Sajad H. Sandilo
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op donderdag 12 december 2013 om 12.30 uur
door
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 voorzitter Prof. dr. ir. A. W. Heemink Technische Universiteit Delft, promotor Dr. ir. W. T. van Horssen Technische Universiteit Delft, copromotor Prof. dr. A. K. Abramian Russian Academy of Sciences, Russia Prof. dr. S. Kaczmarczyk The University of Northampton, United Kingdom Prof. dr. A. V. Metrikine Technische Universiteit Delft Prof. dr. ir. C. Vuik Technische Universiteit Delft Prof. 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 of Technology 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 described in 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 the Delft University of Technology, The Netherlands.
ISBN 978-94-6186-237-2
Copyright c© 2013 by S. H. Sandilo e-post:sajadsandilo@yahoo.com
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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 Gulestan my wife Raheela my daughter Athina and my sisters Anita, Sonia and Fozia
Contents
1 Introduction 1 1.1 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Analytical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 On Boundary Damping for an Axially Moving Tensioned Beam 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 15 2.3 The energy and the boundedness of solutions . . . . . . . . . . . . . . . . 16 2.4 Application of the two timescales perturbation method . . . . . . . . . . . 17 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 On Variable Length Induced Vibrations of a Vertical String 29 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 31 3.3 The case l(t) = l0 + vt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Application of the two-timescales perturbation method . . . . . . . 34 3.4 The case l(t) = l0 +β sin(ωt) . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4.1 Application of the two-timescales perturbation method . . . . . . . 42 3.4.2 The case ω = π
l0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.3 The case ω = π l0 + εσ . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.4 The energy of the infinite dimensional system . . . . . . . . . . . . 45 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
vii
4 On a Cascade of Autoresonances in an Elevator Cable System 47 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 The governing equations of motion . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Interior layer analysis for the single ordinary differential equation . . . . . 54 4.4 A three timescales perturbation method . . . . . . . . . . . . . . . . . . . 57 4.5 Approximations of the solutions of the initial-boundary value problem . . . 66 4.6 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
F The infinite dimensional system 79
G An unexpected timescale of order 1√ ε 81
Bibliography 83
Summary 89
Samenvatting 91
Chapter 1 Introduction
Men who wish to know about the world must learn about it in its particular details.
Heraclitus
1.1 Historical background
Vibrations occur frequently in a variety of many physical or mechanical structures such as 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 and mining 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 a physical structure. Structural failure can occur because of large dynamic stresses developed during 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 of a 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 machine and 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 completely due to an 18.9 m/s wind-flow induced 0.23 Hz torsional oscillation of the bridge deck. The collapse of the bridge is sometimes characterized as in physics and in applied mathematics text 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 The Netherlands 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 these vibrations. Vibrations are initiated when an inertia element is displaced from its equilibrium position due 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 physical system 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, this transfer 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 to mitigate 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 oscillation amplitudes various types of boundary damping can be applied (see, (Cox and Zuazua 1995), (Darmawijoyo and van Horssen 2002), (Darmawijoyo and van Horssen 2003), (Sandilo and van Horssen 2012), and (Zarubinskaya and van Horssen 2006b)). For instance, in the 1930’s the 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 and the cable were sometimes improperly matched, and as a result serious damage to the cables at the points of attachment of the dampers, that is, at their clamp occurred (Hagedorn and Seemann 1998). Nowadays, various types of passive dampers applied at the boundary have been considered extensively ((Rao 1993), and (Wang et al. 1993)). The beam or string is weakly damped because the boundary damping parameters are small, but these dampers can produce 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 viscous damper have been studied in (Zarubinskaya and van Horssen 2006a). In physical or mechanical systems oscillations around equilibrium positions can be described by 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 by Galileo Galilei, Jean le Rond d’Alembert, Leonhard Euler, Daniel Bernoulli, Joseph Louis Lagrange and Joseph Fourier. Mathematically, string vibrations are modeled by a wave equation which is an important second order partial differential equation for the description of waves or vibrations. When bending stiffness becomes more important, the description of mechanical vibrations is represented by a fourth-order partial differential equation often known 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 complicated physical 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 commonly used models for such type of axially moving continua. They are classified in the category of one dimensional continuous systems and consequently the displacement field depends on time 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 and DasGupta 2007), (Meirovitch 1997), and (Weaver et al. 1990)). The pioneering work of axially moving continua is ascending to Willard L. Miranker (see, (Miranker 1960)) who considered a model for the transverse vibrations of a tape moving between a pair of pulleys by using a variational procedure. By means of energy-type integrals, it was shown that the energy of that portion of the tape between the pulleys is not conserved, but that there is a periodic transfer of energy into and out of the system. Further, this work was carried on by 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 perturbation method was introduced in (Suweken and van Horssen 2003a) with a low and time-varying velocity and, a two timescales perturbation method and a Laplace transform method were used in (Ponomareva and van Horssen 2007) with velocity to be time-varying and to be of the same order of magnitude as the wave speed to investigate the applicability of Galerkin’s truncation 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 be applied 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 less problems in applying the truncation method. Transversal vibrations of a moving beam with viscous damping were studied in (Pakdemirli and Oz 2008). Using Hamiltonian dynamics analysis, an axially translating elastic Euler-Bernoulli