin((m 1πσ ατ where p = α τ + 1, q = α τ, r = ατ. (

70 4.3. A PERTURBATION APPROACH TO CONSTRUCT APPROXIMATIONS 57 E(t (a t (b Figure 4.: m = 1, ɛ =.1, V =.8, α =.5, δ =.1. (a Approximated energy. (b Energy of the system. For the particular initial conditions (4.48 and (4.49, we obtain from Eq.(4.54 a (1 + cos((m 1πσ ατ + c sin((m 1πσ ατ f (σ, τ = R, (4.56 p + q cos((m 1πσ ατ + r sin((m 1πσ ατ where, R = 1 (4α τ + 1, a = ατ, c = (1 4α τ, (4.57 and where p, q, r are given in Eq.(4.55. And so, the first order approximation v is given by a (1 + cos((m 1πσ ατ + c sin((m 1πσ ατ v (σ, ξ, τ = R p + q cos((m 1πσ ατ + r sin((m 1πσ ατ a (1 + cos((m 1πξ + ατ c sin((m 1πξ + ατ p + q cos((m 1πξ + ατ r sin((m 1πξ + ατ. (4.58 We can rewrite Eq.(4.58 in terms of the variables x and t as a (1 + cos µ 1 + c sin µ 1 v (x, t, τ = R a (1 + cos λ 1 c sin λ 1, (4.59 p + q cos µ 1 + r sin µ 1 p + q cos λ 1 r sin λ 1 where µ 1 = { (m 1π(x t ατ } and λ 1 = { (m 1π(x + t + ατ }. Case (b: δ = α In this case, the general solution of Eq.(4.43 and Eq.(4.45 (or equivalently Eq.(4.4 subject to the initial condition f σ (σ, = 1 (φ (σ ψ (σ is given by f σ (σ, τ = (φ (σ ψ (σ, (4.6 p q cos((m 1πσ + ατ + r sin((m 1πσ + ατ 57

71 58 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES amplitude.1 amplitude.1.. v o t (a u t (b Figure 4.3: m = 1, ɛ =.1, V =.8, α =.5, δ =α, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u E(t (a t (b Figure 4.4: m = 1, ɛ =.1, V =.8, α =.5, δ =α. (a Approximated energy. (b Energy of the system. 58

72 4.3. A PERTURBATION APPROACH TO CONSTRUCT APPROXIMATIONS 59.3 v o.3 u.. amplitude.1 amplitude t (a t (b Figure 4.5: m = 1, ɛ =.1, V =.8, α =.5, δ =-α, x =.5. (a The unstable first order asymptotic approximation (v. (b The unstable numerical solution (u. where p, q and r are given in Eq.(4.55. The solution of Eq.(4.6 can be obtained using the particular initial values (4.48 and (4.49. With these initial values, the solution of Eq.(4.6 becomes f (σ, τ = 1 a (1 + cos((m 1πσ + ατ sin((m 1πσ + ατ. (4.61 p q cos((m 1πσ + ατ + r sin((m 1πσ + ατ And so, the first order approximation v can be written in the form v (σ, ξ, τ = 1 a (1 + cos((m 1πσ + ατ sin((m 1πσ + ατ p q cos((m 1πσ + ατ + r sin((m 1πσ + ατ a (1 + cos((m 1πξ ατ + sin((m 1πξ ατ. p q cos((m 1πξ ατ r sin((m 1πξ ατ (4.6 The first order approximation v in terms of x and t is given by v (x, t, τ = 1 a (1 + cos µ sin µ a (1 + cos λ + sin λ, (4.63 p q cos µ + r sin µ p q cos λ r sin λ where µ = { (m 1π(x t + ατ } and λ = { (m 1π(x + t ατ } Case 3: δ > α In this case, the general solution of Eq.(4.43 and Eq.(4.45 (or equivalently Eq.(4.4 subject to the initial condition f σ (σ, = 1 (φ (σ ψ (σ is given by f σ (σ, τ = (4α δ (φ (σ ψ (σe iτ δ 4α, (4.64 p3 + q 3 cos((m 1πσ δτ r 3 i sin((m 1πσ δτ 59

73 6 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES E(t (a t (b Figure 4.6: m = 1, ɛ =.1, V =.8, α =.5, δ =-α. (a Approximated energy. (b Energy of the system. where p 3 = α (e iτ δ 4α δ e iτ δ 4α, q 3 = αδ(e iτ δ 4α 1, r 3 = α δ 4α (1 e iτ δ 4α. (4.65 With the initial values (4.48 and (4.49, the solution of Eq. (4.64 becomes a 3 (1 + cos((m 1πσ δτ + c 3 sin((m 1πσ δτ f (σ, τ = R 3 { i ( p 3 + q 3 cos((m 1πσ δτ }, ( r 3 sin((m 1πσ δτ where a 3 = δ δ 4α (e iτ δ 4α 1, c 3 = i(α + δ { δ(e iτ δ 4α + 1 4αe iτ δ 4α }, (α δe iτ δ 4α R 3 =, 4 (α(e iτ δ 4α + 1 δe iτ δ 4α where p 3, q 3 and r 3 are given in Eq. (4.65. And so, the first order approximation v in real form is given by v (σ, ξ, τ = R a 3 (1 + cos((m 1πσ δτ + c 3 sin((m 1πσ δτ 3 { (p } 3 + q 3 cos((m 1πσ δτ + r 3 sin((m 1πσ δτ a 3 (1 + cos((m 1πξ + δτ c 3 sin((m 1πξ + δτ { (p } 3 + q 3 cos((m 1πξ + δτ r 3 sin((m 1πξ + δτ, (4.67 (4.68 6

