On parametric transversal vibrations of axially moving strings
Ali, Rajab DOI 10.4233/uuid:392e166c-9a34-4631-949f-ce128cfb4b14 Publication date 2016 Document Version Final published version Citation (APA)Ali, R. (2016). On parametric transversal vibrations of axially moving strings. https://doi.org/10.4233/uuid:392e166c-9a34-4631-949f-ce128cfb4b14 Important note
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axially moving strings
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen
op donderdag 27 October 2016 om 12:30 uur
door
Rajab ALI
Master of Science in Mathematics, The University of Sindh, Jamshoro. geboren te Sanghar, Sindh-Pakistan.
copromotor: Dr. ir. W. T. van Horssen
Composition of the doctoral committee: Rector Magnificus, Chairman
Prof. dr. ir. A. W. Heemink, Promotor, Delft University of Technology Dr. ir. W. T. van Horssen, Copromotor, Delft University of Technology Independent members:
Prof. dr. ir. C. W. Oosterlee, Delft University of Technology Prof.dr.ir. Peter Steeneken, Delft University of Technology
Prof.dr.J.Molenaar, Wageningen University, The Netherlands Prof. dr. A. K. Abramian, Russian Academy of Sciences, Russia Prof.dr.Igor V. Andrianov, RWTH Aachen University, Aachen
This thesis has been completed in fulfillment of the requirements of the Delft University of Technology, for the award of the PhD degree. The research described in this thesis was carried out in the section of Mathematical Physics at the Delft Institute of Applied Mathematics (DIAM) Delft University of Technology, The Netherlands. This research was supported by Quaid-e-Awam University Nawabshah Sindh Pakistan under the faculty development program of Higher Educa-tion Commission (HEC) of Pakistan and the Delft University of Technology, the Netherlands. ISBN 978-94-6186-722-3
Copyright c 2016 by Rajab Ali e-mail: R.Ali@tudelft.nl
All rights reserved. No part of this publication may be reproduced in any form or by any means of electronic, mechanical, including photocopying, recording or otherwise, without the prior written permission from the author.
Aristotle
Dedicated to my Parents and Teachers with gratitude and love
List of Figures . . . vi
List of Tables . . . viii
1 Introduction 1 1.1 Background . . . 1
1.2 Aim of the Research . . . 4
1.3 The Mathematical Model . . . 4
1.4 Applied Mathematical Methods . . . 5
1.5 Outline of the Thesis . . . 7
2 On resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity1 9 2.1 Introduction. . . 9
2.2 Equations of motion . . . 11
2.3 Harmonically varying velocity about a low mean speed . . . 12
2.3.1 Application of the two timescales perturbation method. . . 13
2.3.2 A general resonance case: Ω = mπ (m is positive odd integer) . . . 14
2.3.3 Application of the truncation method . . . 15
2.3.4 Analysis of the infinite dimensional system (2.28) . . . 16
2.3.5 Near resonance case : Ω = mπ + εδ. . . 17
2.3.6 Analysis of the infinite dimensional system(2.38) . . . 18
2.4 Harmonically varying velocity about a relatively high constant mean speed . . . . 20
2.4.1 Application of the two timescales perturbation method. . . 20
2.4.2 Ω = k∗π(1 − V2 0), a general resonance case. . . 22
2.4.3 Application of the truncation method . . . 24
2.4.4 Analysis of the infinite dimensional system (2.67) . . . 25
2.5 Conclusions and remarks. . . 28
3 On the asymptotic approximation of the solution of an equation for a non-constant axially moving string2 31 3.1 Introduction . . . 31
1This chapter is slightly revised version of published article [1]- “On resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity”, J. Sound Vib., vol. 344, p: 1-17, February 2015
2
This chapter is slightly revised version of published article [2]-“On the asymptotic approximation of the solution of an equation for a non-constant axially moving string”, J. Sound Vibr., vol. 367, p:203-218, January 2016.
3.2 Equations of motion . . . 32
3.3 The construction of asymptotic approximations . . . 34
3.3.1 The non-resonant case (up to O(ε) on timescales of 1ε) . . . 37
3.3.2 Ω = (2m − 1)π, a pure resonance case. . . 37
3.3.3 Specific initial conditions . . . 39
3.3.4 Energy of the system . . . 40
3.4 Results and discussion . . . 40
3.5 Conclusion . . . 41
4 On parametric stability of a non-constant axially moving string near-resonances3 47 4.1 Introduction . . . 47
4.2 Equations of motion . . . 49
4.3 A perturbation approach to construct approximations . . . 50
4.3.1 Ω = (2m − 1)π + εδ, a detuned resonance case . . . 52
4.3.2 Case 1: |δ|< 2α . . . 55
4.3.3 Case 2: |δ|= 2α . . . 56
4.3.4 Case 3: |δ|> 2α . . . 59
4.4 Stability analysis . . . 61
4.5 Conclusion . . . 63
5 Conclusions and Future Work 65
A Infinite dimensional system of coupled ODEs (Eq.(2.28)) 69
B Infinite dimensional system (Eq.(2.33)) 71
C Proof of Integral (Eq.(3.46)) 73
D The infinite dimensional system (Eq.(3.47)) 77
E The infinite dimensional system (Eq. (4.40)) 81
Bibliography 85
Summary 91
Samenvatting 93
Acknowledgments 95
List of publications and presentations 97
About the author 99
3
This chapter is slightly revised version of accepted article [3]-“On parametric stability of a non-constant axially moving string near-resonances”, J. Vibr. Acoust., vol.139, p: 0110051-01100512, October 2016.
1.1 A moving horizontal conveyor belt system.. . . 2
1.2 A moving cable car system. . . 3
1.3 A schematic model of moving belt system between two fixed points. . . 4
3.1 Integration in the characteristic ξ (or σ ) direction. . . 36
3.2 m = 1, V0= 0.8, α = 0.5, = 0.01, x = 0.05. (a) The unstable first order approxi-mation w0. (b) The unstable numerical solution u. . . 42
3.3 m = 1, V0= 2, α = 1, = 0.01, x = 0.05. (a) The unstable first order approximation w0. (b) The unstable numerical solution u.. . . 42
3.4 m = 1, V0= 0.8, α = 0.5, = 0.02, x = 0.05. (a) The unstable first order approxi-mation w0. (b) The unstable numerical solution u. . . 43
3.5 m = 1, V0= 2, α = 1, = 0.02, x = 0.05. (a) The unstable first order approximation w0. (b) The unstable numerical solution u.. . . 43
3.6 Logarithmic scale of energy with = 0.01, V0= 0.8, α = 0.5. (a) Approximated energy of system. (b) Energy of the system. . . 44
3.7 Logarithmic scale of energy with = 0.01, V0=2, α = 1. (a) Approximated energy of system. (b) Energy of the system. . . 44
3.8 Logarithmic scale of energy with = 0.02,V0= 0.8, α = 0.5. (a) Approximated energy of system. (b) Energy of the system. . . 45
3.9 Logarithmic scale of energy with = 0.02, V0= 2, α = 1. (a) Approximated energy of system. (b) Energy of the system. . . 45
4.1 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =0.01, x = 0.05. (a) The unstable first order asymptotic approximation (v0). (b) The unstable numerical solution (u). . . 56
4.2 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =0.01. (a) Approximated energy. (b) Energy of the system. . . 57
4.3 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =2α, x = 0.05. (a) The unstable first order asymptotic approximation (v0). (b) The unstable numerical solution (u). . . 58
4.4 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =2α. (a) Approximated energy. (b) Energy of the system. . . 58
4.5 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =-2α, x = 0.05. (a) The unstable first order asymptotic approximation (v0). (b) The unstable numerical solution (u). . . 59
4.6 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =-2α. (a) Approximated energy. (b) Energy of the system. . . 60
4.7 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =15, x = 0.05. (a) The stable first order
asymptotic approximation (v0). (b) The stable numerical solution (u). . . 61
4.8 m = 1, = 0.01, V0= 0.8, α = 0.5, δ =15. (a) Approximated energy. (b) Energy of
the system. . . 62
4.9 Approximate instability region in the (α, Ω)-plane for ε = 0.1 : the amplitude α of the velocity fluctuation of the belt versus the frequency Ω of the velocity fluctuation of the belt. The boundaries of the instability region are given by Ω = (2m−1)π +εδ with |δ|= 2α and m = 1, 2, .... . . 62
2.1 Approximations of the eigenvalues of the truncated system (2.28) for m = 3 and m = 5. . . 16
2.2 Approximations of the eigenvalues of the truncated system (2.67) for k∗ = 2. . . . 25
2.3 Approximations of the eigenvalues of the truncated system (2.67) for k∗ = 4. . . . 26
2.4 Approximations of the eigenvalues of the truncated system (2.67) for k∗ = 3 and k∗ = 5. . . 27
Chapter
1
Introduction
Imagination is more important than knowledge. Albert Einstein
1.1
Background
Mechanical structures such as belts, cables and tapes are often referred to as axially moving ma-terials or axially moving continua, due to their larger dimension in axial direction in comparison to the other two directions. Axially moving continua have wide range of applications in many branches of engineering, such as aerospace, architectural, civil, chemical, automotive and mechan-ical engineering. Despite the numerous engineering applications of such systems, vibrations have limited their stability and applications because of excessive amplitudes resulting mainly from res-onances. Vibrations, most commonly, transverse vibrations are often caused by the eccentricity of a support roller, irregular speed of the driving motor, material non-uniformity; and/or vibrations developed by natural phenomena such as an earthquake and wind-forces.