cantilever beam featuring time-variant velocity 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 the forced vibration of a simply supported axially moving viscoelastic beam. In Refs. ((Zhang and Zu 1998a), and (Zhang and Zu 1998b)) authors attempted to describe the mechanical energy dissipation using a viscoelastic model for the belt, and utilized the perturbation tech- niques to predict the nonlinear response. These viscoelastic studies provided a systematic methodology to incorporate material damping in the analysis. Most of the aforementioned studies are restricted to only horizontal translating continua. Nowadays, the vertically mov- ing strings and beams frequently appear in research literature. For example, rope and cable systems are used to carry payloads in lift installations, including building elevators and mine hoists, represent typically non-stationary systems. The vertically moving systems are more complicated 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 the same height were first studied in the middle of the 19th century, and a historical discussion was given in (Irvine 1981). The earlier work of string with a mass-spring system emulating an elevator goes back to ((Yamamoto et al. 1978), and (Terumichi et al. 1997)). In (Zhu and Ni 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; they also 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) to dissipate the vibratory energy of a translating medium with arbitrary varying length. The effects of bending stiffness and boundary conditions on the dynamic response of elevator cables were examined in (Zhu and Xu 2003). Recently, a linear model is developed for calculating 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 elevator cable-car systems. Sometimes it happens that studying external or boundary excitations of these 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 to a 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 developed in (Bohm and Foldy 1946) for particle accelerators. The autoresonance is thought of as a universal phenomenon which occurs in a wide range of oscillating physical systems from astronomical 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 atomic and 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 of the 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 are possible 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. By solving beam-like or wave-like equations, important information on the vibrational behavior of a physical system can be found. The key to solving modern problems is mathematical modeling. 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 of magnitude (i.e., smallness or largeness) of different elements of the system by comparing them with each other as well as with the basic elements of the system. This process is called nondimesnionalization or making the variables dimensionless. Consequently, one should always introduce dimensionless variables before attempting to make any approxima- tions. Therefore, expressing the equations in dimensionless form brings out the important dimensionless parameters that govern the behavior of the physical system. The smallness of the dimensionless parameter, say ε , in the governing system (consisting of differential equations, 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 perturbation or asymptotic methods are applied explicit expressions that describe the structural motion can 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 be found 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 moving continua and, free and forced vibrations of the vertical axially moving continua, which are described by continuous or distributed-parameter models. Translating media with constant length can model such low- and high-speed slender members as conveyor belts ((Ponomareva and van Horssen 2007), and (Ponomareva and van Horssen 2009)), chair lifts, aerial cable tramways, pipes carrying water, oil or gas ((Oz and Boyaci 2000), and (Kuiper and Metrikine 2004)), 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, and transport 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 (Stolte and Benson 1992), satellite tethers (Misra and Modi 1982), flexible appendages (Tsuchiya 1983), lift cranes, mining hoists ((Kaczmarczyk 1997), and (Kaczmarczyk and Ostachowicz 2003)), and cable-driven robots exhibit time-varying length, space-time-varying tension and constant or time-varying velocity. The understanding of the vibrations of an axially moving continuous 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 in initial-boundary value problems for (wave-) string-like or beam-like equations depending on the bending stiffness. But they all have something in common, namely, the dimension in so-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 problems of interaction of structures with flows are both of great technological and theoretical interest (see, for instance, (Padoussis 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 into two categories, that of a string-like type or that of a beam-like type, depending on the bending stiffness. 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 conveyor belts or axially moving elevator cables can be derived by using Hamilton’s principle (see, for instance, (Miranker 1960), or (Suweken 2003)). For the conveyor belt system a beam-like equation is considered in horizontal direction where pretension of the belt and longitudinal axial velocity are assumed to be constant. For the elevator cable system a string-like equation is considered in vertical direction where axial longitudinal velocity in vertical direction is assumed to be constant or time-varying, and the tension in the cable varies through a spatial coordinate and time. The transversal vibrations of the conveyor belt system (with constant velocity 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 horizontal direction, 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 assumed that 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 end x = 0 and is attached to a spring-dashpot system at other end x = L, therefore, the boundary conditions 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 is the constant distance between the pulleys. The transversal vibrations of the elevator cable system 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 a string-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 lower end, where l(t) is the time-varying length of the string, g is the acceleration due to gravity and V is the longitudinal acceleration due to attached mass. The elevator car is modeled as a rigid body of mass m attached at the lower end of the cable and, the suspension of the car against the guide rails is assumed to be rigid. The time-varying length l(t) is given by l(t) = l0 +Vt or by l(t) = l0 + β sin(ωt), where l0 is the initial string length, V is the constant string velocity, β is the length variation parameter, ω is the angular frequency of length variation and l0 > |β |. It is assumed that the cable is fixed or that it is externally excited by a harmonic force due to wind or storm in the horizontal direction at its upper end x = 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(t)), u(l(t), t) = 0, (1.4)