74 4.4. STABILITY ANALYSIS amplitude.5 amplitude v o t (a.15 u t (b Figure 4.7: m = 1, ɛ =.1, V =.8, α =.5, δ =15, x =.5. (a The stable first order asymptotic approximation (v. (b The stable numerical solution (u. where R 3 = (α δ ( α cos(τ δ 4α δ, a 3 = δ δ 4α sin(τ δ 4α, c 3 = (α + δ α δ cos(τ δ 4α, p 3 = δ 4α cos(τ δ 4α, q 3 = αδ cos(τ δ 4α 1, r 3 = α δ 4α sin(τ δ 4α. (4.69 The first order approximation (v in terms of x and t can be written as v (x, t, τ = R a 3 (1 + cos ζ + c 3 sin ζ 3 (p 3 + q 3 cos ζ + r 3 sin ζ a 3 (1 + cos η c 3 sin η (p 3 + q 3 cos η r 3 sin η, (4.7 where ζ = ((m 1π(x t δτ, η = ((m 1π(x + t + δτ, and where R 3, a 3, c 3, p 3, q 3, and r 3 are given as in Eq.( Stability analysis In this section, we present the (instability of the solution of Eqs.(4.9-(4.11 with respect to the particular (harmonic initial conditions (4.48 and (4.49 in the neighborhood of the first resonance, that is, with m = 1. The (instability of the system solely depends on the values of the detuning parameter δ. For δ < α: the unstable displacement-response and the energy of the system versus time t at the fixed spatial value x (i.e. x =.5 are shown in Figure. 4.1 and Figure. 4., respectively. Figure. 4.1(a depicts the displacement-response (v of the system obtained from the first order asymptotic approximation, while Figure. 4.1(b shows the displacement-response (u of the system found by using the finite difference numerical method of the governing equations of motion (4.9-(4.11. It can be seen in Figure. 4.1 that the displacement-response of the system increases exponentially as time t increases. Additionally, the unstable displacement-response of the system (v obtained by using the asymptotic approximation (v and the numerical u found by using the finite difference numerical method are in fully agreement on long timescales, that is, on a timescale of order ε 1. The energy graphs for δ < α are shown in Figure. 4.(a 61

75 6 CHAPTER 4. ON PARAMETRIC STABILITY OF A NON-CONSTANT AXIALLY MOVING STRING NEAR-RESONANCES E(t (a t (b Figure 4.8: m = 1, ɛ =.1, V =.8, α =.5, δ =15. (a Approximated energy. (b Energy of the system. α π 3π 5π Ω Figure 4.9: Approximate instability region in the (α, Ω-plane for ε =.1 : the amplitude α of the velocity fluctuation of the belt versus the frequency Ω of the velocity fluctuation of the belt. The boundaries of the instability region are given by Ω = (m 1π + εδ with δ = α and m = 1,,... 6

76 4.5. CONCLUSION 63 and Figure. 4.(b. In Figure. 4.(a, the approximated energy of the system obtained from the first order approximation v is shown, while the numerical simulation of the energy E(t = 1 1 (u t+v u x +u xdx, where V is the velocity function of axially moving strings computed using the finite difference method is depicted in Figure. 4.(b. Both the energy graphs demonstrate that the energy of the system increases as the time t progresses. Moreover, it can clearly be seen from the energy graphs that there is an excellent agreement between the analytically and the numerically obtained energies. For δ = α: two cases are separately discussed. The unstable displacement-response and the energy of the system for δ = α are shown in the Figure. 4.3 and Figure While for δ = α, the unstable displacement-response and energy graphs are illustrated in the Figure. 4.5 and Figure It is apparent in the figures that the displacement-responses and the energies of the system increase as time t increases. In a similar way, the analytically and the numerically obtained energies are also in fully agreement for δ = α. For δ > α: the stable displacement-response and the energy of the system versus time t are shown in Figure. 4.7 and Figure In this case, both the amplitude- response and the energy of the system are shown to be bounded. Furthermore, the analytically and numerically obtained results for the displacement-response and the energy of the system can also be seen in full agreement. It, however, should be observed that δ = α is the boundary between stable and unstable behavior of the system (Figure Conclusion In this chapter, the parametric stability of an axially moving string subject to a time-varying velocity in the neighborhood of resonances has been investigated. The time-dependent velocity is assumed to have a (low constant mean value with a small harmonic variation. It turned out that the system can give rise to resonances only if the fluctuation frequency approaches the odd multiple of the lowest natural frequency of the system. The first order approximation of the solution of the governing equations of motion has been found by means of the two timescales perturbation method in conjunction with the method of characteristic coordinates. The amplitude of response and the energy of the axially moving string system have been computed explicitly for particular (harmonic initial conditions and for various values of the detuning parameter δ. It is found that the energy and the displacement-response of the system are unbounded when δ α, and bounded when δ > α. Furthermore, the analytical approximations for the energy and for the displacement-response of the system are also fully in agreement with the numerical approximations of the solution obtained by using the finite difference method for the original partial differential equation. Moreover, the approximated energies of the system computed in terms of the infinite dimensional system in Ref.1 are also identical with the energies found in this chapter for all three detuning cases. In addition, the infinite dimensional system of coupled ordinary differential equations obtained in Ref. 1 is also verified through the Eq. (4.4 in this chapter. 63

77

78 Chapter 5 Conclusions and Future Work In this thesis we have examined the dynamic (instability of parametric vibrations in an axially moving string (e.g. conveyor belt system with a non-constant axial speed and with fixed end conditions. The time-dependent axial velocity is assumed to be a harmonically varying function about a constant mean speed. The transverse vibrations of an axially moving string is mathematically modeled as a second order linear homogeneous partial differential equation with variable coefficients. The axially moving string undergoes a parametric vibration due to a periodical variation in the belt axial speed. It is found that the system can be unstable (i.e. large transverse vibrations occur in case of mean speed V of O(ε only when the velocity fluctuation frequencies are equal (or close to an odd multiple of the lowest natural frequency of the system. However, for the case of mean speed of O(1, this instability is found when the fluctuation frequencies Ω are equal (or close to kπ(1 V, where k Z+. It is also shown that the truncation method is inappropriate in constructing the accurate approximations for string-like models on long timescales of order ε 1. The linear transverse vibrations and parametric stability of a non-constant axially moving string in terms of the energy and response of the individual vibrational modes are discussed in Chapter. The velocity of the string is assumed to be a harmonically varying function about a constant mean speed, that is, V (t = V + α sin(ωt, where V is a constant mean speed, α is a amplitude of fluctuation, Ω is a fluctuation frequency and are all positive constants. Two different cases for the mean speed V, such as V = O(ε and V = O(1, are taken into consideration to investigate the dynamic (instability of the system. In the former case of mean speed, the natural frequencies of the system are found constant up to O(ε, while in the latter case, the velocity dependent natural frequencies are obtained. For the low mean speed case, the governing partial differential equation is first discretized using the Fourier sine series and then the two timescales perturbation method is used. It turned out that the system can give rise to resonances only when the system s fluctuation frequencies are odd multiple of the lowest natural frequency of the system, that is Ω = (m 1π, where m Z +. Using this approach, an infinite dimensional system of coupled ordinary differential equations is obtained which is in fact difficult to solve due to infinitely many interactions between the vibrational modes. It is found that the one-mode or even multi-mode approximations lead to inaccurate approximations of the solutions of the governing equations of motion on long timescales. However, all explicit approximations for the energy of the system are computed in terms of the infinite dimensional systems on long timescales. In other words, it is observed that the motion of the system at resonances tends to increase the system energy (i.e. yields the unbounded behavior, while mode truncation provides the mixture of stable and/or unstable solutions due 65

