On Aspects of Asymptotics for - TU Delft

On Aspects of Asymptotics forAxially Moving Continua

Sajad H. Sandilo

On Aspects of Asymptotics forAxially Moving Continua

Proefschrift

ter verkrijging van de graad van doctoraan 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 verdedigenop donderdag 12 december 2013 om 12.30 uur

door

Sajad Hussain SANDILO,

Master of Science in Applied Mathematics,Quaid-i-Azam University Islamabad, Pakistan

geboren te Larkana, Sindh-Pakistan.

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. A. W. Heemink

Copromotor:Dr. ir. W. T. van Horssen

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. dr. ir. A. W. Heemink Technische Universiteit Delft, promotorDr. ir. W. T. van Horssen Technische Universiteit Delft, copromotorProf. dr. A. K. Abramian Russian Academy of Sciences, RussiaProf. dr. S. Kaczmarczyk The University of Northampton, United KingdomProf. dr. A. V. Metrikine Technische Universiteit DelftProf. dr. ir. C. Vuik Technische Universiteit DelftProf. dr. W. D. Zhu University of Maryland, Baltimore County, USA

Delft University of Technology

This thesis has been completed in fulfillment of the requirements of the Delft University ofTechnology for the award of the Ph.D. degree. The research described in this thesis was car-ried out at Mathematical Physics Department, Delft Institute of Applied Mathematics, Fac-ulty of Electrical Engineering, Mathematics and Computer Science. The research describedin this thesis was supported by Quaid-e-Awam University Nawabshah Sindh-Pakistan un-der the Faculty Development Program of Higher Education Commission of Pakistan and theDelft University of Technology, The Netherlands.

ISBN 978-94-6186-237-2

Copyright c© 2013 by S. H. Sandiloe-post:[email protected]

All rights reserved. No part of the material protected by this copyright notice may be re-produced or utilized in any form or by any means, electronic or mechanical, including pho-tocopying, recording, or by any information storage and retrieval system, without writtenpermission from the author.

Printed in The Netherlands by Sieca Repro.

All that we are is the result of what we have thought.The mind is everything. What we think we become.

Gautama Buddha

To my parents Qurban Ali and Gulestanmy wife Raheelamy daughter Athinaand my sisters Anita, Sonia and Fozia

Contents

1 Introduction 11.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Analytical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 On Boundary Damping for an Axially Moving Tensioned Beam 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 152.3 The energy and the boundedness of solutions . . . . . . . . . . . . . . . . 162.4 Application of the two timescales perturbation method . . . . . . . . . . . 172.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 On Variable Length Induced Vibrations of a Vertical String 293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 313.3 The case l(t) = l0 + vt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Application of the two-timescales perturbation method . . . . . . . 343.4 The case l(t) = l0 +β sin(ωt) . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.1 Application of the two-timescales perturbation method . . . . . . . 423.4.2 The case ω = π

l0. . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.3 The case ω = πl0+ εσ . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.4 The energy of the infinite dimensional system . . . . . . . . . . . . 453.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

vii

4 On a Cascade of Autoresonances in an Elevator Cable System 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 504.3 Interior layer analysis for the single ordinary differential equation . . . . . 544.4 A three timescales perturbation method . . . . . . . . . . . . . . . . . . . 574.5 Approximations of the solutions of the initial-boundary value problem . . . 664.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

A Real and positive eigenvalues 69

B Large eigenvalues and the damping parameter 71

C The WKBJ (Liouville-Green)-approximation 73

D Resonant terms 75

E Application of the truncation method 77

F The infinite dimensional system 79

G An unexpected timescale of order 1√ε 81

Bibliography 83

Summary 89

Samenvatting 91

Summary (in Russian) 93

Summary (in Chinese) 95

About the author 97

Acknowledgements 99

List of publications and presentations 101

Chapter 1Introduction

Men who wish to know about the world must learn about it inits particular details.

Heraclitus

1.1 Historical background

Vibrations occur frequently in a variety of many physical or mechanical structures suchas tall or high-rise buildings, cable-stayed or suspension bridges, electric power lines,

conveyor belts, elevator cables, pipes transporting fluids, crane cables, heavy lift cranes andmining hoists. Winds, earth quakes, and traffic can be sources that act on mechanical struc-tures. These loads can induce, sometimes large, structural or mechanical vibrations. De-pending on the nature and magnitude of the load, these vibrations can cause damage to aphysical structure. Structural failure can occur because of large dynamic stresses developedduring earth quakes or even wind-induced vibration. During an earth quake, mutual pound-ing between adjacent buildings may occur. Very often vibrations are not a desirable state ofa physical structure. The most serious effect of the vibration, especially in the case of ma-chinery, is that sufficiently high alternating stresses can produce fatigue failure in machineand structural parts. Less serious effects include increased wear of parts and general mal-functioning of apparatus. If uncontrolled, the vibration can lead to catastrophic situations.For instance, in 1940 the Tacoma Narrows suspension bridge in USA collapsed completelydue to an 18.9 m/s wind-flow induced 0.23 Hz torsional oscillation of the bridge deck. Thecollapse of the bridge is sometimes characterized as in physics and in applied mathematicstext books as a classical example of resonance. This collapse, and the research that followed,led to an increased understanding of wind and structure interactions. More examples of un-desirable oscillations are the oscillations of the cable stays of the Erasmus bridge in TheNetherlands during stormy and rainy weather. The main goal of the applied mathematicians,mechanical or civil engineers and, physicists is to understand and to avoid or to reduce thesevibrations.Vibrations are initiated when an inertia element is displaced from its equilibrium positiondue to an energy imparted to the system through an internal or external source. A restoring

2 1. Introduction

Figure 1.1: An example of a horizontally translating cable car system

force or moment pulls the element back towards equilibrium. During this process a physicalsystem experiences the transfer of its potential energy to the kinetic energy and the kinetic en-ergy back to the potential energy, alternatively. In the absence of nonconservative forces, thistransfer of energy is continual, causing the system to oscillate about its equilibrium position.As vibrations can damage structures and can result in human discomfort, it is important tomitigate structural or mechanical vibrations. If a nonconservative (damping) force is present,the system is damped and some energy is dissipated in each cycle of vibration. Damping de-vices are widely used to control structural or mechanical motion. To suppress the oscillationamplitudes various types of boundary damping can be applied (see, (Cox and Zuazua 1995),(Darmawijoyo and van Horssen 2002), (Darmawijoyo and van Horssen 2003), (Sandilo andvan Horssen 2012), and (Zarubinskaya and van Horssen 2006b)). For instance, in the 1930’sthe Stockbridge damper and similar devices have been used successfully to damp out ca-ble vibrations. However, it was noted that the dynamic characteristics of the damper andthe cable were sometimes improperly matched, and as a result serious damage to the cablesat the points of attachment of the dampers, that is, at their clamp occurred (Hagedorn andSeemann 1998). Nowadays, various types of passive dampers applied at the boundary havebeen considered extensively ((Rao 1993), and (Wang et al. 1993)). The beam or string isweakly damped because the boundary damping parameters are small, but these dampers canproduce strong (or uniform) damping effects. Dampers can also be connected to an inter-mediate point of the beam (Main and Jones 2007). The vibrations of a beam with a viscousdamper have been studied in (Zarubinskaya and van Horssen 2006a).In physical or mechanical systems oscillations around equilibrium positions can be describedby mathematical models, such as, linear (nonlinear) wave equations or by linear (nonlinear)beam equations. Historically, the early work on cable dynamics goes back as early as the

1.1. Historical background 3

Figure 1.2: An example of a moving conveyor belt system

eighteenth century, when vibrating strings such as that of musical instrument were studied byGalileo Galilei, Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, Joseph LouisLagrange and Joseph Fourier. Mathematically, string vibrations are modeled by a waveequation which is an important second order partial differential equation for the descriptionof waves or vibrations. When bending stiffness becomes more important, the descriptionof mechanical vibrations is represented by a fourth-order partial differential equation oftenknown as Euler-Bernoulli beam equation. Nowadays, these equations are modified with in-clusion of axial transport velocity of strings or beams and applied to far more complicatedphysical systems often known as axially moving continua or continuous systems. The travel-ing, tensioned Euler-Bernoulli beam and the traveling flexible string are the most commonlyused models for such type of axially moving continua. They are classified in the categoryof one dimensional continuous systems and consequently the displacement field depends ontime and on a single spatial co-ordinate. A wide range of scientific books and journal pa-pers are devoted to these problems. In order to solve challenging problems for vibrations,a number of techniques and methods has been developed (see, for instance, (Hagedorn andDasGupta 2007), (Meirovitch 1997), and (Weaver et al. 1990)). The pioneering work ofaxially moving continua is ascending to Willard L. Miranker (see, (Miranker 1960)) whoconsidered a model for the transverse vibrations of a tape moving between a pair of pulleysby using a variational procedure. By means of energy-type integrals, it was shown that theenergy of that portion of the tape between the pulleys is not conserved, but that there is aperiodic transfer of energy into and out of the system. Further, this work was carried onby Mote and Ulsoy (see, (Mote 1972), and (Ulsoy et al. 1978)), who investigated the vi-bration of a band saw and introduced the gyroscopic items into axially translating system.The summary work of axially moving continua was presented in ((Wickert and Mote 1988),

4 1. Introduction

(Wickert and Mote 1989), and (Wickert and Mote 1990)). A two timescales perturbationmethod was introduced in (Suweken and van Horssen 2003a) with a low and time-varyingvelocity and, a two timescales perturbation method and a Laplace transform method wereused in (Ponomareva and van Horssen 2007) with velocity to be time-varying and to be ofthe same order of magnitude as the wave speed to investigate the applicability of Galerkin’struncation method. It was found in both research articles that there are infinitely many in-teraction between vibration modes, and that the Galerkin’s truncation method can not beapplied in order to obtain asymptotic results on long timescales. A similar analysis was pre-sented in (Andrianov and Awrejcewicz 2006). An associated beam-like equation was stud-ied in (Suweken and van Horssen 2003b) and it was shown in this paper that there are lessproblems in applying the truncation method. Transversal vibrations of a moving beam withviscous damping were studied in (Pakdemirli and Oz 2008). Using Hamiltonian dynamicsanalysis, an axially translating elastic Euler-Bernoulli cantilever beam featuring time-variantvelocity was investigated in (Wang et al. 2009). The multiple scales method was presented in(Yang and Chen 2006) for obtaining the near- and exact-resonant steady-state response of theforced vibration of a simply supported axially moving viscoelastic beam. In Refs. ((Zhangand Zu 1998a), and (Zhang and Zu 1998b)) authors attempted to describe the mechanicalenergy dissipation using a viscoelastic model for the belt, and utilized the perturbation tech-niques to predict the nonlinear response. These viscoelastic studies provided a systematicmethodology to incorporate material damping in the analysis. Most of the aforementionedstudies are restricted to only horizontal translating continua. Nowadays, the vertically mov-ing strings and beams frequently appear in research literature. For example, rope and cablesystems are used to carry payloads in lift installations, including building elevators and minehoists, represent typically non-stationary systems. The vertically moving systems are morecomplicated than horizontally moving systems due to time-varying length and space-time-varying tension. The dynamics of cables or chains suspended between two positions at thesame height were first studied in the middle of the 19th century, and a historical discussionwas given in (Irvine 1981). The earlier work of string with a mass-spring system emulatingan elevator goes back to ((Yamamoto et al. 1978), and (Terumichi et al. 1997)). In (Zhu andNi 2000) authors studied the linear dynamics of a cantilever beam with an arbitrary vary-ing length where the tension from their axially moving acceleration was incorporated; theyalso studied the dynamic stability from the energy view point. An active control methodol-ogy using a pointwise control force and/or moment was developed in (Zhu et al. 2001) todissipate the vibratory energy of a translating medium with arbitrary varying length. Theeffects of bending stiffness and boundary conditions on the dynamic response of elevatorcables were examined in (Zhu and Xu 2003). Recently, a linear model is developed forcalculating the natural frequencies, mode shapes, and dynamic responses of stationary el-evator traveling and compensation cables in (Zhu and Ren 2013b), by the same authors in((Zhu and Ren 2013a), and (Ren and Zhu 2013)) have developed methodology and appli-

1.1. Historical background 5

Figure 1.3: An example of a vertically moving cable-mass system

cation to study the longitudinal, transverse, and their coupled vibrations of moving elevatorcable-car systems. Sometimes it happens that studying external or boundary excitations ofthese space-time-varying mechanical systems interesting phenomena of autoresonance oc-cur. By autoresonance is meant the growth of the amplitude of oscillations of a solution toa linear (nonlinear) equation of motion under action of a small externally oscillating force.This phenomena of autoresonance is the subject of chapter 4 of this thesis. The autoreso-nance concept was first taken into account in (McMillan 1945), and was furthered developedin (Bohm and Foldy 1946) for particle accelerators. The autoresonance is thought of as auniversal phenomenon which occurs in a wide range of oscillating physical systems fromastronomical to atomic one (see, for instance, (Fajans and Friedland 2001)). Many new ap-plications of the autoresonance idea and progress in the theory emerged since 1990 in atomicand molecular physics (see, for instance, (Meerson and Friedland 1990), (Liu et al. 1995),and (Maeda et al. 2007)), nonlinear dynamics (Meerson and Yariv 1991), nonlinear waves(Friedland 1998), plasmas ((Fajans et al. 1999), and (Friedland et al. 2006)), fluid dynamics(Borich and Friedland 2008) and optics (Barak et al. 2009). For a contemporary survey ofthe mathematical aspects of autoresonance the reader is referred to (Kalyakin 2008).Mathematical models of vibrating systems are usually divided into two classes: discrete,or lumped-parameter models, and continuous, or distributed-parameter models. In reality,however, systems can contain both discrete and continuous parts. Since exact solutions arepossible to find only for a limited number of problems, formal approximations of the so-lution are usually constructed by means of asymptotic methods or numerical methods. Bysolving beam-like or wave-like equations, important information on the vibrational behaviorof a physical system can be found. The key to solving modern problems is mathematicalmodeling. This process involves keeping certain elements, neglecting some, and approx-

6 1. Introduction

imating yet others. To accomplish this important step, one needs to decide the order ofmagnitude (i.e., smallness or largeness) of different elements of the system by comparingthem with each other as well as with the basic elements of the system. This process iscalled nondimesnionalization or making the variables dimensionless. Consequently, oneshould always introduce dimensionless variables before attempting to make any approxima-tions. Therefore, expressing the equations in dimensionless form brings out the importantdimensionless parameters that govern the behavior of the physical system. The smallnessof the dimensionless parameter, say ε , in the governing system (consisting of differentialequations, initial and boundary conditions) such that for ε = 0 the system is exactly solv-able, forms the basis for what is called “Perturbation or Asymptotic Method”. Perturba-tion methods can be used to construct approximate analytic solutions. When perturbationor asymptotic methods are applied explicit expressions that describe the structural motioncan be found. An introduction to asymptotic methods with basic principles of asymp-totics and its applications, and an overview of traditional and modern approaches can befound in ((Andrianov and Manevitch 2002), (Holmes 1995), (Kevorkian and Cole 1996),(Murdock 1991), (Nayfeh 1973), (Nayfeh 1991), and (Verhulst 2005)).The goal of the present thesis is to study damped vibrations of the horizontal axially movingcontinua and, free and forced vibrations of the vertical axially moving continua, which aredescribed by continuous or distributed-parameter models. Translating media with constantlength can model such low- and high-speed slender members as conveyor belts ((Ponomarevaand van Horssen 2007), and (Ponomareva and van Horssen 2009)), chair lifts, aerial cabletramways, pipes carrying water, oil or gas ((Oz and Boyaci 2000), and (Kuiper and Metrikine2004)), band saw blades and magnetic paper tapes (Thurman and Mote 1969), power trans-mission chains and belts (Wickert and Mote 1988), plastic films, data storage devices, andtransport cables. Translating media such as elevator cables ((Sandilo and van Horssen 2013),(Zhu and Xu 2003), (Zhu and Chen 2005), and (Zhu and Chen 2006)), paper sheets (Stolteand Benson 1992), satellite tethers (Misra and Modi 1982), flexible appendages (Tsuchiya1983), lift cranes, mining hoists ((Kaczmarczyk 1997), and (Kaczmarczyk and Ostachowicz2003)), and cable-driven robots exhibit time-varying length, space-time-varying tension andconstant or time-varying velocity. The understanding of the vibrations of an axially movingcontinuous medium with constant or variable-length is important in design of these systems.Simple models which describe oscillations of axially moving materials can be expressed ininitial-boundary value problems for (wave-) string-like or beam-like equations depending onthe bending stiffness. But they all have something in common, namely, the dimension inso-called “axial” direction is much larger than the dimensions in the other two directions.That is why the width and the thickness can be neglected when modeling such physical phe-nomena. Studying the dynamic behavior of axially moving materials as well as problemsof interaction of structures with flows are both of great technological and theoretical interest(see, for instance, (Paıdoussis 1998), and (Svetlitsky 2005)).

1.2. Mathematical models 7

1.2 Mathematical models

In the classical analysis of axially moving continua the vibrations are usually classified intotwo categories, that of a string-like type or that of a beam-like type, depending on the bendingstiffness. If the bending stiffness is neglected then the system is classified as string-like,otherwise it is classified as beam-like. The equations of motion for axially moving conveyorbelts or axially moving elevator cables can be derived by using Hamilton’s principle (see, forinstance, (Miranker 1960), or (Suweken 2003)). For the conveyor belt system a beam-likeequation is considered in horizontal direction where pretension of the belt and longitudinalaxial velocity are assumed to be constant. For the elevator cable system a string-like equationis considered in vertical direction where axial longitudinal velocity in vertical direction isassumed to be constant or time-varying, and the tension in the cable varies through a spatialcoordinate and time. The transversal vibrations of the conveyor belt system (with constantvelocity V ) can be modeled mathematically as a beam-like equation:

ρA(utt +2Vuxt +V 2uxx

)− (T (x, t)ux)x +EIuxxxx = 0, (1.1)

where u is a function of the spatial coordinate x and the time t, which models the displace-ment of the beam in the vertical direction, ρ is the mass density of the beam, A is the cross-sectional area of the beam, V is the constant axial velocity of the beam in the horizontaldirection, T (x, t) = T is the constant non-zero pretension of the beam, E is the Young’s mod-ulus of elasticity, and I is the moment of inertia with respect to the beam axis. It is assumedthat the belt always moves forward in one direction therefore V > 0 condition will be im-posed. It is assumed in chapter 2 of this thesis that the beam is simply supported at one endx = 0 and is attached to a spring-dashpot system at other end x = L, therefore, the boundaryconditions for (1.1) are given by

u(0, t) = uxx(0, t) = uxx(L, t) = 0,EIuxxx(L, t) = Tux(L, t)+ ku(L, t)+δ (ut(L, t)+Vux(L, t)),

(1.2)

where k is the stiffness of the spring, δ is the damping coefficient of the dashpot and L isthe constant distance between the pulleys. The transversal vibrations of the elevator cablesystem are considered in chapter 3 and in chapter 4 of this thesis (with constant or time-varying velocity and space-time-varying tension), and can be modeled mathematically as astring-like equation:

ρ(utt +2Vuxt +V ux +V 2uxx

)− (T (x, t)ux)x = 0, (1.3)

where u(x, t) is the horizontal displacement of the string, ρ is the mass density of the string,V is the longitudinal constant or time-varying velocity of the string in the vertical direction,and T (x, t) = mg+ ρ(l(t)− x)g−mV − ρ(l(t)− x)V , is the space-time-varying tension in

8 1. Introduction

string arising from its own weight and the heavy car mass attached to the string at its lowerend, where l(t) is the time-varying length of the string, g is the acceleration due to gravityand V is the longitudinal acceleration due to attached mass. The elevator car is modeledas a rigid body of mass m attached at the lower end of the cable and, the suspension ofthe car against the guide rails is assumed to be rigid. The time-varying length l(t) is givenby l(t) = l0 +Vt or by l(t) = l0 + β sin(ωt), where l0 is the initial string length, V is theconstant string velocity, β is the length variation parameter, ω is the angular frequency oflength variation and l0 > |β |. It is assumed that the cable is fixed or that it is externallyexcited by a harmonic force due to wind or storm in the horizontal direction at its upper endx = 0 and that it is fixed at its lower end x = l(t), therefore the boundary conditions for (1.3)are given by

u(0, t) = 0 (or αsin(Ωt)), u(l(t), t) = 0, (1.4)

where α is the excitation amplitude of the tall building or the structure at its top and Ω is theoscillation rate at the top. The general initial conditions for (1.1) and (1.3) are given by

u(x,0) = f (x), ut(x,0) = h(x), (1.5)

where f (x) is the initial belt or cable displacement from equilibrium, and h(x) is the initialbelt or cable velocity. Equations (1.1) and (1.3) are linear equations of motion, derived byusing the energy function or Hamiltonian of the system with application of Hamilton’s prin-ciple. Although nonlinear equations are more accurate, it is always important to study firstweakly perturbed linear equations to get mathematical and physical insights. Linear differen-tial equations with variable coefficients are in some sense equivalent to nonlinear differentialequations in measure of difficulty to solve them and aspects of their complicated solutions.This thesis focuses on linear equations of motion with constant or variable coefficients, andtheir complicated dynamical aspects.

1.3 Analytical approximations

In many branches of science and engineering, for instance, fluid mechanics, solid mechanics,elasticity, aerodynamics, quantum mechanics, electromagnetism, and mathematical physicsexact solutions can be found only for a limited number of differential equations. Real phys-ical or applied problems are subjected to an essential influence of space- and time-varyingparameters, nonlinearities, and complicated boundary conditions. To find an exact solutionin these cases usually seems impossible. Hence, applied mathematicians, engineers, andphysicists are forced to determine approximate solutions of the problems they are facing.Approximated solutions can sometimes be constructed in order to obtain information aboutbehavior of such physical systems. Approximate analytic solutions for the beam-like and the

1.3. Analytical approximations 9

string-like equations can be constructed by using perturbation (asymptotic) methods. In theapplied perturbation scheme it is assumed that the solution of the problem can be expandedin a power series in ε , where ε is a small dimensionless parameter. If a naive expansion isused, that is, if it is assumed that the solution can be written as

u(x, t;ε) = u0(x, t)+ εu1(x, t)+ ε2u2(x, t)+ · · · , (1.6)

it may turn out that u0, u1, u2, and so on, may contain terms growing in t, εt, ε2t, · · · . Ofcourse, the approximation is still valid for small values of t, but it is not valid anymore forlarge values of t. These unbounded terms are called secular terms. To avoid these secularterms it is convenient to introduce new time variables t0 = t, t1 = εt, t2 = ε2t, and so on. Toremove secular terms occurring in ui for i = 0,1,2, · · · , it is then assumed that the approxi-mation of u is a function of x, t0, t1, t2, and so on. Then u is expanded in a perturbation seriesin ε , that is,

u(x, t;ε) = u0(x, t0, t1, t2, · · ·)+ εu1(x, t0, t1, t2, · · ·)+ε2u2(x, t0, t1, t2, · · ·)+ · · · ,

(1.7)

and that all ui’s for i = 0,1,2, · · · , are determined in such a way that no secular or unboundedterms occur. It is assumed that the functions ui are O(1). This method is called the multiple-timescales perturbation method or sometimes called the method of multiple scales. Some-times it happens that for an equation unexpected timescales occur. If this happens thenthe function u also depends explicitly on these timescales including others. This phenom-ena will be discussed in more detail in chapter 4, where an unexpected timescale of order

1√ε will occur, and therefore, a three timescales perturbation method will be used to obtain

asymptotic approximations of the solution. The idea of the multiscale method is to intro-duce several scaled variables directly into the intended approximation and then choose thedependence of the approximate solution on the variables so as to obtain a uniformly validapproximation to the exact solution. This is done by introducing a fast-scale, a slow-scaleand, even sometimes slower-scale variables for the independent space or time variable, andsubsequently treating these variables as if they are independent. In the solution process ofthe perturbation problem the resulting additional freedom introduced by the new indepen-dent variables is used to remove unbounded or secular terms. This freedom puts constraintson the subsequent approximate solutions, which are called solvability conditions. The firstscheme to address this problem is what Milton D. van Dyke (van Dyke 1975) refers to asthe method of strained coordinates. The idea of explicitly using scaled variables in a per-turbation procedure goes back at least one hundred years to the work of astronomer AndersLindstedt in period 1882-1883, who introduced and used such scaled variables to eliminatesecular (resonant, unbounded) terms in perturbation expansions in celestial mechanics. Thework of Lindstedt was carried further by Henry Poincare (Poincare 1892), and in his fa-mous treatise on celestial mechanics, Poincare also credits the basic idea for this method toLindstedt. Perhaps due to the inaccessibility of Lindstedt’s 1882 paper, some subsequent

10 1. Introduction

authors have referred to this as Poincare’s method. Actually, the basic idea was used evenearlier in 1847 by George Stokes in his study of periodic solutions for water waves. Vari-ous similar methods were later rediscovered by such authors as Edmund Whittaker in 1914,Erwin Schrodinger in 1926 and James Lighthill in 1949, in the contexts of various differ-ent applications. The method of multiple timescales was also independently invented byAlexey Krylov and Nikolay Bogoliubov in 1935. Another paper in this school was by G.Kuzmak (Kuzmak 1959), but they did not pursue the idea as “they thought multiple tim-ing was not a good method”, see (Verhulst 2005). The method of multiple-timescales wasindependently discovered by Jirair Kevorkian and Julian Cole (Kevorkian and Cole 1996),James Alan Cochran (Cochran 1962), and John Mahony (Mahony 1962), and was promotedby Ali Hasan Nayfeh (see, (Nayfeh 1973), and (Nayfeh 1991)) to study various oscillationproblems, which is now the more standard approach. Using this method, Jirair Kevorkianingeniously solved a number of difficult problems. For more than two decades workingat Mathematical Physics Group of Delft University of Technology, Wim T. van Horssenand his research students are using multiple timescales and other asymptotic methods fre-quently to obtain approximate analytical solutions to very complicated ordinary, partial, dif-ference and even functional equations, see for instance, ((van Horssen 2001), (Sandilo andvan Horssen 2012), (van Horssen and ter Brake 2009), and (Rafei and van Horssen 2009)).Throughout this thesis, the multiple timescales perturbation method will be used to constructapproximations of the solutions of the initial-boundary value problems.