where α is the excitation amplitude of the tall building or the structure at its top and is the oscillation 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 initial belt or cable velocity. Equations (1.1) and (1.3) are linear equations of motion, derived by using 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 first weakly perturbed linear equations to get mathematical and physical insights. Linear differen- tial equations with variable coefficients are in some sense equivalent to nonlinear differential equations 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, and their 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 physics exact 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-varying parameters, nonlinearities, and complicated boundary conditions. To find an exact solution in these cases usually seems impossible. Hence, applied mathematicians, engineers, and physicists are forced to determine approximate solutions of the problems they are facing. Approximated solutions can sometimes be constructed in order to obtain information about behavior 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 the applied perturbation scheme it is assumed that the solution of the problem can be expanded in a power series in ε , where ε is a small dimensionless parameter. If a naive expansion is used, 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, · · · . Of course, the approximation is still valid for small values of t, but it is not valid anymore for large values of t. These unbounded terms are called secular terms. To avoid these secular terms it is convenient to introduce new time variables t0 = t, t1 = εt, t2 = ε2t, and so on. To remove 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 series in 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 unbounded terms 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 then the 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 the dependence of the approximate solution on the variables so as to obtain a uniformly valid approximation to the exact solution. This is done by introducing a fast-scale, a slow-scale and, even sometimes slower-scale variables for the independent space or time variable, and subsequently treating these variables as if they are independent. In the solution process of the perturbation problem the resulting additional freedom introduced by the new indepen- dent variables is used to remove unbounded or secular terms. This freedom puts constraints on the subsequent approximate solutions, which are called solvability conditions. The first scheme to address this problem is what Milton D. van Dyke (van Dyke 1975) refers to as the 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 Anders Lindstedt in period 1882-1883, who introduced and used such scaled variables to eliminate secular (resonant, unbounded) terms in perturbation expansions in celestial mechanics. The work 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 to Lindstedt. 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 even earlier 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 by Alexey 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 was independently discovered by Jirair Kevorkian and Julian Cole (Kevorkian and Cole 1996), James Alan Cochran (Cochran 1962), and John Mahony (Mahony 1962), and was promoted by Ali Hasan Nayfeh (see, (Nayfeh 1973), and (Nayfeh 1991)) to study various oscillation problems, which is now the more standard approach. Using this method, Jirair Kevorkian ingeniously solved a number of difficult problems. For more than two decades working at Mathematical Physics Group of Delft University of Technology, Wim T. van Horssen and 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 and van 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 construct approximations 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 moving tensioned beam equation will be considered. The axial velocity of the beam is assumed to be 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 are attached, where the damping generated by the dashpot is assumed to be small. The equations of motion of the moving conveyor belt will be derived by using Hamilton’s principle. The energy of the initial-boundary value problem and the boundedness of the solutions will be shown. A multiple timescales perturbation method is used to construct formal asymptotic approximations of the solutions of the initial-boundary value problem, and it will be shown that 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 be considered. The equations of motion of the vertically translating system will be derived by the 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 the transversal 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 given mathematical models a rigid body is attached to the lower end of the cable and suspension of the car against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the cable is small compared to nominal wave velocity and that the cable mass is small compared to car mass, and for harmonically length variations small oscillation amplitudes are assumed. A multiple timescales perturbation method is used to 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 length that there are infinitely many values of ω that can cause internal resonances. In this chapter the resonance case ω = π
l0 is 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 all cases in order to obtain approximations valid on long timescales. In chapter 4, the forced responses of a vertically translating string with a time-varying length and a space-time-varying tension will be considered. The problem is used as a simple model to describe the forced vibrations of an elevator cable for which the length changes linearly in time. The vertical velocity of the cable is assumed to be constant and relatively small compared to nominal wave velocity, and the cable mass is small compared to car mass. In given 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 externally excited at the upper end by the displacement of the building in horizontal direction from the equilibrium. This external excitation has a constant amplitude of order ε , where ε is a dimensionless small parameter. The fascinating phenomena of autoresonance occurs when a perturbed system is captured into (dynamic) resonance. This autoresonance phenomena and the time of autoresonant growth of amplitude of the modes of fast oscillations will be discussed 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 shown that there exists an unexpected new timescale of order 1√
ε . For this reason, a three timescales perturbation method is used to construct formal asymptotic approximations of the solutions of 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 up and from these solutions it can be seen that as the mode number k increases the amplitudes decrease in size.
Published as: S. H. Sandilo and W. T. van Horssen – ”On Boundary Damping for an Axially Moving Tensioned Beam”, American Society of Mechanical Engineers, Journal of Vibration and Acoustics, vol. 134, no. 1, art. no. 11005, February 2012
Chapter 2 On Boundary Damping for an Axially Moving
Tensioned Beam
It is through science that we prove, but through intuition that we 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, where the 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 the vertical vibrations of a conveyor belt, for which the velocity is assumed to be constant and relatively small compared to the wave speed. A multiple time-scales perturbation method is used to construct formal asymptotic approximations of the solutions, and it is shown how different oscillation modes are damped.