Severe vibrations are, in general, undesirable attribute in mechanical structures as it reduces efficiency, creating unwanted sound vibrations and greatly affecting human comfort. In addition, it is evident that the severity of the vibrations in the systems can also lead to operational and maintenance problems, including an increase of the energy consumption of the system and even damage to structures [4]. The Tacoma Narrow suspension bridge in USA collapsed on 7 November 1940 due to 42 mile-per-hour wind, is a classic example of a structural failure. The main objec-tive of the mechanical or civil engineers, physicists and applied mathematicians is to understand and mitigate the structural and mechanical vibrations through testing (experimentally and ana-lytically) and predicting the dynamic responses of structural or mechanical systems at multiple operational conditions.
Axially moving continua can be modeled as flexible strings and beams, depending on the bending stiffness of the belt. If the bending stiffness is negligible, the systems are in the class of string-like [5, 6, 7, 8, 9]; otherwise, they are in the class of beam-like [10, 11, 12,13, 14]. In addition to this, these systems behave linearly [11, 15, 16, 17] at very small amplitudes, while nonlinearities in these systems occur at large amplitudes [18,19,20,21].
Several studies have been devoted to dynamic investigations of axially moving systems. The relevant research on the vibration analysis of axially moving materials undertook in the last decade of nineteenths century, when Skutch [22] published his study in German language on the transverse vibrations of a traveling elastic string moving through two pinholes and determined its fundamental
Figure 1.1: A moving horizontal conveyor belt system.
resonance frequency by superposition of two waves propagating in opposite directions. In the early 1950s, one of the first papers in the English-language on this subject was published by Sack [23] to investigate the vertical vibrations of an undamped string subject to a harmonic excitation which travels under constant tension at a constant speed over fixed supports. Archibald and Emslie [24] derived the same equations of motion for the traveling strings by using a variational approach and examined the linear transverse oscillations in the string with a constant speed along the longitudinal direction where one or both of its ends are sinusoidally excited. Mahalingam [25] studied the parametric transverse displacement of the power transmission chains under tension fluctuations. Transversal vibrations of a moving strip subjected to constant speed and tension are examined by Thurman and Mote [26]. Swope and Ames [27] determined the response of the transverse vibrations of a translating string with a constant speed and examined the properties of wave propagation in the string. Recently, Gaiko and van Horssen [28] analyzed the transverse oscillations of a traveling string under a constant speed and examined the effects of damping at boundaries. While, the transverse vibrations in an axially moving string under internal damping is examined in Refs. [29,30]. In addition, Akaya and van Horssen [31] studied the various damping properties of waves in a semi-infinite string by means of D’Almebert’s method.
All above-mentioned researchers considered the axial speed of the moving system to be con-stant. Axially moving system moving with a constant axial speed, however, have limited practical applications. In reality, there are various components of belt systems which can vary the axial speed and cause the accelerations and decelerations in motion, e.g. the eccentricity of pulleys and other belt imperfections. The systems, therefore, often exhibit transverse vibrations due to axial variation speed termed as parametric vibrations.
The time-dependent velocity of axially moving continua consists of a constant mean speed along with some (small) harmonic variations. These harmonic variations in the time-dependent velocity may cause parametric resonances (instability) which adversely effect the response of the system and may lead to failure of the system. In other words, parametric resonance is generally excited by the time-varying coefficients (i.e. speed variation) in the governing equations of motion. Resonant vibrations in mechanical structures occur when the excitation frequency (frequency of velocity variations) coincides with one of the natural frequencies of the system. In order to achieve an optimal design of the systems, the major task is to evaluate and assess all possible resonance frequencies and to keep them away from the natural frequencies of the system. The most fundamental and crucial step in analyzing the oscillatory motion in structural or mechanical systems is the development of mathematical models.
Figure 1.2: A moving cable car system.
be described in terms of the second-order (string-like) and fourth-order (beam-like) (non)linear hyperbolic partial differential equations with variable-coefficients. Such partial differential equa-tions include Coriolis acceleration component, because the axially moving continua belongs to the class of distributed gyroscopic systems [32]. Since closed-form (analytic) solutions of partial differential equations with Coriolis acceleration term are in general not available; several other numerical and approximate techniques have been employed to facilitate the vibration analysis, and to investigate the (in)stability of an axially moving continua.
There are comprehensive studies on the mathematical analysis of parametric transverse vi-brations of axially moving systems subject to the speed variation. Mote [33] used the Laplace transform technique to investigate the parametric vibrations of an axially moving string with re-spect to the time-varying speed driven harmonically at one end by replacing the varying speed by its time-averaged values. Wickert and Mote [34] developed a modal analysis solution to the linear transverse vibrations of axially moving strings and beams under arbitrary excitations and initial conditions. They obtained the natural frequencies using the Green’s function method. Zhu and Guo [35] determined the free and forced responses of a translating string with an arbitrarily velocity profile by using the method of characteristic transformations. Van Horssen [5] employed the method of Laplace transforms to determine the exact responses of an axially moving string due to arbitrary lateral vibration of supports. Zhu and Zheng [36] investigated parametric instability of a translating string in terms of exact response with a sinusoidally varying length by means of characteristic transformations. Pellicano et al. [37] used the Galerkin’s truncation method to investigate the parametric instability in power transmission belts. Zhu et al. [38] used the characteristic transformation without discretizing the governing partial differential equations to investigate the parametric instability behavior in terms of bounded displacement and unbounded vibratory energy of a translating string under sinusoidally varying speed.
The method of multiple scales is a powerful tool to investigate the parametric vibrations of axially moving strings under speed variation. The development of an approximate analytic method, i.e. the method of multiple scales in investigating the axially moving system, started in the early 1970s (see for instance [39]). Pakdemirli and co-investigators [40, 41, 42] used the method of multiple scales based on the truncation method to construct the approximations of the governing partial differential equation of an axially moving string subject to the time-dependent velocity. Ghayesh et al. [8, 43] approximated the nonlinear transversal vibrations of an axially moving string by means of the perturbation method in association with the Galerkin truncation method. Chen et al. [9, 44, 45, 46] examined the parametric instability in the axially moving string by using the method of multiple scales based on the truncation method. Parker and Lin
Figure 1.3: A schematic model of moving belt system between two fixed points.