1.4 Outline of the thesis

In this chapter 1, a brief introduction to the subject has been given.In chapter 2, an initial-boundary value problem for a linear-homogeneous axially movingtensioned beam equation will be considered. The axial velocity of the beam is assumed tobe constant and relatively small compared to the wave speed. One end of the beam is as-sumed to be simply supported and to the other end of the beam a spring and a dashpot areattached, where the damping generated by the dashpot is assumed to be small. The equationsof motion of the moving conveyor belt will be derived by using Hamilton’s principle. Theenergy of the initial-boundary value problem and the boundedness of the solutions will beshown. A multiple timescales perturbation method is used to construct formal asymptoticapproximations of the solutions of the initial-boundary value problem, and it will be shownthat all oscillation modes are damped.In chapter 3, the free transversal responses of the vertically translating media with time-varying length, space-time-varying tension, and constant or time-varying velocity will beconsidered. The equations of motion of the vertically translating system will be derived bythe application of the modified Hamilton’s principle. The translating media are modeled as

1.4. Outline of the thesis 11

taut strings with fixed boundaries. The problem is used as a simple model to describe thetransversal vibrations of an elevator cable, for which the length changes linearly in time,or for which the length changes harmonically about a constant mean length. In the givenmathematical models a rigid body is attached to the lower end of the cable and suspensionof the car against the guide rails is assumed to be rigid. For linearly length variations it isassumed that the axial velocity of the cable is small compared to nominal wave velocity andthat the cable mass is small compared to car mass, and for harmonically length variationssmall oscillation amplitudes are assumed. A multiple timescales perturbation method is usedto construct formal asymptotic approximations of the solutions to show the complicated dy-namical behavior of the cable. It turns out for the case with the harmonically varying lengththat there are infinitely many values of ω that can cause internal resonances. In this chapterthe resonance case ω = π

l0is investigated and a detuning case for this value is studied. It will

also be shown that the Galerkin’s truncation method can not be applied to this problem in allcases in order to obtain approximations valid on long timescales.In chapter 4, the forced responses of a vertically translating string with a time-varying lengthand a space-time-varying tension will be considered. The problem is used as a simple modelto describe the forced vibrations of an elevator cable for which the length changes linearlyin time. The vertical velocity of the cable is assumed to be constant and relatively smallcompared to nominal wave velocity, and the cable mass is small compared to car mass. Ingiven mathematical model a rigid body is attached to the lower end of the cable and suspen-sion of the car against the guide rails is assumed to be rigid. The elevator cable is externallyexcited at the upper end by the displacement of the building in horizontal direction fromthe equilibrium. This external excitation has a constant amplitude of order ε , where ε is adimensionless small parameter. The fascinating phenomena of autoresonance occurs whena perturbed system is captured into (dynamic) resonance. This autoresonance phenomenaand the time of autoresonant growth of amplitude of the modes of fast oscillations will bediscussed in detail. It will also be shown that order ε boundary excitations result in order√

ε solution responses. By performing interior layer analysis systematically, it will be shownthat there exists an unexpected new timescale of order 1√

ε . For this reason, a three timescalesperturbation method is used to construct formal asymptotic approximations of the solutionsof the initial-boundary value problem. It will also be shown that there is a cascade of au-toresonances when all modal solutions to the initial-boundary value problem are summed upand from these solutions it can be seen that as the mode number k increases the amplitudesdecrease in size.

Published as: S. H. Sandilo and W. T. van Horssen – ”On Boundary Damping for an Axially Moving TensionedBeam”, American Society of Mechanical Engineers, Journal of Vibration and Acoustics, vol. 134, no. 1, art. no.11005, February 2012

Chapter 2On Boundary Damping for an Axially Moving

Tensioned Beam

It is through science that we prove, but through intuition thatwe discover.

Henry Poincare

Abstract

In this chapter, an initial-boundary value problem for a linear-homogeneous axially mov-ing tensioned beam equation is considered. One end of the beam is assumed to be simply-supported and to the other end of the beam a spring and a dashpot are attached, wherethe damping generated by the dashpot is assumed to be small. In this chapter only bound-ary damping is considered. The problem can be used as a simple model to describe thevertical vibrations of a conveyor belt, for which the velocity is assumed to be constantand relatively small compared to the wave speed. A multiple time-scales perturbationmethod is used to construct formal asymptotic approximations of the solutions, and it isshown how different oscillation modes are damped.

2.1 Introduction

Many engineering devices can be represented by an axially moving continua. The un-derstanding of the vibrations of an axially moving continuous medium is important in

the design of conveyor belts (see, for instance, (Suweken and van Horssen 2003a), (Suwekenand van Horssen 2003b), (Ponomareva and van Horssen 2007), (Pakdemirli and Oz 2008),and (Ponomareva and van Horssen 2009)), elevator cables ((Zhu et al. 2001), (Zhu andChen 2005), and (Zhu and Chen 2006)), aerial cable tramways, overhead transmission lines(Darmawijoyo and van Horssen 2002), power-transmission chains (Mahalingam 1957), plas-tic films, pipes transporting fluids ((Oz and Boyaci 2000), and (Kuiper and Metrikine 2004)),high speed magnetic paper tapes, fiber textiles (Chen 2005), band-saws (Ulsoy et al. 1978),data storage devices, chair lifts, and even models of DNA and proteins (Xu 2006) wherethe axial transport of mass can be associated with transverse vibrations. Simple modelswhich describe these oscillations can be expressed in the initial-boundary value problems

14 2. On Boundary Damping for an Axially Moving Tensioned Beam

for wave equations (van Horssen and Ponomareva 2005) or for beam equations (Oz andPakdemirli 1999). The main goal of applied mathematicians, mechanical and civil engineersand, physicists is to reduce the vibrations in these devices because they cause damage to thestructure.Investigating transverse vibrations of such systems is a challenging subject which has beenstudied for many years by many researchers and is still of great interest today. A great dealof research has been done on the transverse vibrations of such systems where linear andnonlinear models have been taken into account. Many contributions on an axially movingcontinuum can be found in the literature. The interest in studying axially moving systemsis also motivated by the increased use of pipelines conveying water, cooling water in nu-clear power plants, oil, gas and dangerous liquids in chemical plants since early 1950s. Afundamental work was done in (Wickert and Mote 1990), where the moving string and themoving beam with the effect of tension for simply supported and clamped boundary con-ditions was investigated. Using a similar model, the authors in (Chakraborty et al. 1999)investigated the free and forced responses of a traveling beam. The governing equations ofcoupled longitudinal and transverse vibrations of an axially moving strip were first obtainedin (Thurman and Mote 1969). After this work on moving strip, the transversal vibrations ofa moving material were studied in ((Wickert 1992), and (Pellicano and Vestroni 2000)). In(Miranker 1960), author took a model for the transverse vibrations of a tape moving betweena pair of pulleys and by using a variational procedure, derived the equations of motion anddiscussed both the constant and the time-dependent tape velocity. In (Spelsberg-Korspeteret al. 2008) authors considered an axially moving beam in frictional contact with pads andstudied the mechanical behavior caused by friction and interpreted damping and nonconser-vative forces as perturbations. In (Chen and Ding 2010) the steady-state transverse responsein coupled planar nonlinear vibrations of an axially moving viscoelastic beam was analyzed.Very recently in (Bagdatli et al. 2011) an axially moving beam supported at both ends whichalso has an intermediate support has been investigated. In all these studies the axial transportvelocity is assumed to be either constant or time-varying.In this study, transverse vibrations of an axially moving beam are investigated and explicitasymptotic approximations of the solutions will be constructed, which are valid on a longtime-scale as, for instance, described in ((Nayfeh 1973), and (Kevorkian and Cole 1996)).A stretched beam will be considered which is simply-supported at one end and attached toa spring-dashpot-system at other end. It will also be shown in this chapter that the use ofboundary damping can be used effectively to suppress the oscillation amplitudes. To ourknowledge, the use of boundary damping and the explicit construction of approximations ofoscillations for these types of problems have not been previously investigated.The present chapter is organized as follows. Section 2.2 establishes the governing equationsof motion. Section 2.3 will discuss the energy of the initial-boundary value problem and theboundedness of the solutions will be shown. From this energy analysis it can not be con-

2.2. The governing equations of motion 15

cluded whether the energy of the belt system decreases or not. For that reason, in section2.4, formal approximations for the solutions of the initial-boundary value problem are con-structed by using a two-timescales perturbation method and these solutions will be analyzed.Finally, in section 2.5, some conclusions will be drawn and some remarks will be made.

2.2 The governing equations of motion

To obtain the equations of motion, Hamilton’s principle will be used (Miranker 1960). Con-sider a uniform axially moving beam of mass-density ρ , cross-sectional area A, moment ofinertia I, flexural rigidity EI, and uniform tension T . A stretched beam is simply-supportedat x = 0 and attached to a spring-dashpot-system at x = L. The beam travels at the uniformconstant transport speed V between two supports that are a distance L apart as shown inFigure 2.1. It is assumed that V , ρ , T , k (the stiffness of the spring), and c (the dampingcoefficient of dashpot) are all positive constants. Furthermore, we only consider the verti-cal displacement u(x, t) of the beam, where x is the position along the beam, and t is thetime. Gravity and other external forces are neglected. The equation describing the verticaldisplacement of the beam is given by

x = Lx = 0

u(x,t)

x

kc

V

Figure 2.1: The moving belt system between two supports

utt +2Vuxt +(V 2−C2)uxx +EIρA

uxxxx = 0, t > 0, 0 < x < L. (2.1)

The boundary and the initial conditions for (2.1) are given by

u(0, t) = uxx(0, t) = uxx(L, t) = 0, t > 0, (2.2)

EIuxxx(L, t) = Tux(L, t)+ ku(L, t)+ c(ut(L, t)+Vux(L, t)), t > 0, (2.3)

u(x,0) = f (x), and, ut(x,0) = g(x), 0 < x < L, (2.4)

where the wave speed C =√

TρA , and where f (x) and g(x) represent the initial displacement

and the initial velocity of the belt, respectively. The axial speed V of the belt is assumed to


be small compared to wave speed C, to be constant, and O(ε), that is, V = εV , where ε isdimensionless small parameter. The damping coefficient c is also assumed to be of O(ε),that is, c = εδ . To put the equations in a nondimensional form the following dimensionlessquantities are used: u∗(x, t) = u(x,t)

L , x∗ = xL , V ∗ = V

C , t∗ = CL t, µ = EI

ρAC2L2 , k∗ = kLρAC2 ,

δ ∗ = δρAC , f ∗(x) = f (x)

L , and g∗(x) = g(x)C . Then the equation (2.1) in a nondimensional form

becomesutt −uxx +µuxxxx =−2εVuxt − ε2V 2uxx, t > 0, 0 < x < 1, (2.5)

with the boundary conditions

u(0, t;ε) = uxx(0, t;ε) = uxx(1, t;ε) = 0, t > 0, (2.6)

and,

µuxxx(1, t;ε) = ux(1, t;ε)+ ku(1, t;ε)+ εδ (ut(1, t;ε)+ εVux(1, t;ε)), t > 0, (2.7)

and the initial conditions

u(x,0;ε) = f (x), and, ut(x,0;ε) = g(x), 0 < x < 1. (2.8)

The asterisks indicating the dimensionless quantities are omitted in (2.5) through (2.8) andhenceforth.In this chapter, the initial-boundary value problem (2.5)-(2.8) for u(x, t) will be studied andformal approximations (that is, functions that satisfy the differential equation and the initialand the boundary values up to some order in ε) will be constructed.

2.3 The energy and the boundedness of solutions

In this section, we shall derive the energy of the moving beam as modeled by the tensionedbeam equation

utt −uxx +µuxxxx +2εVuxt + ε2V 2uxx = 0, t > 0, 0 < x < 1. (2.9)

By multiplying (2.9) with (ut + εVux), we obtain after long but elementary calculations

12 (ut + εVux)

2 + 12 (u

2x +µu2

xx)

t+−ux(ut + εVux)−µuxx(uxt + εVuxx)

+µuxxx(ut + εVux)x = 0.(2.10)

Integrating (2.10) with respect to x from x = 0 to x = 1, and then by integrating the so-obtained equation with respect to t from t = 0 to t = t, we obtain

∫ 10

(12 (ut + εVux)

2 + 12 (u

2x +µu2

xx))|tt=0dx =

∫ t0(ux(ut + εVux)+µuxx(uxt + εVuxx)−µuxxx(ut + εVux) |1x=0dt.

(2.11)

2.4. Application of the two timescales perturbation method 17

The total mechanical energy E(t) in the span (0,1) is the sum of the kinetic and the potentialenergy of the moving beam and the potential energy of the spring, that is,

E(t) =∫ 1

0

(12(ut + εVux)

2 +12(u2

x +µu2xx)

)dx+

12

ku2(1, t). (2.12)

Then, by using (2.11) and (2.12), and the boundary conditions (2.6) and (2.7), it follows thatthe time-rate of change of the total mechanical energy is

dEdt =−εδ (ut(1, t)+ εVux(1, t))2− ku(1, t)(εVux(1, t))

+(µuxxx(0, t)−ux(0, t))(εVux(0, t)).(2.13)

In (2.13), the temporal variation of the total mechanical energy of a traveling beam equalsthe net rate of work done on the beam, and the expression (2.13) has simple physical inter-pretations. The first term stands for the influence of damping due to a dashpot (δ ) at x = 1,the second term stands for the rate of work done by the spring force at x = 1 with a trans-verse velocity component εVux(1, t), and the third term explains that with the simple supportcondition at x = 0, material particles enter and exit the span at x = 0 with the transversevelocity εVux(0, t), and the shear force µuxxx(0, t)− ux(0, t) does work on the particle in-stantaneously located at the support. From (2.13), it can be concluded that dE

dt ≤ εαE, whereα is a constant independent of ε . Hence, E(t)≤ E(0)eεαt . From this energy estimate it fol-lows that u(x, t) is bounded for times t of order ε−1 when E(0) is bounded; however, it cannot be concluded whether the energy of the belt system decreases or not. For that reason weapproximate the solution of the initial-boundary value problem in the section 2.4. For moredetailed descriptions of the energetics of a translating continua, the reader is referred to Refs.((Wickert and Mote 1989), (Zhu and Ni 2000), (Chen 2006), and (Chen and Zu 2004)).

2.4 Application of the two timescales perturbation method

In this section, an approximation of the solution of the initial-boundary value problem (2.5)-(2.8) will be constructed. A two-timescales perturbation method will be used. Conditionssuch as t > 0, t ≥ 0, 0 < x < 1 will be dropped for abbreviation. Expand the solution in aTaylor series with respect to ε straightforwardly, that is,

u(x, t;ε) = u0(x, t)+ εu1(x, t)+ ε2u2(x, t)+ · · · . (2.14)

It is assumed that the functions ui(x, t) are O(1). The approximation of the solution ofthe problem will contain secular terms; that is, unbounded terms in t. Since the ui(x, t)are assumed to be O(1), and because the solutions are bounded on timescales of O(ε−1),secular terms should be avoided when approximations are constructed on long timescales of


O(ε−1). That is why a two-timescales perturbation method is applied. For a more completeoverview of this perturbation method the reader is referred to (Nayfeh 1973) or (Kevorkianand Cole 1996). By using such a two-timescales perturbation method the function u(x, t;ε)is supposed to be a function of x, t, and τ = εt. For that reason, we put

u(x, t;ε) = y(x, t,τ;ε). (2.15)

The following transformations are needed for the time derivatives

ut = yt + εyτ , (2.16)

utt = ytt +2εytτ + ε2yττ . (2.17)

Substitution of (2.15)-(2.17) into the problem (2.5)-(2.8) yields an initial-boundary valueproblem for y(x, t,τ)

(ytt − yxx +µyxxxx)+2ε(ytτ +V yxt)+ ε2(yττ +2V yxτ +V 2yxx) = 0, (2.18)

y(0, t,τ;ε) = yxx(0, t,τ;ε) = yxx(1, t,τ;ε) = 0, (2.19)

µyxxx(1, t,τ;ε)− yx(1, t,τ;ε)− ky(1, t,τ;ε)− εδ (yt(1, t,τ;ε)+εyτ(1, t,τ;ε)+ εV yx(1, t,τ;ε)) = 0,

(2.20)

y(x,0,0;ε) = f (x), (2.21)

yt(x,0,0;ε)+ εyτ(x,0,0;ε) = g(x). (2.22)

Using a two-timescales perturbation method it is usually assumed that not only the solutionu(x, t;ε) will depend on two time-scales, but also that u(x, t;ε) = y(x, t,τ;ε) can be approxi-mated by the formal expansion

y(x, t,τ;ε) = y0(x, t,τ)+ εy1(x, t,τ)+ ε2 · · · . (2.23)

It is reasonable to assume this solution form because the partial differential equation and theboundary conditions analytically depend on ε . Substituting (2.23) into (2.18)-(2.22), andafter equating the coefficients of like powers in ε , it follows from the problem for y(x, t,τ)that the O(1)-problem is

y0tt − y0xx +µy0xxxx = 0, (2.24)

y0(0, t,τ) = y0xx(0, t,τ) = 0, (2.25)

y0xx(1, t,τ) = 0, (2.26)

µy0xxx(1, t,τ)− y0x(1, t,τ)− ky0(1, t,τ) = 0, (2.27)

y0(x,0,0) = f (x), (2.28)


y0t (x,0,0) = g(x), (2.29)

and that the O(ε)-problem is

y1tt − y1xx +µy1xxxx =−2V y0xt −2y0tτ , (2.30)

y1(0, t,τ) = y1xx(0, t,τ) = 0, (2.31)

y1xx(1, t,τ) = 0, (2.32)

µy1xxx(1, t,τ)− y1x(1, t,τ)− ky1(1, t,τ) = δy0t (1, t,τ), (2.33)

y1(x,0,0) = 0, (2.34)

y1t (x,0,0) =−y0τ (x,0,0), (2.35)

Since the partial differential equation (2.24) and the boundary conditions (2.25)-(2.27) arelinear and homogeneous, the method of separation of variables can be applied. We look forspecial product solutions of the form

y0(x, t,τ) = φ(x)g(t,τ). (2.36)

By substituting (2.36) into (2.24) and by dividing the so-obtained equation by φ(x)g(t,τ), itfollows that

gtt(t,τ)g(t,τ)

=φ ′′(x)φ(x)

−µφ (4)(x)φ(x)

=−λ . (2.37)

A separation constant −λ is introduced so that the time-dependent part of the product solu-tion oscillates if λ > 0. The eigenvalues turn out to be real and positive (see Appendix A).We thus obtain two equations from (2.37): a time-dependent part

gtt(t,τ)+λg(t,τ) = 0, (2.38)

and a space-dependent part

φ (4)(x)− 1µ

φ ′′(x)− λµ

φ(x) = 0. (2.39)

The four homogeneous boundary conditions (2.25)-(2.27) imply that

φ(0) = φ ′′(0) = φ ′′(1) = µφ ′′′(1)−φ ′(1)− kφ(1) = 0. (2.40)

Thus, (2.39) and (2.40) form a boundary value problem. Instead of first reviewing the solu-tion of (2.39) and (2.40), let us analyze the time-dependent equation (2.38). In Appendix A,it has been shown that λ > 0. And so, the general solution of (2.38) is a linear combinationof sines and cosines in t,

g(t,τ) = σ1(τ)cos√

λ t +σ2(τ)sin√

λ t, (2.41)


and it oscillates with frequency√

λ . The values of λ determine the natural frequencies of theoscillations of a vibrating belt. Now by analyzing the boundary-value problem, we can usethe fact that the eigenvalues λ are real and positive. The characteristic equation for (2.39) isgiven by

γ4− γ2

µ− λ

µ= 0, (2.42)

and the solutions of (2.39) are given by

φ(x) = c1sinh(αx)+ c2cosh(αx)+ c3sin(βx)+ c4cos(βx), (2.43)

where c1, c2, c3, and c4 are constants, and where

β =

√−1+

√1+4λ µ

2µ, and, α =

√β 2 +

1µ. (2.44)

Applying the boundary conditions (2.40), we observe that the nontrivial solutions are found(when c2 = c4 = 0) and when

fµ,k(λ ) = µα2β 2(αcosh(α)sin(β )−β sinh(α)cos(β ))−αβ (αsinh(α)cos(β )+βcosh(α)sin(β ))− ksin(β )sinh(α)(α2 +β 2)

= 0.(2.45)

For given values of k and µ the eigenvalues λm = µβ 4m +β 2

m can be numerically computedfrom (2.45). In Table 2.1 some of these eigenvalues are presented for some fixed values ofk and µ . From (2.39), (2.40) and (2.45) the m-th eigenfunction φm(x) corresponding to them-th eigenvalue (λm) can be determined and is given by (up to a multiplicative constant)

φm(x) = θmsinh(αmx)+ sin(βmx), (2.46)

where, θm = β 2msin(βm)

α2msinh(αm)

, βm =

√−1+√

1+4λmµ2µ , and αm =

√β 2

m + 1µ .

The general solution of the O(1)-problem (2.24)-(2.27) for y0(x, t,τ) is now given by

y0(x, t,τ) =∞

∑m=1

(Am(τ)cos(

√λmt)+Bm(τ)sin(

√λmt)

)φm(x), (2.47)

where φm(x) is given by (2.46), and where Am(τ) and Bm(τ) are still arbitrary functionswhich can be used to avoid secular terms in y1(x, t,τ). To prove the orthogonality of theeigenfunctions given by (2.46), we need to use Green’s formula and operator notations. Wedefine the linear operator L = d4

dx4 − 1µ

d2

dx2 , such that L(φ) = λµ φ , where φ(x) satisfies the

boundary conditions given in (2.40); for details the reader is referred to (Haberman 2004). It


Table 2.1: Some eigenvalues λi which are roots of (2.45) with λi the i-th root

λ µ = 0.001 µ = 0.01 µ = 0.1 µ = 1

k = 1

λ1 4.13205 4.25996 4.95805 5.79525

λ2 24.68662 28.76105 54.15242 268.92735

λ3 67.44960 95.67926 324.65675 2571.66438

λ4 136.98595 244.43954 1225.60427 11006.46538

λ5 239.83409 538.15563 3400.33222 32002.42493

λ10 1637.47770 8066.66186 72248.27763 714061.89600

λ25 39720.69922 342894.17678 3374613.36828 33691805.21440

λ50 597499.42934 5755328.62344 57333618.60889 573116518.47183

k = 10

λ1 8.26224 8.84812 13.67452 27.26977

λ2 34.26368 43.10528 90.48572 308.15032

λ3 81.38438 123.13015 367.04961 2608.70728

λ4 154.79829 282.39881 1265.23389 11042.89713

λ5 261.40539 580.19837 3438.37666 32038.64875

λ10 1674.08333 8105.43489 72284.61492 714097.93038

λ25 39759.43522 342930.55039 3374649.40673 33691841.21826

λ50 597536.20660 5755364.70709 57333654.61732 573116554.47267

can easily be shown that two different eigenfunctions belonging to two different eigenvaluesare orthogonal with respect to the inner product, as defined by

< φm(x),φn(x)>=∫ 1

0φm(x)φn(x)dx. (2.48)

Hence, by using the superposition principle, the inner product (2.48), and the initial values(2.28) and (2.29), we finally obtain the solution of the problem (2.24)-(2.29), yielding

y0(x, t,τ) =∞

∑m=1

(Am(τ)cos(

√λmt)+Bm(τ)sin(

√λmt)

)φm(x), (2.49)


where Am(0) and Bm(0) are given by

Am(0) =1

ζm

∫ 1

0f (x)φm(x)dx, (2.50)

√λmBm(0) =

1ζm

∫ 1

0g(x)φm(x)dx, (2.51)

and where,

ζm =∫ 1

0φ 2

m(x)dx. (2.52)

Now the solution of the O(ε)-problem (2.30)-(2.35) will be determined. The problem (2.30)-(2.35) has an inhomogeneous boundary condition. To solve the O(ε)-problem, we first con-vert the problem into a problem with homogeneous boundary conditions by introducing thefollowing transformation:

y1(x, t,τ) = w(x, t,τ)+(

x4−2x3

12µ +2+ k

)h(t,τ), (2.53)

where,h(t,τ) = δy0t(1, t,τ), (2.54)

and where w(x, t,τ) satisfies the same homogeneous boundary conditions as for y0(x, t,τ).Substituting (2.53) into (2.30)-(2.35), we obtain

wtt −wxx +µwxxxx =−2V y0xt −2y0tτ −(

x4−2x3

12µ+2+k

)htt(t,τ)

+(

12x2−12x−24µ12µ+2+k

)h(t,τ),

(2.55)

w(0, t,τ) = wxx(0, t,τ) = wxx(1, t,τ) = 0, (2.56)

µwxxx(1, t,τ)−wx(1, t,τ)− kw(1, t,τ) = 0, (2.57)

w(x,0,0) =−(

x4−2x3

12µ +2+ k

)h(0,0), (2.58)

wt(x,0,0) =−y0τ(x,0,0)−(

x4−2x3

12µ +2+ k

)ht(0,0), (2.59)

where h(t,τ) is given by (2.54), and where h(0,0), and ht(0,0) are given by

h(0,0) = δg(1), (2.60)

ht(0,0) = δ ( f ′′(1)−µ f (4)(1)). (2.61)

To solve (2.55)-(2.59), w(x, t,τ) is written in the following eigenfunction expansion

w(x, t,τ) =∞

∑m=1

wm(t,τ)φm(x). (2.62)


By substituting (2.62) into the partial differential equation (2.55), we obtain

∑∞m=1(wmtt (t,τ)+wm(t,τ)λm)φm(x) =−2V y0xt −2y0tτ −

(x4−2x3

12µ+2+k

)htt(t,τ)

+(

12x2−12x−24µ12µ+2+k

)h(t,τ).