79 66 CHAPTER 5. CONCLUSIONS AND FUTURE WORK to non-convergence of eigenvalues which confirms the non-applicability of truncation method. Additionally, the energy of the system is found to be bounded in the neighborhood of resonances when δ > α, and unbounded when δ α, where δ is the detuning parameter. On the other hand, in the case of the mean speed of O(1, the resonances in the system occur only when the speed fluctuation frequencies Ω are equal (or close to kπ(1 V, where k Z+. In this case, the two timescales perturbation method is applied directly to the governing equations of motion in conjunction with the Laplace transform method. Using this technique, a complicated infinite dimensional system of coupled ODEs with infinitely many mode interactions is obtained. It is found that mode-truncation does not produce accurate approximations of the solutions of initial boundary value problem on long timescales, while the energy of the system yields accurate approximations on long timescales i.e. grows exponentially when time t increases. The linear transverse resonant response and the energy of an axially moving string subject to the time dependent axial speed are examined in Chapter 3. The axial speed is assumed to be a harmonic function about a (low constant mean speed, that is, mean speed is of O(ε. The amplitude-response with respect to the specific initial conditions is determined by using the two timescales perturbation method in combination with the method of characteristic coordinates. For a low mean speed, it is found that the resonances in the system can occur only when the fluctuation frequencies are equal (or close to an odd multiple of the lowest natural frequency of the system, i.e. Ω = (m 1π, where m Z +. By means of this approach, a partial differential equation in characteristic coordinates σ and ξ is obtained, rather than the coupled infinite dimensional system of ODEs as was observed in Chapter. All approximations valid on long timescales, that is, on a timescale of O(ε 1 are computed for the amplitude-response and the energy of the system. Unlike in Chapter, it is found that both the amplitude-response and the energy of the system yield unstable behavior at higher resonances on long timescales. In addition, these results are also found to be in full agreement with the numerical finite difference method. Indeed, it is shown that the energy of the system found in Refs.1 is analogue with the energy obtained in Chapter 3. Moreover, it is also proved that the elimination of secular terms yields the same infinite dimensional system of coupled ordinary differential equation for resonance case which was studied in 1. Meanwhile, for the non-resonant problem, it was observed in 48 that the amplitude-response and the energy of the system are constant and bounded up to order ε on long timescales of O(ε 1 ; the bounded amplitude-response of the system, however, for this non-resonant case is also discussed in this work using the method of characteristic coordinates. The stability of the parametric transverse vibrations of an axially moving string with a harmonic axial speed variations in the neighborhood of resonances (i.e. in the neighborhood of Ω = (m 1π, where m Z + has been investigated in Chapter 4. The asymptotic approximations of the solutions of the initial-boundary value problem, which are valid on long timescales of order ε 1 are constructed using the two timescales perturbation method in combination with the method of characteristic coordinates. Using this approach, a partial differential equation in characteristic coordinates σ and ξ is obtained. Three cases are observed for solving the partial differential equation subject to the specific harmonic initial conditions. For the case δ < α: the amplitude-response and the energy of the system yield unstable behavior on long timescales. For the case δ = α: the amplitude-response and the energy of the system are unbounded. For the case δ > α: the amplitude-response and the energy of the system are found to be bounded, where δ is detuning parameter. Additionally, the results obtained for amplitude-response and the energy of the system in this chapter are verified with a finite difference numerical method and are also in full agreement with the results obtained in Chapter for all detuning resonance conditions. 66

80 67 Future Work The method of characteristic coordinates can have numerous advantageous over the various mathematical techniques such as Fourier-mode expansion method, the Laplace transform method and so on. However we enlist some major areas for the application of the method of characteristics in the vibration analysis of axially moving strings for possible future research. The linear parametric transversal vibrations of an axially moving string with mean speed V of O(1 and fixed end conditions is solved in this thesis (Chapter by means of the Laplace transform method. The energy of the system is computed through this method, while the much more interesting phenomena present in the infinite dimensional system is seen lost when the finite dimensional approximations are taken into account in terms of the mode-truncations. Like the low mean speed case (Chapter 3 and 4, the method of characteristic coordinates is still applicable to the O(1-mean speed case to compute the energy and the amplitude-response of the system for some specific initial conditions, but special attention has to be paid to extending all functions in x outside the interval, 1. The second interesting problem is to study the nonlinear transverse vibrations of an axially moving string subject to low mean speed and fixed end conditions. In the literature, nonlinear problems describing the transversal vibrations of a non-constantly axially moving string have been studied by using the so-called Galerkin s truncation method. It will, however, be interesting to see the behavior of the system in terms of the energy and the displacement-response by applying the method of characteristic coordinates. 67

81

82 Appendix A Infinite dimensional system of coupled ODEs (Eq.(.8 In this appendix, we will show that the removal of secular terms in the approximation for w(x, t; ε in section (.3. leads to the infinite dimensional system of coupled ordinary differential equations: da k = α dt 1 m db k = α dt 1 m (m kb (m k (k + mb (k+m (k mb (k m, (m ka (m k + (k + ma (k+m + (k ma (k m, (A.1 for k Z + and m is a positive odd integer. This can be derived as follows. The introduction of an approximation of w k (t, t 1 ; ε (i.e. Eq. (. in section (.3.1 into Eq. (.1 yields the O(1-problem and O(ε-problem. The solution of the O(1-problem is given as: w k (t, t 1 = A k (t 1 cos(kπt + B k sin(kπt, (A. where A k and B k are arbitrary functions and can be used to prevent secular terms in the solution of the O(ε equation. Plugging Eq. (A. with its derivatives with respect to t and t 1 into O(ε-problem (i.e. in Eq. (.5 yields + n±k is odd + n±k is odd w k1 t { + (kπ dak w k1 = kπ dt 1 4nk n k αω cos(ωt 8nk n k (V + α sin(ωt sin(kπt db } k cos(kπt dt 1 { } A n (t 1 cos(nπt + B n (t 1 sin(nπt nπ { A n (t 1 sin(nπt + B n (t 1 cos(nπt }. (A.3 The right hand side of Eq. (A.3 clearly exhibit resonant behavior when the fluctuation frequency (Ω is equal (or close to an odd multiple of the lowest natural frequency of the system, that is, when Ω = mπ, where m is positive odd integer and therefore gives rise to secular terms. Substitution of Ω = mπ into the right hand side of Eq.(A.3 yields: 69