[47] studied the parametric vibrations of a translating string using the perturbation method. In most of the references mentioned above, the method of multiple scales based on the trun-cation method has been used without any valid justifitrun-cation of mode-truntrun-cation to approximate the transverse vibrations of an axially moving string. However, one should be careful in using the truncation method for like problems. The non-applicability of truncation method for string-like problems, for instance, has been proven with mathematical justification in Refs. [48,49,50]. The detailed comments on this mathematical justification for the truncation method are presented in Chapter 2and Chapter 3of this thesis.
1.2
Aim of the Research
Most of the previous studies on the subject of stability analysis of an axially moving string focused only on determination of the instability for the lower resonances and the (non)applicability of the truncation method. The dynamic (in)stability of an axially moving string with time-dependent velocities at lower resonance frequencies have been extensively studied (see for instance in Refs.[2,3,48,49,50,51]); however, their stability analysis for higher resonances has developed slowly. In understanding the dynamic (in)stability in the axially moving system, all resonance frequencies must be taken into account.
The main goal of the present thesis is to gain a better insight into the linear dynamics of axially moving strings which are fixed between two supports. The transverse vibrations of conveyor belt systems, for instance, can be modeled by these string-like problems. The dynamic (in)stability in the parametric transverse vibrations of the axially moving string under speed variations will mainly be studied at higher resonances and the (non)applicability of truncation method will also be discussed in detail.
1.3
The Mathematical Model
In this thesis, a belt moving with an axial velocity V between two fixed points that are a distance L apart, as shown in Figure.1.3. The transverse displacement of the belt can be modeled as an axially moving string. The mathematical model of an axially moving string is based on the following assumptions:
• The density ρ (mass m per unit length) of the string is assumed to be constant. • The string is perfectly elastic and offers no resistance to bending.
• The string is tightly stretched, which implies that the tension T0 in the string is regarded
as a constant. In addition, the tension T0 is assumed as large compared to the weight of the
string so that the effect of gravity is negligible.
• Effect due to internal viscosity of the string is also neglected.
• Only the motion of portion of the string between the pulleys are determined. • It is assumed that no external forces are acting on the string.
• The displacement slope is small, | ux | 1, which ensures that the transverse displacement
is small compared to the length of the string.
Using these assumptions the governing equations of motion for an axially moving string can be written in the form [48,52]:
utt+ 2V uxt+ Vtux+ (V2− c2)uxx= 0, t > 0, 0 < x < L, (1.1)
where
u(x, t) : the displacement in the transversal (lateral) direction x : the axial position along the string
t : the time
L : the distance between the pulleys c : wave speed
V : the time-varying velocity of a belt ρ : the mass density of a belt
T0 : the tension of a belt
and where c = qT0
ρ. The time-dependent velocity V is assumed to be a harmonically varying
function about a constant mean speed, that is, V (t) = V0 + εα sin(Ωt) where the mean speed
V0 = O(ε) or V0 = O(1) with ε (i.e. 0 < ε 1) and V0, α, Ω are all positive constants. The
terms utt, uxt, uxx on the left hand side of Eq. (1.1) represent, respectively, the accelerations of
local inertia, Coriolis acceleration associated with the rotation, and the centrifugal acceleration associated with the curvature. The velocity fluctuation frequency Ω may give rise to all kinds of resonances in the system. Furthermore, the belt is fixed at both ends, boundary constraints for the belt system are
u(0, t) = 0, u(L, t) = 0, t > 0, (1.2) and the initial conditions are given by
u(x, 0) = φ(x), and, ut(x, 0) = ψ(x), 0 < x < L, (1.3)
where φ(x) and ψ(x) represent, respectively, the initial displacement and velocity of the belt system.
1.4
Applied Mathematical Methods
In this thesis, the two timescales perturbation method, the Fourier-mode expansion method, the Laplace transform method and the method of characteristic coordinates will be explored. The
motion of an axially moving string is described by a second order homogeneous linear partial differential equations with variable coefficients. To obtain the solution in closed (analytic) form is usually not possible, an analytic-approximate method, such as, the method of multiple scales, is often used to construct approximations for the solutions of these differential equations. For a complete overview of the method of multiple scales, the interested reader is referred to standard texts such as [39, 53, 54]. In order to apply the multiple timescales perturbation method, the following steps are used:
The original initial-boundary value problem is first converted into a perturbation problem by introducing a small dimensionless parameter ε (where 0 < ε 1). In this thesis we will convert the original problem into a perturbation problem by the inclusion of the velocity function V (t) into the governing equations of motion.
The solution of the perturbation problem is then presumed in the form:
u(x, t) = u(x, t; ε). (1.4) The assumed solution close to ε = 0 is expanded into a power series in ε as follows
u(x, t; ε) = u0(x, t) + εu1(x, t) + ε2u2(x, t) + ..., (1.5)
where all u0is for i = 0, 1, 2, ... are O(1) on time and space scales. The straight forward expansion (1.5) may break down when the O(εi+1) term dominates the O(εi), for some i = 0, 1, 2, ... as the time t becomes large because of resonances leading to unbounded (so-called secular) terms. These secular terms should be avoided, because these terms may cause non-uniformities and inconsistencies in the asymptotic solutions. To develop an asymptotic solution of the perturbation problem valid for all time t up to O(ε−1), additional different timescales t0= t, t1 = εt, t2= ε2t,...,
should be introduced. The solution u(x, t; ε) is then assumed to be a function of these timescales: u(x, t; ε) = w(x, t0, t1, ...). This new function can be expanded in power series in ε:
w(x, t0, t1; ε) = w0(x, t0, t1, ...) + εw1(x, t0, t1, ...) + ..., (1.6)
where wi = O(1) for i = 0, 1, 2, ..., and explicit computations of all wi guarantee the elimination of
secular terms. Based on the expansion (1.6), the following transformations for the time derivatives are introduced: d(.) dt = ∂(.) ∂t0 + ε∂(.) ∂t1 + ε2∂(.) ∂t2 + ..., (1.7) d2(.) dt2 = ∂2(.) ∂t20 + 2ε ∂2(.) ∂t0∂t1 + ε2∂ 2(.) ∂t21 + 2 ∂2(.) ∂t1∂t2 + ..., (1.8) and so on. Substitution of (1.6)-(1.8) into a perturbation problem and equating the coefficients of like powers of small parameter ε yields the sequence of differential equations, which can be solved sequentially (if possible). Such ordinary and/or partial differential equations can also be solved by using the method of Laplace transform. The interested reader can view the method of Laplace transform in Refs. [55,56,57].