(2.63)

Now, by expanding(

x4−2x3

12µ+2+k

)and

(12x2−12x−24µ

12µ+2+k

)into a series of eigenfunctions φm(x),

we obtainx4−2x3

12µ +2+ k=

∞

∑m=1

cmφm(x), (2.64)

12x2−12x−24µ12µ +2+ k

=∞

∑m=1

dmφm(x), (2.65)

where,

cm =1

ζm

∫ 1

0

(x4−2x3

12µ +2+ k

)φm(x)dx, (2.66)

dm =1

ζm

∫ 1

0

(12x2−12x−24µ

12µ +2+ k

)φm(x)dx. (2.67)

and where ζm is given by (2.52). By multiplying both sides of (2.63) with φn(x), then byintegrating the so-obtained equation from x = 0 to x = 1, and by using the orthogonalityproperties of the eigenfunctions, we obtain

wntt (t,τ)+wn(t,τ)λn =−2Tntτ (t,τ)− cnhtt(t,τ)+dnh(t,τ)− 2Vζn

Tnt (t,τ)Θnn

− 2Vζn

∑∞m=1m 6=n

Tmt (t,τ)Θmn,(2.68)

where,

Tm(t,τ) = Am(τ)cos(√

λmt)+Bm(τ)sin(√

λmt), (2.69)

and,

Θmn =∫ 1

0φ ′m(x)φn(x)dx. (2.70)

From (2.54) with (2.49), it follows that h(t,τ) and htt(t,τ) can be written as

h(t,τ) =∞

∑m=1

δTmt (t,τ)φm(1), (2.71)

htt(t,τ) =∞

∑m=1

δTmttt (t,τ)φm(1). (2.72)


Thus, (2.68) with (2.71) and (2.72) can be expressed as

wntt (t,τ)+wn(t,τ)λn =(2A′n(τ)+

(2Vζn

Θnn− (cnλn +dn)δφn(1))

An(τ))√

λnsin(√

λnt)−(2B′n(τ)+

(2Vζn

Θnn− (cnλn +dn)δφn(1))

Bn(τ))√

λncos(√

λnt)+

∑∞m=1m 6=nδ (dnTmt (t,τ)− cnTmttt (t,τ))φm(1)− 2V

ζnTmt (t,τ)Θmn.

(2.73)

The right hand side of (2.73) contains terms which are the solutions of the homogeneous partcorresponding to (2.73). These terms will give rise to unbounded terms, the so-called secularterms, in the solution wn(t,τ) of (2.73). Since it is assumed that y0(x, t,τ), y1(x, t,τ), · · · arebounded on time-scales of O(ε−1), it follows that these secular terms should be avoided. In(2.69), the functions An(τ) and Bn(τ) are still undetermined. These functions will be used toavoid secular terms in the solution of (2.73). In order to remove secular terms, it now easilyfollows from (2.73) that An(τ) and Bn(τ) have to satisfy

A′n(τ)+(

Vζn

Θnn−12(cnλn +dn)δφn(1)

)An(τ) = 0, (2.74)

B′n(τ)+(

Vζn

Θnn−12(cnλn +dn)δφn(1)

)Bn(τ) = 0. (2.75)

The solutions of (2.74) and (2.75) are given by

An(τ) = An(0)e(− V

ζnΘnn+

12 (cnλn+dn)δφn(1)

)τ, (2.76)

Bn(τ) = Bn(0)e(− V

ζnΘnn+

12 (cnλn+dn)δφn(1)

)τ, (2.77)

where An(0) and Bn(0) are given by (2.50) and (2.51), respectively. By using (2.66) and(2.67) with (2.39) and (2.46) it can easily be shown that Θnn

ζn= − 1

2 (cnλn + dn)φn(1). Thus,(2.69) with (2.76) and (2.77) can be re-written as

Tn(t,τ) = e−Θnnζn

(V+δ )τ(

An(0)cos(√

λnt)+Bn(0)sin(√

λnt)). (2.78)

Now, by substituting τ = εt into −Θnnζn

(V +δ )τ and by dividing the so-obtained result by t,it follows that the damping for each oscillation mode can be approximated by

Γn =−εΘnn

ζn(V +δ ), (2.79)

where ζn is given by (2.52) and Θnn =12 φ 2

n (1)> 0. In Table 2.2 and in Table 2.3 numericalapproximations of λn, Θnn, ζn, and the damping parameter Γn are given for different values


Table 2.2: Numerical approximations of Θnn, ζn, Θnnζn

, and Γn for k = 1

k = 1n λn Θnn ζn

Θnnζn

Γn(damping parameter)µ = 0.0011 4.13205 040564 0.59793 0.67841 -0.67841 (V +δ )ε2 24.68662 0.50368 0.52101 0.96673 -0.96673 (V +δ )ε3 67.44960 0.55464 0.51041 1.08666 -1.08666 (V +δ )ε4 136.98595 0.61296 0.50854 1.20534 -1.20534 (V +δ )ε5 239.83409 0.68046 0.50868 1.33771 -1.33771 (V +δ )ε

10 1637.47770 0.99062 0.50860 1.94772 -1.94772 (V +δ )ε25 39720.69922 1.06288 0.50144 2.11965 -2.11965 (V +δ )ε50 597499.42934 1.01969 0.50020 2.03858 -2.03858 (V +δ )εµ = 0.011 4.25996 0.43698 0.60378 0.72374 -0.72374 (V +δ )ε2 28.76105 0.69738 0.54328 1.28366 -1.28366 (V +δ )ε3 95.67926 0.93156 0.53738 1.73352 -1.73352 (V +δ )ε4 244.43954 1.06460 0.52806 2.01605 -2.01605 (V +δ )ε5 538.15563 1.10515 0.51879 2.13027 -2.13027 (V +δ )ε


10 72248.27763 1.00658 0.50032 2.01187 -2.01187 (V +δ )ε25 3374613.36828 1.00090 0.50002 2.00174 -2.00174 (V +δ )ε50 57333618.60889 1.00021 0.50000 2.00042 -2.00042 (V +δ )εµ = 11 5.79525 1.34103 0.95897 1.39841 -1.39841 (V +δ )ε2 268.92735 1.05693 0.51685 2.04494 -2.04494 (V +δ )ε3 2571.66438 1.01508 0.50266 2.01942 -2.01942 (V +δ )ε4 11006.46538 1.00656 0.50083 2.00976 -2.00976 (V +δ )ε5 32002.42493 1.00361 0.50036 2.00577 -2.00577 (V +δ )ε

10 714061.89600 1.00067 0.50003 2.00121 -2.00121 (V +δ )ε25 33691805.21440 1.00009 0.50000 2.00017 -2.00017 (V +δ )ε50 573116518.47183 1.00002 0.50000 2.00004 -2.00004 (V +δ )ε


Table 2.3: Numerical approximations of Θnn, ζn, Θnnζn

, and Γn for k = 10

k = 10n λn Θnn ζn

Θnnζn

Γn(damping parameter)µ = 0.0011 8.26224 0.03851 0.54627 0.07049 -0.07049 (V +δ )ε2 34.26368 0.13351 0.53827 0.24804 -0.24804 (V +δ )ε3 81.38438 0.25275 0.53087 0.47610 -0.47610 (V +δ )ε4 154.79829 0.38129 0.52598 0.72491 -0.72491 (V +δ )ε5 261.40539 0.51567 0.52321 0.98559 -0.98559 (V +δ )ε



10 72284.61491 1.01385 0.50051 2.02562 -2.02562 (V +δ )ε25 3374649.40672 1.00131 0.50002 2.00253 -2.00253 (V +δ )ε50 57333654.61732 1.00026 0.50000 2.00052 -2.00052 (V +δ )εµ = 11 27.26977 1.12354 1.11675 1.00608 -1.00608 (V +δ )ε2 308.15032 1.31972 0.57508 2.29484 -2.29484 (V +δ )ε3 2608.70728 1.06540 0.50822 2.09633 -2.09633 (V +δ )ε4 11042.89713 1.02338 0.50209 2.03823 -2.03823 (V +δ )ε5 32038.64875 1.01114 0.50079 2.01911 -2.01911 (V +δ )ε

10 714097.93038 1.00140 0.50005 2.00260 -2.00260 (V +δ )ε25 33691841.21826 1.00013 0.50000 2.00025 -2.00025 (V +δ )ε50 573116554.47267 1.00003 0.50000 2.00005 -2.00005 (V +δ )ε


of k and µ . From these Tables it can be seen that the damping parameter Γn for large valuesof n tends to −2(V +δ )ε . In Appendix B, it also has been analytically shown that Γn tendsto this value for large n. Now, from (2.73) with (2.74)-(2.78), we obtain

wntt (t,τ)+wn(t,τ)λn =∞

∑m=1m 6=n

e−Θmmζm

(V+δ )τ(

Ω1mncos(√

λmt)+Ω2mnsin(√

λmt)), (2.80)

where, Ω1mn and Ω2mn are given by

Ω1mn =

(δφm(1)(dn + cnλm)−

2Vζn

Θmn

)√λmBm(0) (2.81)

Ω2mn =

(δφm(1)(−dn− cnλm)+

2Vζn

Θmn

)√λmAm(0). (2.82)

The solution of (2.80) is given by

wn(t,τ) = Dn(τ)cos(√

λnt)+En(τ)sin(√

λnt)+

∑∞m=1m6=n

1λn−λm

e−Θmmζm

(V+δ )τ(Ω1mncos(√

λmt)+Ω2mnsin(√

λmt)),

(2.83)

where Dn(τ) and En(τ) are still arbitrary functions which can be used to avoid secular termsin y2(x, t,τ), and where Ω1mn and Ω2mn are given by (2.81) and (2.82), respectively. Thus,(2.62) with (2.83) can be expressed as

w(x, t,τ) = ∑∞m=1Dm(τ)cos(

√λmt)+Em(τ)sin(

√λmt)+

∑∞n=1n6=m

1λm−λn

e−Θnnζn

(V+δ )τ(Ω1nmcos(√

λnt)+Ω2nmsin(√

λnt))φm(x),

(2.84)

where, by using the inner product (2.48) and the initial conditions (2.58) and (2.59) with(2.60) and (2.61), we obtain

Dm(0) =1

ζm

∫ 1

0

(2x3− x4

12µ +2+ k

)δg(1)φm(x)dx−

∞

∑n=1n6=m

1λm−λn

Ω1nm, (2.85)

√λmEm(0) = 1

ζm

∫ 10

(2x3−x4

12µ+2+k

)δ ( f ′′(1)−µ f (4)(1))φm(x)dx

+ Vζm

Am(0)Θmm−∑∞n=1n6=m

√λn

λm−λnΩ2nm.

(2.86)

Hence, the solution y1(x, t,τ) of (2.30)-(2.35) now follows from (2.53), (2.54), (2.62) and(2.84), yielding

y1(x, t,τ) =∞

∑m=1wm(t,τ)+ cm

∞

∑p=1

Hp(t,τ)φm(x), (2.87)


where,

Hp(t,τ) = δ(Bp(τ)

√λpcos(

√λpt)−Ap(τ)

√λpsin(

√λpt)

)φp(1), (2.88)

where wm(t,τ) is given by (2.83), where Am(τ) and Bm(τ) are given by (2.76) and (2.77)respectively, and where cm is given by (2.66). It can be observed that y1(x, t,τ) still containsinfinitely many undetermined functions Dm(τ) and Em(τ), m = 1,2, · · · . These functionscan be used to avoid secular terms in the solution of y2(x, t,τ). At this moment, we are notinterested in the higher order approximations. For this reason, we will take Dm(τ) = Dm(0)and Em(τ) = Em(0), where Dm(0) and Em(0) are given by (2.85) and (2.86), respectively.So far, we have constructed a formal approximation y(x, t,τ) = y0(x, t,τ)+ εy1(x, t,τ) foru(x, t), where y0(x, t,τ) and y1(x, t,τ) are continuously twice differentiable with respect to t,four times with respect to x, and infinitely many times with respect to τ .

2.5 Conclusions

In this chapter an initial-boundary value problem for a linear homogeneous axially movingtensioned beam equation with non-classical boundary condition has been studied. One end ofthe beam is assumed to be simply-supported, whereas the other end of the beam is assumedto be attached to a spring-dashpot system. Formal asymptotic approximations of the exactsolution have been constructed using a two-timescales perturbation method. By using theenergy integral, it has been shown that the solutions are bounded for times t of O( 1

ε ). Somedamping rates of the modes are given in Table 2.2 and in Table 2.3 for different values of theparameters µ and k. It has been shown that the damping parameter (Γn) essentially dependson two system parameters V and δ > 0. The most optimal way to place a damper dependson the direction of the axial velocity V . If a damper is placed at x = L and the belt moveswith velocity V < 0, then to have damping in the system we should have δ > −V , whereasif a damper is placed at x = L and the belt moves with velocity V > 0, then we have alwaysdamping since V +δ > 0. To have always damping (regardless of the sign of V ) δ should belarger than |V |. For a nonmoving belt (V = 0) only a dashpot (δ ) is responsible to generatedamping in the system. In the absence of a dashpot, oscillations can still be damped whenV > 0 and the spring with stiffness k is placed at x = L. For the damping parameter Γn it alsohas been analytically and numerically shown that all solutions (up to O(ε)) are uniformlydamped when δ +V > 0.

Submitted as: S. H. Sandilo and W. T. van Horssen – ”On Variable Length Induced Vibrations of a VerticalString”, Elsevier, Journal of Sound and Vibration, February 2013.

Chapter 3On Variable Length Induced Vibrations of a

Vertical String

To myself I am only a child playing on the beach, while vastoceans of truth lie undiscovered before me.

Isaac Newton

Abstract

The purpose of this chapter is to study the free lateral responses of vertically translat-ing media with variable length and tension, subject to general initial conditions. Thetranslating media are modeled as taut strings with fixed boundaries. The problem can beused as a simple model to describe the lateral vibrations of an elevator cable, for whichthe length changes linearly in time, or for which the length changes harmonically abouta constant mean length. In this chapter an initial-boundary value problem for a linear,axially moving string equation is formulated. In the given model a rigid body is attachedto the lower end of the cable, and the suspension of the car against the guide rails isassumed to be rigid. For linearly length variations it is assumed that the axial velocityof the cable is small compared to nominal wave velocity and cable mass is small com-pared to car mass, and for harmonically length variations small oscillation amplitudesare assumed and it is also assumed that the cable mass is small compared to total massof the cable and the car. A multiple-timescales perturbation method is used to constructformal asymptotic approximations of the solutions to show the complicated dynamicalbehavior of the cable. It will also be shown that the Galerkin’s truncation method cannot be applied to this problem in all cases to obtain approximations valid on long timescales.

3.1 Introduction

Many engineering devices are represented by axially moving continua. Translating me-dia with constant length can model such low- and high-speed slender members as con-

veyor belts (see, (Sandilo and van Horssen 2012), (Pakdemirli and Oz 2008), (Ponomarevaand van Horssen 2009), and (Suweken and van Horssen 2003a)), chair lifts, power-transmissionchains, pipes transporting fluids ((Kuiper and Metrikine 2004), and (Oz and Boyaci 2000)),

30 3. On Variable Length Induced Vibrations of a Vertical String

aerial tramways, magnetic paper tapes, band saws and transport cables. In many applica-tions, systems including elevator cables (see, (Zhu and Ni 2000), (Zhu et al. 2001), (Zhu andXu 2003), and (Zhu and Chen 2005)), paper sheets (Stolte and Benson 1992), satellite teth-ers, flexible appendages, cranes and mine hoists ((Kaczmarczyk and Ostachowicz 2003), and(Kaczmarczyk 1997)), and cable-driven robots exhibit variable-length and transport speedduring operation. The traveling, tensioned Euler-Bernoulli beam and the traveling flexiblestring are the most commonly used models for such types of axially moving continua. Theyare classified in the category of one-dimensional continuous systems and consequently thedisplacement field depends on time and on a single spatial co-ordinate. The last few decadeshave seen an extensive research effort on the dynamics of translating media, where moststudies were restricted to cases with constant span length and transport velocity.Vibrations of horizontal and vertical translating strings and beams have been studied by manyresearchers. The forced response of translating media with variable length and tension wasanalyzed in (Zhu and Chen 2005). The effects of bending stiffness and boundary condi-tions on the dynamic response of elevator cables were examined in (Zhu and Xu 2003). Bytransforming the governing partial differential equation to ordinary differential equations, in(Carrier 1949) the response of a translating string with varying-length was first studied. In(Tabarrok et al. 1974) the dynamics of a translating beam with varying-length was first stud-ied and the equations of motion of a simple cantilever beam model utilizing Newton’s sec-ond law were derived. Among the earliest known considerations, the authors in (Vesnitskiiand Potapov 1975) found for some special cases the exact solution of one-dimensional me-chanical systems of variable length. For earlier work on strings with mass-spring systemsemulating an elevator, the reader is referred to ((Yamamoto et al. 1978), and (Terumichiet al. 1997)). In these two studies a constant transport velocity was assumed. The naturalfrequencies associated with the longitudinal vibration of a stationary cable and a car sys-tem were calculated in (Chi and Shu 1991). General stability characteristics of horizontallyand vertically translating strings and beams with arbitrary varying-length and with variousboundary conditions were investigated in (Zhu and Ni 2000). An active control methodol-ogy using a pointwise control force and/or moment was developed in (Zhu et al. 2001) todissipate the vibratory energy of a translating medium with arbitrary varying length.To improve the design of elevators, one of the major tasks is to develop a better understandingof elevator cable dynamics and new methods to effectively reduce the vibration and noise.The dynamics of vertical media with variable-length, -velocity and -tension is the subjectof this chapter. Due to small allowable vibrations the lateral and vertical cable vibrationsin elevators can be assumed to be uncoupled and only lateral cable vibrations in elevatorsare considered here. The elevator car is modelled as a rigid body of mass m attached at thelower end of the cable, and the suspension of the car against the guide rails is assumed tobe rigid, where external excitation is not considered at the boundaries. This is considered tobe a basic and simple model of an elevator cable from the practical viewpoint. The initial-


boundary value problems will be studied, and explicit asymptotic approximations of thesolutions, which are valid on a long time-scale, will be constructed as for instance describedin ((Nayfeh 1991), and (Kevorkian and Cole 1996)). Two cases for varying-length will beconsidered (i) l(t) = l0 + vt, where l0 is the initial cable length and v denotes the constantcable velocity, and (ii) l(t) = l0 + β sin(ωt), where β defines a length variation parameterand ω signifies the angular frequency of length variation and l0 > |β |. Regarding both casesof varying-length different dimensionless parameters will be used to obtain dimensionlessequations of motion. For the first case, it is assumed that v

√ρ

mg = O(ε) and ρLm = O(ε),

where ρ is the cable mass density, m is the car mass, g is the acceleration due to gravity and,L is the maximum length of the cable. For this case the exact solution of the initial-boundaryvalue problem has been approximated up to O(ε) and the free response of the elevator sys-tem is obtained in closed form solutions. For the second case, it is assumed that β

L = O(ε)and ρL

m+ρL = O(ε), where L is the maximum cable length and L |β |. For this case, itwill be shown that the Galerkin’s truncation method can not be applied for the parameter|α| ≤ 2 due to the distribution of energy among all vibration modes. To our knowledge, theexplicit construction of approximations of oscillations for these types of problems have notbeen given before.The outline of the chapter is as follows. In section 3.2, the generalized Hamilton’s principleis used to derive a model for an elevator suspension system in which the hoisting rope and thecar are included. In section 3.3 and section 3.4, a two-timescales perturbation method is ap-plied to construct formal asymptotic approximations for the solutions of the initial-boundaryvalue problems. It turns out for the case with the harmonically varying length that there areinfinitely many values of ω that can cause internal resonances. In this chapter we only inves-tigate the resonance case ω = π

l0and we also study a detuning case for this value. Finally, in

section 3.5, we make some remarks and draw some conclusions.


The vertically translating cable in elevators has no sag and will be modelled as a taut stringwith fixed boundaries in horizontal direction, as shown in Figure 3.1. The elevator car ismodelled as a rigid body of mass m attached at the lower end x = l(t), and suspension of thecar against the guide rails is assumed to be rigid. During its motion the cable of density ρhas a variable length l(t) and an axial velocity v(t) = l(t), where the over dot denotes timedifferentiation. The cable is assumed to be inextensible with an arbitrarily prescribed trans-lational velocity v(t), where t is time. A positive or negative transport velocity designatesextension or retraction of the cable, respectively. The lateral and longitudinal vibrations ofelevator cables are assumed to be uncoupled. In this chapter longitudinal vibrations will notbe considered. Relative to the fixed coordinate system as shown in Figure 3.1, the lateral dis-


l (t)

v

m

g

u (x,t)

x

Figure 3.1: Vertically translating string with an attached rigid mass at x = l(t).

placement of the cable particle instantaneously located at spatial position x at time t, where0≤ x≤ l(t), is described by u(x, t).The equations of motion for vertical string with variable-length and tension are obtained byusing Hamilton’s principle. The total kinetic energy associated with the lateral vibration ofthe string of length l(t) with end mass is

T =12

ρ∫ l(t)

0

(Du(x, t)

Dt

)2

dx+12

m(

Du(l(t), t)Dt

)2

, (3.1)

where

DDt≡ ∂

∂ t+ v

∂∂x

, (3.2)

defines the differentiation with respect to the motion, and ρ is the mass per unit length. Thepotential energy for the cable of length l(t) is

V =12

∫ l(t)

0P(x, t)

(∂u(x, t)

∂x

)2

dx, (3.3)


where P(x, t) is the axial force. The axial force in the vertically translating string in Figure3.1, arising from its own weight and longitudinal acceleration, is

P(x, t) = mg+ρ(l(t)− x)g−m ˙v(t)−ρ(l(t)− x) ˙v, (3.4)

where g is the acceleration due to gravity. Note that the axial force is tensile and compressiveduring deceleration ( ˙v < 0) and acceleration ( ˙v > 0) of the string, respectively, and vanishesduring its uniform motion ( ˙v = 0). Substituting (3.1) and (3.3) into Hamilton’s principle,

∫ t2

t1(δT −δV )dt = 0, (3.5)

and then applying the standard variational techniques, we obtain the following equation ofmotion (with the appropriate boundary and initial conditions),

ρD2u(x, t)

Dt2 − ∂∂x

(P(x, t)

∂u(x, t)∂x

)= 0, t > 0, 0 < x < l(t), (3.6)

u(0, t) = u(l(t), t) = 0, t > 0, (3.7)

u(x,0) = f (x), and ut(x,0) = h(x), 0 < x < l(0), (3.8)

where the lettered subscript for u denotes partial differentiation, and where

D2

Dt2 ≡∂ 2

∂ t2 +2v∂ 2

∂x∂ t+ v2 ∂ 2

∂x2 + ˙v∂∂x

. (3.9)

A new independent non-dimensional spatial coordinate ξ = xl(t) is introduced and the time-

varying spatial domain [0, l(t)] for x is converted to a fixed domain [0,1] for ξ . The newdependent variable is u(ξ , t) = u(x, t). The partial derivatives of u(x, t) with respect to x andt are related to those of u(ξ , t) with respect to ξ and t. Thus, we have

ux =1

l(t) uξ , uxx =1

l2(t) uξ ξ , ut = ut − vξl(t) uξ ,

uxt =1

l(t) uξ t − vξl2(t) uξ ξ − v

l2(t) uξ ,

utt = utt − 2vξl(t) uξ t +

v2ξ 2

l2(t) uξ ξ − ξ (l(t) ˙v−2v2)l2(t) uξ ,

(3.10)

where the subscripts denote partial differentiations.In the next two sections we will study two cases of varying length. In section 3.3 we willstudy the case l(t) = l0 + vt, where l0 and v are constants. In section 3.4 the case l(t) = l0 +β sin(ωt) will be studied, where l0, β , and ω are constants. In both cases small parameterswill occur, and a multiple time-scales perturbation method will be used to construct accurateapproximations of the solutions of the initial-boundary value problems for u(ξ , t), which arevalid on long time-scales.