83 7 APPENDIX A. INFINITE DIMENSIONAL SYSTEM OF COUPLED ODES (EQ.(.8 w k1 t + (kπ dak w k1 = kπ sin(kπt db k cos(kπt dt 1 dt 1 (k + m +αkπ cos(kπt (k + m A (k m (k+m + (m k A (m k (k m + (m k A (m k (k + m +αkπ sin(kπt (k + m B (k m (k+m + (m k B (m k (k m (m k B (m k (k + m m(k + m A (k m (k+m + m(m k A (m k (k m m(m k A (m k (k + m m(k + m B (k m (k+m + m(m k B (m k (k m + m(m k B (m k +4αkπ cos(kπt +4αkπ sin(kπt + non-secular terms. (A.4 Since cos(kπt and sin(kπt are the part of homogeneous solutions of w k1, we must set the coefficients of cos(kπt and sin(kπt in the right hand side of Eq.(A.4 equal to zero in order to remove the secular terms, one obtains (A.1. 7

84 Appendix B Infinite dimensional system (Eq.(.33 In this appendix we will show that: 1 d dt k=1 1 (X k + Y k = m (Xk + Y k=1 k. (B.1 It follows from Eqs. (.31 and (.3 that: 1 k=1 d dt 1 (X k + Y k = k=1 Y k Ẏ k + X k Ẋ k, (B. and 1 d (Xk dt + Y k = m (X (k+m Y k Y (k+m X k 1 k=1 k=1 ( + (1 X (m 1 Y 1 Y (m 1 X 1 ( + ( X (m Y Y (m X. ( + (m X Y (m Y X (m ( + (m 1 X 1 Y (m 1 Y 1 X (m 1. (B.3 Differentiating Eq. (B.3 with respect to t 1, and using Eq. (.3, we obtain: 71

85 7 APPENDIX B. INFINITE DIMENSIONAL SYSTEM (EQ.(.33 1 d dt k=1 1 (Xk + Y k = m Ẋ(k+m Y k + ẎkX (k+m Ẏ(k+mX k ẊkY (k+m k=1 + (1 Ẋ(m 1Y 1 Ẏ1X (m 1 Ẏ(m 1X 1 Ẋ1Y (m 1 + ( Ẋ(m Y ẎX (m Ẏ(m X ẊY (m. + (m ẊY (m Ẏ(m X ẎX (m Ẋ(m Y + (m 1 Ẋ1Y (m 1 Ẏ(m 1X 1 Ẏ1X (m 1 Ẋ(m 1Y 1 (B.4 = 1 d dt k=1 1 (Xk + Y k = m (k + m(xk + Y k m k=1 r=m+1 (r m(x r + Y + m(m 1(X 1 + Y 1 + m(m (X + Y. + m(x (m + Y (m + m(x (m 1 + Y (m 1 r = 1 d dt k=1 1 (X k + Y k = m k=m+1 (X k + Y k + m (X 1 + Y 1 + m (X + Y (B.5. + m (X (m + Y (m + m (X (m 1 + Y (m 1 = 1 So (B.1 has been proved. d dt k=1 1 (X k + Y k = m (Xk + Y k=1 k. (B.6 7

86 Appendix C Proof of Integral (Eq.(3.46 In this appendix, it will be shown that I = g ξ ( (n 1π(ξ + σ cos sin ( (m 1π(ξ σ dξ sin((m 1πσf σ. (C.1 First observe that I = 1 g ξ sin((n + m 1πξ cos((n mπσ + cos((n + m 1πξ sin((n mπσ sin((n mπξ cos((n + m 1πσ cos((n mπξ sin((n + m 1πσ dξ. (C. In order to simplify (C., the following Fourier series representation for the function w (σ, ξ, τ is used. where w (σ, ξ, τ = ñ=1 f (σ, τ = 1 g (ξ, τ = 1 g ξ (ξ, τ = ( Añ (τ cos(ñπt + Bñ (τ sin(ñπt sin(ñπx ñ=1 ñ=1 ñ=1 f (σ, τ + g (ξ, τ, (C.3 ( Añ (τ sin(ñπσ + Bñ (τ cos(ñπσ, (C.4 ( Añ (τ sin(ñπξ Bñ (τ cos(ñπξ, (C.5 ñπ ( Añ (τ cos(ñπξ + Bñ (τ sin(ñπξ. (C.6 73

87 74 APPENDIX C. PROOF OF INTEGRAL (EQ.(3.46 Substitution of (C.6 into (C. yields I = 1 ñ=1 ñπ ( cos((n mπσ Añ (τ +Bñ (τ ( + sin((n mπσ Añ (τ +Bñ (τ ( cos((n + m 1πσ Añ (τ cos(ñπξ sin((n + m 1πξdξ sin(ñπξ sin((n + m 1πξdξ cos(ñπξ cos((n + m 1πξdξ sin(ñπξ cos((n + m 1πξdξ +Bñ (τ ( sin((n + m 1πσ Añ (τ +Bñ (τ cos(ñπξ sin((n mπξdξ sin(ñπξ sin((n mπξdξ cos(ñπξ cos((n mπξdξ sin(ñπξ cos((n mπξdξ, (C.7 = (n + m 1π { } A 4 (n+m 1 (τ sin((n mπσ + B (n+m 1 (τ cos((n mπσ (n mπ { } A (n m (τ sin((n + m 1πσ + B (n m (τ cos((n + m 1πσ 4 (m nπ 4 { A (m n (τ sin((n + m 1πσ B (m n (τ cos((n + m 1πσ}. (C.8 74