2.1 Introduction
M any 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), (Suweken and 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 and Chen 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) where the axial transport of mass can be associated with transverse vibrations. Simple models which 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 and Pakdemirli 1999). The main goal of applied mathematicians, mechanical and civil engineers and, physicists is to reduce the vibrations in these devices because they cause damage to the structure. Investigating transverse vibrations of such systems is a challenging subject which has been studied for many years by many researchers and is still of great interest today. A great deal of research has been done on the transverse vibrations of such systems where linear and nonlinear models have been taken into account. Many contributions on an axially moving continuum can be found in the literature. The interest in studying axially moving systems is 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. A fundamental work was done in (Wickert and Mote 1990), where the moving string and the moving 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 of coupled longitudinal and transverse vibrations of an axially moving strip were first obtained in (Thurman and Mote 1969). After this work on moving strip, the transversal vibrations of a 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 between a pair of pulleys and by using a variational procedure, derived the equations of motion and discussed both the constant and the time-dependent tape velocity. In (Spelsberg-Korspeter et al. 2008) authors considered an axially moving beam in frictional contact with pads and studied the mechanical behavior caused by friction and interpreted damping and nonconser- vative forces as perturbations. In (Chen and Ding 2010) the steady-state transverse response in 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 which also has an intermediate support has been investigated. In all these studies the axial transport velocity is assumed to be either constant or time-varying. In this study, transverse vibrations of an axially moving beam are investigated and explicit asymptotic approximations of the solutions will be constructed, which are valid on a long time-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 to a spring-dashpot-system at other end. It will also be shown in this chapter that the use of boundary damping can be used effectively to suppress the oscillation amplitudes. To our knowledge, the use of boundary damping and the explicit construction of approximations of oscillations for these types of problems have not been previously investigated. The present chapter is organized as follows. Section 2.2 establishes the governing equations of motion. Section 2.3 will discuss the energy of the initial-boundary value problem and the boundedness 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 section 2.4, 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 of inertia I, flexural rigidity EI, and uniform tension T . A stretched beam is simply-supported at x = 0 and attached to a spring-dashpot-system at x = L. The beam travels at the uniform constant transport speed V between two supports that are a distance L apart as shown in Figure 2.1. It is assumed that V , ρ , T , k (the stiffness of the spring), and c (the damping coefficient 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 the time. Gravity and other external forces are neglected. The equation describing the vertical displacement of the beam is given by
x = Lx = 0
utt +2Vuxt +(V 2−C2)uxx + EI ρA
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 = √
T ρA , 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
16 2. On Boundary Damping for an Axially Moving Tensioned Beam
be small compared to wave speed C, to be constant, and O(ε), that is, V = εV , where ε is dimensionless 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 dimensionless quantities are used: u∗(x, t) = u(x,t)
L , x∗ = x L , V ∗ = V
C , t∗ = C L t, µ = EI
ρAC2L2 , k∗ = kL ρAC2 ,
δ ∗ = δ ρAC , f ∗(x) = f (x)
L , and g∗(x) = g(x) C . Then the equation (2.1) in a nondimensional form
becomes utt −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) and henceforth. In this chapter, the initial-boundary value problem (2.5)-(2.8) for u(x, t) will be studied and formal approximations (that is, functions that satisfy the differential equation and the initial and 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 tensioned beam 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 {
1 2 (ut + εVux)
2 + 1 2 (u
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
∫ 1 0
(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 potential energy of the moving beam and the potential energy of the spring, that is,
E(t) = ∫ 1
ku2(1, t). (2.12)
Then, by using (2.11) and (2.12), and the boundary conditions (2.6) and (2.7), it follows that the time-rate of change of the total mechanical energy is
dE dt =−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 equals the 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 support condition at x = 0, material particles enter and exit the span at x = 0 with the transverse velocity 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, where α 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 can not be concluded whether the energy of the belt system decreases or not. For that reason we approximate the solution of the initial-boundary value problem in the section 2.4. For more detailed 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. Conditions such as t > 0, t ≥ 0, 0 < x < 1 will be dropped for abbreviation. Expand the solution in a Taylor 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 of the 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
18 2. On Boundary Damping for an Axially Moving Tensioned Beam
O(ε−1). That is why a two-timescales perturbation method is applied. For a more complete overview of this perturbation method the reader is referred to (Nayfeh 1973) or (Kevorkian and 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)
Substitution of (2.15)-(2.17) into the problem (2.5)-(2.8) yields an initial-boundary value problem 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 solution u(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 the boundary conditions analytically depend on ε . Substituting (2.23) into (2.18)-(2.22), and after 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)
y0(x,0,0) = f (x), (2.28)
y0t (x,0,0) = g(x), (2.29)
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) are linear and homogeneous, the method of separation of variables can be applied. We look for special 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), it follows that
gtt(t,τ) g(t,τ)
=−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)− 1 µ
φ ′′(x)− λ µ
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 combination of sines and cosines in t,
g(t,τ) = σ1(τ)cos √
λ t +σ2(τ)sin √
λ t, (2.41)
20 2. On Boundary Damping for an Axially Moving Tensioned Beam
and it oscillates with frequency √
λ . The values of λ determine the natural frequencies of the oscillations of a vibrating belt. Now by analyzing the boundary-value problem, we can use the fact that the eigenvalues λ are real and positive. The characteristic equation for (2.39) is given by
γ4− γ2
µ − λ
φ(x) = c1sinh(αx)+ c2cosh(αx)+ c3sin(βx)+ c4cos(βx), (2.43)
where c1, c2, c3, and c4 are constants, and where
β =
√ −1+
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 4 m +β 2
m can be numerically computed from (2.45). In Table 2.1 some of these eigenvalues are presented for some fixed values of k and µ . From (2.39), (2.40) and (2.45) the m-th eigenfunction φm(x) corresponding to the m-th eigenvalue (λm) can be determined and is given by (up to a multiplicative constant)
φm(x) = θmsinh(αmx)+ sin(βmx), (2.46)
α2 msinh(αm)
√ β 2
m + 1 µ .