3.3 The case l(t) = l0 + vt

To put the equations (3.6)-(3.8) into a non-dimensional form, the following dimensionlessparameters will be used:u∗ = u

L , x∗ = xL , t∗ = t

L

√mgρ , l∗ = l

L , l∗0 =l0L , v = v

√ρ

mg , µ = ρLm , f ∗ = f

L , h∗ = h√

ρmg , where

L is the maximum length of string. The equations of motion in non-dimensional form thenbecome ( ˙v = 0):

∂ 2u(x,t)∂ t2 +2v ∂ 2u(x,t)

∂x∂ t +(v2−µ(l(t)− x)−1

) ∂ 2u(x,t)∂x2 +µ ∂u(x,t)

∂x = 0, 0 < x < l(t), t > 0,(3.11)

u(0, t) = u(l(t), t) = 0, t > 0, (3.12)

u(x,0) = f (x),∂u(x,0)

∂ t= h(x), 0 < x < l0, (3.13)

where the asterisks indicating the dimensionless variables and parameters are omitted in(3.11)-(3.13) and henceforth. By using dimensionless parameters in (3.10) and then by sub-stituting the so-obtained transformations into (3.11)-(3.13) yields an initial-boundary valueproblem for u(ξ , t):

utt +2v(1−ξ )

l(t) uξ t +(

v2ξ 2

l2(t) −2v2ξl2(t) +

v2

l2(t) −1

l2(t) −µ(1−ξ )

l(t)

)uξ ξ

+(

µl(t) +

2v2(ξ−1)l2(t)

)uξ = 0, 0 < ξ < 1, t > 0,

(3.14)

u(0, t) = u(1, t) = 0, t > 0, (3.15)

u(ξ ,0) = f (ξ ), and ut(ξ ,0)−v(0)l(0)

ξ uξ (ξ ,0) = h(ξ ), 0 < ξ < 1, (3.16)

where f (ξ ) = f (x), and h(ξ ) = h(x) for t = 0.An approximation of the solution of the initial-boundary value problem (3.14)-(3.16) willbe constructed by using a two-timescales perturbation method for the case when the time-varying length l(t) = l0 + εt, where ε is a small non-dimensional parameter, that is, ε = v =

v√

ρmg . So, it is assumed that v

√ρ

mg is small. The approximation will be constructed for

t = O(ε−1). It is also assumed that the mass of the cable is small compared to the mass ofthe car. For this reason, we rewrite the parameter µ = ρL

m by µ = εµ0.

3.3.1 Application of the two-timescales perturbation method

If we expand the solution in a Taylor’s series with respect to ε straightforwardly, that is,

u(ξ , t;ε) = u0(ξ , t)+ ε u1(ξ , t)+ ε2u2(ξ , t) · · · , (3.17)

3.3. The case l(t) = l0 + vt 35

the approximation of the solution of the problem will contain secular terms, that is, un-bounded terms in t. To avoid secular terms a two-timescales perturbation method will beapplied. Using such a two-timescales perturbation method the function u(ξ , t;ε) is supposedto be a function of ξ , the fast time t+ =

∫ t0

dsl(s) , and the slow time t = εt, where the fast time

t+ =∫ t

0ds

l(s) is justified in the Appendix C. Thus, the function u(ξ , t;ε) can be written interms of ξ , t+, and t as

u(ξ , t;ε) = y(ξ , t+, t;ε). (3.18)

For the new variables t+ and t, the partial differential operator with respect to t can beexpressed as

∂∂ t

=1

l(t)∂

∂ t++ ε

∂∂ t

, (3.19)

and,∂ 2

∂ t2 =1

l2(t)∂ 2

∂ t+2 + ε(

2l(t)

∂ 2

∂ t+∂ t− 1

l2(t)∂

∂ t+

)+ ε2 ∂ 2

∂ t2 , (3.20)

where l(t) = l0 + t and dl(t)dt = 1. Substitution of (3.18)-(3.20) into (3.14)-(3.16) and observ-

ing that v = ε , yields an initial-boundary value problem for y(ξ , t+, t;ε):

∂ 2y∂ t+2 − ∂ 2y

∂ξ 2 =

ε(−2l(t) ∂ 2y

∂ t+∂ t +2(ξ −1) ∂ 2y∂ξ ∂ t+ −µ0 l(t) ∂y

∂ξ +µ0 l(t)(1−ξ ) ∂ 2y∂ξ 2 +

∂y∂ t+

)

+O(ε2), t+ > 0, t > 0, 0 < ξ < 1,

(3.21)

y(0, t+, t;ε) = y(1, t+, t;ε) = 0, t+ > 0, t > 0, (3.22)

y(ξ ,0,0;ε) = f (ξ ), 0 < ξ < 1, (3.23)

1l0

∂y(ξ ,0,0;ε)∂ t+

+ ε(

∂y(ξ ,0,0;ε)∂ t

− ξl0

∂y(ξ ,0,0;ε)∂ξ

)= h(ξ ). (3.24)

Using a two-timescales perturbation method it is usually assumed that not only the solu-tion u(ξ , t;ε) will depend on two-timescales, but also that u(ξ , t;ε) = y(ξ , t+, t;ε) can beapproximated by a formal expansion in ε , that is,

y(ξ , t+, t;ε) = y0(ξ , t+, t)+ εy1(ξ , t+, t)+ ε2 · · · . (3.25)

It is reasonable to assume this solution form because the partial differential equation and theboundary conditions depend analytically on ε . Substituting (3.25) into (3.21)-(3.24) and afterequating the coefficients of like powers in ε , it follows from the problem for y(ξ , t+, t;ε) thatthe O(1)-problem is:

∂ 2y0

∂ t+2 −∂ 2y0

∂ξ 2 = 0, t+ > 0, 0 < ξ < 1, (3.26)


y0(0, t+, t) = y0(1, t+, t) = 0, t+ > 0, t > 0, (3.27)

y0(ξ ,0,0) = f (ξ ), and,1l0

∂y0(ξ ,0,0)∂ t+

= h(ξ ), 0 < ξ < 1, (3.28)

and that the O(ε)-problem is:

∂ 2y1∂ t+2 − ∂ 2y1

∂ξ 2 =

−2l(t)(

∂ 2y0∂ t+∂ t +

(1−ξ )l(t)

∂ 2y0∂ξ ∂ t+ + µ0

2∂y0∂ξ −

µ0(1−ξ )2

∂ 2y0∂ξ 2 − 1

2l(t)∂y0∂ t+

),

t+ > 0, t > 0, 0 < ξ < 1,

(3.29)

y1(0, t+, t) = y1(1, t+, t) = 0, t+ > 0, t > 0, (3.30)

y1(ξ ,0,0) = 0,1l0

∂y1(ξ ,0,0)∂ t

=−∂y0(ξ ,0,0)∂ t

+ξl0

∂y0(ξ ,0,0)∂ξ

. (3.31)

It is observed that the O(1)-problem is a well-known initial-boundary value problem and canbe solved by using the method of separation of variables, where the boundary-value problemonly has positive eigenvalues λn which are given by (nπ)2, n = 1,2,3, · · · . For details thereader is referred to (Haberman 2004). The solution of the O(1)-problem is given by

y0(ξ , t+, t) =∞

∑n=1

(An0(t)cos(

√λnt+)+Bn0(t)sin(

√λnt+)

)φn(ξ ), (3.32)

where An0(t) and Bn0(t) are still arbitrary functions of t which can be used to avoid secularterms in y1(ξ , t+, t), and where φn(ξ ) is given by

φn(ξ ) = sin(√

λnξ ). (3.33)

Two different eigenfunctions belonging to two different eigenvalues are orthogonal with re-spect to following inner product as defined by

< φn(ξ ),φm(ξ )>=∫ 1

0φn(ξ )φm(ξ )dξ . (3.34)

Thus, using the superposition principle, the inner product (3.34), and the initial values (3.28),An0(0) and Bn0(0) are given by

An0(0) =1ζn

∫ 1

0f (ξ )φn(ξ )dξ , (3.35)

√λnBn0(0) =

l0ζn

∫ 1

0h(ξ )φn(ξ )dξ , (3.36)

where,

ζn =∫ 1

0φ 2

n (ξ )dξ =12. (3.37)

3.3. The case l(t) = l0 + vt 37

Now the solution of the O(ε)-problem will be determined. To solve (3.29)-(3.31), y1(ξ , t+, t)is written in the following eigenfunction expansion

y1(ξ , t+, t) =∞

∑n=1

wn(t+, t)φn(ξ ), (3.38)

where wn(t+, t) are the generalized Fourier coefficients. Substituting (3.38) into the partialdifferential equation (3.29), we obtain

∑∞n=1

(wnt+ t

(t+, t)+λnwn(t+, t))

φn(ξ ) =

−2l(t)(

y0t+ t+ (1−ξ )

l(t)y0ξ t+

+ µ02 y0ξ −

µ0(1−ξ )2 y0ξ ξ − 1

2l(t)y0t+

).

(3.39)

By multiplying both sides of (3.39) with φm(ξ ), then by integrating the so-obtained equationfrom ξ = 0 to ξ = 1, and by using the orthogonality properties of the eigenfunctions, weobtain

wmt+t+(t+, t)+λmwm(t+, t) =(

−2l(t)√

λmB′m0(t)+(

2ζm

√λm ˆΘmm− 2

ζm

√λmΘmm + l′(t)

√λm

)Bm0(t)

−(

µ0ζm

l(t)Θmm +µ0λm l(t)− µ0ζm

λm l(t)Θmm

)Am0(t)

)cos(√

λmt+)

+(

2l(t)√

λmA′m0(t)−(

2ζm

√λmΘmm− 2

ζm

√λmΘmm + l′(t)

√λm

)Am0(t)

−(

µ0ζm

l(t)Θmm +µ0λm l(t)− µ0ζm

λm l(t)Θmm

)Bm0(t)

)sin(√

λmt+)

+2∑∞n=1n 6=m

Tnt+

(t+, t)Θnm−Tnt+(t+, t)Θnm−µ0 l(t)Tn(t+, t)Θnm

+µ0λn l(t)ΘnmTn(t+, t),

(3.40)

where,

Θnm =∫ 1

0φ ′n(ξ )φm(ξ )dξ , Θnm =

∫ 1

0ξ φ ′n(ξ )φm(ξ )dξ , (3.41)

Θnm =∫ 1

0ξ φn(ξ )φn(ξ )dξ , (3.42)

and,Tn(t+, t) = An0(t)cos(

√λnt+)+Bn0(t)sin(

√λnt+). (3.43)

It can elementarily be shown that ζm = 12 , Θmm = 0, Θmm = − 1

4 , and Θmm = 14 , therefore

from (3.40) it follows that

wmt+t+(t+, t)+λmwm(t+, t) =(−2l(t)

√λmB′m0(t)− 1

2 µ0λm l(t)Am0(t))

cos(√

λmt+)+(2l(t)√

λmA′m0(t)− 12 µ0λm l(t)Bm0(t)

)sin(√

λmt+)+2∑∞

n=1n6=m

Tnt+

(t+, t)Θnm−Tnt+(t+, t)Θnm−µ0 l(t)Tn(t+, t)Θnm

+µ0λn l(t)ΘnmTn(t+, t).

(3.44)


The right hand side of (3.44) contains terms which are solutions of the homogeneous partcorresponding to (3.44). These terms will give rise to unbounded terms, the so-called secularterms, in the solution wm(t+, t) of (3.44). Since it is assumed that y0(ξ , t+, t),y1(ξ , t+, t), · · ·are bounded on timescales of O( 1

ε ), so these secular terms should be avoided. In (3.32)the functions An0(t) and Bn0(t) are still undetermined. These functions will be used to avoidsecular terms in the solution of (3.44). In order to remove secular terms, it now easily followsfrom (3.44) that Am0(t) and Bm0(t) have to satisfy

A′m0(t)−14

µ0√

λmBm0(t) = 0, (3.45)

B′m0(t)+14

µ0√

λmAm0(t) = 0. (3.46)

To find the solutions of the system of equations (3.45)-(3.46), we use polar coordinates, thatis,

Am0(t) = rm0(t)cos(ψm0(t)), and, Bm0(t) = rm0(t)sin(ψm0(t)), (3.47)

where,

rm0(t) =√

A2m0(t)+B2

m0(t), and, ψm0(t) = tan−1(

Bm0(t)Am0(t)

). (3.48)

By substituting (3.47) in (3.45) and (3.46), we get the following two ordinary differentialequations in rm0(t) and ψm0(t):

r′m0(t) = 0, ψ ′m0(t)+14

µ0√

λm = 0. (3.49)

Thus, (3.49) yields the solutions

rm0(t) = rm0(0), and, ψm0(t) =−14

µ0√

λmt +ψm0(0), (3.50)

where,

rm0(0) =√

A2m0(0)+B2

m0(0), and, ψm0(0) = tan−1(

Bm0(0)Am0(0)

). (3.51)

Thus, Am0(t) and Bm0(t) are given by (3.47)-(3.51). Now, from (3.44) with (3.45)-(3.51), weget

wmt+t+(t+, t)+λmwm(t+, t) =

∞

∑n=1n6=m

(A∗nm(t)cos(

√λnt+)+B∗nm(t)sin(

√λnt+)

), (3.52)

where,

A∗nm(t) = 2(√

λn(Θnm−Θnm)Bn0(t)+µ0 l(t)(λnΘnm−Θnm)An0(t)), (3.53)

3.3. The case l(t) = l0 + vt 39

and,

B∗nm(t) = 2(√

λn(−Θnm +Θnm)An0(t)+µ0 l(t)(λnΘnm−Θnm)Bn0(t)). (3.54)

The solution of (3.52) is given by

wm(t+, t) = Dm1(t)cos(√

λmt+)+Em1(t)sin(√

λmt+)+∑∞

n=1n6=m

1λm−λn

(A∗nm(t)cos(

√λnt+)+B∗nm(t)sin(

√λnt+)

), (3.55)

where Dm1(t) and Em1(t) are still arbitrary functions which can be used to avoid secular termsin y2(ξ , t+, t), and where A∗nm(t) and B∗nm(t) are given by (3.53) and (3.54), respectively.Thus, (3.38) with (3.55) can be expressed as:

y1(ξ , t+, t) = ∑∞n=1

Dn1(t)cos(

√λnt+)+En1(t)sin(

√λnt+)+

∑∞m=1m 6=n

1λn−λm

(A∗mn(t)cos(

√λmt+)+B∗mn(t)sin(

√λmt+)

)sin(nπξ ),

(3.56)

where by using the initial conditions (3.31) and the inner product (3.34), we obtain

Dn1(0) =∞

∑m=1m 6=n

1λn−λm

A∗mn(0), (3.57)

and,

√λnEn1(0) =−l0A′n0(0)−

12

An0(0)+∞

∑m=1m6=n

(2Am0(0)Θmn−

√λm

λn−λmB∗mn(0)

). (3.58)

It is observed that y1(ξ , t+, t) still contains infinitely many undetermined functions Dn1(t)and En1(t), n = 1,2,3, · · · . These functions can be used to avoid secular terms in the solutionof y2(ξ , t+, t). At this moment, we are not interested in the higher order approximations.For this reason, we will take Dn1(t) = Dn1(0) and En1(t) = En1(0), where Dn1(0) and En1(0)are given by (3.57) and (3.58), respectively. So far, we have constructed a formal approxi-mation y(ξ , t+, t;ε) = y0(ξ , t+, t)+εy1(ξ , t+, t) of u(ξ , t), where y0(ξ , t+, t) and y1(ξ , t+, t)are twice continuously differentiable with respect to ξ and t, and infinitely many times withrespect to t. Hence, the solution in terms of x and t up to O(ε) is given by

u(x, t;ε) = u0(x, t)+ εu1(x, t), (3.59)

where u0(x, t) and u1(x, t) are given by

u0(x, t) = ∑∞n=1 rn0(0)cos

(nπ∫ t

0ds

l(s) −ψn0(εt))sin( nπx

l(t)

)

= ∑∞n=1 rn0(0)cos

( nπε ln(l0 + εt)− 1

4 nπµ0εt +ψn0(0))sin( nπx

l0+εt

),

(3.60)


u1(x, t) = ∑∞n=1

[Dn1(0)cos

(nπ∫ t

0ds

l(s)

)+En1(0)sin

(nπ∫ t

0ds

l(s)

)+

∑∞m=1m6=n

rm0(0)λn−λm

C∗mncos

(nπ∫ t

0ds

l(s) −ψm0(εt))+D∗mnsin

(nπ∫ t

0ds

l(s) −ψm0(εt))]×

sin( nπxl(t) ),

(3.61)

where,C∗mn = 2µ0l(t)(λmΘmn−Θmn), D∗mn = 2

√λm(Θmn− Θmn), (3.62)

and where rn0(0), ψn0(0) and ψn0(εt), are given by (3.50)-(3.51), and where, An0(0), Bn0(0),Dn1(0) and En1(0) are given by (3.35), (3.36), (3.57) and (3.58), respectively. By using thetechniques as developed in (van Horssen 1988) it can be shown that u0(x, t) as given by(3.60) is an O(ε) accurate approximation of the exact solution of the initial-boundary valueproblem (3.14)-(3.16) for times t of order 1

ε .

3.4 The case l(t) = l0 +β sin(ωt)

To put the equations (3.6)-(3.8) into non-dimensional form, the following dimensionlessparameters will be used:u∗ = u

L , x∗ = xL , t∗ = t

L

√g(m

ρ +L), µ = ρLm+ρL , l∗(t) = l(t)

L , v∗ = v√g(m+ρL

ρ ), γ = m

ρL , where L

is the maximum length of the cable. Thus, the equations of motion in non-dimensional formbecome:

∂ 2u∂ t2 − ∂ 2u

∂x2 +2v ∂ 2u∂x∂ t +

(v2 +µ− (l(t)− x)µ + γ v

) ∂ 2u∂x2

+(µ + v

) ∂u∂x = 0, 0 < x < l(t), t > 0,

(3.63)

u(0, t) = u(l(t), t) = 0, t > 0, (3.64)

u(x,0) = f (x),∂u(x,0)

∂ t= h(x), 0 < x < l(0). (3.65)

The asterisks indicating the dimensionless parameters are omitted in (3.63)-(3.65) and hence-forth. Introducing again ξ = x

l(t) , and by using dimensionless parameters in (3.10), and thenby substituting the so-obtained transformations into (3.63)-(3.65) yields an initial-boundaryvalue problem for u(ξ , t) = u(x, t):

utt − 1l2(t) uξ ξ +

2v(1−ξ )l(t) uξ t +

(v2(ξ−1)2

l2(t) + µl2(t) −

(1−ξ )µl(t) + γ v

l2(t)

)uξ ξ

+(

2v2(ξ−1)l2(t) + v(1−ξ )

l(t) + µl(t)

)uξ = 0, 0 < ξ < 1, t > 0,

(3.66)

u(0, t) = u(1, t) = 0, t > 0, (3.67)

u(ξ ,0) = f (ξ ), and ut(ξ ,0) = h(ξ )+v(0)ξl(0)

f ′(ξ ), 0 < ξ < 1, (3.68)

3.4. The case l(t) = l0 +β sin(ωt) 41

where f (ξ ) = f (x), and h(ξ ) = h(x) for t = 0. Now, we consider the case l(t) = l0 +εβ0sin(ωt), where l0, β0 and ω are positive constants, and ε is small non-dimensional pa-rameter. It is also assumed that µ = εµ0 and g ˙v. For this reason, −ρ(l(t)− x) ˙v isneglected in (3.4) to obtain dimensionless equations of motion. Now by using Taylor’s se-ries in ε , we observe that 1

l(t) =1l0− εβ0sin(ωt)

l20

+O(ε2), and 1l2(t) =

1l20− 2εβ0sin(ωt)

l30

+O(ε2).

Therefore, the equations (3.66)-(3.68) become up to O(ε):

utt − 1l20

uξ ξ =

ε[

2β0(ξ−1)ωcos(ωt)l0

uξ t +

((1−ξ )µ0

l0− µ0

l20+ γβ0ω2sin(ωt)

l20

− 2β0sin(ωt)l30

)uξ ξ

+(

β0(1−ξ )ω2sin(ωt)l0

− µ0l0

)uξ

],

(3.69)

u(0, t) = u(1, t) = 0, (3.70)

u(ξ ,0) = f (ξ ), and ut(ξ ,0) = h(ξ )+εωξ

l0f ′(ξ ). (3.71)

To satisfy the boundary conditions all functions should be expanded in Fourier-sin-series.Therefore, by substituting u(ξ , t;ε) = ∑∞

n=1 un(t;ε)sin(nπξ ) in (3.69), we get

∑∞n=1

(un(t;ε)+( nπ

l0)2un(t;ε)

)sin(nπξ ) =

ε ∑∞n=1

[2nπβ0(ξ−1)ωcos(ωt)

l0un(t;ε)cos(nπξ )

−(nπ)2(

(1−ξ )µ0l0− µ0

l20+ γβ0ω2sin(ωt)

l20

− 2β0sin(ωt)l30

)un(t;ε)sin(nπξ )

−(nπ)((1−ξ )β0ω2sin(ωt)

l0− µ0

l0

)un(t;ε)cos(nπξ )

].

(3.72)

Now, by multiplying (3.72) with sin(kπξ ), and by integrating the so-obtained equation fromξ = 0 to ξ = 1 and by using the orthogonality properties of the eigenfunctions, we obtain

uk(t;ε)+( kπl0)2uk(t;ε) =

ε[− β0ω

l0cos(ωt)uk(t;ε)+

( µ0l20− µ0

2l0

)(kπ)2uk(t;ε)

−(

γβ0ω2

l20− 2β0

l30

)sin(ωt)(kπ)2uk(t;ε)+ β0ω2

2l0sin(ωt)uk(t;ε)

+∑∞n=1,n6=k

4nπβ0ω

l0cos(ωt)un(t;ε)

(f1(n,k)− f2(n,k)

)

+ 2µ0l0(nπ)2un(t;ε) f3(n,k)− 2µ0

l0(nπ)un(t;ε) f2(n,k)

+ 2β0ω2

l0(nπ)sin(ωt)un(t;ε)

(f2(n,k)− f1(n,k)

)],

(3.73)


where f1(n,k), f2(n,k), and f3(n,k) are given by

f1(n,k) =∫ 1

0 ξ cos(nπξ )sin(kπξ )dξ=− (n3−n2k−nk2+k3)(−1)n+k−(n3+n2k−nk2−k3)(−1)−n+k

2π(n+k)2(−n+k)2 ,(3.74)

f2(n,k) =∫ 1

0 cos(nπξ )sin(kπξ )dξ=−−2k+(−n+k)(−1)n+k+(n+k)(−1)−n+k

2π(n+k)(−n+k) ,(3.75)

f3(n,k) =∫ 1

0 ξ sin(nπξ )sin(kπξ )dξ=− 4nk−(n+k)2(−1)−n+k+(−n+k)2(−1)n+k

2π2(n+k)2(−n+k)2 .(3.76)

3.4.1 Application of the two-timescales perturbation method

We consider again equation (3.73). The application of a straight-forward expansion methodto solve (3.73) will result in the occurrence of so-called secular terms which causes theapproximations to become unbounded on long time-scales. To remove those secular terms,we introduce two time-scales t+ = t and t = εt. The introduction of these two time-scalesdefines the following transformations: uk(t;ε) = wk(t+, t;ε), duk

dt = ∂wk∂ t+ + ε ∂wk

∂ t and d2ukdt2 =

∂ 2wk∂ t+2 + 2ε ∂ 2wk

∂ t+∂ t + ε2 ∂ 2wk∂ t2 . By substituting above mentioned transformations into (3.73) we

obtain

∂ 2wk∂ t+2 +2ε ∂ 2wk

∂ t+∂ t +( kπl0)2wk =

ε[− β0ω

l0cos(ωt+) ∂wk

∂ t+ +( µ0

l20− µ0

2l0

)(kπ)2wk

−(

γβ0ω2

l20− 2β0

l30

)sin(ωt+)(kπ)2wk +

β0ω2

2l0sin(ωt+)wk

+∑∞n=1,n6=k

4nπβ0ω

l0cos(ωt+) ∂wn

∂ t+(

f1(n,k)− f2(n,k))

+ 2µ0l0(nπ)2wn f3(n,k)− 2µ0

l0(nπ)wn f2(n,k)

+ 2β0ω2

l0(nπ)sin(ωt+)wn

(f2(n,k)− f1(n,k)

)],

(3.77)

where f1(n,k), f2(n,k), and f3(n,k) are given by (3.74)-(3.76), respectively. Assuming thatwk(t+, t;ε) = wk0(t+, t)+ εwk1(t+, t)+ ε2 · · · , then in order to remove the secular terms up


to O(ε), we have to solve the following problems:

O(1) : ∂ 2wk0∂ t+2 +

(kπl0

)2wk0 = 0,

O(ε) : ∂ 2wk1∂ t+2 +

(kπl0

)2wk1 =−2 ∂ 2wk0

∂ t+∂ t +( µ0l20− µ0

2l0)(kπ)2wk0

+∑∞n=1,n 6=k

[4nπβ0ω

l0cos(ωt+) ∂wn0

∂ t+ ( f1(n,k)− f2(n,k))

− 2(nπ)β0ω2

l0sin(ωt+)wn0( f1(n,k)− f2(n,k))

]+N.S.T.