The method of characteristic coordinates was discovered by the French mathematician Jean Le Rond d’Alembert in 18th century to study the wave equation. In solving the equations of motion governing transverse vibrations of axially moving strings, the method of characteristic coordinates is of great interest, in the sense that it avoids the problem of convergence and computational difficulties of infinite series (as has been noticed in [1,48,49,50]) in the Fourier series approach. A crucial requirement for the applicability of the method of characteristic coordinates to the governing equations of motion for the axially moving strings over the finite domain is that the
initial-boundary value problem be first replaced into an initial value problem. This replacement requires for small V (t) an odd and 2-periodic extension of the dependent variable u(x, t) as well as the initial values φ(x) and ψ(x) on the infinite interval −∞ < x < ∞ (method of reflection), so that the Dirichlet type boundary conditions are satisfied. Having transformed the initial-boundary value problem to an initial-value problem, the solution of the governing equations of motion is assumed to be a function of the characteristic coordinates: u(x, t) = v(σ, ξ), where σ = x − t, ξ = x + t. The expansion of the assumed solution close to ε = 0 into a power series in the small parameter ε yields v(σ, ξ) = v0(σ, ξ) + εv1(σ, ξ) + .... This straightforward expansion causes
secular terms. To obtain secular-free approximations valid on long timescales, the two timescales perturbation method is used. By introducing σ = x − t, ξ = x + t and τ = εt (0t0 a fast-scale over which oscillations occur and 0τ0 a slow-scale over which amplitudes evolve ); the solution is then assumed in the form:
u(x, t) = w(σ, ξ, τ ) ≡ w0(σ, ξ, τ ) + εw1(σ, ξ, τ ) + .... (1.9)
The introduction of σ, ξ, and τ leads to the following transformations: ∂(.) ∂t = − ∂(.) ∂σ + ∂(.) ∂ξ + ε ∂(.) ∂τ , (1.10) ∂2(.) ∂t2 = ∂2(.) ∂σ2 + ∂2(.) ∂ξ2 − 2 ∂2(.) ∂σ∂ξ + 2ε ∂2(.) ∂ξ∂τ − ∂2(.) ∂σ∂τ + ε2∂ 2(.) ∂τ2 , (1.11) ∂(.) ∂x = ∂(.) ∂σ + ∂(.) ∂ξ , (1.12) ∂2(.) ∂x2 = ∂2(.) ∂σ2 + ∂2(.) ∂ξ2 + 2 ∂2(.) ∂σ∂ξ, (1.13) ∂2(.) ∂x∂t = − ∂2(.) ∂σ2 + ∂2(.) ∂ξ2 + ε ∂2(.) ∂σ∂τ + ∂2(.) ∂ξ∂τ . (1.14) Substitution of Eqs. (1.9)-(1.14) into the perturbation problem and equating coefficients of like powers of ε produces the sequence of differential equations, which can be solved one by one.
1.5
Outline of the Thesis
The thesis is organized as follows.
In Chapter 2 an initial-boundary value problem is considered for the linear transverse vibra-tions of an axially moving string. Accurate asymptotic approximavibra-tions valid on long timescales of O(ε−1), are constructed by means of the two timescales perturbation method in combination with either a Fourier-mode expansion method or the method of Laplace transforms. All explicit approximations of the energy of the system are computed up to O(ε) and the (non)applicability of the truncation method is discussed.
In Chapter 3 the linear transverse parametric vibrations of an axially moving string under speed variations are studied. The displacement-response subject to the specific harmonic initial conditions is determined analytically at higher resonances by using the two timescales pertur-bation method in combination with the method of characteristic coordinates. Also explicit ap-proximations for the energy of the system up to O(ε) are computed. The results obtained for the displacement-response and the energy of the system on long timescales are verified with a
numerical finite difference method.
Chapter 4 deals with the analysis of the parametric (in)stability of an axially moving string in the neighborhood of resonance frequencies. Explicit approximations for the amplitude-response and the energy of the system up to O(ε) are computed near resonances by means of the two timescales perturbation method in combination with the method of characteristic coordinates. It has been shown for what values of the detuning parameter the amplitude-response and the energy of the system are (un)stable. All approximations obtained for the amplitude-response and the energy of the system are verified using the numerical finite difference method.
In Chapter 5 conclusions about the parametric (in)stability of an axially moving string with time-dependent speed are presented, and some possibilities for future research related to this work are discussed.
Chapter
2
On resonances and the applicability of
Galerkin’s truncation method for an axially
moving string with time-varying velocity
1
Nature laughs at the difficulties of integration. Pierre-Simon Laplace As was described in the introduction, an axially moving string system may experience (in)stabilities due to harmonic variations in the axial speed. This time-varying velocity of the axially moving system consists of a constant mean value along with some small harmonic variations. These har-monic variations in the time-dependent velocity may cause a resonance (instability) in the system. In this chapter the (in)stability of an axially moving string subject to the time-dependent velocity at all resonant frequencies is examined. The two timescales perturbation method in conjunction with the Fourier-mode expansion method and the Laplace transform method is employed to the equations of motion in search of infinite mode approximate solutions. All explicit approximations for the energy of the system are computed. In addition, it is shown that mode-truncation does not yield accurate approximations on long timescales, that is, on a timescale of order ε−1.
2.1
Introduction
Axially moving strings can represent many engineering devices such as serpentine belts, aerial ca-bles, power transmission belts, plastic films, magnetic tapes, paper sheets and textile fibers. Roll eccentricity of pulleys, small variations in the driving force of the pulley and other belt imperfec-tions can lead to varying belt speed. In other words, (un)known sources can lead to varying belt speed, and so leading to undesirable oscillations in such systems. In engineering, understanding the transverse vibrations of axially moving strings is important for the safe design, construction and operation of a variety of machines and structures. Analysis of transverse vibrations of ax-ially moving strings is a challenging subject, which has been studied for many years by several researchers and still of interest today. Much research has been done in this area to study the linear vibrations of axially moving strings, which was reviewed in Refs. [23,24,25]. In all of these
in-1
This chapter is slightly revised version of published article [1]- “On resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity”, J. Sound Vib., vol. 344, p: 1-17, February 2015
vestigations, it is found that linear analysis is applicable to small amplitude vibrations. The axial speed plays a significant role on the dynamics of axially moving strings. Both the constant and time-varying axial speed cases have been examined in the literature [6,7,12,21,48,49,58]. While most of the studies deal with constant axial velocity, some paper addressed the influence of speed variation on the vibrations, see in Refs. [16,41,59,60]. Miranker [52] was the first to derive the equations of motion for a string traveling with time-dependent axial velocity. Several techniques have been employed to analyze the governing equations of motion for axially moving strings with constant and time-varying speed. Thurman and Mote [26] used the Linsted perturbation method and the averaging method to study linear and non-linear models of an axially moving string, and found that the influence of non-linearities increases with the increase of transport speed. Mote [33] studied the problem of an axially accelerating string driven harmonically at one end, obtained approximate solutions and investigated the stability of resulting constant coefficient equation by means of the Laplace transform method. The transverse vibrations of traveling strings and the moving beam were studied by Wickert and Mote [34] by using the Green’s function method; they obtained modal functions of string and found that orthonormal modal function become singular when the string transports at critical speed. Pakdemirli and Boyaci [61] used the Lindstedt-Poincar technique and the two timescales perturbation method, respectively, to approximate the transverse oscillations of the beam-like and string-like problems subject to constant axial speed. However, in that paper the approximation is truncated to a single mode of vibration for string-like problem and it is shown that the frequencies and amplitudes of vibration may grow or decay in time. While, the same problem was studied by van Horssen [5] using the Laplace transform techniques and found that the truncation to a single mode of vibration leads to inaccurate results on long timescales. A two timescales perturbation method was also used in Refs. [41, 42] to investigate the stability in transverse vibrations of an axially moving string with a time varying speed. The truncation method is then used to approximate the solutions. In these papers, the approximation is truncated to a single mode of vibration. However, the solution of the partial differential equation can consist of infinitely many interactions between vibration modes. So this truncation can cause inaccurate results on long timescales (see for instance Refs.[11, 62, 63]). Recently, Chen[9] used the method of multiple scale to investigate the approximate analytical solution of the non-linear transverse vibration of axially accelerating strings with a harmonically varying velocity about a constant mean speed and with longitudinally varying tension and found that the exact internal resonances among the first three frequencies exists, while the effects of in-finite exact internal resonances may be neglectable. On the other hand, Sandilo and van Horssen [50] studied the axially moving string with harmonically varying length about constant mean length, and found that the Galerkin’s truncation method can not be applied to obtain asymptotic approximations on long time-scales.