88 75 I = cos((m 1πσ cos((m 1πσ (n + m 1π { A 4 (n+m 1 sin((n + m 1πσ } +B (n+m 1 cos((n + m 1πσ (n mπ { A 4 (n m sin((n mπσ } +B (n m cos((n mπσ sin((m 1πσ + cos((m 1πσ (m nπ { A 4 (m n sin((m nπσ } +B (m n cos((m nπσ (n + m 1π { A 4 (n+m 1 cos((n + m 1πσ } B (n+m 1 sin((n + m 1πσ (C.9 sin((m 1πσ sin((m 1πσ (n mπ { A 4 (n m cos((n mπσ } B (n m sin((n mπσ (m nπ { A 4 (m n cos((m nπσ B (m n sin((m nπσ}. For further simplification, we take m = 1, m =, and so on. For m = 1: nπ { } I = cos(πσ A n (τ sin(nπσ + B n (τ cos(nπσ 4 (n 1π { } A 4 (n 1 (τ sin((n 1πσ + B (n 1 (τ cos((n 1πσ nπ { } sin(πσ A n (τ cos(nπσ B n (τ sin(nπσ 4 + (n 1π { A 4 (n 1 (τ cos((n 1πσ B (n 1 (τ sin((n 1πσ}, (C.1 75

89 76 APPENDIX C. PROOF OF INTEGRAL (EQ.(3.46 for m = : = sin(πσ = cos(3πσ + nπ { } A n (τ cos(nπσ B n (τ sin(nπσ sin(πσf σ, (n + 1π { } A 4 (n+1 (τ sin((n + 1πσ + B (n+1 (τ cos((n + 1πσ (n π 4 ( nπ 4 ( nπ 4 { } A (n (τ sin((n πσ + B (n (τ cos((n πσ { } A ( n (τ sin(( nπσ + B ( n (τ cos(( nπσ (n + 1π { } sin(3πσ A 4 (n+1 (τ cos((n + 1πσ B (n+1 (τ sin((n + 1πσ (n π { } + A 4 (n (τ cos((n πσ B (n (τ sin((n πσ { + A ( n (τ cos(( nπσ B ( n (τ sin(( nπσ}, (C.11 (C.1 = sin(3πσ nπ { } A n (τ cos(nπσ B n (τ sin(nπσ sin(3πσf σ. (C.13 In general, we obtain I = g ξ which justifies (C.1. ( (n 1π(ξ + σ cos sin ( (m 1π(ξ σ dξ sin((m 1πσf σ, (C.14 76

90 Appendix D The infinite dimensional system (Eq.(3.47 If we substitute the Fourier series for the function f into Eq.(3.47, we will obtain the same infinite dimensional system of coupled ordinary differential equations, which were obtained by authors using Fourier-series expansion (see 1. da k dτ db k dτ = α m (m kb (m k (k + m B (k+m (k m B (k m = α m (m ka (m k + (k + m A (k+m + (k m A (k m,, (D.1 where m is the positive odd integer, that is, m = m 1, for m Z + and fixed and k = 1,, 3,... Eq.(3.47 is given by f στ α sin((m 1πσf σ + α (m 1π cos((m 1πσf σσ =. (D. To show that this partial differential equation leads to the same infinite dimensional system of coupled ordinary differential equations as given in 1, the Fourier series for f and g are helpful. We know that w (σ, ξ, τ = f (σ, τ + g (ξ, τ, (D.3 where f (σ, τ = f στ (σ, τ = f σσ (σ, τ = nπ A n (τ Substitution of Eqs. (D.4-(D.6 into Eq.(D. yields sin(nπσ + B n(τ cos(nπσ, (D.4 ( A n(τ cos(nπσ B n(τ sin(nπσ, (D.5 n π ( A n (τ sin(nπσ + B n (τ cos(nπσ. (D.6 77

91 78 APPENDIX D. THE INFINITE DIMENSIONAL SYSTEM (EQ.(3.47 α + (m 1π nπ ( A n(τ cos(nπσ B n(τ sin(nπσ = nπ ( α A n (τ cos(nπσ B n (τ sin(nπσ sin((m 1πσ n π ( A n (τ sin(nπσ + B n (τ cos(nπσ cos((m 1πσ. (D.7 Multiplying Eq.(D.7 with sin(kπσ and integrating w.r.t σ over the period, and using the orthogonality relations for the sine functions on < σ <, we obtain n A n(τ = α + α (m 1 cos(nπσ sin(kπσdσ B n(τ n A n (τ B n (τ n A n (τ +B n (τ sin(nπσ sin(kπσdσ cos(nπσ sin((m 1πσ sin(kπσdσ sin(nπσ sin((m 1πσ sin(kπσdσ sin(nπσ cos((m 1πσ sin(kπσdσ cos(nπσ cos((m 1πσ sin(kπσdσ, (D.8 or equivalently, kb k (τ = α (k m + 1A (k m+1 (τ + (m 1 ka (m 1 k (τ (k + m 1A (k+m 1 (τ + α (k m + 1 A m 1 (k m+1 (τ + (k + m 1 A (k+m 1 (τ (m 1 k A (m 1 k (τ, (D.9 = db k dτ = α (m 1 ka m 1 (m 1 k + (k + m 1A (k+m 1 +(k m + 1A (k m+1. (D.1 Next, by multiplying Eq.(D.7 with cos(kπσ and integrating w.r.t σ over the period, and 78

92 79 using the orthogonality relations, we obtain n A n(τ cos(nπσ cos(kπσdσ B n(τ = α + α (m 1 n A n (τ B n (τ n A n (τ +B n (τ sin(nπσ cos(kπσdσ cos(nπσ sin((m 1πσ cos(kπσdσ sin(nπσ sin((m 1πσ cos(kπσdσ sin(nπσ cos((m 1πσ cos(kπσdσ cos(nπσ cos((m 1πσ cos(kπσdσ, (D.11 or equivalently, ka k (τ = α (k m + 1B (k m+1 (τ (k + m 1B (k+m 1 (τ (m 1 kb (m 1 k (τ + α (k m + 1 B m 1 (k m+1 (τ + (k + m 1 B (k+m 1 (τ +(m 1 k B (m 1 k (τ, (D.1 = da k dτ = α (m 1 kb m 1 (m 1 k (k + m 1B (k+m 1 (k m + 1B (k m+1. (D.13 From Eqs.(D.1 and (D.13, we have da k dτ db k dτ = α m (m kb (m k (k + m B (k+m (k m B (k m = α m (m ka (m k + (k + m A (k+m + (k m A (k m,, (D.14 where m is the positive odd integer, that is, m = m 1, for m = 1,, 3,... Hence (D.1 is proved to be equivalent with (D.. 79