(3.78)

The O(1)-problem has as solution

wk0(t+, t) = Ak0(t)cos(

kπt+

l0

)+Bk0(t)sin

(kπt+

l0

), (3.79)

where Ak0 and Bk0 are still arbitrary functions of t and they can be used to avoid secularterms in the solution of the O(ε)-problem. From the O(ε)-problem it can readily be seenthat there are infinitely many values of ω that can cause internal resonance. These values aremultiples of π

l0. In the next two sections the case π

l0and the detuning case ω = π

l0+ εσ will

be studied in detail.

3.4.2 The case ω = πl0

In Appendix D it has been shown for ω = πl0

what equations Ak0(t) and Bk0(t) have to satisfysuch that the approximations of the solution of the problem do not contain secular terms. Itturns out that Ak0 and Bk0 have to satisfy:

dAk0dt1

=−(k−1)A(k−1)0 +(k+1)A(k+1)0 +(l0−2)µ0l0

2β0kBk0,

dBk0dt1

=−(k−1)B(k−1)0 +(k+1)B(k+1)0− (l0−2)µ0l02β0

kAk0,(3.80)

where t1 =β0π2l2

0t, and k = 1,2,3, · · · . In Appendix E, we found that if system (3.80) is trun-

cated at any order then the eigenvalues of the truncated system are always purely imaginaryor zero. In this case we shall show that the results obtained by applying the truncation methodare not valid on time-scales of order ε−1 in all cases. Now, by putting kAk0(t1) = Xk0(t1) andkBk0(t1) = Yk0(t1), the system (3.80) becomes:

dXk0dt1

= k[−X(k−1)0 +X(k+1)0 +αYk0],dYk0dt1

= k[−Y(k−1)0 +Y(k+1)0−αXk0],(3.81)

for k = 1,2,3, · · · , and X00 = Y00 = 0, and where α = (l0−2)µ0l02β0

. Accordingly we also have:

Xk0Xk0 = k[−Xk0X(k−1)0 +Xk0X(k+1)0 +αXk0Yk0],

Yk0Yk0 = k[−Yk0Y(k−1)0 +Yk0Y(k+1)0−αXk0Yk0],(3.82)


where the over dot represents the derivative with respect to t1. In Appendix F, it has beenshown that the system (3.82) can be reduced as

d2

dt21

∞

∑k=1

(X2

k0 +Y 2k0)+(α2−4)

∞

∑k=1

(X2

k0 +Y 2k0)= D1, (3.83)

where D1 is a constant. And so, by putting ∑∞k=1(X2

k0 +Y 2k0

)=W (t1), we finally get

d2W (t1)dt2

1+(α2−4)W (t1) = D1, (3.84)

where D1 =W ′′(0)+(α2−4)W (0). Now elementary calculations yield,

for |α|< 2 : W (t1) =C1cosh(

t1√

4−α2)+C2sinh

(t1√

4−α2)+ D1

α2−4 ,

for |α|= 2 : W (t1) =C1 +C2t1 + 12 D1t2

1 ,

for |α|> 2 : W (t1) =C1cos(

t1√

α2−4)+C2sin

(t1√

α2−4)+ D1

α2−4 ,

where C1, C2 and D1 are constants of integration and can be determined by using the initialconditions. It can be seen that the above solutions exhibit interesting features. For |α| < 2,W (t1) (and so the energy) increases exponentially due to hyperbolic functions. For |α|= 2,W (t1) increases polynomially due to algebraic functions. And finally, for |α| > 2, W (t1)is bounded due to trigonometric functions. So, for |α| ≤ 2, W (t1) is unbounded in t1 andincreases as t1 increases. This behavior is different from the behavior of Ak0(t1) and Bk0(t1)as obtained by applying the truncation method (see also Appendix E). If we apply the trun-cation method, we merely obtain sine and cosine functions for Ak0 and Bk0 while the energyincreases. This means that the approximations obtained by applying the truncation methodto system (3.80 ) are not accurate on long time-scales, that is, on time-scales of order ε−1.For |α|> 2, the energy is bounded and it is an open problem whether it is allowed to applythe truncation method or not.

3.4.3 The case ω = πl0+ εσ

In this case we will consider the detuning from ω = πl0

, that is, we will study the case ω =πl0+ εσ where σ = O(1). In order to avoid secular terms in the approximation, it can be

shown (the calculations are similar to those in section 3.4.2 and can be followed by AppendixD) that Ak0(t1) and Bk0(t1) have to satisfy

dAk0dt1

= [(k+1)A(k+1)0− (k−1)A(k−1)0]cos(σ t1)−[(k+1)B(k+1)0 +(k−1)B(k−1)0]sin(σ t1)+αkBk0,

dBk0dt1

= [(k+1)B(k+1)0− (k−1)B(k−1)0]cos(σ t1)+[(k+1)A(k+1)0 +(k−1)A(k−1)0]sin(σ t1)−αkAk0,

(3.85)


for k = 1,2,3, · · · . It can be noticed that for σ = 0 we can obtain the same system (3.80)again. Following the similar calculations as given in Appendix F, we obtain

d2W (t1)dt2

1+((σ −α)2−4)W (t1) = D1, (3.86)

where D1 =W ′′(0)+((σ−α)2−4)W (0) and can be determined by using initial conditions.By elementary calculations, we obtain solutions of (3.86) as follows

for |σ −α|< 2 :

W (t1) =C1cosh(

t1√

4− (σ −α)2)+C2sinh

(t1√

4− (σ −α)2)+ D1

α2−4 ,

for |σ −α|= 2 :W (t1) =C1 +C2t1 + 1

2 D1t21 ,

for |σ −α|> 2 :

W (t1) =C1cos(

t1√

(σ −α)2−4)+C2sin

(t1√(σ −α)2−4

)+ D1

α2−4 ,

where C1, C2, and D1 are constants of integration and can be determined by using the initialconditions. The interesting features of these solutions are, that for |σ −α| ≤ 2, W (t1) (andso the energy) is unbounded. For |σ −α| > 2, W (t1) remains bounded due to sin and cosfunctions. It is important to note that for σ large enough i.e., (σ −α)2 > 4, the system isalways stable.

3.4.4 The energy of the infinite dimensional system

The non-dimensional energy E(t) of the elevator cable system can also be approximated byusing the function W (t). Since

u(ξ , t) = ∑∞k=1 uk(t)sin(kπξ )

= ∑∞k=1

[Ak0(t)cos

(kπt+

l0

)+Bk0(t)sin

(kπt+

l0

)]sin(kπξ )+O(ε),

it follows that the energy E(t) satisfies

E(t) = 12∫ 1

0

(u2

t+ + 1l20

u2ξ

)dξ +O(ε)

= π2

4l20

∑∞k=1 k2

[(−Ak0(t)sin

(kπt+

l0

)+Bk0(t)cos

(kπt+

l0

))2

+(

Ak0(t)cos(

kπt+l0

)+Bk0(t)sin

(kπt+

l0

))2]+O(ε)

= π2

4l20

∑∞k=1[(kAk0)

2 +(kBk0)2]+O(ε)

= π2

4l20

∑∞k=1[X

2k0 +Y 2

k0]+O(ε)

= π2

4l20W (t)+O(ε).


So, the energy of the infinite dimensional system (3.80) increases for |α| ≤ 2, and remainsbounded for |α|> 2.

3.5 Conclusions

In this chapter, the generalized Hamilton’s principle is employed to derive the governingequations of motion under free vibration for an axially moving string. A system composedof a string with time-varying length, fixed at the upper and lower ends, is assumed to be asimple, analytical model of an elevator system. The elevator system dynamics vary with timeduring the system operation. This variation is caused by the varying-length of the hoist ropes.The non-dimensional rate of variation defined by the quantity ε has been used as a smallparameter to assess the slow variability of the component length. The equations of motionhave been solved approximately for lengths varying linearly in time where it is assumedthat v

√ρ

mg = O(ε) and ρLm = O(ε), and for lengths varying harmonically about a constant

mean value with small fluctuation amplitudes assuming that βL = O(ε) and ρL

m+ρL = O(ε).A two-timescales perturbation method has been used to obtain the analytical approximatesolutions for a vertically translating string with linearly length variation on a time-scale oforder ε−1 and in search of infinite mode approximate solutions. For the case l(t) = l0 + vt,analytic approximations of the solutions have been obtained up to O(ε) and it is shownthat the free response of the elevator system has in fact a complicated dynamical behavior.For the case l(t) = l0 + β sin(ωt) it has been shown that each truncated system of ODEshas only purely imaginary (or zero) eigenvalues but the energy of the infinite dimensionalsystem grows without bound for the parameter |α| ≤ 2. Therefore, the truncation methodcan not be applied for |α| ≤ 2 to obtain asymptotic approximations on long time-scales, thatis, on time-scales of order ε−1. It is also shown that the infinite dimensional system remainsbounded for |α| > 2 and whether it is allowed to apply the Galerkin’s truncation method ornot remains an open problem (as well as for the other frequencies ω = mπ

l0with m ∈ N and

m > 1). For detuning large enough, i.e., (σ −α)2 > 4, the system will always remain stable.

Submitted as: S. H. Sandilo and W. T. van Horssen – ”On a Cascade of Autoresonances in an Elevator CableSystem”, Springer, Nonlinear Dynamics, September 2013.

Chapter 4On a Cascade of Autoresonances in an Elevator

Cable System

The two operations of our understanding, intuition anddeduction, on which alone we have said we must rely in theacquisition of knowledge.

Rene Descartes

Abstract

The aim of this chapter is to study autoresonance phenomena in a space-time-varyingmechanical system. The maximal amplitude of the autoresonant solution and the time ofautoresonant growth of the amplitude of the modes of fast oscillations are determined. Avertically translating string with a time-varying length and a space-time-varying tensionare considered. The problem can be used as a simple model to describe transversal vi-brations of an elevator cable for which the length changes linearly in time. The slowlytime-varying length is given by l(t) = l0+εt, where l0 is a constant and ε is a dimension-less small parameter. It is assumed that the axial velocity of the cable is small comparedto nominal wave velocity and the cable mass is small compared to car mass. The elevatorcable is excited sinusoidally at the upper end by the displacement of the building in thehorizontal direction from its equilibrium position caused by wind forces. This externalexcitation has a constant amplitude of order ε . It is shown that order ε amplitude ex-citations at the upper end result in order

√ε solution responses. Interior layer analysis

has been provided systematically to show that there exists an unexpected timescale oforder 1√

ε . For this reason, a three-timescales perturbation method is used to constructasymptotic approximations of the solutions of the initial-boundary value problem.

4.1 Introduction

Strings and beams, with constant length can model such low- and high-speed slendermembers as conveyor belts (see, (Suweken and van Horssen 2003a), (Ponomareva and

van Horssen 2009), and (Sandilo and van Horssen 2012)), chair lifts, pipes transporting liq-uids or gases ((Kuiper and Metrikine 2004), and (Oz and Boyaci 2000)), aerial tramways,magnetic paper tapes, band saws, and transport cables. Strings and beams, with time-variable

48 4. On a Cascade of Autoresonances in an Elevator Cable System

lengths, are used as elements of machines and structures. For example, in many engineeringapplications, systems such as elevator cables (see, (Zhu and Xu 2003), (Zhu and Chen 2005),and (Terumichi et al. 1997)), paper sheets (Stolte and Benson 1992), crane and mining hoists((Kaczmarczyk 1997), and (Kaczmarczyk and Ostachowicz 2003)), and cable-driven robotsexhibit time-varying-length, space-time-varying tension and constant or time-varying veloc-ity during system operation. The traveling flexible string and the traveling tensioned Euler-Bernoulli beam are the most commonly used models for such type of axially moving con-tinuous systems. They are classified in the category of one-dimensional continuous systemsand consequently the displacement field depends on time and on a single spatial coordinate.The understanding of continuous systems described by linear (nonlinear) partial differentialequations with space- and/or time-varying parameters comprises a great challenge becauseof the variety and complexity of possible solutions of the underlying PDEs.Vibrations of horizontal and vertical translating strings and beams have been studied by manyresearchers. The forced response of translating media with variable length and tension wasanalyzed in (Zhu and Chen 2005). The effects of bending stiffness and the boundary condi-tions on the dynamic response of elevator cables were examined in (Zhu and Xu 2003). Bytransforming the governing partial differential equation to ordinary differential equations, in(Carrier 1949) the response of a translating string with varying-length was first studied. In(Tabarrok et al. 1974) the dynamics of a translating beam with varying-length was studiedand the equations of motion of a simple cantilever beam model utilizing Newton’s secondlaw were derived. Among the earliest known considerations, the authors in (Vesnitskii andPotapov 1975) found the exact solution of one-dimensional mechanical systems of variablelength for some special cases. For earlier work on strings with mass-spring systems emulat-ing an elevator, the reader is referred to ((Terumichi et al. 1997) and (Yamamoto et al. 1978)).In both of these studies a constant transport velocity was assumed. The natural frequenciesassociated with the longitudinal vibration of a stationary cable and a car system were cal-culated in (Chi and Shu 1991). General stability characteristics of horizontally and verti-cally translating strings and beams with arbitrary varying-length and with various boundaryconditions were investigated in (Zhu and Ni 2000). An active control methodology using apointwise control force and/or moment was developed in (Zhu et al. 2001) to dissipate the vi-bratory energy of a translating medium with arbitrary varying length. Recently, in ((Sandiloand van Horssen 2011), and (Sandilo and van Horssen 2013)) the authors have studied avertical translating string with time-varying length for a constant and for a time-varying ax-ial velocity. For the case l(t) = l0 + εβ0sin(ωt), it has been shown that there are infinitelymany values of ω giving rise to internal resonances in the elevator system. It was also shownin (Sandilo and van Horssen 2013) that Galerkin’s truncation method can not be applied inorder to obtain asymptotic results on long timescales (that is, on timescales of order ε−1).To improve the design of elevators, one of the major tasks is to develop a better under-standing of elevator cable dynamics and to construct new methods of solution to effectively

4.1. Introduction 49

reduce the vibration and noise. The dynamics of vertical media with time-varying length,time-space-varying tension, and a constant velocity is the subject of this chapter. The systemis composed of a string with time-varying length and attached to a mass at its lower end.The string is hung vertically and is excited sinusoidally at the upper end by a horizontal dis-placement of the building from its equilibrium due to a wind force. From the practical viewpoint, this is considered to be a basic model of an elevator system. Due to small allowablevibrations the transversal and the vertical cable vibrations in elevators can be assumed to beuncoupled and only transversal cable vibrations are considered here. The elevator car is theaccessible carrier for passengers in the elevator which is modelled as a rigid body of massm attached at the lower end of the cable, and the suspension of the car against the guiderails is assumed to be rigid. We consider a linearly varying-length l(t) = l0 + vt and theexternal excitations u(0, t) = αsin(Ωt) at the upper end, where l0 is the initial cable length,v denotes the constant axial velocity of the cable, u is the horizontal displacement functionof the cable, α is the oscillation-amplitude of the building from equilibrium and Ω signifiesthe frequency of this excitation. Small parameters occur when dimensionless equations ofmotion are obtained. It is assumed that v

√ρ

mg = O(ε), ρLm = O(ε) and α

L = O(ε), whereρ is the cable mass density, m is the car mass, g is the acceleration due to gravity, L is themaximum length of the cable and ε is a small dimensionless parameter. The exact solution ofthe initial- boundary value problem will be approximated up to O(ε) and the approximationsof the forced response of the elevator system will be obtained in closed form solutions.Interesting phenomena arise when there occurs a passage through resonance. When sys-tem’s natural frequency parameter (for instance, a first mode) gets closer to the excitationfrequency parameter (some fixed value which is less than or equal to natural frequency pa-rameter) a significant amplitude growth begins. Thereafter the amplitude of the oscillationwill be such that the linear wave frequency matches the excitation frequency and this re-sults in the occurrence of a linear jump of order 1√

ε in the amplitude of the system. Thisphenomena can be seen in Figure G.1 and Figure G.2. This is a fascinating phenomenon ofautoresonance, where a perturbed system is captured into resonance. By the autoresonanceis meant the growth of the amplitude of oscillations of a solution to a linear (nonlinear)equation of motion under action of a small external oscillating force. The autoresonance isthought of as universal phenomenon which occurs in a wide range of oscillating physical sys-tems from astronomical to atomic one (see, for instance, (Fajans and Friedland 2001)). Theautoresonance concept goes back to (McMillan 1945), and was further developed in (Bohmand Foldy 1946) for particle accelerators. The term “phase stability principle” was used todescribe the phenomenon in these early studies. Many new applications of the autoresonanceidea and progress in the theory emerged since 1990 in atomic and molecular physics (see, forinstance, (Meerson and Friedland 1990), (Liu et al. 1995), and (Maeda et al. 2007)), nonlin-ear dynamics (Meerson and Yariv 1991), nonlinear waves (Friedland 1998), plasmas ((Fajanset al. 1999), and (Friedland et al. 2006)), fluid dynamics (Borich and Friedland 2008) and


optics (Barak et al. 2009). For a contemporary survey of the mathematical aspects of au-toresonance the reader is referred to (Kalyakin 2008). In this chapter, it is shown that anO(ε)-amplitude excitation gives rise to O(

√ε) solution responses. The equation of motion

is solved approximately and amplitudes are obtained in terms of Fresnel Sine and Cosinefunctions in the resonance zone. The solution can be simplified to the sum of two terms:one, a very small amplitude oscillation at the linear mode frequency, and two, an oscilla-tion at the excitation frequency. This excitation oscillation starts with the same amplitudeas the linear mode oscillation, but increases in time until an energy exchange between thenatural mode of vibration to the forced vibration has been completed. This occurs approxi-mately at the arbitrary times t = 1

ε ln(

kπΩl0

)and the system reaches the maximum amplitude

of length 1√ε as can be observed in Figure G.1 and Figure G.2. The occurrence of unexpected

timescale of order 1√ε suggests to introduce a three-timescales perturbation method. The

initial-boundary value problem will be studied, and the explicit asymptotic approximationsof the solutions, which are valid on a long timescale, will be constructed as for instance de-scribed in ((Nayfeh 1991), (Kevorkian and Cole 1996), and (Murdock 1991)). There will bea cascade of autoresonances when we sum up all the modal solutions to the initial-boundaryvalue problem and from these solutions it can also be seen that as mode number k increasesthe amplitudes decrease in size. To our knowledge, the explicit construction of asymptoticapproximations of oscillations for these type of problems have not been given before.The outline of the chapter is as follows. In section 4.2, the generalized Hamilton’s princi-ple is used to build the model of the elevator suspension system in which the hoisting ropeand the car are included. In section 4.3, we perform interior layer analysis systematicallyand obtain an unexpected timescale of order 1√

ε . For this reason, in section 4.4, a threetimescales perturbation method is applied to construct formal asymptotic approximations ofthe solutions of the initial value problem. In section 4.5, approximations of the solutions ofthe initial-boundary value problem will be obtained. Finally, in section 4.6, we make someremarks and draw some conclusions.


The vertically translating cable will be modeled as a taut string with harmonic excitationsat x = 0 and fixed at x = l(t) in horizontal direction. The elevator car is modeled as a rigidbody of mass m attached at the lower end x = l(t), and suspension of the car against the guiderails is assumed to be rigid. During its motion the cable of density ρ has a variable lengthl(t) and an axial velocity v(t) = l(t), where the over dot denotes time differentiation. Thecable is assumed to be inextensible with an arbitrarily prescribed translational velocity v(t),where t is time. A positive or negative transport velocity designates extension or retractionof the cable, respectively. The transversal and longitudinal vibrations of elevator cables are


assumed to be uncoupled. In this chapter longitudinal vibrations will not be considered.Relative to the fixed coordinate system as shown in Figure 4.1, the transversal displacementof the cable particle instantaneously located at spatial position x at time t, where 0≤ x≤ l(t),is described by u(x, t).

l (t)

v

m

g

x

u (x,t)

u(0, t) = α sin(Ωt)

Figure 4.1: Vertically translating string with boundary excitations and attached mass.

The equation of motion for a vertically moving string with time-varying length and space-time-varying tension is obtained by using Hamilton’s principle, see for instance (Miranker1960):

ρ(utt +2vuxt + ˙vux + v2uxx)− (P(x, t)ux)x = 0, t > 0, 0 < x < l(t), (4.1)

u(0, t) = αsin(Ωt), u(l(t), t) = 0, t > 0, (4.2)

u(x,0) = f (x), and ut(x,0) = h(x), 0 < x < l(0), (4.3)

where the lettered subscript for u denotes partial differentiation, P(x, t) = mg + ρ(l(t)−x)g−m ˙v(t)− ρ(l(t)− x) ˙v is an axial force arising from its own weight and longitudinalacceleration, α is the excitation amplitude at the upper end, Ω is the excitation frequency, gis the acceleration due to gravity, and where f (x) and h(x) represent the initial displacement


and the initial velocity, respectively. Note that the axial force is tensile and compressiveduring deceleration ( ˙v < 0) and acceleration ( ˙v > 0) of the string, respectively, and vanishesduring its uniform motion ( ˙v = 0). In this chapter, we only consider the linearly lengthvariation in time case, i.e., l(t) = l0 + vt, where l0 is an initial string length, v is a constantvelocity and t is time. To put the equations (4.1)-(4.3) into a non-dimensional form thefollowing dimensionless parameters will be used:u∗ = u

L , x∗ = xL , t∗ = t

L

√mgρ , l∗ = l

L , l∗0 = l0L , v = v

√ρ

mg , µ = ρLm , α∗ = α

L , Ω∗ = L√

ρmg Ω,

f ∗ = fL , h∗ = h

√ρ

mg , where L is the maximum length of the string. The equations of motionin non-dimensional form then become:

utt −uxx =−2vuxt −(v2−µ(l(t)− x)

)uxx−µux, t > 0, 0 < x < l(t), (4.4)

u(0, t) = αsin(Ωt), u(l(t), t) = 0, t > 0, (4.5)

u(x,0) = f (x), ut(x,0) = h(x), 0 < x < l0, (4.6)

where µ , α , Ω and l0 are positive constants, and where the asterisks (indicating the dimen-sionless variables and parameters) are omitted in (4.4) through (4.6) and henceforth. Theproblem (4.4)-(4.6) has an inhomogeneous boundary condition. To solve the problem (4.4)-(4.6), we first convert the problem into a problem with homogeneous boundary conditionsby introducing the following transformation

u(x, t) = u(x, t)+(

1− xl(t)

)αsin(Ωt), (4.7)

where u(x, t) satisfies the homogeneous boundary conditions. For dimensionless parametersv, µ , and α we have made the following assumptions: that longitudinal velocity is smallcompared to nominal wave velocity, that cable mass is small compared to car mass, and thatthere are small oscillation amplitudes at x = 0. For this reason, we can write v = v

√ρ

mg = ε ,µ = εµ0, and, α = εα0, where ε is dimensionless small parameter. It is also assumed thatboth initial conditions are O(ε), that is, f (x) = ε f (x), and h(x) = ε h(x). Thus, by usingthese assumptions in (4.4)-(4.7), we get the equation of motion for u(x, t;ε) as given by

utt − uxx = ε(−2uxt +µ0(l(t)− x)uxx−µ0ux +α0Ω2

(1− x

l(t)

)sin(Ωt)

)

+O(ε2), t > 0, 0 < x < l(t),(4.8)

u(0, t;ε) = u(l(t), t;ε) = 0, t > 0, (4.9)

u(x,0;ε) = ε f (x)+O(ε2), 0 < x < l0, (4.10)

ut(x,0;ε) = ε(

h(x)−α0Ω(

1− xl0

))+O(ε2), 0 < x < l0, (4.11)


where l(t) = l0 + εt. A new independent non-dimensional spatial coordinate ξ = xl(t) is

introduced and the time-varying spatial domain [0, l(t)] for x is converted to a fixed domain[0,1] for ξ . The new dependent variable is u(ξ , t;ε) = u(x, t;ε). The partial derivatives ofu(x, t;ε) with respect to x and t are related to those of u(ξ , t;ε) with respect to ξ and t, see forinstance (Zhu and Chen 2005). Thus, we obtain following equations of motion for u(ξ , t;ε),

utt − 1l2(t) uξ ξ = ε

(2(ξ−1)

l(t) uξ t +µ0(1−ξ )

l(t) uξ ξ − µ0l(t) uξ +α0Ω2(1−ξ )sin(Ωt)

)

+O(ε2), t > 0, 0 < ξ < 1,(4.12)

u(0, t;ε) = u(1, t;ε) = 0, t > 0, (4.13)

u(ξ ,0;ε) = ε f (ξ )+O(ε2), 0 < ξ < 1, (4.14)

ut(ξ ,0;ε) = ε(

h(ξ )−(

α0Ω(1−ξ )− ξl0

f ′(ξ )))

+O(ε2), 0 < ξ < 1, (4.15)

where f (ξ ) = f (x), and h(ξ ) = h(x) for t = 0. To satisfy the boundary conditions (4.13) allfunctions should be expanded in Fourier-sin-series. Therefore, by substituting u(ξ , t;ε) =∑∞

n=1 un(t;ε)sin(nπξ ) in (4.12)-(4.15), we get

∑∞n=1

(un(t;ε)+

(nπl(t)

)2un(t;ε)

)sin(nπξ )

= ε ∑∞n=1

[2nπ(ξ−1)

l(t) un(t;ε)cos(nπξ )

−(nπ(1−ξ )sin(nπξ )+ cos(nπξ )) nπµ0l(t) un(t;ε)+α0(1−ξ )Ω2sin(Ωt)

]

+O(ε2),

(4.16)

∞

∑n=1

un(0;ε)sin(nπξ ) = ε f (ξ )+O(ε2),∞

∑n=1

unt (0;ε)sin(nπξ ) = ε q(ξ )+O(ε2), (4.17)

where q(ξ ) =(

h(ξ )−α0Ω(1−ξ )+ ξl0

f ′(ξ ))

. Now, by multiplying (4.16) and (4.17) with

sin(kπξ ), then by integrating the so-obtained equation from ξ = 0 to ξ = 1, and then byusing the orthogonality properties of the eigenfunctions, we obtain

uk(t;ε)+(

kπl(t)

)2uk(t;ε)

= ε(− 1

l(t) uk(t;ε)− µ0(kπ)2

2l(t) uk(t;ε))+ 2α0Ω2

kπ sin(Ωt)+

∑∞n=1,n 6=k

[4nπl(t) ( f1(n,k)− f2(n,k))un(t;ε)+(nπ f3(n,k)− f2(n,k))

2nπµ0l(t) un(t;ε)

]

+O(ε2),(4.18)

uk(0;ε) = 2εF +O(ε2), uk(0;ε) = 2εQ+O(ε2), (4.19)


where, F =∫ 1

0 f (ξ )sin(kπξ )dξ , Q =∫ 1

0 q(ξ )sin(kπξ )dξ , and where f1, f2, and f3 are givenby

f1(n,k) =∫ 1

0ξ cos(nπξ )sin(kπξ )dξ , f2(n,k) =

∫ 1

0cos(nπξ )sin(kπξ )dξ , (4.20)

f3(n,k) =∫ 1

0ξ sin(nπξ )sin(kπξ )dξ . (4.21)

In the left hand side of (4.18) the coefficient of uk(t;ε) is time-varying, to make this coeffi-cient “constant” we define a new time variable that is a measure of the period of oscillation.The transformation t+ =

∫ t0

dsl(s) is assumed, where t+ is non-negative, smooth and increases

as t increases. By using this transformation in (4.18) and (4.19), we get the equation foruk(t+;ε) as given by

d2ukdt+2 +(kπ)2uk = ε

(− 1

2 l(εt+)µ0(kπ)2uk +2α0Ω2

kπ l2(εt+)sin(

Ωl0ε (eεt+ −1)

)+

∑∞n=1,n6=k

(4nπ( f1(n,k)− f2(n,k)) dun

dt+ +(nπ f3(n,k)− f2(n,k))2nπµ0 l(εt+)un

))

+O(ε2),(4.22)

and the initial conditions as given by

uk(0;ε) = 2εF +O(ε2),duk(0;ε)

dt+= 2εl0Q+O(ε2), (4.23)

where l(εt+) = l0eεt+ . The equations (4.22) and (4.23) will be studied further in section 4.3and in section 4.4.