In this chapter, the transverse vibrations of axially moving strings with time dependent velocity varying harmonically about low constant mean speed and about relatively high constant mean speed will be studied. In case of low speed, the governing equation of motion is first discretized by using the Fourier sine series, and then the two timescales perturbation method is employed, while in the case of relatively high constant speed, the Laplace transform method is used followed by the two timescales perturbation method to obtain the analytical approximate solution. The Laplace transform method is, however, well-described in elementary text books on partial differ-ential equations see for instance Ref. [56]. Additionally, It will be shown that the truncation method cannot be applied to string-like problems in all axial speed cases.
This chapter is organized as follows. Section2.2presents the mathematical formulation of the ax-ially moving string system. The governing equations of motion with the time-dependent velocity at a low mean speed is studied in section 2.3. In section 2.4, the governing equations of motion
is examined with the harmonically varying velocity about a relatively high constant mean speed. Finally, the conclusions of this chapter are presented in section2.5.
2.2
Equations of motion
The schematic representation of an axially moving string with length L, moving with velocity V is shown in Figure 1.3. We will assume that the string is fixed at x = 0 and x = L, where L is the distance between the two supports (that is, it is assumed that there is no displacement of the string in vertical direction at the supports). The equations of motion describing an axially moving string is obtained either through the application of Hamilton’s principle see Ref. [62] or Newton’s second law of motion see Ref.[27].
Let t be the time, x be the spatial coordinate along the longitude of motion, V be the time-varying axial speed of the string and u(x, t) be the transversal displacement of the string at spatial coordinate x and time t. Then the equation describing the transversal displacement of the string in vertical direction is given by
utt+ 2V uxt+ Vtux+ (V2− c2)uxx= 0, t > 0, 0 < x < L, (2.1)
where c is the wave speed due to a pretension of the string, and c = q
T0
ρ, in which T0 is assumed
to be non-zero constant tension in the string, and ρ is the (constant) mass of the string per unit length. The boundary and initial conditions are given by
u(0, t) = 0, u(L, t) = 0, t > 0, (2.2) and
u(x, 0) = φ(x), and, ut(x, 0) = ψ(x), 0 < x < L, (2.3)
where φ(x) and ψ(x) represents the initial displacement and initial velocity of the string respec-tively. To put the equation in non-dimensional form, we introduce the following dimensionless quantities: x∗= x L, V ∗ = V c, t ∗ = ct L, u ∗ (x, t) = u(x, t) L , Ω∗ = LΩ c , φ ∗(x) = φ(x) L , ψ ∗(x) = ψ(x) c . (2.4)
Substitution of Eq. (2.4) into the initial-boundary value problem (2.1)-(2.3) yields the dimen-sionless form of equation of motion:
utt− uxx= −Vtux− 2V uxt− V2uxx, t ≥ 0, 0 < x < 1, (2.5)
with non-dimensional boundary conditions
u(0, t) = 0, u(1, t) = 0, t > 0, (2.6) and initial conditions
u(x, 0) = φ(x), and ut(x, 0) = ψ(x), 0 < x < 1. (2.7)
In the following sections we present the methods to obtain an accurate approximations of the solutions of the governing equations of motion (2.5)-(2.7) for an axially moving string at low and relatively high mean speed of the system.
2.3
Harmonically varying velocity about a low mean speed
In this section, the methods for solving the initial-boundary value problems (2.5)-(2.7) with har-monically varying velocity about a low mean speed are discussed. We assume that the velocity is a harmonically varying function about a constant mean speed
V (t) = ε(V0+ α sin(Ωt)), (2.8)
where V0 and α are, respectively, the mean speed and the amplitude of order ε, and Ω is the
velocity fluctuation frequency of O(1), and ε is a dimensionless small parameter with 0 < ε 1. It is also assumed that V0 > |α|, which guarantees that the belt will always move forward in one
direction. In addition, belt velocity V is assumed to be small compared to the wave speed c of the belt in this case; consequently, the term V2uxx will have no contribution to the solutions up
to O(ε) on time-scales of order 1ε. If the velocity function (2.8) is substituted into Eq. (2.5), one obtains the following perturbation problem:
utt− uxx = ε h − αΩ cos(Ωt)ux− 2 V0+ α sin(Ωt)uxt i − ε2hV0+ α sin(Ωt) i2 uxx, (2.9)
with boundary conditions
u(0, t; ε) = 0, u(1, t; ε) = 0, t > 0, (2.10) and initial conditions
u(x, 0; ε) = φ(x), and ut(x, 0; ε) = ψ(x), 0 < x < 1. (2.11)
The following Fourier series expansion (that is, an eigenfunction expansion) [48] u(x, t) =
∞
X
n=1
un(t; ε) sin(nπx), (2.12)
for u(x, t) is assumed, so that the Dirichlet type boundary conditions (2.10) are satisfied. Substi-tution of Eq. (2.12) into the initial-boundary value problem (2.9)-(2.11) leads to
∞ X n=1 [¨un+(nπ)2un] sin(nπx) = −ε ∞ X n=1 nπ h
αΩ cos(Ωt)un+2 V0+α sin(Ωt) ˙un
i
cos(nπx)+O(ε2). (2.13) The following orthogonality properties for the set of functions sin(nπx) on 0 < x < 1 should be observed: Z 1 0 sin(nπx) sin(kπx)dx = 0 f or n 6= k, 1/2 f or n = k, (2.14)
Z 1 0 cos(nπx) sin(kπx)dx = ( 0 f or n ± k is even, −(n2−k2k2)π f or n ± k is odd. (2.15) Multiplying Eq. (2.13) by sin(kπx) on both sides and then integrating w.r.t.‘x’from x = 0 to x = 1, and by using Eqs. (2.14)and (2.15) we obtain:
¨ uk+ (kπ)2uk= ε ∞ X n=1 n±k is odd nk n2− k2 h
4αΩ cos(Ωt)un+ 8 V0+ α sin(Ωt) ˙un
i
+ O(ε2),
(2.16)
where uk must satisfy the following initial conditions:
uk(0) = 2 Z 1 0 φ(x) sin(kπx)dx, u˙k(0) = 2 Z 1 0 ψ(x) sin(kπx)dx. (2.17) Equation (2.16) is an infinite dimensional system of ordinary differential equations, which cannot be solved exactly. In what follows, we will apply the two timescales perturbation method to obtain O(ε) accurate approximations of the solutions of (2.16) for different values of Ω on the timescales of O(ε−1).