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
) φm(x), (2.47)
where φm(x) is given by (2.46), and where Am(τ) and Bm(τ) are still arbitrary functions which can be used to avoid secular terms in y1(x, t,τ). To prove the orthogonality of the eigenfunctions given by (2.46), we need to use Green’s formula and operator notations. We define the linear operator L = d4
dx4 − 1 µ
d2
dx2 , such that L(φ) = λ µ φ , where φ(x) satisfies the
boundary conditions given in (2.40); for details the reader is referred to (Haberman 2004). It
2.4. Application of the two timescales perturbation method 21
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
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 eigenvalues are orthogonal with respect to the inner product, as defined by
< φm(x),φn(x)>= ∫ 1
0 φm(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
) φm(x), (2.49)
22 2. On Boundary Damping for an Axially Moving Tensioned Beam
where Am(0) and Bm(0) are given by
Am(0) = 1
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 the following transformation:
y1(x, t,τ) = w(x, t,τ)+ (
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
) h(t,τ),
(2.55)
w(x,0,0) =− (
) 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)
2.4. Application of the two timescales perturbation method 23
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
) h(t,τ).
(2.63)
we obtain x4−2x3
= ∞
) φm(x)dx. (2.67)
and where ζm is given by (2.52). By multiplying both sides of (2.63) with φn(x), then by integrating the so-obtained equation from x = 0 to x = 1, and by using the orthogonality properties of the eigenfunctions, we obtain
wntt (t,τ)+wn(t,τ)λn =−2Tntτ (t,τ)− cnhtt(t,τ)+dnh(t,τ)− 2V ζn
Tnt (t,τ)Θnn
− 2V ζn
Tmt (t,τ)Θmn, (2.68)
0 φ ′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) = ∞
δTmttt (t,τ)φm(1). (2.72)
24 2. On Boundary Damping for an Axially Moving Tensioned Beam
Thus, (2.68) with (2.71) and (2.72) can be expressed as
wntt (t,τ)+wn(t,τ)λn =( 2A′n(τ)+
( 2V ζn
Bn(τ) )√
λncos( √
λnt)+
∑∞ m=1 m 6=n {δ (dnTmt (t,τ)− cnTmttt (t,τ))φm(1)− 2V
ζn Tmt (t,τ)Θmn}.
(2.73)
The right hand side of (2.73) contains terms which are the solutions of the homogeneous part corresponding to (2.73). These terms will give rise to unbounded terms, the so-called secular terms, in the solution wn(t,τ) of (2.73). Since it is assumed that y0(x, t,τ), y1(x, t,τ), · · · are bounded 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 to avoid secular terms in the solution of (2.73). In order to remove secular terms, it now easily follows from (2.73) that An(τ) and Bn(τ) have to satisfy
A′n(τ)+ (
V ζn
) An(τ) = 0, (2.74)
) Bn(τ) = 0. (2.75)
An(τ) = An(0)e ( − V
) 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− Θnn ζn
Now, by substituting τ = εt into −Θnn ζn
(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 =−ε Θnn
ζn (V +δ ), (2.79)
where ζn is given by (2.52) and Θnn = 1 2 φ 2
n (1)> 0. In Table 2.2 and in Table 2.3 numerical approximations of λn, Θnn, ζn, and the damping parameter Γn are given for different values
2.4. Application of the two timescales perturbation method 25
Table 2.2: Numerical approximations of Θnn, ζn, Θnn ζn
, and Γn for k = 1
k = 1 n λn Θnn ζn
Θnn ζn
Γn(damping parameter) µ = 0.001 1 4.13205 040564 0.59793 0.67841 -0.67841 (V +δ )ε 2 24.68662 0.50368 0.52101 0.96673 -0.96673 (V +δ )ε 3 67.44960 0.55464 0.51041 1.08666 -1.08666 (V +δ )ε 4 136.98595 0.61296 0.50854 1.20534 -1.20534 (V +δ )ε 5 239.83409 0.68046 0.50868 1.33771 -1.33771 (V +δ )ε
10 1637.47770 0.99062 0.50860 1.94772 -1.94772 (V +δ )ε 25 39720.69922 1.06288 0.50144 2.11965 -2.11965 (V +δ )ε 50 597499.42934 1.01969 0.50020 2.03858 -2.03858 (V +δ )ε µ = 0.01 1 4.25996 0.43698 0.60378 0.72374 -0.72374 (V +δ )ε 2 28.76105 0.69738 0.54328 1.28366 -1.28366 (V +δ )ε 3 95.67926 0.93156 0.53738 1.73352 -1.73352 (V +δ )ε 4 244.43954 1.06460 0.52806 2.01605 -2.01605 (V +δ )ε 5 538.15563 1.10515 0.51879 2.13027 -2.13027 (V +δ )ε
10 8066.66186 1.05297 0.50288 2.09390 -2.09390 (V +δ )ε 25 342894.17678 1.00876 0.50017 2.01683 -2.01683 (V +δ )ε 50 5755328.62344 1.00212 0.50002 2.00417 -2.00417 (V +δ )ε µ = 0.1 1 4.95805 0.71497 0.71021 1.00670 -1.00670 (V +δ )ε 2 54.15242 1.18143 0.58159 2.03137 -2.03137 (V +δ )ε 3 324.65675 1.10561 0.52146 2.12021 -2.12021 (V +δ )ε 4 1225.60427 1.05542 0.50757 2.07934 -2.07934 (V +δ )ε 5 3400.33222 1.03271 0.50341 2.05142 -2.05142 (V +δ )ε
10 72248.27763 1.00658 0.50032 2.01187 -2.01187 (V +δ )ε 25 3374613.36828 1.00090 0.50002 2.00174 -2.00174 (V +δ )ε 50 57333618.60889 1.00021 0.50000 2.00042 -2.00042 (V +δ )ε µ = 1 1 5.79525 1.34103 0.95897 1.39841 -1.39841 (V +δ )ε 2 268.92735 1.05693 0.51685 2.04494 -2.04494 (V +δ )ε 3 2571.66438 1.01508 0.50266 2.01942 -2.01942 (V +δ )ε 4 11006.46538 1.00656 0.50083 2.00976 -2.00976 (V +δ )ε 5 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 )ε 25 33691805.21440 1.00009 0.50000 2.00017 -2.00017 (V +δ )ε 50 573116518.47183 1.00002 0.50000 2.00004 -2.00004 (V +δ )ε