4.3 Interior layer analysis for the single ordinary differen-tial equation

In this section, a simplified version of (4.22) is considered. To study (4.22) a perturbationapproach can be used. In this approach secular terms play an important role. For that reasonwe now first exclude the nonsecular terms in (4.22), and we consider the following oscillator-type equation with weak forcing

d2ukdt+2 +(kπ)2uk = ε

(− 1

2 l(εt+)µ0(kπ)2uk +2α0Ω2

kπ l2(εt+)sin(

Ωl0ε (eεt+ −1)

))

+O(ε2),(4.24)

When ε = 0, the solution of (4.24) can be written as

uk(t+) = Akcos(kπt+)+Bksin(kπt+), (4.25)

4.3. Interior layer analysis for the single ordinary differential equation 55

where Ak and Bk are constants, which are sometimes referred to as parameters. It followsfrom (4.25) that

u′k(t+) =−(kπ)Aksin(kπt+)+(kπ)Bkcos(kπt+). (4.26)

When ε 6= 0, we assume that the solution of (4.24) is still given by (4.25) but with time-varying Ak and Bk. In other words, we consider (4.25) as a transformation from uk(t+) toAk(t+) and Bk(t+). Using this view, we note that we have introduced two unknown functionsAk(t+) and Bk(t+) for the unknown function uk(t+). Hence, we can impose one condition.Out of all possible conditions, we choose to impose the condition that u′k(t

+) is given by(4.26) also when Ak and Bk are time-varying. Differentiating (4.25) with respect to t+ andrecalling that Ak and Bk are functions of t+, we have

u′k(t+) =−(kπ)Aksin(kπt+)+(kπ)Bkcos(kπt+)+A′kcos(kπt+)+B′ksin(kπt+). (4.27)

Comparing (4.27) with (4.26), we conclude that

A′kcos(kπt+)+B′ksin(kπt+) = 0. (4.28)

Differentiating (4.26) with respect to t+, we obtain

u′′k (t+) =−(kπ)2Akcos(kπt+)− (kπ)2Bksin(kπt+)− (kπ)A′ksin(kπt+)+(kπ)B′kcos(kπt+).

(4.29)Substituting for uk and u′′k from (4.25) and (4.29) into (4.24), we have

A′ksin(kπt+)−B′kcos(kπt+) = 12 ε l(εt+)µ0(kπ)(Akcos(kπt+)+Bksin(kπt+))−2εα0Ω2

(kπ)2 l2(εt+)sin(

Ωl0ε (eεt+ −1)

).

(4.30)We note that (4.28) and (4.30) constitute a system of two first order equations for A′k and B′k.Thus, by solving for A′k and B′k, we obtain

A′k =14 ε l(εt+)µ0(kπ)(Aksin(2kπt+)−Bkcos(2kπt+))+ 1

4 ε l(εt+)µ0(kπ)Bk−2εα0Ω2

(kπ)2 l2(εt+)sin(kπt+)sin(

Ωl0ε (eεt+ −1)

),

(4.31)

B′k = − 14 ε l(εt+)µ0(kπ)(Akcos(2kπt+)+Bksin(2kπt+))− 1

4 ε l(εt+)µ0(kπ)Ak+2εα0Ω2

(kπ)2 l2(εt+)cos(kπt+)sin(

Ωl0ε (eεt+ −1)

).

(4.32)Thus, the original second-order equation (4.24) for uk(t+) has been replaced by the twofirst-order equations (4.31) and (4.32) for Ak and Bk. We emphasize that no approximationshave been made in deriving (4.31) and (4.32). We observe that in (4.31) and (4.32), the firstterms are fast varying terms and average out when an averaging method is applied. Thesecond terms 1

4 ε l(εt+)µ0(kπ)Bk and − 14 ε l(εt+)µ0(kπ)Ak are slowly varying terms and do


not depend on any value of Ω. What happens with the third terms in (4.31) and (4.32)? Toanswer this question, let τ = εt+, φ = kπt+, and ψ = Ωl0

ε (eτ −1). This implies that, τ ′ = ε ,φ ′ = kπ , and ψ ′ = Ωl0eτ . Thus, we get the following system of equations from (4.31) and(4.32):


4 ε l(τ)µ0(kπ)Bk−2εα0Ω2

(kπ)2 l2(τ)sin(φ)sin(ψ),

B′k =− 14 ε l(εt+)µ0(kπ)(Akcos(2kπt+)+Bksin(2kπt+))− 1

4 ε l(τ)µ0(kπ)Ak+2εα0Ω2

(kπ)2 l2(τ)cos(φ)sin(ψ),

τ ′ = ε, τ(0) = 0,φ ′ = kπ, φ(0) = 0,ψ ′ = Ωl0eτ , ψ(0) = 0.

(4.33)

By using trigonometric identities in the first two equations in system (4.33), we get


4 ε l(τ)µ0(kπ)Bk+εα0Ω2

(kπ)2 l2(τ)(cos(φ +ψ)− cos(φ −ψ)) ,


4 ε l(τ)µ0(kπ)Ak+εα0Ω2

(kπ)2 l2(τ)(sin(φ +ψ)− sin(φ −ψ)) ,

τ ′ = ε, τ(0) = 0,φ ′ = kπ, φ(0) = 0,ψ ′ = Ωl0eτ , ψ(0) = 0.

(4.34)

Resonance in system (4.34) can be expected when φ ′= 0, or when ψ ′= 0, or when φ ′+ψ ′=0, or when φ ′−ψ ′ = 0. But since φ ′ > 0, ψ ′ > 0, and φ ′+ψ ′ > 0, resonance only will occurwhen φ ′−ψ ′ = 0⇔ kπ −Ωl0eτ = 0⇔ τ − ln( kπ

Ωl0) = 0. Outside this resonance manifold

we can average over φ and ψ , and we obtain the following set of averaged equations:

A′k =14 ε l(εt+)µ0(kπ)Bk,

B′k =− 14 ε l(εt+)µ0(kπ)Ak.

(4.35)

The solution of system (4.35) is given by

Ak(t+) = rk(t+)cos(qk(t+)), Bk(t+) = rk(t+)sin(qk(t+)), (4.36)

whererk(t+) = rk(0) =

√A2

k(0)+B2k(0), (4.37)

qk(t+) =−14

l(εt+)µ0(kπ)+14

l0µ0(kπ)+ tan−1(

Bk(0)Ak(0)

). (4.38)

To study the situation in the resonance zone we introduce Φ = φ −ψ , and rescale τ asτ − ln

(kπΩl0

)= δ (ε)τ with τ = O(1). Therefore, Φ′ = φ ′ −ψ ′ = kπ −Ωl0eτ = kπ −

4.4. A three timescales perturbation method 57

Ωl0eln(

kπΩl0

)+δ (ε)τ

= kπ(1−eδ (ε)τ) =−kπδ (ε)τ +O(δ 2(ε)). By introducing this rescaling,we can re-write the system (4.34) as follows


4 ε l(τ)µ0(kπ)Bk+εα0Ω2

(kπ)2 l2(τ)(cos(φ +ψ)− cos(Φ)) ,


4 ε l(τ)µ0(kπ)Ak+εα0Ω2

(kπ)2 l2(τ)(sin(φ +ψ)− sin(Φ)) ,

τ ′ = εδ (ε) , τ(0) =− 1

δ (ε) ln(

kπΩl0

),

Φ′ =−kπδ (ε)τ +O(δ 2(ε)), Φ(0) = 0,φ ′ = kπ, φ(0) = 0,ψ ′ = kπeδ (ε)τ , ψ(0) = 0,

(4.39)

where l(τ) = kπΩ eδ (ε)τ . The balance occurs when ε

δ (ε) = δ (ε) and this implies for the aver-aging procedure in the resonance zone that δ (ε) =

√ε . And so, τ ′ =

√ε , Φ′ =−kπ

√ετ +

O(ε), and ψ ′ = kπe√

ετ . Thus, Φ′ = −kπ(

εt+− ln( kπ

Ωl0

))+O(ε), and this implies that

Φ(t+)=− 12 (kπ)εt+2

+O(ε)+constant. Hence in the resonance zone, we can write cos(Φ)=

cos(− 1

2 kπεt+2+O(ε)+ constant

), and sin(Φ) = sin

(− 1

2 kπεt+2+O(ε)+ constant

).

In the next section, the occurrence of this unexpected timescale of O(ε−12 ) can be utilized

to introduce a three timescales perturbation method to study (4.22) in detail and constructasymptotic approximations of the solution of the initial-boundary value problem (4.1)-(4.3).

4.4 A three timescales perturbation method

In the previous section it was shown that resonance occurs around time t+ is 1ε ln(

kπΩl0

),

therefore, we rescale t+ by defining t+ = t + 1ε ln(

kπΩl0

). Thus (4.22) and (4.23) can be

re-written in t as follows

d2ukdt2 +(kπ)2uk

= ε(− 1

2 l(ε t)µ0(kπ)2uk +2α0Ω2

kπ l2(ε t)sin(

kπε eε t − Ωl0

ε

)+

∑∞n=1,n 6=k

(4nπ( f1(n,k)− f2(n,k)) dun

dt +(nπ f3(n,k)− f2(n,k))2nπµ0 l(ε t)un

))

+O(ε2),

(4.40)

uk(a;ε) = 2εF +O(ε2),duk(a;ε)

dt= 2εl0Q+O(ε2), (4.41)


where l(ε t) = kπΩ eε t and a = − 1

ε ln(

kπΩl0

). In this section, we study (4.40) in detail. The

application of the straight-forward expansion method to solve (4.40) will result in the occur-rence of so-called secular terms which cause the approximations of the solutions to becomeunbounded on long timescales. It is shown in the previous section and also in Appendix Gthat the O(ε) excitation at x = 0 produces an unexpected new timescale of order 1√

ε . For this

reason, to remove secular terms, we introduce three timescales t0 = t, t1 =√

ε t, and t2 = ε t.By using such a three timescales perturbation method the function uk(t;

√ε) is supposed to

be a function of t0, t1 and t2. Thus, the introduction of these three timescales defines thefollowing transformations:

uk(t;√

ε) = wk(t0, t1, t2;√

ε), dukdt = ∂wk

∂ t0+√

ε ∂wk∂ t1

+ ε ∂wk∂ t2

,d2ukdt2 = ∂ 2wk

∂ t20+2√

ε ∂ 2wk∂ t0∂ t1

+ ε(

2 ∂ 2wk∂ t0∂ t2

+ ∂ 2wk∂ t2

1

)+2ε√

ε ∂ 2wk∂ t1∂ t2

.(4.42)

By substituting (4.42) into (4.40) and (4.41), we obtain the following equations up to O(ε√

ε):

∂ 2wk∂ t2

0+(kπ)2wk +2

√ε ∂ 2wk

∂ t0∂ t1+ ε(

2 ∂ 2wk∂ t0∂ t2

+ ∂ 2wk∂ t2

1

)+2ε√

ε ∂ 2wk∂ t1∂ t2

= ε(− 1

2 l(t2)µ0(kπ)2wk +2α0Ω2

kπ l2(t2)sin(

kπε eεt0 − Ωl0

ε

)

+∑∞n=1,n6=k

(4nπ( f1(n,k)− f2(n,k))

(∂wn∂ t0

+√

ε ∂wn∂ t1

)

+(nπ f3(n,k)− f2(n,k))2µ0nπ l(t2)wn

)).

(4.43)

wk(a,b,c;√

ε) = 2εF , (4.44)

∂wk

∂ t0(a,b,c;

√ε)+√

ε∂wk

∂ t1(a,b,c;

√ε)+ ε

∂wk

∂ t2

(a,b,c;

√ε) = 2εl0Q, (4.45)

where a, b and c are defined as follows

a =−1ε

ln(

kπΩl0

), b =− 1√

εln(

kπΩl0

), c =−ln

(kπΩl0

). (4.46)

Notice that if kπ = Ωl0, the constants a, b and c will be zero. By using a three timescalesperturbation method wk(t0, t1, t2;

√ε) can be approximated by the formal asymptotic expan-

sion

wk(t0, t1, t2;√

ε)=√

εwk0(t0, t1, t2)+εwk1(t0, t1, t2)+ε√

εwk2(t0, t1, t2)+O(ε2) · · · . (4.47)

It is reasonable to assume this solution form because the function wk(t0, t1, t2;√

ε) analyt-ically depends on

√ε and the powers of it and because we are interested to construct an

approximation of the solution of (4.4)-(4.6), when the initial conditions and the boundaryexcitation are of O(ε) (see also the previous section). By substituting (4.47) into (4.43)-(4.45), and after equating the coefficients of like powers of

√ε , it follows from the problem


for wk(t0, t1, t2;√

ε) that the O(√

ε)-problem is

∂ 2wk0

∂ t20

+(kπ)2wk0 = 0, (4.48)

wk0(a,b,c) = 0,∂wk0

∂ t0(a,b,c) = 0, (4.49)

and that the O(ε)-problem is

∂ 2wk1

∂ t20

+(kπ)2wk1 =−2∂ 2wk0

∂ t0∂ t1+

2α0Ω2

kπl2(t2)sin

(kπε

eεt0 − Ωl0ε

), (4.50)

wk1(a,b,c) = 2F ,∂wk1

∂ t0(a,b,c) =−∂wk0

∂ t1(a,b,c)+2l0Q, (4.51)

and that the O(ε√

ε)-problem is

∂ 2wk2∂ t2

0+(kπ)2wk2 =−2 ∂ 2wk1

∂ t0∂ t1−2 ∂ 2wk0

∂ t0∂ t2− ∂ 2wk0

∂ t21− 1

2 µ0(kπ)2 l(t2)wk0+

∑∞n=1,n 6=k

(4nπ( f1(n,k)− f2(n,k))

∂wn0∂ t0

+(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)wn0

),

(4.52)

wk2(a,b,c) = 0,∂wk2

∂ t0(a,b,c) =−∂wk0

∂ t2(a,b,c)− ∂wk1

∂ t1(a,b,c). (4.53)

The O(√

ε)-equation has as solution

wk0(t0, t1, t2) = Ak0(t1, t2)cos(kπt0)+Bk0(t1, t2)sin(kπt0), (4.54)

where Ak0, and Bk0 are still unknown functions of the slow variables t1 and t2, and they can beobtained by avoiding secular terms from the O(ε)- and the O(ε

√ε)-problems, respectively.

By using the initial conditions (4.49), it can be observed that Ak0(b,c) = Bk0(b,c) = 0. Nowwe shall solve the O(ε)-problem outside and inside the resonance zone. By using (4.54), theO(ε)-equation outside and inside the resonance manifold can be written as

∂ 2wk1∂ t2

0+(kπ)2wk1 = 2kπ ∂Ak0

∂ t1sin(kπt0)−2kπ ∂Bk0

∂ t1cos(kπt0)+

2α0Ω2

kπ l2(t2)sin(


ε

).

(4.55)

Observe that outside the resonance zone the last term in (4.55) does not give rise to secularterms in wk1. In order to remove secular terms outside this resonance zone, it follows from(4.55) that Ak0 and Bk0 have to satisfy

∂Ak0

∂ t1= 0,

∂Bk0

∂ t1= 0, (4.56)


where (4.56) has as solutions

Ak0(t1, t2) = Ck0(t2), Bk0(t1, t2) = Dk0(t2), (4.57)

and where Ck0 and Dk0 are unknown functions of the slower variable t2 and can be obtainedby removing secular terms from the O(ε

√ε)-equation. Since Ak0(b,c) = Bk0(b,c) = 0,

this implies that, Ck0(c) = Dk0(c) = 0. Now we consider the O(ε)-equation inside theresonance zone and observe that inside the resonance zone the last term in (4.55) givesrise to secular terms in wk1. We can write sin

(kπε eεt0 − Ωl0

ε

)as sin

(kπε e√

ε√

εt0 − Ωl0ε

)=

sin(

kπε e√

εt1 − Ωl0ε

). By expanding e

√εt1 around zero and neglecting higher order terms, we

get sin(


ε

)= sin

(kπt0 + 1

2 kπt21 +φ

), where φ = kπ

ε −Ωl0ε is a phase. Note that

the phase φ explicitly depends on the mode number k and ε . And when kπ = Ωl0 the phaseφ = 0. Now, by using the trigonometric identity sin(a+ b) = sin(a)cos(b)+ cos(a)sin(b),and re-arranging terms, we can re-write (4.55) inside the resonance zone as

∂ 2wk1∂ t2

0+(kπ)2wk1 =

(2kπ ∂Ak0

∂ t1+ 2α0Ω2

kπ l2(t2)cos( 12 kπt2

1 +φ))

sin(kπt0)+(−2kπ ∂Bk0

∂ t1+ 2α0Ω2

kπ l2(t2)sin( 12 kπt2

1 +φ))

cos(kπt0).(4.58)

The right hand side of (4.58) contains terms which are the solutions of the homogeneous partcorresponding to (4.58). These terms will give rise to unbounded terms, the so-called secularterms, in the solution wk1 of (4.58). In order to remove secular terms, it now easily followsfrom (4.58) that Ak0 and Bk0 have to satisfy

∂Ak0

∂ t1+

α0Ω2

(kπ)2 l2(t2)cos(1

2kπt2

1 +φ)= 0, (4.59)

∂Bk0

∂ t1− α0Ω2

(kπ)2 l2(t2)sin(1

2kπt2

1 +φ)= 0. (4.60)


Ak0(t1, t2) =α0Ω2√

k(kπ)2l2(t2)

(sin(φ)FresnelS(

√kt1)− cos(φ)FresnelC(

√kt1))+Ck0(t2),

(4.61)

Bk0(t1, t2) =α0Ω2√

k(kπ)2l2(t2)

(cos(φ)FresnelS(

√kt1)+ sin(φ)FresnelC(

√kt1))+Dk0(t2),

(4.62)where Ck0 and Dk0 are unknown functions of t2 and to be determined from the O(ε

√ε)-

problem. Thus, outside and inside the resonance zone, the homogeneous solution wk1 isgiven by

wk1(t0, t1, t2) = Ak1(t1, t2)cos(kπt0)+Bk1(t1, t2)sin(kπt0), (4.63)


where Ak1 and Bk1 are unknown functions of t1 and t2, and can be determined from higherorder problems. By using the initial conditions (4.51), the values of Ak1(b,c) and Bk1(b,c)are given by the following equations,

kπAk1(b,c) = 2kπFcos(kπa)−2l0Qsin(kπa)+ ∂Ak0(b,c)∂ t1

cos(kπa)sin(kπa)+∂Bk0(b,c)

∂ t1sin2(kπa),

(4.64)

kπBk1(b,c) = 2kπFsin(kπa)+2l0Qcos(kπa)− ∂Bk0(b,c)∂ t1

cos(kπa)sin(kπa)−∂Ak0(b,c)

∂ t1cos2(kπa),

(4.65)

where, ∂Ak0(b,c)∂ t1

and ∂Bk0(b,c)∂ t1

are zero outside the resonance manifold. These values are non-zero inside the resonance manifold, and are given by

∂Ak0(b,c)∂ t1

=−α0Ω2

(kπ)2 l20cos

(12

kπb2 +φ), (4.66)

∂Bk0(b,c)∂ t1

=α0Ω2

(kπ)2 l20sin

(12

kπb2 +φ). (4.67)

By using (4.54) and (4.63) with (4.57), the O(ε√

ε)-equation outside the resonance zone canbe written as follows

∂ 2wk2∂ t2

0+(kπ)2wk2 =

(2kπ ∂Ak1

∂ t1+2kπ dCk0

dt2− 1

2 (kπ)2 l(t2)µ0Dk0(t2))

sin(kπt0)−(2kπ ∂Bk1

∂ t1+2kπ dDk0

dt2+ 1

2 (kπ)2 l(t2)µ0Ck0(t2))

cos(kπt0)+

∑∞n=1,n6=k

(−4(nπ)2( f1(n,k)− f2(n,k))Cn0(t2)+

(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)Dn0(t2))

sin(nπt0)+(4(nπ)2( f1(n,k)− f2(n,k))Dn0(t2)+

(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Cn0(t2)))

cos(nπt0),

(4.68)

To avoid secular terms in the solution wk2 in (4.68), the following conditions have to beimposed

∂Ak1

∂ t1+

dCk0

dt2− 1

4(kπ)l(t2)µ0Dk0(t2) = 0, (4.69)

∂Bk1

∂ t1+

dDk0

dt2+

14(kπ)l(t2)µ0Ck0(t2) = 0. (4.70)

If we solve (4.69) and (4.70) for Ak1 and Bk1 and integrate (4.69) and (4.70) w.r.t. t1, weobserve that the solutions will be unbounded in t1 due to terms which are only dependingon t2. Therefore, to have secular free solutions, the following conditions have to be imposedindependently

∂Ak1

∂ t1= 0,

∂Bk1

∂ t1= 0, (4.71)


dCk0

dt2− 1

4(kπ)l(t2)µ0Dk0(t2) = 0,

dDk0

dt2+

14(kπ)l(t2)µ0Ck0(t2) = 0. (4.72)


Ak1(t1, t2) = Ck1(t2), Bk1(t1, t2) = Dk1(t2), (4.73)

Ck0(t2) = rk0(t2)cos(ψk0(t2)), Dk0(t2) = rk0(t2)sin(ψk0(t2)), (4.74)

where Ck1 and Dk1 are undetermined functions of t2 and can be obtained from the O(ε2)-

problem, and where rk0(t2) = rk0(c) =√

C2k0(c)+ D2

k0(c) and ψk0(t2) = − 14 kπµ0 l(t2) +

tan−1(

Dk0(c)Ck0(c)

). Since Ck0(c) = Dk0(c) = 0, this implies that, (4.74) yields zero solutions

outside the resonance zone. Now we solve the O(ε√

ε)-equation inside the resonance zone.By using (4.54) with (4.61) and (4.62), (4.63) and then by combining all the terms, we get

∂ 2wk2∂ t2

0+(kπ)2wk2

=(

2kπ ∂Ak1∂ t1

+2kπ dCk0dt2

+4kπ l2(t2)Fk(t1)− l2(t2)d2Gkdt2

1−

12 (kπ)2 l(t2)µ0Dk0(t2)− 1

2 (kπ)2 l2(t2)µ0Gk(t1))

sin(kπt0)−(2kπ ∂Bk1

∂ t1+2kπ dDk0

dt2+4kπ l2(t2)Gk(t1)+ l2(t2)

d2Fkdt2

1+

12 (kπ)2 l(t2)µ0Ck0(t2)+ 1

2 (kπ)2 l2(t2)µ0Fk(t1))

cos(kπt0)+

∑∞n=1,n6=k

(−4(nπ)2( f1(n,k)− f2(n,k))(Fn(t1)+Cn0(t2))+

(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Gn(t1)+Dn0(t2)))

sin(nπt0)+(4(nπ)2( f1(n,k)− f2(n,k))(Gn(t1)+Dn0(t2))+

(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Fn(t1)+Cn0(t2)))

cos(nπt0),

(4.75)

where Fk(t1) and Gk(t1) are the combination of Fresnel Sine and Cosine functions and aregiven by the following relations

Fk(t1) =α0Ω2√

k(kπ)2

(sin(φ)FresnelS(

√kt1)− cos(φ)FresnelC(

√kt1)), (4.76)

Gk(t1) =α0Ω2√

k(kπ)2

(cos(φ)FresnelS(

√kt1)+ sin(φ)FresnelC(

√kt1)). (4.77)

To avoid secular terms in the solution wk2 in (4.75), the following conditions have to beimposed

∂Ak1∂ t1

+2Fk(t1)l2(t2)− 12kπ l2(t2)

d2Gkdt2

1− 1

4 (kπ)l2(t2)µ0Gk(t1)+dCk0dt2− 1

4 (kπ)l(t2)µ0Dk0(t2) = 0,(4.78)


∂Bk1∂ t1

+2Gk(t1)l2(t2)+ 12kπ l2(t2)

d2Fkdt2

1+ 1

4 (kπ)l2(t2)µ0Fk(t1)+dDk0dt2

+ 14 (kπ)l(t2)µ0Ck0(t2) = 0.