2.3.1 Application of the two timescales perturbation method
This section presents the application of a two timescales perturbation method for constructing formal approximations of the solution of the infinite dimensional system of ordinary differential equations (2.16). For a more complete overview of this perturbation method (i.e. the method of multiple scales) the reader is referred to Refs. [53,64,65]. A straight-forward expansion in ε may have secular terms which signals the nonuniform validity of approximations for large values of t. In order to get rid of these unbounded (so-called secular) terms, we will use the two timescales perturbation method. These secular terms may occur on the right-hand side of Eq. (2.16). To avoid these terms we introduce the fast time-scale t0 = t and slow time-scale t1 = εt, and assume
that uk(t) can be expanded in a formal power series in ε, that is,
uk(t; ε) = wk(t0, t1; ε). (2.18)
In terms of the new variables, the following transformations are needed for the time derivatives: duk dt = ∂wk ∂t0 + ε∂wk ∂t1 , (2.19) d2uk dt2 = ∂2wk ∂t2 0 + 2ε ∂ 2w k ∂t0∂t1 + ε2∂ 2w k ∂t2 1 . (2.20)
Substitution of Eqs. (2.19) and (2.20) into Eq. (2.16) yields:
∂2wk ∂t2 0 +2ε ∂ 2w k ∂t0∂t1 +(kπ)2wk= ε ∞ X n=1 n±k is odd nk n2− k2 h 4αΩ cos(Ωt)wn+8 V0+α sin(Ωt) ∂ wn ∂t0 i +O(ε2). (2.21)
An approximation of wk(t0, t1; ε) is sought in the form
wk(t0, t1; ε) = wk0(t0, t1) + εwk1(t0, t1) + . . . . (2.22)
Substitution of Eq. (2.22) into Eq. (2.21) and then equalization of the coefficients of ε0 and ε in the resulting equation leads to the O(1)-problem and O(ε)-problem for wk0 and wk1 :
O(1) : ∂2wk0 ∂t2 0 + (kπ)2wk0 = 0, (2.23) wk0(0, 0) = 2 Z 1 0 φ(x) sin(kπx)dx, ∂ ∂t0 wk0(0, 0) = 2 Z 1 0 ψ(x) sin(kπx)dx, (2.24) for k = 1,2,3,. . . . O(ε) : ∂2wk1 ∂t20 + (kπ) 2w k1 = −2 ∂2wk0 ∂t0∂t1 + ∞ X n=1 n±k is odd nk n2− k2 h 4αΩ cos(Ωt0)wn0+ 8(V0+ α sin(Ωt0)) ∂wn0 ∂t0 i , (2.25) wk1(0, 0) = 0, ∂ ∂t0 wk1(0, 0) = − ∂ ∂t1 wk0(0, 0), (2.26)
for k = 1,2,3,. . . . The solution of the O(1)-problem can be written as follows:
wk0(t0, t1) = Ak0(t1) cos(kπt0) + Bk0(t1) sin(kπt0), (2.27)
where the amplitudes Ak0 and Bk0 are functions of slow time t1, which are so determined as to
make the solution of the O(ε)-problem for wk1(t0, t1) free of secular terms. As stated above, it
is assumed that wk0(t0, t1) and wk1(t0, t1), ... are bounded on timescales of O(ε−1); these secular
(unbounded) terms may destroy the accuracy of the approximations on long timescales, so they should be avoided. By plugging Eq.(2.27) into Eq.(2.25), the resulting equation (see appendix.A) will give rise to resonances when Ω = (k + n)π, Ω = (k − n)π or Ω = (n − k)π with k ± n is odd. In other words, secular terms in the solution of the O(ε)-problem will occur when Ω is an odd multiple of π. In what follows the following two resonance cases will be considered to prevent these secular (unbounded) terms.
case 1: Ω = mπ (m is positive odd integer)
case 2: Ω ∼ mπ (as opposed to Ω = mπ) is referred to as near-resonance.
We will consider these cases for m > 1, where m is a positive odd integer. The solution of the problem for m = 1 is studied in [48].
2.3.2 A general resonance case: Ω = mπ (m is positive odd integer)
In this case, it is assumed that the velocity fluctuation frequency (Ω) of the axially moving string is equal to mth times the natural frequency of the string, where m is a positive odd integer, that is, Ω = mπ. In order to preclude the appearance of secular terms in wk1(t0, t1), we substitute
Ω = mπ into O(ε)-problem (2.25); the following conditions must be applied to Ak0(t1) and Bk0(t1) (see appendix.A): dAk0 d ¯t1 =h(k + m)B(k+m)0+ (k − m)B(k−m)0− (m − k)B(m−k)0i, dBk0 d ¯t1
= −h(k + m)A(k+m)0+ (k − m)A(k−m)0+ (m − k)A(m−k)0i,
(2.28)
where ¯t1 = αtm1 and k = 1, 2, 3, .... And for non positive indices k, the functions Ak0 and Bk0 are
defined to be zero. For convenience, we will drop the bar from ¯t1. System (2.28) is an infinite
dimensional system of coupled ordinary differential equations (ODEs). The solution of the system (2.28) for Ak0 and Bk0 can yield the amplitude-response and the energy of the system. However,
it can clearly be seen from the system that there are infinitely many interactions between the vibration modes, and cannot be easily solved. In the subsequent sections, the system (2.28) will be analyzed by using the truncation method and also the energy of the belt system will be computed in terms of infinite dimensional system.
2.3.3 Application of the truncation method
In this section the infinite dimensional system of coupled ordinary differential equations (2.28) will be studied by using the truncation method, that is, the infinite dimensional system will be truncated to a finite dimensional one (that is, only a finite number of vibration modes are considered). For the case m = 1, it is shown in Ref. [48] that the truncated system (2.28) has only purely imaginary eigenvalues and/or zero eigenvalues. In this study, we will consider the case m > 1, where m is a positive odd integer. We will truncate the system for m = 3 and m = 5 and use just some first few modes and neglect the higher order modes. For example, truncating the infinite dimensional system (2.28) for m = 3 to the first four modes, we obtain:
˙ X = AX (2.29) where X = A10 B10 A20 B20 A30 B30 A40 B40 , and A = 0 0 0 −2 0 0 0 4 0 0 −2 0 0 0 −4 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 .
This system has eigenvalues ±√2i, ±√2i and 0, 0, 0, 0. Using the computer software package Maple, the eigenvalues of system (2.28) have been computed up to 10 modes for m = 3 and m = 5 and are listed in Table2.1.
From Table2.1, it can be seen that the eigenvalues of the truncated system are always either reals or purely imaginary or complex with both the positive and negative real part. It is well known in mathematics that in this case no conclusion can be drawn for the infinite dimensional system.
m = 3 No.of
modes
eigenvalues of matrix A (all multiplicity 2) Dimension eigenspace of A 1 0 2 2 ±√2 4 3 0, ±√2 6 4 0, 0, ±√2i 8 5 0, ±0.40 + 2.48i, ±0.40 − 2.48i 10 6 ±0.40 + 2.48i, ±0.40 − 2.48i, ±4.24i 12 7 0, ±4.24i, ±2.89i, ±5.64i 14 8 0, 0, ±4.24i, ±2.89i, ±5.64i 16 9 0, 0, 0, ±2.89i, ±5.64i, ±8.48i 18 10 0, 0, ±2.08i, ±2.55i, ±8.48i, ±9.96i 20
m = 5 1 0 2 2 0, 0 4 3 0, ±√6 6 4 ±2, ±√6 8 5 0, ±2, ±√6 10 6 0, 0, ±√6,±√2i 12 7 0, 0, 0, ±√2i, ±2√2i 14 8 0, 0, ±√2i, ±1.08 + 4.14i, ±1.08 − 4.14i 16 9 0, ±1.08 + 4.14i, ±1.08 − 4.14i, ±2.64i, ±5.57i 18 10 ±1.08 + 4.14i, ±1.08 − 4.14i, ±2.64i, ±5.57i, ±7.07i 20
Table 2.1: Approximations of the eigenvalues of the truncated system (2.28) for m = 3 and m = 5.
2.3.4 Analysis of the infinite dimensional system (2.28)
In the preceding section, the infinite dimensional system (2.28) was truncated to finite dimensional one and was shown that the mode approximation leads to the stable and/or unstable solution. In the following, we shall compute the energy of the belt system and demonstrate that the results obtained by applying the truncation method are not valid on long timescales, that is, on a time-scales of O(ε−1) in all cases. By introducing Xk0(t1) = kAk0(t1) and Yk0(t1) = kBk0(t1), system
(2.28) yields: (dX k0 dt1 = k[−Y(m−k)0+ Y(k+m)0+ Y(k−m)0], dYk0 dt1 = −k[X(m−k)0+ X(k+m)0+ X(k−m)0], (2.30) for k = 1, 2, 3, . . . , and the functions Xk0 and Yk0 are zero for non positive indices k. Then it can
be deduced that: (
Xk0X˙k0 = k[−Xk0Y(m−k)0+ Xk0Y(k+m)0+ Xk0Y(k−m)0],
Yk0Y˙k0= −k[Yk0X(m−k)0+ Yk0X(k+m)0+ Yk0X(k−m)0].