26 2. On Boundary Damping for an Axially Moving Tensioned Beam
Table 2.3: Numerical approximations of Θnn, ζn, Θnn ζn
, and Γn for k = 10
k = 10 n λn Θnn ζn
Θnn ζn
Γn(damping parameter) µ = 0.001 1 8.26224 0.03851 0.54627 0.07049 -0.07049 (V +δ )ε 2 34.26368 0.13351 0.53827 0.24804 -0.24804 (V +δ )ε 3 81.38438 0.25275 0.53087 0.47610 -0.47610 (V +δ )ε 4 154.79829 0.38129 0.52598 0.72491 -0.72491 (V +δ )ε 5 261.40539 0.51567 0.52321 0.98559 -0.98559 (V +δ )ε
10 1674.08333 1.07661 0.51557 2.08817 -2.08817 (V +δ )ε 25 39759.43522 1.09627 0.50183 2.18453 -2.18453 (V +δ )ε 50 597536.20660 1.02435 0.50022 2.04779 -2.04779 (V +δ )ε µ = 0.01 1 8.84812 0.04469 0.54874 0.08145 -0.08145 (V +δ )ε 2 43.10528 0.23295 0.55903 0.41669 -0.41669 (V +δ )ε 3 123.13015 0.67117 0.57597 1.16529 -1.16529 (V +δ )ε 4 282.39881 1.16336 0.57122 2.03663 -2.03663 (V +δ )ε 5 580.19837 1.35568 0.54803 2.47373 -2.47373 (V +δ )ε
10 8105.43489 1.11801 0.50481 2.21472 -2.21472 (V +δ )ε 25 342930.55038 1.01277 0.50021 2.02468 -2.02468 (V +δ )ε 50 5755364.70709 1.00261 0.50002 2.00513 -2.00513 (V +δ )ε µ = 0.1 1 13.67452 0.13025 0.61250 0.21266 -0.21266 (V +δ )ε 2 90.48572 1.31507 0.73154 1.79761 -1.79761 (V +δ )ε 3 367.04961 1.46710 0.57577 2.54806 -2.54806 (V +δ )ε 4 1265.23389 1.20961 0.52082 2.32250 -2.32250 (V +δ )ε 5 3438.37666 1.10508 0.50784 2.17603 -2.17603 (V +δ )ε
10 72284.61491 1.01385 0.50051 2.02562 -2.02562 (V +δ )ε 25 3374649.40672 1.00131 0.50002 2.00253 -2.00253 (V +δ )ε 50 57333654.61732 1.00026 0.50000 2.00052 -2.00052 (V +δ )ε µ = 1 1 27.26977 1.12354 1.11675 1.00608 -1.00608 (V +δ )ε 2 308.15032 1.31972 0.57508 2.29484 -2.29484 (V +δ )ε 3 2608.70728 1.06540 0.50822 2.09633 -2.09633 (V +δ )ε 4 11042.89713 1.02338 0.50209 2.03823 -2.03823 (V +δ )ε 5 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 )ε 25 33691841.21826 1.00013 0.50000 2.00025 -2.00025 (V +δ )ε 50 573116554.47267 1.00003 0.50000 2.00005 -2.00005 (V +δ )ε
2.4. Application of the two timescales perturbation method 27
of k and µ . From these Tables it can be seen that the damping parameter Γn for large values of n tends to −2(V +δ )ε . In Appendix B, it also has been analytically shown that Γn tends to this value for large n. Now, from (2.73) with (2.74)-(2.78), we obtain
wntt (t,τ)+wn(t,τ)λn = ∞
e− Θmm ζm
(V+δ )τ (
1mn =
wn(t,τ) = Dn(τ)cos( √
λnt)+En(τ)sin( √
λmt) ) ,
(2.83)
where Dn(τ) and En(τ) are still arbitrary functions which can be used to avoid secular terms in y2(x, t,τ), and where 1mn and 2mn are given by (2.81) and (2.82), respectively. Thus, (2.62) with (2.83) can be expressed as
w(x, t,τ) = ∑∞ m=1{Dm(τ)cos(
√ λmt)+Em(τ)sin(
(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
+ V ζm
√ λn
(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) = ∞
Hp(t,τ)}φm(x), (2.87)
28 2. On Boundary Damping for an Axially Moving Tensioned Beam
where,
) 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 contains infinitely many undetermined functions Dm(τ) and Em(τ), m = 1,2, · · · . These functions can be used to avoid secular terms in the solution of y2(x, t,τ). At this moment, we are not interested 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) for u(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 moving tensioned beam equation with non-classical boundary condition has been studied. One end of the beam is assumed to be simply-supported, whereas the other end of the beam is assumed to be attached to a spring-dashpot system. Formal asymptotic approximations of the exact solution have been constructed using a two-timescales perturbation method. By using the energy integral, it has been shown that the solutions are bounded for times t of O( 1
ε ). Some damping rates of the modes are given in Table 2.2 and in Table 2.3 for different values of the parameters µ and k. It has been shown that the damping parameter (Γn) essentially depends on two system parameters V and δ > 0. The most optimal way to place a damper depends on the direction of the axial velocity V . If a damper is placed at x = L and the belt moves with velocity V < 0, then to have damping in the system we should have δ > −V , whereas if a damper is placed at x = L and the belt moves with velocity V > 0, then we have always damping since V +δ > 0. To have always damping (regardless of the sign of V ) δ should be larger than |V |. For a nonmoving belt (V = 0) only a dashpot (δ ) is responsible to generate damping in the system. In the absence of a dashpot, oscillations can still be damped when V > 0 and the spring with stiffness k is placed at x = L. For the damping parameter Γn it also has been analytically and numerically shown that all solutions (up to O(ε)) are uniformly damped when δ +V > 0.
Submitted as: S. H. Sandilo and W. T. van Horssen – ”On Variable Length Induced Vibrations of a Vertical String”, Elsevier, Journal of Sound and Vibration, February 2013.