(4.79)

If we solve the (4.78) and (4.79) for Ak1 and Bk1 and integrate (4.78) and (4.79) w.r.t. t1, weobserve that the solutions will be unbounded in t1 due to presence of terms which are onlydepending on t2, and since the integral of Fk(t1) = t1Fk(t1)+bounded terms and the integralof Gk(t1) = t1Gk(t1)+ bounded terms. Therefore to have secular free solutions for Ak1 andBk1, the following conditions have to be imposed independently

∂Ak1

∂ t1− 1

2kπl2(t2)

d2Gk(t1)dt2

1= 0, (4.80)

∂Bk1

∂ t1+

12kπ

l2(t2)d2Fk(t1)

dt21

= 0, (4.81)

dCk0

dt2− 1

4(kπ)µ0 l(t2)Dk0(t2)+2l2(t2)Fk(t1)−

14

kπµ0 l2(t2)Gk(t1) = 0, (4.82)

dDk0

dt2+

14(kπ)µ0 l(t2)Ck0(t2)+2l2(t2)Gk(t1)+

14

kπµ0 l2(t2)Fk(t1) = 0. (4.83)


Ak1(t1, t2) =α0Ω2

2(kπ)3 l2(t2)sin(1

2kπt2

1 +φ)+Ck1(t2), (4.84)

Bk1(t1, t2) =α0Ω2

2(kπ)3 l2(t2)cos(1

2kπt2

1 +φ)+Dk1(t2), (4.85)

where Ck1 and Dk1 are arbitrary functions of t2 and can be determined by removing secularterms from the O(ε2)-equation. And, Ck1(c) and Dk1(c) are given by

Ck1(c) = Ak1(b,c)−α0Ω2

2(kπ)3 l20sin

(12

kπb2 +φ), (4.86)

Dk1(c) = Bk1(b,c)−α0Ω2

2(kπ)3 l20cos

(12

kπb2 +φ), (4.87)

where Ak1(b,c) and Bk1(b,c) are given by (4.64) and (4.65). Now, the solution of the (4.82)and (4.83) are given by

Ck0(t2) = c1cos(1

4kπµ0 l(t2)

)+ c2sin

(14

kπµ0 l(t2))+ l2(t2)Fk(t1), (4.88)

Dk0(t2) =−c1sin(1

4kπµ0 l(t2)

)+ c2cos

(14

kπµ0 l(t2))+ l2(t2)Gk(t1), (4.89)


where c1 and c2 are constants of integration and can be determined by using (4.49), (4.54),(4.61) and (4.62). Thus, the values of c1 and c2 are given by

c1 = 2l20Gk(b)sin

(14

kπµ0l0

)−2l2

0 Fk(b)cos(

14

kπµ0l0

), (4.90)

c2 =−2l20 Fk(b)sin

(14

kπµ0l0

)−2l2

0Gk(b)cos(

14

kπµ0l0

). (4.91)

Now, from (4.68) with (4.69) and (4.70), we obtain the following non-homogeneous equation

∂ 2wk2

∂ t20

+(kπ)2wk2 =∞

∑n=1,n6=k

(Θ1nk(t2)sin(nπt0)+Θ2nk(t2)cos(nπt0)

), (4.92)

where Θ1nk(t2) and Θ2nk(t2) are given by

Θ1nk(t2) =−4(nπ)2( f1(n,k)− f2(n,k))Cn0(t2)+(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)Dn0(t2),(4.93)

Θ2nk(t2) = 4(nπ)2( f1(n,k)− f2(n,k))Dn0(t2)+(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Cn0(t2)).(4.94)

Thus, the solution of (4.92) is given by

wk2(t0, t1, t2) = Ak2(t1, t2)cos(kπt0)+Bk2(t1, t2)sin(kπt0)+

∑∞n=1,n 6=k

1(kπ)2−(nπ)2

(Θ1nk(t2)sin(nπt0)+Θ2nk(t2)cos(nπt0)

),

(4.95)

where Ak2(t1, t2) and Bk2(t1, t2) are still arbitrary functions which can be used to avoid secularterms in the solution wk3(t0, t1, t2), and where Θ1nk(t2) and Θ2nk(t2) are given by (4.93) and(4.94), respectively. From (4.75) with (4.78) and (4.79), we get

∂ 2wk2

∂ t20

+(kπ)2wk2 =∞

∑n=1,n6=k

(Γ1nk(t1, t2)sin(nπt0)+Γ2nk(t1, t2)cos(nπt0)

), (4.96)

where, Γ1nk(t1, t2) and Γ2nk(t1, t2) are given by

Γ1nk(t1, t2) =−4(nπ)2( f1(n,k)− f2(n,k))(Fn(t1)+Cn0(t2))+(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Gn(t1)+Dn0(t2)),

(4.97)

Γ2nk(t1, t2) = 4(nπ)2( f1(n,k)− f2(n,k))(Gn(t1)+Dn0(t2))+(nπ f3(n,k)− f2(n,k))2nπµ0 l(t2)(Fn(t1)+Cn0(t2)).

(4.98)

And the solution of (4.96) is given by

wk2(t0, t1, t2) = Ak2(t1, t2)cos(kπt0)+Bk2(t1, t2)sin(kπt0)+

∑∞n=1,n 6=k

1(kπ)2−(nπ)2

(Γ1nk(t1, t2)sin(nπt0)+Γ2nk(t1, t2)cos(nπt0)

),

(4.99)


where Ak2(t1, t2) and Bk2(t1, t2) are still arbitrary functions which can be used to avoid secularterms in the solution wk3(t0, t1, t2), and where Γ1nk(t1, t2) and Γ2nk(t1, t2) are given by (4.97)and (4.98), respectively. By using the initial conditions (4.53), we get Ak2(b,c) and Bk2(b,c)as follows

Ak2(b,c) =(− c1µ0l0

4 sin( 1

4 kπµ0l0)+ c2µ0l0

4 cos( 1

4 kπµ0l0)+ 4

kπ l20 Fk(b)+

α0bΩ2

2(kπ)3 l20cos

( 12 kπb2 +φ

))cos(kπa)sin(kπa)+(

− c1µ0l04 cos

( 14 kπµ0l0

)− c2µ0l0

4 sin( 1

4 kπµ0l0)+ 4

kπ l20Gk(b)−

α0bΩ2

2(kπ)3 l20sin

( 12 kπb2 +φ

))sin2(kπa)−

∑∞n=1,n6=k

1(kπ)2−(nπ)2×(

Γ1nk(b,c)(sin(nπa)cos(kπa)− n

k cos(nπa)sin(kπa)+

Γ2nk(b,c)(cos(nπa)cos(kπa)+ n

k sin(nπa)sin(kπa)))

,

(4.100)

Bk2(b,c) =(

c1µ0l04 sin

( 14 kπµ0l0

)− c2µ0l0

4 cos( 1

4 kπµ0l0)− 4

kπ l20 Fk(b)−

α0bΩ2

2(kπ)3 l20cos

( 12 kπb2 +φ

))cos2(kπa)+(

c1µ0l04 cos

( 14 kπµ0l0

)+ c2µ0l0

4 sin( 1

4 kπµ0l0)− 4

kπ l20Gk(b)+

α0bΩ2

2(kπ)3 l20sin

( 12 kπb2 +φ

))cos(kπa)sin(kπa)−

∑∞n=1,n 6=k

1(kπ)2−(nπ)2×(

Γ1nk(b,c)(sin(nπa)sin(kπa)+ n

k cos(nπa)cos(kπa)+

Γ2nk(b,c)(cos(nπa)sin(kπa)− n

k sin(nπa)cos(kπa)))

,

(4.101)

where c1 and c2 are given by (4.90) and (4.91), Fk(b) and Gk(b) can be obtained by (4.76) and(4.77), and where Γ1nk(b,c) and Γ2nk(b,c) can be obtained by (4.97) and (4.98), respectively.Thus, the O(

√ε)-problem is completely determined by (4.54), (4.61), (4.62), (4.88) and

(4.89). The O(ε)-problem is determined by (4.63), (4.84) and (4.85). Note that in (4.84) and(4.85), Ck1 and Dk1 are undetermined functions of t2 and in (4.96) Ak2(t1, t2) and Bk2(t1, t2)are yet also undetermined functions. All these unknown functions can be determined fromthe O(ε2)-problem. At this moment, we are not interested in the higher order approxima-tions. For this reason, we will take Ck1(t2) =Ck1(c), Dk1(t2) = Dk1(c), Ak2(t1, t2) = Ak2(b,c)and Bk2(t1, tt2) = Bk2(b,c), where Ck1(c) and Dk1(c) are given by (4.86) and (4.87), andwhere Ak2(b,c) and Bk2(b,c) are given by (4.100) and (4.101), respectively.


4.5 Approximations of the solutions of the initial-boundaryvalue problem

In the previous section approximations of the solutions of the initial value problem have beenconstructed by using a three timescales perturbation method. In this section we shall con-struct approximations of the solutions of the initial-boundary value problem (4.12)-(4.15).In the previous section, we have constructed a formal approximation wk(t0, t1, t2;

√ε) =√

εwk0(t0, t1, t2)+εwk1(t0, t1, t2)+ε√

εwk2(t0, t1, t2) for uk(t;√

ε), where wk0(t0, t1, t2) is givenby (4.54), (4.61), (4.62), (4.88)-(4.91), wk1(t0, t1, t2) is given by (4.63), (4.84) and (4.85), andwhere wk2(t0, t1, t2) is given by (4.99) with (4.97) and (4.98). In the original t-coordinate thesolution wk0(t0, t1, t2) can be written in following form

uk0(t) = wk0(t0, t1, t2)

= Ak0(t)cos(

kπ∫ t

0ds

l(s) − 1ε ln(

kπΩl0

))+Bk0(t)sin

(kπ∫ t

0ds

l(s) − 1ε ln(

kπΩl0

))

= Ak0(t)cos(

kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))+Bk0(t)sin

(kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

)),

(4.102)where Ak0(t) and Bk0(t) are given by

Ak0(t) =2α0Ω2√

k(kπ)2 (l0 + εt)2[sin(φ)FresnelS

(√kε ln(

Ω(l0+εt)kπ

))−

cos(φ)FresnelC(√

kε ln(

Ω(l0+εt)kπ

))]

+c1cos( 1

4 kπµ0(l0 + εt))+ c2sin

( 14 kπµ0(l0 + εt)

),

(4.103)

Bk0(t) =2α0Ω2√

k(kπ)2 (l0 + εt)2[cos(φ)FresnelS

(√kε ln(

Ω(l0+εt)kπ

))+

sin(φ)FresnelC(√

kε ln(

Ω(l0+εt)kπ

))]

−c1sin( 1

4 kπµ0(l0 + εt))+ c2cos

( 14 kπµ0(l0 + εt)

),

(4.104)

where c1 and c2 are given by (4.88) and (4.89). And the solution wk1(t0, t1, t2) can be writtenas

uk1(t) = wk1(t0, t1, t2)

= Ak1(t)cos(

kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))+Bk1(t)sin

(kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

)),

(4.105)where Ak1(t) and Bk1(t) are given by

Ak1(t) =α0Ω2

2(kπ)3 (l0 + εt)2sin(

12

kπεt2 +φ)+Ck1

(ln(

Ωl0kπ

)), (4.106)

Bk1(t) =α0Ω2

2(kπ)3 (l0 + εt)2cos(

12

kπεt2 +φ)+Dk1

(ln(

Ωl0kπ

)), (4.107)

4.6. Conclusions 67

where Ck1 and Dk1 can easily be obtained from (4.86) and (4.87) by using (4.64)-(4.67). Andfinally, the solution wk2(t0, t1, t2) is given in t-coordinate as follows

uk2(t) = wk2(t0, t1, t2)

= Ak2(t)cos(

kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))+Bk2(t)sin

(kπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))+

∑∞n=1,n6=k

1(kπ)2−(nπ)2

(Γ1nk(t)sin

(nπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))+

Γ2nk(t)cos(

nπε ln

(l0+εt

l0

)− 1

ε ln(

kπΩl0

))),

(4.108)where Ak2(t) and Bk2(t) can be obtained from the higher order approximations and at thismoment we are not interested to construct higher order approximations therefore we assumethese functions to be constants and these constant values are given by (4.100) and (4.101),respectively, and where Γ1nk(t) and Γ2nk(t) can easily be obtained from (4.97) and (4.98) byusing the values of slow scales t1 and t2. Thus, the solution u(ξ , t;ε) is given by

u(ξ , t;ε) =∞

∑n=1

un(t;ε)sin(nπξ ) =∞

∑n=1

(√

εun0(t)+εun1(t)+ε√

εun2(t))sin(nπξ ). (4.109)

Finally, in x and t coordinates, the solution of initial-boundary values problem (4.8)-(4.11)is given by

u(x, t;ε) =∞

∑n=1

un(t;ε)sin(nπξ ) =∞

∑n=1

(√εun0(t)+ εun1(t)+ ε

√εun2(t)

)sin(

nπxl0 + εt

).

(4.110)

4.6 Conclusions

In this chapter an initial boundary value problem for a linear axially moving string equationwith time-varying length, space-time-varying tension and a constant velocity has been stud-ied. The upper end of the string is excited sinusoidally, whereas the lower end of the string isassumed to be fixed. The problem can be used as a model for the transversal vibrations of anelevator cable system. By performing interior layer analysis systematically, we have shownthat there exists a new timescale of order 1√

ε . For this reason, formal asymptotic approxi-mations of the exact solution have been constructed by using a three timescales perturbationmethod. It has also been shown that for the O(ε)-boundary excitations there are in factO(√

ε)-solution responses. A set of new problems have arisen in studying the elevator cablesystem under weak boundary excitations which is related to the investigation of a physicalphenomenon known as autoresonance. This interesting phenomenon occurs when there is apassage through (dynamic) resonance. In the neighbourhood of times t = 1

ε ln(

kπΩl0

)there


is an interior layer in which the approximate solutions have been found from the differen-tial equation. These solutions correspond to resonance capture. Apart from these solutions,there exist solutions with constant amplitudes outside the interior layer. When Ωl0 ≤ kπfor k ∈ Z+, there occurs a large but finite linear jump of order

√ε which is bigger than the

original excitation of O(ε). It can also be observed that the smaller the values of ε the largerthe jumps. When Ωl0 = kπ for any choice of k, autoresonance will occur at arbitrary timet = 0 and then system will stabilize after times of order 1√

ε . When kπΩl0

= 1+O(ε) then theresonance zones will occur on timescales of O(1). Note that autoresonance will not occur foroscillation mode k when Ωl0 > kπ , this is because of absence of passage through resonance.From mechanical point of view, it is interesting to observe that when increasing length fitsthe eigenfrequency, and that excitation frequency matches the eigenfrequency the systemgets into resonance. After sometime the system gets out of resonance until again lengthfits the eigenfrequency and same phenomenon of resonance repeats. This phenomena ofautoresonances have been described analytically by asymptotic approximations as expectedmechanically. It has been shown that as k increases the amplitudes decrease and solutionsbecome smaller and smaller with larger k. Since there are infinite k modes and asymptoticapproximations consist of infinite modal solutions, therefore, there is a cascade of autores-onances to the solution of the initial-boundary value problem. As results in this chapterdemonstrate, mathematical models of autoresonance can be effectively investigated analyt-ically. On the one hand, we would like to attract the attention of applied mathematicians,engineers, and physicists to this class of problems in the theory of differential equations withvariable-coefficients under weak forcing. And on the other hand, we would like to point outthe possibility of effective analysis of such linear (nonlinear) differential equations to appliedspecialists dealing with oscillatory systems. As a final comment, it is worth to mention thatfor O(

√ε)-boundary or external excitations there will be O(1)-solution responses.

Appendix AReal and positive eigenvalues

Suppose that λ is a complex eigenvalue and φ(x) is the corresponding eigenfunction (also allowed tobe complex since the differential equation defining the eigenfunction would be complex):

L(φ)− λµ

φ = 0, (A.1)

where the operator L is defined as L = d4

dx4 −(

1µ

)d2

dx2 . We introduce the notation ‘¯’ for the complexconjugate. So, the complex conjugate of (A.1) is:

L(φ)− λµ

φ = 0, (A.2)

assuming that the coefficient µ is real; hence, µ = µ . The complex conjugate of L(φ) is exactly Loperating on the complex conjugate of φ , L(φ) = L(φ). Thus,

L(φ)− λµ

φ = 0. (A.3)

Multiplying (A.1) by φ and (A.3) by φ , and then by subtracting the so-obtained equations, yielding

φL(φ)−φL(φ)+(λ −λ )

µφφ = 0. (A.4)

Now integrating (A.4) from x = 0 to x = 1, we obtain (by using the boundary conditions (2.40))

(λ −λ )µ

∫ 1

0φφdx = 0. (A.5)

Since φφ = |φ |2 > 0, the integral in (A.5) is > 0. Observe that the integral can only be equal to zero ifφ ≡ 0, which is prohibited since φ is an eigenfunction. Thus, (A.5) implies that λ = λ , and hence, λis real. The eigenfunctions can always be chosen to be real.Often in physical problems, the sign of λ is quite important. In vibration problems, only positive λcorresponds to the usually expected oscillations. Thus, we expect that λ > 0. The Rayleigh quotientcan be derived from the differential equation,

φ (4)(x)− 1µ

φ ′′(x)− λµ

φ(x) = 0, (A.6)

by multiplying (A.6) by φ , and then by integrating the so-obtained equation from x = 0 to x = 1,yielding ∫ 1

0φ(

φ (4)− 1µ

φ ′′)

dx− λµ

∫ 1

0φ 2dx = 0. (A.7)

70 A. Real and positive eigenvalues

Since∫ 1

0 φ 2dx > 0 and µ is positive, we can solve for λ :

λ =

∫ 10 φ(µφ (4)−φ ′′)dx

∫ 10 φ 2dx

. (A.8)

Integration by parts yields an expression involving the function φ evaluated at the boundary:

λ =kφ 2(1)+

∫ 10 (µφ ′′2 +φ ′2)dx∫ 1

0 φ 2dx, (A.9)

is known as the Rayleigh quotient. Since k and µ are positive constants, it is obvious that λ ≥ 0. Thus,λ can only be equal to zero when φ(1) ≡ 0, φ ′′(x) ≡ 0, and φ ′(x) ≡ 0, implying that φ(x) ≡ 0. Sinceφ(x) ≡ 0 is not an eigenfunction, it now follows that λ > 0. And so, all eigenvalues λ are real andpositive.

Appendix BLarge eigenvalues and the damping parameter

The equation (2.45) is a transcendental equation. It can not be solved exactly. However, a graphicaltechnique can be used to obtain information about the eigenvalues. In order to graph the solution of atranscendental equation, an artificial coordinate ξ is introduced. Thus, we write

ξ = tan(β ), (B.1)

and also,

ξ =

(β 5µ +2β 3 + β

µ

)tanh

(√β 2 + 1

µ

)

(β 4µ

√β 2 + 1

µ − k(2β 2 + 1µ )tanh

(√β 2 + 1

µ

)) , (B.2)

where α has been replaced by√

β 2 + 1µ in (2.45), and where k and µ are positive constants. Now the

0

1

ξ

β4π3π2ππ 5π

Figure B.1: Graphical determination of the eigenvalues when µ = 1 and k = 1.

solutions of (B.1) and (B.2) (i.e., their points of intersection) correspond to the solutions of (2.45). Itcan be seen that (B.1) is a tangent function in β . We know that the tangent function is periodic withperiod π; it is zero at β = 0,π,2π , etc.; and it approaches ±∞ as β approaches π

2 , 3π2 , 5π

2 , and so on.The intersection of the two curves is sketched in Figure B.1 for µ = 1 and k = 1. There is an infinitenumber of intersection points; each corresponds to a positive eigenvalue. We exclude β = 0 since wehave assumed throughout that λ > 0 (since λ = µβ 4 +β 2). For β → ∞ on the right hand side of (B.2)has an asymptote 1. Therefore, we are able to obtain the following approximate (asymptotic) formulafor βn as solution of (2.45):

βn ∼π4+(n−1)π (B.3)

as n→∞. Thus, for large n we can write

λn ∼ µ(π

4+(n−1)π

)4+(π

4+(n−1)π

)2. (B.4)

72 B. Large eigenvalues and the damping parameter

Now, it can be shown analytically that the quantity Θnnζn→2 as n→∞. From (2.70) and (2.52) with

(2.46), Θnn and ζn can be expressed as

Θnn =12

(2β 2

n + 1µ

β 2n + 1

µ

)2

(sin(βn))2 , (B.5)

ζn =sin2(βn)coth(

√β 2

n +1µ )

2βn(1+ 1β2n µ

)52

− sin2(βn)

2(1+ 1β2n µ

)2sinh2(√

β 2n +

1µ )− sin(βn)cos(βn)

βn

+2sin2(βn)coth(

√β 2

n +1µ )

βn

√1+ 1

β2n µ(2+ 1

β2n µ)− 2sin(βn)cos(βn)

βn(1+ 1β2n µ

)(2+ 1β2n µ

)sinh(√

β 2n +

1µ )

+ 12 ,

(B.6)

respectively. The expression on the right hand side of (B.5) tends to 1 and the expression on the righthand side of (B.6) tends to 1

2 , as n→ ∞. Thus, the damping parameter Γn for n→ ∞ satisfies

Γn→−2(V +δ )ε. (B.7)

Appendix CThe WKBJ (Liouville-Green)-approximation

If we consider the equations (3.14)-(3.15) and assume that v and µ are of the O(ε), we obtain up to theO(ε) for t = O(ε−1) that

∂ 2u(ξ , t)∂ t2 − 1

l2(t)∂ 2u(ξ , t)

∂ξ 2 = 0, (C.1)

u(0, t) = u(1, t) = 0. (C.2)

By separation of variables,u(ξ , t) = φ(ξ )g(t), (C.3)

we find the eigenfunctions φn(ξ ) (given by (3.33)) corresponding to eigenvalues n2π2 with n ∈ Z+.From (C.1) it follows that the time-dependent equation is given by

d2gn(t)dt2 +

n2π2

l2(t)gn(t) = 0. (C.4)

Now, if l(t) = l0 + εt and t = εt, then the transformation gn(t) = gn(t) yields

gn(t) = ε g′n(t), and, gn(t) = ε2g′′n(t), (C.5)

where the overdot denotes differentiation with respect to t, and the prime with respect to the argumentt. Then, (C.4) becomes

ε2g′′n(t)+n2π2

l2(t)gn(t) = 0, (C.6)

where l(t) = l0 + t. Since n2π2

l2(t)> 0, we expect that (C.6) has oscillatory solutions. Now for (C.6), we

assume that the solution can be written in the form of an asymptotic series expansion

gn(t)∼ exp

(1δ

∞

∑n=0

δ nSn(t)

). (C.7)

Substituting (C.7) into (C.6), yields

ε2

1

δ 2

(∞

∑n=0

δ nS′n(t)

)2

+1δ

∞

∑n=0

δ nS′′n(t)

=−n2π2

l2(t). (C.8)

To leading order equation (C.8) can be approximated by

ε2

δ 2 S′20 +2ε2

δS′0S′1 +

ε2

δS′′0 =−n2π2

l2(t). (C.9)

74 C. The WKBJ (Liouville-Green)-approximation

In the limit δ → 0, the dominant balance is given by

ε2

δ 2 S′20 ∼−n2π2

l2(t). (C.10)

So, δ should be proportional to ε . Setting δ equal to ε and comparing powers, we get as O(ε0)-problem:

S′20 (t) =−n2π2

l2(t), (C.11)

which has as solution

S0(t) =±inπ∫ t

0

dsl(s)

+ k1, (C.12)

where s is a dummy variable, and k1 an arbitrary constant of integration. Looking at the first-orderpowers of ε in (C.9) we get as O(ε1)-problem:

2S′0(t)S′1(t)+S′′0(t) = 0, (C.13)

which has as solutionS1(t) =−

12

ln |S′0(t)|+ k2, (C.14)

where k2 is an arbitrary constant of integration. We now have an approximation to the solution of (C.6).Thus, the first-order WKBJ-approximation of gn(t) will be:

Γn1(t)cos(

nπε

∫ t

0

dsl(s)

)+Γn2(t)sin

(nπε

∫ t

0

dsl(s)

), (C.15)

where,

Γn1(t) = an

(nπl(t)

)− 12

, and, Γn2(t) = bn

(nπl(t)

)− 12

, (C.16)

where an and bn are constants of integration. The solution of (C.4) is now given by

gn(t) = Γn1(εt)cos(

nπ∫ t

0

dsl(s)

)+Γn2(εt)sin

(nπ∫ t

0

dsl(s)

)+O(ε). (C.17)

Thus, it can be seen that the solution (C.17) of the time-dependent equation (C.4) obtained by WKBJ(Liouville-Green)-approximation is in accordance with the time-dependent solution (3.32) obtained byusing the transformation t+ =

∫ t0

dsl(s) .