By adding both the equations in (2.31), and by taking the sum from k = 1 to ∞, it follows that: 1 2 ∞ X k=1 d dt1 (Xk02 + Yk02) =m ∞ X k=1 (X(k+m)0Yk0− Y(k+m)0Xk0) + (1)(−X(m−1)0Y10− Y(m−1)0X10) + (2)(−X(m−2)0Y20− Y(m−2)0X20) .. . + (m − 2)(−X20Y(m−2)0− Y20X(m−2)0) + (m − 1)(−X10Y(m−1)0− Y10X(m−1)0). (2.32)
By differentiating Eq. (2.32) with respect to t1 on both sides, we obtain (see appendix B):
1 2 ∞ X k=1 d2 dt21(X 2 k0+ Yk02) = 2m2 ∞ X k=1 (Xk02 + Yk02), (2.33) and then by puttingP∞
k=1(Xk02 + Yk02) = W (t1) into Eq. (2.33) yields:
d2W (t1)
dt21 − 4m
2W (t
1) = 0. (2.34)
The solution of (2.34) is:
W (t1) = C1e2mt1 + C2e−2mt1, (2.35)
where C1 and C2 are arbitrary constants and can be determined by using the initial conditions.
The energy of belt system can be approximated using the function W (t1) (see in Ref. [48]). For
C1 6= 0, W (t1) (so the energy) increases exponentially if t1 increases. Thus W (t1) is unbounded
in t1 and increases as t1 increases. This behavior is different from the behavior of Ak0 and
Bk0 as obtained by applying the truncation method. If we apply the truncation method, we
obtain the mixture behavior to the system (2.28) due to combination of real, complex and purely imaginary eigenvalues. In other words, truncation method yields both the stable and unstable approximations for the (unstable) solution. This implies that the approximations obtained by applying the truncation method to system (2.28) are not accurate on long time scales, that is, on time-scales of order ε−1. In what follows, we shall investigate the (in)stability of an axially moving system in the neighborhood of resonances.
2.3.5 Near resonance case : Ω = mπ + εδ
In the previous section, the instability of the axially moving string was examined at resonances. In this subsection, we shall investigate the stability of the system near the resonances, that is, Ω ∼ mπ, where m is a positive odd integer. To do so, we express the nearness of Ω by employing the relation:
Ω = mπ + εδ, (2.36) where δ is a detuning parameter of O(1) and ε is a small dimensionless parameter, that is, 0 < ε 1. Plugging Eq. (2.36) into Eq. (2.25) yields for the O(ε) for wk1:
∂2wk1 ∂t20 + (kπ) 2w k1 = −2 ∂2wk0 ∂t0∂t1 + ∞ X n=1 n±k is odd nk n2− k2 h 4α mπ + εδ cos (mπ + εδ)t0wn0 + 8V0+ α sin (mπ + εδ)t0 ∂wn0 ∂t0 i . (2.37) In order to prevent the secular terms (by carrying out the same operations as described in section
2.3.2), Ak0 and Bk0 have to satisfy:
dAk0 d ¯t1 = (k + m) h A(k+m)0sin(δ ¯t1) + B(k+m)0cos(δ ¯t1) i −(k − m)hA(k−m)0sin(δ ¯t1) − B(k−m)0cos(δ ¯t1) i −(m − k)hA(m−k)0sin(δ ¯t1) + B(m−k)0cos(δ ¯t1) i , dBk0 d ¯t1 = (k + m) h − A(k+m)0cos(δ ¯t1) + B(k+m)0sin(δ ¯t1) i −(k − m)hA(k−m)0cos(δ ¯t1) + B(k−m)0sin(δ ¯t1) i −(m − k)hA(m−k)0cos(δ ¯t1) − B(m−k)0sin(δ ¯t1) i , (2.38)
where ¯t1 = αtm1 and k = 1, 2, 3, . . . , and the functions Ak0 and Bk0 are defined to be zero for k ≤ 0.
For convenience, we will again drop the bar from ¯t1. It should be noticed that for δ = 0, we can
again obtain system (2.28). The system (2.38) is an infinite dimensional system of coupled ODEs, which is in fact not easy to solve for the functions Ak0and Bk0. In the following, we shall compute
the energy of the system near the resonances and will demonstrate the behavior of the system for different values of the detuning parameter δ.
2.3.6 Analysis of the infinite dimensional system(2.38)
By introducing Xk0(t1) = kAk0(t1) and Yk0(t1) = kBk0(t1), system (2.38) yields:
dXk0 dt1 = (k) h
X(k+m)0sin(δt1) + Y(k+m)0cos(δt1) − X(k−m)0sin(δt1)
+Y(k−m)0cos(δt1) − X(m−k)0sin(δt1) − Y(m−k)0cos(δt1)
i ,
dYk0
dt1 = (k)
h
− X(k+m)0cos(δt1) + Y(k+m)0sin(δt1) − X(k−m)0cos(δt1)
−Y(k−m)0sin(δt1) − X(m−k)0cos(δt1) + Y(m−k)0sin(δt1)
i ,
(2.39)
for k = 1, 2, 3, ..., and the functions Xk0, Yk0 are zero for k ≤ 0. Then it can be deduced from
(2.39) that: Xk0X˙k0= (k) h Xk0X(k+m)0sin(δt1) + Xk0Y(k+m)0cos(δt1) −Xk0X(k−m)0sin(δt1) + Xk0Y(k−m)0cos(δt1) −Xk0X(m−k)0sin(δt1) − Xk0Y(m−k)0cos(δt1) i , Yk0Y˙k0= (k) h − Yk0X(k+m)0cos(δt1) + Yk0Y(k+m)0sin(δt1)
−Yk0X(k−m)0cos(δt1) − Yk0Y(k−m)0sin(δt1)
−Yk0X(m−k)0cos(δt1) + Yk0Y(m−k)0sin(δt1)
i .
By adding both equations in (2.40), and then by taking the sum from k = 1 to ∞, we obtain: 1 2 ∞ X k=1 d dt1 (Xk02 + Yk02) = m ∞ X k=1
[(Yk0X(k+m)0− Xk0Y(k+m)0)cos(δt1) − (Xk0X(k+m)0+ Yk0Y(k+m)0)sin(δt1)]
+ (1)[(Y10Y(m−1)0− X10X(m−1)0)sin(δt1) − (X10Y(m−1)0+ Y10X(m−1)0)cos(δt1)]
+ (2)[(Y20Y(m−2)0− X20X(m−2)0)sin(δt1) − (X20Y(m−2)0+ Y20X(m−2)0)cos(δt1)]
.. .
+ (m − 2)[(Y20Y(m−2)0− X20X(m−2)0)sin(δt1) − (Y20X(m−2)0+ X20Y(m−2)0)cos(δt1)]
+ (m − 1)[(Y10Y(m−1)0− X10X(m−1)0)sin(δt1) − (Y10X(m−1)0+ X10Y(m−1)0)cos(δt1)].