Chapter 3 On Variable Length Induced Vibrations of a
Vertical String
To myself I am only a child playing on the beach, while vast oceans 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. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral 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 this chapter an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the cable, and the suspension of the car against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of 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 amplitudes are assumed and it is also assumed that the cable mass is small compared to total mass of the cable and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the cable. It will also be shown that the Galerkin’s truncation method can not be applied to this problem in all cases to obtain approximations valid on long time scales.
3.1 Introduction
M any 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), (Ponomareva and van Horssen 2009), and (Suweken and van Horssen 2003a)), chair lifts, power-transmission chains, 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 and Xu 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 speed during operation. The traveling, tensioned Euler-Bernoulli beam and the traveling flexible string are the most commonly used models for such types of axially moving continua. They are classified in the category of one-dimensional continuous systems and consequently the displacement field depends on time and on a single spatial co-ordinate. The last few decades have seen an extensive research effort on the dynamics of translating media, where most studies were restricted to cases with constant span length and transport velocity. Vibrations of horizontal and vertical translating strings and beams have been studied by many researchers. The forced response of translating media with variable length and tension was analyzed 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). By transforming 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 (Vesnitskii and 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 systems emulating an elevator, the reader is referred to ((Yamamoto et al. 1978), and (Terumichi et al. 1997)). In these two studies a constant transport velocity was assumed. The natural frequencies 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 horizontally and vertically translating strings and beams with arbitrary varying-length and with various boundary 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) to dissipate 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 understanding of 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 subject of this chapter. Due to small allowable vibrations the lateral and vertical cable vibrations in elevators can be assumed to be uncoupled and only lateral cable vibrations in elevators are considered here. The elevator car is modelled as a rigid body of mass m attached at the lower end of the cable, and the suspension of the car against the guide rails is assumed to be rigid, where external excitation is not considered at the boundaries. This is considered to be a basic and simple model of an elevator cable from the practical viewpoint. The initial-
3.2. The governing equations of motion 31
boundary value problems will be studied, and explicit asymptotic approximations of the solutions, which are valid on a long time-scale, will be constructed as for instance described in ((Nayfeh 1991), and (Kevorkian and Cole 1996)). Two cases for varying-length will be considered (i) l(t) = l0 + vt, where l0 is the initial cable length and v denotes the constant cable velocity, and (ii) l(t) = l0 + β sin(ωt), where β defines a length variation parameter and ω signifies the angular frequency of length variation and l0 > |β |. Regarding both cases of varying-length different dimensionless parameters will be used to obtain dimensionless equations of motion. For the first case, it is assumed that v
√ ρ
mg = O(ε) and ρL m = 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-boundary value 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, it will 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, the explicit construction of approximations of oscillations for these types of problems have not been given before. The outline of the chapter is as follows. In section 3.2, the generalized Hamilton’s principle is used to derive a model for an elevator suspension system in which the hoisting rope and the car 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-boundary value problems. It turns out for the case with the harmonically varying length that there are infinitely many values of ω that can cause internal resonances. In this chapter we only inves- tigate the resonance case ω = π
l0 and we also study a detuning case for this value. Finally, in
section 3.5, we make some remarks and draw some conclusions.