Appendix DResonant terms

To avoid secular terms in the approximation for u(ξ , t;ε) we will show in this appendix that the func-tions Ak0(t) and Bk0(t) have to satisfy

dAk0dt

=−(k−1)A(k−1)0 +(k+1)A(k+1)0 +(l0−2)µ0l0

2β0kBk0,

dBk0dt

=−(k−1)B(k−1)0 +(k+1)B(k+1)0− (l0−2)µ0l02β0

kAk0,(D.1)

for k = 1,2,3, · · · . This can be derived as follows. After introducing a fast and a slow time in section3.4.1, we obtain the equation (3.78) with ω = π

l0 . The solution of the O(1) problem is wk0(t+, t) =

Ak0(t)cos(

kπt+l0

)+Bk0(t)sin

(kπt+

l0

), where Ak0 and Bk0 can be determined from the O(ε) equation

by removing terms in the right hand side of this equation that cause resonance terms in wk1(t+, t). Thefirst term on the right hand side of the O(ε) equation causing secular terms is

−2∂ 2wk0

∂ t+∂ t= 2

kπl0

[dAk0

dtsin(

kπt+

l0

)− dBk0

dtcos(

kπt+

l0

)]. (D.2)

The second term on the right hand side of the O(ε) equation causing secular terms is(

µ0l20− µ0

2l0

)(kπ)2wk0

=(

µ0l20− µ0

2l0

)(kπ)2

(Ak0(t)cos

(kπt+

l0

)+Bk0(t)sin

(kπt+

l0

)).

(D.3)

The third term on the right hand side of the O(ε) equation causing secular terms is

∑∞n=1,n 6=k

4nπβ0ωl0 cos(ωt+) ∂wn0

∂ t+ ( f1(n,k)− f2(n,k))

= 2β0π2

l30

(k(k−1)2

2k−1 A(k−1)0− k(k+1)2

2k+1 A(k+1)0

)sin(

kπt+l0

)

− 2β0π2

l30

(k(k−1)2

2k−1 B(k−1)0− k(k+1)2

2k+1 B(k+1)0

)cos(

kπt+l0

)

+ terms not giving rise to secular terms in wk1,

(D.4)

where f1(n,k) and f2(n,k) are given by (3.74) and (3.75), respectively. The fourth term in the righthand side of the O(ε) equation causing secular terms is

−∑∞n=1,n 6=k

2nπβ0ω2

l0 sin(ωt+)wn0( f1(n,k)− f2(n,k))

= β0π2

l30

(k(k−1)2k−1 A(k−1)0 +

k(k+1)2k+1 A(k+1)0

)sin(

kπt+l0

)

− β0π2

l30

(k(k−1)2k−1 B(k−1)0 +

k(k+1)2k+1 B(k+1)0

)cos(

kπt+l0

)

+ terms not giving rise to secular terms in wk1.

(D.5)

Finally, collecting all the terms on the right hand side of the O(ε)-equation containing sin(

kπt+l0

)and

containing cos(

kπt+l0

)and then setting their coefficients equal to zero in order to remove secular terms,

we obtain (D.1).

Appendix EApplication of the truncation method

In this appendix we will find an approximation of the solution of the system (3.80) by using Galerkin’struncation method. So, we will use just some first few modes and neglect the higher order modes.Consider the system (3.80)

dAk0dt1

=−(k−1)A(k−1)0 +(k+1)A(k+1)0 +αkBk0,

dBk0dt1

=−(k−1)B(k−1)0 +(k+1)B(k+1)0−αkAk0,(E.1)

for k = 1,2,3, · · · , where α =(l0−2)µ0l0

2β0and α has fixed sign. Truncating the infinite dimensional

system (E.1) to a single mode, that is, k = 1, we get the following system of ODEs

dA10

dt1= 2A20 +αB10,

dB10

dt1= 2B20−αA10, (E.2)

then neglecting A20 and B20 we can obtain the matrix A =

[0 α−α 0

]. Now we define B = PAP−1

where P = (pii) for i = 1,2 is a diagonal matrix and P−1 is an inverse matrix of P. So, A and B are

similar matrices. Therefore we can write B =

[0 α p11

p22

−α p22p11

0

], and then, −BT =

[0 α p22

p11

−α p11p22

0

].

Now if we choose p11p22

= p22p11

for p11 being arbitrary, then p222 = p2

11 and the matrix B = −BT , whereT stands for transpose of matrix B. Similarly, truncating the infinite dimensional system (E.1) to 2modes, that is, k = 1,2, we obtain the following system of ODEs

dA10dt1

= 2A20 +αB10,dB10dt1

= 2B20−αA10,

dA20dt1

=−A10 +2αB20,dB20dt1

=−B10−2αA20,(E.3)

we obtain the following matrix A =

0 α 2 0−α 0 0 2−1 0 0 20 −1 −2α 0

, then by using P = (pii), and P−1 = ( 1

pii)

for i = 1,2,3,4 such that B = PAP−1, we obtain B and BT given by

B =

0 α p11p22

2p11p33

0−α p22

p110 0 2p22

p44

− p33p11

0 0 2α p33p44

0 − p44p22

− 2α p44p33

0

, and, −BT =

0 α p22p11

p33p11

0−α p11

p220 0 p44

p22

− 2p11p33

0 0 2α p44p33

0 − 2p22p44

− 2α p33p44

0

78 E. Application of the truncation method

No.ofmodes

Eigenvalues of matrix A Eigen-spaceof A

1 ±i 22 0,0,±3i 43 ±0.75i,±1.54i,±5.20i 64 ±0.67i,±1.46i,±3.25i,±7.54i 85 0,0,±2.07i,±2.19i,±5.15i,±9.96i 106 ±0.60i,±1.23i,±2.95i,±3.67i,±7.19i,±12.46i 127 ±0.57i,±1.20i,±2.60i,±3.73i,±5.43i,±9.33i,±15.00i 148 0,0,±1.77i,±1.81i,±4.14i,±4.53i,±7.30i,±11.55i, ±17.58i 169 ±0.53i,±1.09i,±2.46i,±3.12i,±5.34i,±5.81i,±9.28i,±13.83i,

±20.20i18

10 ±0.52i,±1.07i,±2.30i,±3.13i,±4.62i,±6.17i,±7.59i,±11.34i,±16.16i,±22.84i

20

11 0,0,±1.61i,±1.63i,±3.65i,±3.82i,±6.24i,±7.00i,±9.46i,±13.46i,±18.54i,±25.50i

22

12 ±0.50i,±1.01i,±2.21i,±2.82i,±4.53i,±5.12i,±7.85i,±7.95i,±11.40i,±15.64i,±20.96i,±28.12i

24

13 ±0.48i,±0.99i,±2.12i,±2.82i,±4.17i,±5.26i,±6.70i,±8.71i,±9.75i,±13.41i,±17.86i,±23.41i,±30.88i

26

14 0,0,±1.49i,±1.51i,±3.35i,±3.45i,±5.62i,±6.00i,±8.36i,±9.56i,±11.61i,±15.48i,±20.12i,±25.88i,±33.59i

28

15 ±0.47i,±0.95i,±2.05i,±2.63i,±4.10i,±4.69i,±6.76i,±7.17i,±10.10i,±10.43i,±13.54i,±17.59i,±22.43i,±28.38i,±36.32i

30

Table E.1: Approximations of the eigenvalues of the truncated system (E.1) for α = 1.

Now for p11 arbitrary if we choose p222 = p2

11, p233 = 2p2

11, and p244 = 2p2

11 then clearly B =−BT . Ingeneral, by mathematical induction it can be shown that for all (2n×2n) matrices if p11 is arbitrary thenp22, p33, p44, · · · , p2n2n are fixed (apart from their signs). So, for a given n it can be shown that thereexists a non-singular matrix P such that PA = BP with B = −BT . So, A and B are similar matrices,this implies that A and B have the same eigenvalues. From B =−BT it follows that all eigenvalues ofmatrix B are purely imaginary (or zero). Using the computer software package Maple, the eigenvaluesof system (E.1) have been computed up to 15 modes and are listed in Table E.1 for α = 1. Similarresults can be found for other values of α . From the Table E.1, it can be seen that the eigenvalues ofthe truncated system are always purely imaginary. For (3 j−1) number of modes for j = 1,2,3 · · · , weget an additional pair of zero eigenvalues. We have shown that the dimension of eigenspace equals thenumber of eigenvalues. So, all solutions of the truncated system are bounded.

Appendix FThe infinite dimensional system

In this appendix we will show that

d2

dt21

∞

∑k=1

(X2

k0 +Y 2k0

)+(α2−4)

∞

∑k=1

(X2

k0 +Y 2k0

)= D1, (F.1)

where D1 =d2

dt21

∑∞k=1(X2

k0(0)+Y 2k0(0)

)+(α2−4)∑∞

k=1(X2

k0(0)+Y 2k0(0)

). By adding both equations

in system (3.82), and then by taking the sum from k = 1 to ∞ we obtain:

12

ddt1

∞

∑k=1

(X2

k0 +Y 2k0

)=−

∞

∑k=1

(Xk0X(k+1)0 +Yk0Y(k+1)0

). (F.2)

By differentiating (F.2) with respect to t1 and by using system (3.81), we find

12

d2

dt21

∑∞k=1(X2

k0 +Y 2k0)

=−∑∞k=1

(Xk0X(k+1)0 +Xk0X(k+1)0 + Yk0Y(k+1)0 +Yk0Y(k+1)0

)

= ∑∞k=1

(k+1)

(X2

k0 +Y 2k0)− k(

X2(k+1)0 +Y 2

(k+1)0

)

+α(

X(k+1)0Yk0−Xk0Y(k+1)0

)

= ∑∞k=1(k+1)

(X2

k0 +Y 2k0)−∑∞

m=2(m−1)(X2

m0 +Y 2m0)

+α ∑∞k=1

(X(k+1)0Yk0−Xk0Y(k+1)0

)

= 2(X2

10 +Y 210)+∑∞

k=2(k+1)(X2

k0 +Y 2k0)−∑∞

m=2(m−1)(X2

m0 +Y 2m0)

+α ∑∞k=1

(X(k+1)0Yk0−Xk0Y(k+1)0

)

= 2(X2

10 +Y 210)+∑∞

k=2(k+1)− (k−1)(X2

k0 +Y 2k0)

+α ∑∞k=1

(X(k+1)0Yk0−Xk0Y(k+1)0

)

= 2(X2

10 +Y 210)+∑∞

k=2 2(X2

k0 +Y 2k0)+α ∑∞

k=1

(X(k+1)0Yk0−Xk0Y(k+1)0

)

= ∑∞k=1

2(X2

k0 +Y 2k0)+α

(X(k+1)0Yk0−Xk0Y(k+1)0

).

(F.3)

By differentiating (F.3) with respect to t1 and by using system (3.81) and (F.2), we find

12

d3

dt31

∑∞k=1(X2

k0 +Y 2k0)

= 2 ddt1

∑∞k=1(X2

k0 +Y 2k0)

+α ∑∞k=1

(X(k+1)0Yk0 + Yk0X(k+1)0− Xk0Y(k+1)0−Xk0Y(k+1)0

)

= 2 ddt1

∑∞k=1(X2

k0 +Y 2k0)+α2

(Xk0X(k+1)0 +Yk0Y(k+1)0

)

= 2 ddt1

∑∞k=1(X2

k0 +Y 2k0)− α2

2d

dt1∑∞

k=1(X2

k0 +Y 2k0).

(F.4)

80 F. The infinite dimensional system

Thus, (F.4) can be re-written as follows

ddt1

[d2

dt21

∞

∑k=1

(X2

k0 +Y 2k0

)+(α2−4)

∞

∑k=1

(X2

k0 +Y 2k0

)]= 0. (F.5)

And so, integration of (F.5) will result in (F.1).

Appendix G

An unexpected timescale of order 1√ε

For the O(ε) boundary excitations, when the O(ε)-problem is being taken into account we encounterthe following two integrals:

J1 =∫ t+

0sin(ωs)sin

(Ωl0ε

(eεs−1))

ds, J2 =∫ t+

0cos(ωs)sin

(Ωl0ε

(eεs−1))

ds, (G.1)

where sin(ωt+) and cos(ωt+) are oscillatory solutions of the time-dependent homogeneous part and

ω = kπ . We can write sin(

Ωl0ε(eεs−1

))as sin

(Ωl0(

s+ εs2

2

)+h.o.t.

), where s is a dummy variable

and h.o.t. stands for higher order terms in ε . We can keep the first two dominant terms in the argumentof sin and neglect the higher order terms due to fact that they have very small contribution. Now byusing trigonometric identity 2sin(a)sin(b) = cos(a− b)− cos(a+ b) in the first integral in (G.1), wecan write J1 as

J1 =12

∫ t+

0

cos((ω−Ωl0)s−

εΩl0s2

2

)− cos

((ω +Ωl0)s+

εΩl0s2

2

)ds. (G.2)

We consider only the first term on the right hand side of (G.2) and neglect the second term due to fact

that there is no resonance contribution of this term. Now assuming the transformation s =√

2εΩl0 p+

ω−Ωl0εΩl0 , we find

J1 =1√

2εΩl0

∫ t+√

εΩl02 +δ

δcos(− p2 +δ 2

)d p, (G.3)

where δ = Ωl0−ωεΩl0 . (G.3) can be split into

J1 =1√

2εΩl0

(cos(δ 2)

∫ t+√

εΩl02 +δ

δcos(p2)d p+ sin(δ 2)

∫ t+√

εΩl02 +δ

δsin(p2)d p

). (G.4)

The integrals in (G.4) are bounded and are often known as Fresnel integrals. Thus, (G.4) shows thatthe effect of passing through resonance is O(

√ε) for O(ε) excitations. Similarly, it can be shown for

second integral in (G.1) that the effect of passing through resonance is O(√

ε). Small changes in εproduce relatively large changes of the solution as shown in Figure G.1 and Figure G.2 for J1 and J2for different values of ω , Ωl0 and ε .

82 G. An unexpected timescale of order 1√ε

Figure G.1: Single mode behavior of J1 for ω = Ωl0 = π and ε = 0.1.

Figure G.2: Single mode behavior of J2 for ω = 2π , Ωl0 = π and ε = 0.01.

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Summary

In this thesis two models for axially moving continua have been studied: a string-like model and abeam-like model. Mathematically, a string-like model is described by a wave equation and a beam-like model is usually described by the Euler-Bernoulli beam equation. A string-like model has beenused to describe the transversal vibrations of a vertically moving elevator cable system with time-varying length and a beam-like model has been used to describe the transversal vibrations of a constantlength conveyor belt system between two supports. For the string model, a rigid body is attached tothe lower end of the string, and the suspension of this body against the guide rails is assumed to berigid. For the string model, it is assumed that the length changes linearly in time or that the lengthchanges harmonically about a constant mean length. For the linear length-variations it is assumedthat the axial velocity of the string is small compared to nominal wave velocity and the string massis small compared to the mass of the rigid body and, for the harmonically length variations smalloscillation amplitudes are assumed. The case with boundary excitations has also been investigated indetail, and interesting resonance conditions have been found. For the beam model, the axial velocityis assumed to be constant and relatively small compared to the wave speed. The case with boundarydamping has been investigated in detail for the beam equation and interesting damping propertieshave been obtained. The corresponding initial boundary value problems have been formulated, and inall cases formal asymptotic approximations of the analytic solutions have been constructed by usingthe multiple timescales perturbation methods. For the string-like problem, it has been shown thatGalerkin’s truncation method can not be applied in order to obtain asymptotic approximations valid onlong timescales. For boundary excitations the interesting phenomenon of autoresonance occurs whenthere is passage through (dynamic) resonance. The maximal amplitude of the autoresonant solutionand the time of autoresonant growth of the amplitude of the modes of fast oscillations have beendetermined. Interior layer analysis has been provided systematically and it has been shown that thereexists an unexpected timescale of order 1√

ε . For this reason three timescales have been introducedwhen constructing asymptotic results. For the beam-like problem, by using the energy integral, it hasbeen shown that the solutions are bounded for times t of order 1

ε . It has also been analytically andnumerically shown that all solutions (up to order ε) are uniformly damped.

Samenvatting

In dit proefschrift worden twee modellen voor axiaal bewegende continua bestudeerd: een snaarmodelen een balk-model. Een snaar-model wordt wiskundig beschreven door een golfvergelijking en eenbalk-model door een Euler-Bernoulli balkvergelijking. Een snaar-model is gebruikt om transversaletrillingen van een vertikaal bewegend liftkabel systeem met tijd-varierende lengte te beschrijven, eneen balk-model is gebruikt om de transversale trillingen van een transportbandsysteem te beschrijven.Voor het snaar-model is een star lichaam bevestigd aan de onderkant van de snaar, en de ophangingvan dit lichaam tegen de geleider-rails wordt verondersteld star te zijn. Voor het snaar model wordtaangenomen dat de lengte linear in de tijd varieert of dat de lengte harmonisch varieert rond een con-stante lengte. Voor de lineare lengte variaties wordt aangenomen dat de axiale snelheid van de snaarklein is in vergelijking met de nominale golfsnelheid, en dat de snaarmassa klein is ten opzichte van demassa van het starre lichaam. De harmonische lengte variaties worden klein verondersteld te zijn. Hetgeval met rand excitatie is ook in detail bestudeerd, en interessante resonantie condities zijn gevon-den. Voor het balk model wordt aangenomen dat de axiale snelheid constant en relatief klein is tenopzichte van de golfsnelheid. Het geval met rand demping is ook bestudeerd voor de balk-vergelijkingen interessante dempings eigenschappen zijn gevonden. De bijbehordende begin-, randwaardeproble-men zijn geformuleerd, en in alle gevallen zijn formele asymptotische benaderingen van de analytischeoplossingen geconstrueerd door gebruik te maken van de meer-tijdschalen storingsmethode. Voor hetsnaar-probleem is aangetoond dat Galerkin’s truncatie methode niet toegepast kan worden om asymp-totische benaderingen te construeren welke geldig zijn op lange tijdschalen. In het geval met randexci-taties treedt het interessante verschijnsel van autoresonantie op als er een passage door de (dynamische)resonantie plaatsvindt. De maximale amplitude van de oplossing met auto-resonantie, en de tijd(en)dat er autoresonante groei van de amplitudes van trillingsmodes is, zijn bepaald. Grenslaag analyse istoegepast, en aangetoond is dat er een onverwachte tijdschaal van O( 1√

ε ) bestaat. Om die reden zijndrie tijdschalen gıntroduceerd om asymptotische resultaten te construeren. Voor het balk probleem isaangetoond door gebruik te maken van energie integralen dat de oplossingen begrensd zijn voor tijdent van O

( 1ε). Ook is analytisch en numeriek aangetoond dat alle oplossingen (tot op O(ε)) uniform

gedempt zijn.

Summary (in Russian)

Summary (in Chinese)

About the author

Sajad H. Sandilo was born on January 10, 1980 in Larkana, Sindh, Pakistan. InJune 2002 he completed his Bachelor’s Degree in Mathematics with Physics atInstitute of Mathematics and Computer Science, University of Sindh, Jamshoro,Pakistan. In July 2005 he was awarded Master’s Degree in Applied Mathematicsfrom Department of Mathematics, Quaid-i-Azam University Islamabad, Pakistan.From January 2006 to July 2007, he started his career as a Research Associate atInstitute of Mathematics and Computer Science in University of Sindh, and then

from July 2007 to September 2008, he served as a Lecturer at Department of Mathematics in Quaid-e-Awam University of Engineering, Science and Technology Nawabshah, Sindh, Pakistan. In September2008 Sajad H. Sandilo came to Netherlands and started his Ph.D. research program at the MathematicalPhysics Department in Delft Institute of Applied Mathematics of Delft University of Technology. Heworked on the project “Aspects of Asymptotics for Axially Moving Continua” under the supervisionof Dr. ir. Wim T. van Horssen. The research is related to Applied Mechanics, Mechanical Engineeringand Applied Mathematics, and is concerning the problems of free, forced and damped oscillationsof axially moving continuous mechanical systems. During his Ph.D. research work S. H. Sandilo haspresented research results in many local and international conferences held in Netherlands, Italy, UnitedStates of America, and Portugal. He has also visited some other European countries like Germany,Belgium, France, Hungary, Slovakia, Austria, and Czech Republic for scientific and historic reasons.Currently he is a member of American Society of Mechanical Engineers (ASME), European MechanicsSociety (EuroMech), and International Centre for Mechanical Sciences (CISM). His research interestsare asymptotics and perturbation methods for ordinary and partial differential equations, and linear andnonlinear oscillations in structural and mechanical systems.

Acknowledgments

Each person comes into this world with a specific destiny, he has something to fulfill, some messagehas to be delivered, some work has to be completed. You are not here accidentally, you are heremeaningfully. There is a purpose behind you. The whole intends to do something through you.

Osho

Around five years ago, I came to Netherlands for my PhD study at the Mathematical Physics Depart-ment of the Delft University of Technology under supervision of Dr. ir. Wim T. van Horssen. Duringthese five years I did research on current mathematical problems in applied mechanics and mechanicalengineering, specially axially moving continuous mechanical systems. I would like to thank my super-visor Dr. ir. Wim T. van Horssen from whom I learned the interesting and the most important aspectsof research in real physical systems. He strongly drew my attention to view the engineering problemsfrom mathematical and physical perspectives, and guided me in formulating interesting topics in re-search. It is due to him that I went through asymptotics and perturbation methods, and complicatedordinary and partial differential equations which are the core of this thesis. I am very grateful to him forthe freedom and the openness he provided me throughout my research work. I am also thankful to ourgroup leader and promoter prof. dr. ir. Arnold W. Heemink for his hospitality and valuable supportsthroughout the long years of my study. I would like to thank Quaid-e-Awam University Pakistan andDelft University of Technology Netherlands for their financial supports of my research at MathematicalPhysics Department. Plato is right when he says: “The direction in which education starts a man willdetermine his future in life.”

November 15, 2013 Sajad H. SandiloDelft, The Netherlands

List of publications and presentations

Journal Articles

• S. H. Sandilo and W. T. van Horssen, 2012, On Boundary Damping for an Axially Moving Ten-sioned Beam, American Society of Mechanical Engineers, Journal of Vibration and Acoustics,vol. 134, no. 1, art. no. 011005.

• S. H. Sandilo and W. T. van Horssen, 2013, On Variable Length Induced Vibrations of a VerticalString, Elsevier, Journal of Sound and Vibration, submitted.

• S. H. Sandilo and W. T. van Horssen, 2013, On a Cascade of Autoresonances in an ElevatorCable System, Springer, Nonlinear Dynamics, submitted.

Conference Proceedings

• S. H. Sandilo and W. T. van Horssen, 2011, On Boundary Damping for an Axially Moving Beamand On Variable Length Induced Vibrations of an Elevator Cable, Eds. D. Bernardini, G. Regaand F. Romeo, Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC2011),24-29 July, Rome, Italy.

Book Chapter

• S. H. Sandilo and W. T. van Horssen, 2012, On Small Variable Length Induced Oscillations ofa Vertical String, Problems of Nonlinear Dynamics and Condensed Matter Physics dedicatedto the 75th Birthday of Prof. Leonid I. Manevich, Eds. A. I. Manevich, M. A. Mazo and V. V.Smirnov, Publisher. Russian Academy of Sciences, Moscow, Russia.

Technical Presentations

• On Boundary Damping for an Axially Moving Beam and On Variable Length Induced Vibra-tions of an Elevator Cable, European Nonlinear Dynamics Conference (ENOC2011), 24-29 July2011, Rome, Italy.

• On Variable Length Induced Vibrations of an Elevator Cable, Nonlinear Dynamics in NaturalSystems (NDNS+), PhD Days in Analysis and Dynamics, 26-27 April 2012, Lunteren, TheNetherlands.

• On Variable Length Induced Vibrations of a Vertical Translating String, ASME 2012 Inter-national Mechanical Engineering Congress and Exposition, 9-15 November 2012, Houston,Texas, USA.

• On Variable Length Induced Vibrations of a Vertically Moving String, Nonlinear Dynamics inNatural Systems (NDNS+), PhD Days in Analysis and Dynamics, 25-26 April 2013, Lunteren,The Netherlands.

• On Variable Length Induced Oscillations of a Vertically Moving String, ICOVP 2013, 11thInternational Conference on Vibration Problems, 9-12 September 2013, Lisbon, Portugal.

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