93

94 Appendix E The infinite dimensional system (Eq. (4.4 In this appendix, the equivalence between the partial differential equation (4.4 and the infinite dimensional system of ODEs for the Fourier coefficients given in 1, is proved; that is, the equation da k = α dτ m (k + m { A (k+m sin(δτ + B (k+m cos(δτ } (k m { A (k m sin(δτ B (k m cos(δτ } db k dτ (m k { A (m k sin(δτ + B (m k cos(δτ }, = α m (k + m { A (k+m cos(δτ + B (k+m sin(δτ } (k m { A (k m cos(δτ + B (k m sin(δτ } (m k { A (m k cos(δτ B (m k sin(δτ }, (E.1 where m is the positive odd integer, that is, m = m 1, for m Z + and fixed and k = 1,, 3,..., has been proved from the following equation: f στ α sin((m 1πσ δτf σ + α (m 1π cos((m 1πσ δτf σσ =. (E. If we substitute the Fourier series v (σ, ξ, τ = ( A n (τ cos(nπt + B n (τ sin(nπt sin(nπx f (σ, τ + g (ξ, τ, (E.3 where f (σ, τ = 1 ( A n (τ sin(nπσ + B n (τ cos(nπσ, (E.4 81

95 8 APPENDIX E. THE INFINITE DIMENSIONAL SYSTEM (EQ. (4.4 and g (ξ, τ = 1 ( A n (τ sin(nπξ B n (τ cos(nπξ, (E.5 and where σ = x t, ξ = x + t and τ = εt, into Eq.(E., we will obtain the same infinite dimensional system of coupled ordinary differential equations, which were obtained in 1 by using this Fourier-series expansion for the solution, that is, Eq. (E.1. From Eq.(E.4, we have f σ (σ, τ = f στ (σ, τ = f σσ (σ, τ = Substitution of Eqs. (E.6-(E.8 into Eq.(E.5 yields nπ ( A n (τ cos(nπσ B n (τ sin(nπσ, (E.6 nπ ( A n(τ cos(nπσ B n(τ sin(nπσ, (E.7 n π ( A n (τ sin(nπσ + B n (τ cos(nπσ. (E.8 α + (m 1π nπ ( A n(τ cos(nπσ B n(τ sin(nπσ = nπ ( α A n (τ cos(nπσ B n (τ sin(nπσ sin((m 1πσ δτ n π ( A n (τ sin(nπσ + B n (τ cos(nπσ cos((m 1πσ δτ. (E.9 Multiplying Eq.(E.9 with sin(kπσ and integrating with respect to σ over the period, and using the orthogonality properties for the sine functions on < σ <, we obtain n A n(τ = α + α (m 1 cos(nπσ sin(kπσdσ B n(τ n A n (τ B n (τ n A n (τ +B n (τ sin(nπσ sin(kπσdσ cos(nπσ sin((m 1πσ δτ sin(kπσdσ sin(nπσ sin((m 1πσ δτ sin(kπσdσ sin(nπσ cos((m 1πσ δτ sin(kπσdσ cos(nπσ cos((m 1πσ δτ sin(kπσdσ, (E.1 8

96 83 or equivalently, kb k (τ = α { (k m + 1A (k m+1 (τ + (m 1 ka (m 1 k (τ } (k + m 1A (k+m 1 (τ + α { (k m + 1 A m 1 (k m+1 (τ + (k + m 1 A (k+m 1 (τ (m 1 k A (m 1 k (τ} cos(δτ +α { (k m + 1B (k m+1 (τ (m 1 kb (m 1 k (τ } +(k + m 1B (k+m 1 (τ + α { (k m + 1 B m 1 (k m+1 (τ (k + m 1 B (k+m 1 (τ (E.11 = db k dτ = +(m 1 k B (m 1 k (τ} sin(δτ, α (k + m 1 { A m 1 (k+m 1 cos(δτ + B (k+m 1 sin(δτ } (k m + 1 { A (k m+1 cos(δτ + B (k m+1 sin(δτ } (m 1 k { A (m 1 k cos(δτ B (m 1 k sin(δτ }. (E.1 Next, by multiplying Eq.(E.9 with cos(kπσ and integrating with respect to σ over the period, and using the orthogonality properties for the cosine function on the interval < σ <, we obtain n A n(τ cos(nπσ cos(kπσdσ B n(τ sin(nπσ cos(kπσdσ = α + α (m 1 n A n (τ B n (τ n A n (τ +B n (τ cos(nπσ sin((m 1πσ δτ cos(kπσdσ sin(nπσ sin((m 1πσ δτ cos(kπσdσ sin(nπσ cos((m 1πσ δτ cos(kπσdσ cos(nπσ cos((m 1πσ δτ cos(kπσdσ, (E.13 83

97 84 APPENDIX E. THE INFINITE DIMENSIONAL SYSTEM (EQ. (4.4 or equivalently, ka k (τ = α { (k m + 1A (k m+1 (τ (m 1 ka (m 1 k (τ } (k + m 1A (k+m 1 (τ + α { (k m + 1 A m 1 (k m+1 (τ + (k + m 1 A (k+m 1 (τ +(m 1 k A (m 1 k (τ} sin(δτ +α { (k m + 1B (k m+1 (τ (m 1 kb (m 1 k (τ } (k + m 1B (k+m 1 (τ + α { (k m + 1 B m 1 (k m+1 (τ + (k + m 1 B (k+m 1 (τ (E.14 = da k dτ = +(m 1 k B (m 1 k (τ} cos(δτ, α (k + m 1 { A m 1 (k+m 1 sin(δτ + B (k+m 1 cos(δτ } (k m + 1 { A (k m+1 sin(δτ B (k m+1 cos(δτ } (m 1 k { A (m 1 k sin(δτ + B (m 1 k cos(δτ }. (E.15 From Eqs.(E.1 and (E.15, we have da k = α dτ m (k + m { A (k+m sin(δτ + B (k+m cos(δτ } (k m { A (k m sin(δτ B (k m cos(δτ } db k dτ (m k { A (m k sin(δτ + B (m k cos(δτ }, = α m (k + m { A (k+m cos(δτ + B (k+m sin(δτ } (k m { A (k m cos(δτ + B (k m sin(δτ } (m k { A (m k cos(δτ B (m k sin(δτ }, (E.16 where m is the positive odd integer, that is, m = m 1, for m = 1,, 3,... Hence (E. is proved to be equivalent with (E.1. 84