3.2 The governing equations of motion
The vertically translating cable in elevators has no sag and will be modelled as a taut string with fixed boundaries in horizontal direction, as shown in Figure 3.1. The elevator car is modelled as a rigid body of mass m attached at the lower end x = l(t), and suspension of the car against the guide rails is assumed to be rigid. During its motion the cable of density ρ has a variable length l(t) and an axial velocity v(t) = l(t), where the over dot denotes time differentiation. 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 designates extension or retraction of the cable, respectively. The lateral 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 3.1, the lateral dis-
32 3. On Variable Length Induced Vibrations of a Vertical String
l (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, where 0≤ x≤ l(t), is described by u(x, t). The equations of motion for vertical string with variable-length and tension are obtained by using Hamilton’s principle. The total kinetic energy associated with the lateral vibration of the string of length l(t) with end mass is
T = 1 2
∂ ∂x
, (3.2)
defines the differentiation with respect to the motion, and ρ is the mass per unit length. The potential energy for the cable of length l(t) is
V = 1 2
3.2. The governing equations of motion 33
where P(x, t) is the axial force. The axial force in the vertically translating string in Figure 3.1, 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 compressive during deceleration ( v < 0) and acceleration ( v > 0) of the string, respectively, and vanishes during 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 of motion (with the appropriate boundary and initial conditions),
ρ D2u(x, t)
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
. (3.9)
A new independent non-dimensional spatial coordinate ξ = x l(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) = u(x, t). The partial derivatives of u(x, t) with respect to x and t are related to those of u(ξ , t) with respect to ξ and t. Thus, we have
ux = 1
uxt = 1
l2(t) uξ ,
v2ξ 2
(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 will study 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 parameters will occur, and a multiple time-scales perturbation method will be used to construct accurate approximations of the solutions of the initial-boundary value problems for u(ξ , t), which are valid on long time-scales.
34 3. On Variable Length Induced Vibrations of a Vertical String
3.3 The case l(t) = l0 + vt
To put the equations (3.6)-(3.8) into a non-dimensional form, the following dimensionless parameters will be used: u∗ = u
L , x∗ = x L , t∗ = t
L
√ ρ
L , h∗ = h √
ρ mg , where
L is the maximum length of string. The equations of motion in non-dimensional form then become ( v = 0):
∂ 2u(x,t) ∂ t2 +2v ∂ 2u(x,t)
) ∂ 2u(x,t) ∂x2 +µ ∂u(x,t)
u(x,0) = f (x), ∂u(x,0)
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 value problem for u(ξ , t):
utt + 2v(1−ξ )
l(t) uξ t + (
(3.14)
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) will be 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 √
√ 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 of the 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 be applied. Using such a two-timescales perturbation method the function u(ξ , t;ε) is supposed to be a function of ξ , the fast time t+ =
∫ t 0
ds l(s) , and the slow time t = εt, where the fast time
t+ = ∫ t
0 ds
l(s) is justified in the Appendix C. Thus, the function u(ξ , t;ε) can be written in terms 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 be expressed as
∂ ∂ t
= 1
l(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
∂ t+∂ t +2(ξ −1) ∂ 2y ∂ξ ∂ t+ −µ0 l(t) ∂y
∂ξ +µ0 l(t)(1−ξ ) ∂ 2y ∂ξ 2 +
∂y ∂ t+
(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)
1 l0
) = 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 be approximated 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 the boundary conditions depend analytically on ε . Substituting (3.25) into (3.21)-(3.24) and after equating the coefficients of like powers in ε , it follows from the problem for y(ξ , t+, t;ε) that the O(1)-problem is:
∂ 2y0
∂ξ 2 = 0, t+ > 0, 0 < ξ < 1, (3.26)
36 3. On Variable Length Induced Vibrations of a Vertical String
y0(0, t+, t) = y0(1, t+, t) = 0, t+ > 0, t > 0, (3.27)
y0(ξ ,0,0) = f (ξ ), and, 1 l0
∂y0(ξ ,0,0) ∂ t+
∂ 2y1 ∂ t+2 − ∂ 2y1
(3.29)
y1(0, t+, t) = y1(1, t+, t) = 0, t+ > 0, t > 0, (3.30)
y1(ξ ,0,0) = 0, 1 l0
∂y1(ξ ,0,0) ∂ t
=−∂y0(ξ ,0,0) ∂ t
. (3.31)
It is observed that the O(1)-problem is a well-known initial-boundary value problem and can be solved by using the method of separation of variables, where the boundary-value problem only has positive eigenvalues λn which are given by (nπ)2, n = 1,2,3, · · · . For details the reader is referred to (Haberman 2004). The solution of the O(1)-problem is given by
y0(ξ , t+, t) = ∞
) φn(ξ ), (3.32)
where An0(t) and Bn0(t) are still arbitrary functions of t which can be used to avoid secular terms in y1(ξ , t+, t), and where φn(ξ ) is given by
φn(ξ ) = sin( √
λnξ ). (3.33)
< φn(ξ ),φm(ξ )>= ∫ 1
0 φn(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) = 1 ζn
√ λnBn0(0) =
where,
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) = ∞
where wn(t+, t) are the generalized Fourier coefficients. Substituting (3.38) into the partial differential equation (3.29), we obtain
∑∞ n=1
( wnt+ t
2l(t) y0t+
(3.39)
By multiplying both sides of (3.39) with &ph