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104 Summary If you can t explain it simply, You don t understand it well enough. Albert Einstein Axially moving strings such as serpentine belts, aerial cables, power transmission belts, plastic films, magnetic tapes, paper sheets and textile fibers can be met in numerous engineering applications. These include mechanical, aerospace, and automotive engineering fields. Despite their enormous advantages in such diverse fields, the noise and excessive vibrations have limited their applications. For safe design of such systems, it is essential to analyze and predict their vibratory motions. This thesis describes the stability analysis of parametric transversal vibrations in an axially moving string with the time-dependent axial speed and with fixed end conditions. The parametric transverse vibrations of an axially moving string is mathematically modeled as a second order linear homogeneous partial differential equations with variable coefficients. The time-dependent velocity is presumed to be a harmonically varying function about a constant mean speed and relatively small compared to the wave speed. The analytic-approximate method, i.e. the two timescales perturbation method is used in combination with either the Fourier-mode expansion method or the Laplace transform method or the method of characteristic coordinates to construct the formal asymptotic approximations of the solutions of the governing equations of motion on long timescales of order ε 1. The vibratory motion is described in terms of the amplitude-response and the energy of the systems. The results obtained for the amplitude-response and the energy of the system by using the analytic-approximate method are verified through the finite difference numerical technique. For a low speed belt systems, that is, when the mean speed is of O(ε, instability of the parametric vibrations (i.e. resonances occurred only when the velocity fluctuation frequencies in a belt system are equal (or close to an odd multiple of the lowest natural frequency of the system. For all resonances, the amplitude-response and the energy of the system are found to be unbounded, while the individual mode responses either revealed stable and/or unstable behavior of the solutions of the governing equation of motion. While, in the neighborhood of resonances, the energy and the amplitude-response behavior detected to be unstable when δ α, and stable when δ > α, where δ is the detuning parameter. Meanwhile, for the mean speed V of O(1, the system induces resonances only when the velocity fluctuation frequency Ω is equal (or close to kπ(1 V, where k Z+. However, the results for both the amplitude-response and the energy of the system in this case of mean speed are found to be similar to those which were found for the low mean speed case; that is, the energy of the system grows exponentially when the time t increases, while the responses of individual 91

105 9 Summary mode are either found to be stable and/or unstable. It turned out that the amplitude-response and the energy of the system increased at resonances on long timescales. The individual mode responses, on the other hand, have yielded the stable and/or unstable behavior which confirms the non-applicability of the truncation method on long timescales of O(ε 1 for these string-like problems. 9

106 Samenvatting Als je het niet eenvoudig uit kan leggen, begrijp je het niet goed genoeg. Albert Einstein Axiaal bewegende snaren of kabels, zoals aandrijfriemen, liftkabels, kunststoffilms, geluidsbanden, papiervellen en textielvezels, worden regelmatig toegepast in de ingenieurspraktijk. De vakgebieden beslaan onder de werkluigbouwkunde, de luchvaart-en ruimtevaart techniel, en de auto-industrie. Ondanks de grote voordelen van axiaal bewegende kabels in dit scala aan vakgebieden, wordt de toepassing ervan beperkt vanwege geluid en overmatige trilling en. Om een veilig ontwerp van dit soort kabelsystemen te maken, is het essentieel om de trillingen te analyseren en te voorspellen. Dit proefschrift beschrijft de stabiliteitsanalyse van parametrische transversale in een axiaal bewegende kabel die beweegt met een tijdsafhankelijke, axiale snelheid. De transversale trillingen van een axiaal bewegende kabel worden wiskundig gemodelleerd door middel van een tweedeorde lineaire, homogene, partiële differentiaalvergelijking met variabele coëfficiënten. Er wordt aangenomen dat de tijdsafhankelijke snelheid zich gedraagt als een harmonisch variërende functie rond een constante gemiddelde snelheid die relatief klein is ten opzichte van de golfsnelheid. Een analytische benaderingsmethode, te weten de twee-tijdschalen perturbatiemethode, wordt gebruikt in combinatie met een Fourier-eigenfunctie expansiemethode, de Laplace transformatiemethode of de karakteristieke coördinatenmethode om formele asymptotische benaderingen van de oplossingen van de geldende bewegingsvergelijkingen te construeren voor lange tijdschalen van orde ε 1. De trillingen worden beschreven in termen van de amplitude-respons en energie van de systeem. De resulterende amplitude-respons en energie uit de analytische benaderingsmethode zijn geverifieerd met een numerieke eindige-differentiemethode. Voor bandsystemen met een lage snelheid, dat wil zeggen met een gemiddelde snelheid van O(ε, komt instabiliteit van de parametrische trillingen (d.w.z. resonantie alleen voor wanneer de snelheidsfluctuatie gelijk (of bijna gelijk is aan een oneven veelvoud van de laagste natuurlijke frequentie van het systeem. Zowel de amplitude-respons als de energie blijken dan onbegrensd te worden, terwijl de respons in individuele modi stabiel en/of instabiel gedrag van oplossingen van de geldende bewegingsvergelijkingen lieten zien voor alle hogere resonantiefrequenties. In de buurt van resonantie is het gedrag van de energie en amplitude-respons instabiel als δ α en stabiel als δ > α waarbij δ een detuning-parameter is. Voor gemiddelde snelheden V van O(1 het systeem induceert alleen resonantie als de snelheidsfluctuatiefrequentie Ω gelijk (of bijna gelijk is aan kπ(1 V, waarbij k Z+. Echter, in het geval van hoge gemiddelde snelheden zijn de resultaten voor zowel de amplitude respons en de energie van het systeem vergelijkbaar met het geval van lage gemiddelde snelheid; dat wil zeggen 93

107 94 Samenvatting de energie van het systeem neemt exponentieel toe wanneer t toeneemt, terwijl de respons van individuele modi ofwel stabiel ofwel instabiel zijn. Aan de ene kant is het gebleken dat de amplitude-respons en energie van het systeem toeneemt bij resonantie op lange tijdschalen. Aan de andere kant levert de respons van de individuele modi stabiel en/of instabiel gedrag op. Dit bevestigd dat de truncatiemethode niet toepasbaar is op lange tijdschalen van O(ε 1 for dit type kabelproblemen. 94

108 Acknowledgements Those that know, do. Those that understand, teach. Aristotle During my journey as a doctoral candidate in the Delft Institute of Applied Mathematics, TU Delft, Netherlands, I received tremendous support and assistance from a number of individuals and would like to express my gratitude towards all of them who have made this dissertation possible. First and foremost, I would like to express special words of gratitude to my daily supervisor Dr.ir.Wim van Horssen, for believing in me and accepting me as a PhD candidate in his group. I must appreciate him for being an excellent thesis advisor and teaching mentor, who has supported me throughout PhD research project. This was he, who taught me many interesting phenomena in mechanics and introduced me to perturbation methods. Although, he always gave me a liberty to work, at the same time he continued to contribute through his valuable feedback. I am grateful to him for stimulating discussions, useful comments and suggestions for improvement of my dissertation as well as conference and journal papers. His patience, support and never ending enthusiasm has always helped me to overcome many situations and getting this dissertation done on time. In addition, I must admire him as a person. Whenever I had a visit in his office without appointment, he has always responded positively. Besides my daily supervisor, I am fully indebted to my promoter Prof. dr. ir. Heemink for his continuous encouragement, kindness and warm hospitality. I wish to extend my special gratitude to the staff at the mathematical physics group, TU Delft, specially Evelyn and Dorothee, for their prompt help in many of the official matters. My special thanks also goes to Kees Lemmens, whose contribution in computer-related issues is highly appreciated. I take this opportunity to express gratitude to all the department faculty members for all the fun, we have had during the cup of coffee and on various social occasions. The gratitude also goes to the people at the CICAT of TU Delft, specially Theda Olsder for her kind assistance in administratively matters during my PhD journey. I am specially thankful to Yoeri Dijkstra for translating the summary in Dutch language. I like to mention my PhD colleagues, both past and present, at the mathematical physics group, specially Umer Altaf, Sajad, Atif, Hyder, Wei, Fu, Sha, Mohat and others, for their useful assistance and discussions during this course of research. I would like to extend my special thanks to my former and current officemates Tugce, Corine, Robert, Jin, and Wang who contributed making my stay pleasant and fruitful at the department of Applied Mathematics, TU Delft. I would like to mention here also my group-fellows particularly Nick and Tugce for sharing life stories, traveling together to various conferences and discussing research. 95

109 96 Acknowledgments I also thank my Pakistani friends in Delft, Hanan, Faheem Raees, Imran Ashraf, Faisal, Seyab, Fakhar, Akram, Ibrahim, Hamid Mushtaq, Osama, Usman, Zubair, Umer, Noor, Iftikhar Faraz, Tabish, Nauman, Qasim, Fahim Riaz, Irfan, Muneeb, Farooq, Husssam, Fawad, Fahad, Saad and all those whose names I have missed. I must appreciate you guys for all the fun we have had in the last four years. Additionally, my gratitude is also extended to my well wishers back in Pakistan and in different parts of the world, who have always wished me and pray for my success. My special and heartiest thanks in particular go to uncle Habibullah Malookani for his unconditional support in scholarship-related issues. Your support in this regard is indeed the first step of this journey and highly appreciated. Most importantly, I would like to extend my sincerest thanks and appreciations to my beloved family, specially my parents for their endless support, patient and blessings; without their blessings, this work would not have been possible. Last but definitely not least, the financial support from the Quaid-e-Awam University, Nawabshah under faculty development program by Higher Education Commission of Pakistan and TU Delft are highly acknowledged. Rajab Ali September 1, 16 Delft, Netherlands 96

110 List of Publications Journal Papers R. A Malookani and W.T.van Horssen, On parametric stability of a non-constant axially moving string near resonances, Accepted in Journal of Vibration and Acoustics (online (August 16. R. A Malookani and W.T.van Horssen, On the asymptotic approximation of the solution of an equation for a non-constant axially moving string, Journal of Sound and Vibration 367, 3 18 (16. R. A Malookani and W.T.van Horssen, On resonance and applicability of Galerkin s truncation method for an axially moving string with time-varying velocity, Journal of Sound and Vibration 344, 1 17(15. Conference Proceedings R. A Malookani and W.T.van Horssen, On the vibrations of an axially moving string with a timedependent velocity, Proceeding in the International Mechanical Engineering Congress and Exposition (IMECE conference November 13-19, 15, Houston, Texas, USA. R. A Malookani, S. H.Sandilo and W.T.van Horssen, On the applicability of Galerkin s truncation method for string-like problems, the 8 th European Nonlinear Dynamics Conference (ENOC (poster, July 6-11, 14, Vienna Austeria. Presentations On the vibrations of an axially moving string with a time-dependent velocity, International Mechanical Engineering Congress and Exposition (IMECE conference November 13-19, 15, Houston, Texas, USA. On the applicability of Galerkin s truncation method for an axially moving string with a harmonically varying speed, Nonlinear Dynamics in Natural Systems (NDNS+ PhD days in Analysis and Dynamics, 3-4 April 15, Lunteren, The Netherlands. On parametric vibrations of an axially moving string, Nonlinear Dynamics in Natural Systems (NDNS+ PhD days in Analysis and Dynamics, 4-5 April 14, Lunteren, The Netherlands. On the transversal vibrations of a non-constant axially moving string, The National Conference of Recent Advance in Pure and Applied Mathematics (RAPAM, January 16, QUEST, Nawabshah, Pakistan. 97

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112 About the author Rajab A. Malookani was born on February 15, 198 in Sanghar, Sindh, Pakistan. After finishing his schooling, he entered the New Ali Garh College in Tando Adam, where he gained his intermediate education in In 3, he earned his Bachelor of Science degree in Mathematics at the Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Pakistan. From the same institution, he received his Master of Science degree in Mathematics in 4. In 5, Rajab A.Malookani joined as a research assistant in the department of Basic Science and Related Studies (BSRS at Mehran University of Engineering and Technology, Jamshoro, Pakistan. In 8, he joined as a lecturer in the department of Mathematics and Statistics at the Quaid-e-Awam University of Engineering, Science and Technology, Nawabshah, Pakistan. In the January of 1, he was awarded a scholarship under the faculty development program of Higher Education Commission (HEC of Pakistan to pursue PhD. In the spring of 1, he was accepted into the doctoral program in the area of vibrations of conveyor belt systems at the department of Applied Mathematics, Delft University of Technology, The Netherlands. In the September of 1, he moved to the Netherlands and joined the mathematical physics group in the department of Applied Mathematics, Delft University of Technology as a PhD candidate and worked on the project vibrations of conveyor belt systems under the supervision of Dr.ir.Wim T. van Horssen. During his PhD journey, he presented his research in the local as well as international conferences related to the field of dynamics and vibrations. In the mean time, he also reviewed conference and journal papers. His hobbies are reading, traveling, and watching documentaries on astronomy and multi-cultures. In future, Rajab A. Malookani intends to pursue his career in academia. 99