(2.41) By differentiating Eq. (2.41) twice with respect to t1, we obtain:
1 2 ∞ X k=1 d3 dt31(X 2 k0+ Yk02) = − δ2 2 ∞ X k=1 d dt1 (Xk02 + Yk02) + 2m2 ∞ X k=1 d dt1 (Xk02 + Yk02), (2.42) and then by puttingP∞
k=1(Xk02 + Yk02) = W (t1) into Eq. (2.42) yields:
d3W (t1)
dt31 + (δ
2− 4m2)dW (t1)
dt1
= 0. (2.43) Integration of Eq. (2.43) w.r.t. t1, gives:
d2W (t 1)
dt21 + (δ
2− 4m2)W (t
1) = D1, (2.44)
where D1 is a constant of integration. Solution of Eq. (2.44) leads to the following results:
for |δ|< 2m : W (t1) = − D1 4m2− δ2 + D2cosh( p 4m2− δ2t 1) + D3sinh( p 4m2− δ2t 1), for |δ|= 2m : W (t1) = D1+ D2t1+ 1 2D3t 2 1, for |δ|> 2m : W (t1) = D1 δ2− 4m2 + D2cos( p δ2− 4m2t 1) + D3sin( p δ2− 4m2t 1), (2.45)
where D1, D2 , and D3 are constants of integration. The important conclusions of these solutions
are: for |δ|< 2m, W (t1) increases exponentially, for |δ|= 2m, W (t1) increases polynomially; so
W (t1) (the energy of the infinite dimensional system) is unbounded for |δ|≤ 2m and finally for
2.4
Harmonically varying velocity about a relatively high
constant mean speed
The preceding sections considered the transverse vibrations of an axially moving string subject to the harmonically varying velocity about a low constant mean speed. This section discusses the analysis of the governing equations of motion (2.5)-(2.7) with a time-varying velocity about a relatively high constant mean velocity. The lowest resonance case for the problem is studied in [49]. In this study, we generalize the work of [49] to all higher resonance cases. Assuming that the velocity is harmonically varying about a constant mean velocity V0, one writes
V (t) = V0+ εα sin(Ωt), (2.46)
where V0, α and Ω are some positive constants, and ε is a small dimensionless parameter with
0 < ε 1. Unlike the low mean speed case, the mean speed V0 is assumed to be O(1) in this
case. In other words, the mean speed V0 is considered to be of same order as the wave speed c.
Substitution of Eq.(2.46) into an initial-boundary value problem (2.5)-(2.7) yields the following perturbation problem:
utt+ 2V0uxt+ (V02− 1)uxx = ε
− 2α sin(Ωt)uxt− 2V0α sin(Ωt)uxx− αΩ cos(Ωt)ux
+ O(ε2), (2.47) with the boundary and initial conditions (2.6) and (2.7) respectively. In the following subsection, the system (2.47) will be analyzed using the two timescales perturbation method in conjunction with the Laplace transform method to find approximations valid on long timescales, that is, on a timescales of O(ε−1).
2.4.1 Application of the two timescales perturbation method
In this section the application of two timescales perturbation method to the perturbation problem (2.47) is considered. Assuming an expansion for u(x, t) of the form
u(x, t; ε) = v(x, t0, t1) ≡ v0(x, t0, t1) + εv1(x, t0, t1) + . . . , (2.48)
in which t0= t and t1= εt are the usual fast and slow timescales. In terms of the new variables,
the following transformations are needed for the time derivatives: ∂u ∂t = ∂v ∂t0 + ε∂v ∂t1 , (2.49) ∂2u ∂t2 = ∂2v ∂t20 + 2ε ∂2v ∂t0∂t1 + ε2∂ 2v ∂t21· (2.50) Substituting Eqs. (2.48)-(2.50) into Eq. (2.47), separating terms at each order of ε, we obtain the O(1) and O(ε)-problem as follows:
O(1) : ∂2v0 ∂t2 0 + 2V0 ∂2v0 ∂t0∂x − (1 − V02)∂ 2v 0 ∂x2 = 0, (2.51)
O(ε) : ∂2v1 ∂t2 0 + 2V0 ∂2v1 ∂t0∂x + (V02− 1)∂ 2v 1 ∂x2 = −2 ∂2v0 ∂t0∂t1 − 2V0 ∂2v0 ∂t1∂x − 2α sin(Ωt) ∂ 2v 0 ∂t0∂x − 2V0α sin(Ωt) ∂2v0 ∂x2 − αΩ cos(Ωt) ∂v0 ∂x· (2.52) The solution of the O(1) problem can be found by means of the Laplace transform method [49].
v0(x, t0, t1) = ∞ X n=1 h F[1]n(x) An0(t1) cos(ωnt0) − Bn0(t1) sin(ωnt0) +F[2]n(x) An0(t1) sin(ωnt0) + Bn0(t1) cos(ωnt0) i , (2.53) where F[1]n(x) = cos(nπ(V0+ 1)x) − cos(nπ(V0− 1)x), F[2]n(x) = sin(nπ(V0+ 1)x) − sin(nπ(V0− 1)x), (2.54) and ωn= nπ(1−V02), n ∈ Z+are the natural frequencies of the belt system. In Eq. (2.53), An0(t1)
and Bn0(t1) are still arbitrary functions and can be used to eliminate secular terms in the solution
of the O(ε)-problem. Substituting Eqs. (2.53) and (2.54) into Eq. (2.52), the O(ε)-problem becomes ∂2v 1 ∂t20 + 2V0 ∂2v 1 ∂t0∂x + (V02− 1)∂ 2v 1 ∂x2 = ∞ X n=1 h sin(ωnt0)Θn+ cos(ωnt0) eΘn i + ∞ X n=1 h
sin(Ωt0) sin(ωnt0)ϕn+ sin(Ωt) cos(ωnt0)ϕen +cos(Ωt0) sin(ωnt0)ζn+ cos(Ωt) cos(ωnt0)eζn
i , (2.55) where Θn(x, t1) = 2 h∂An0 ∂t1 F[1]nωn− V0 ∂F[2]n ∂x +∂Bn0 ∂t1 F[2]nωn+ V0 ∂F[1]n ∂x i , e Θn(x, t1) = 2 h∂An0 ∂t1 − F[2]nωn− V0 ∂F[1]n ∂x +∂Bn0 ∂t1 F[1]nωn− V0 ∂F[2]n ∂x i , ϕn(x, t1) = 2α h An0 ∂F[1]n ∂x ωn− V0 ∂2F[2]n ∂x2 + Bn0 ∂F[2]n ∂x ωn+ V0 ∂2F[1]n ∂x2 i , e ϕn(x, t1) = 2α h An0 −∂F[2]n ∂x ωn− V0 ∂2F [1]n ∂x2 + Bn0 ∂F[1]n ∂x ωn− V0 ∂2F [2]n ∂x2 i , ζn(x, t1) = αΩ h − An0∂F[2]n ∂x + Bn0 ∂F[1]n ∂x i , and ζen(x, t1) = αΩ h − An0 ∂F[1]n ∂x − Bn0 ∂F[2]n ∂x i . (2.56)
According to Eq. (2.55), if the axial speed variation frequency Ω is equal to (or close to) an integer times a natural frequency ωn of the linear generating system (2.51); resonances may occur. The
first resonance case, that is Ω = π(1 − V02) has been studied in Ref.[49]. In this work, the k∗th resonance case, that is, Ω = k∗π(1 − V02) where k∗ ∈ Z+ will be investigated.
2.4.2 Ω = k∗π(1 − V2
0), a general resonance case
In this case, we assume that Ω = ωk∗, that is, Ω = k∗π(1−V2
0), where k∗ ∈ Z+and fixed. Plugging
Ω = k∗π(1 − V02) into Eq. (2.55) yields ∂2v1 ∂t20 + 2V0 ∂2v1 ∂t0∂x + (V02− 1)∂ 2v 1 ∂x2 = ∞ X n=1 h sin(ωnt0)Θn+ cos(ωnt0) eΘn i +1 2 ∞ X n=1 h cos(ωn−k∗t0) ϕn+ eζn + cos(ωn+k∗t0) − ϕn+ eζn +sin(ωn−k∗t0) − e ϕn+ ζn + sin(ωn+k∗t0) e ϕn+ ζn i , (2.57)
where functions Θ, eΘ, ϕ,ϕ, ζ and ee ζ are given by Eq. (2.56). In Eq. (2.57), it should be observed that ω−n= −ωnand ω0 = 0. In order to find v1at all higher resonances, we will apply the Laplace
transform method to Eq. (2.57). After taking the Laplace transform on both sides of Eq. (2.57), we will calculate the poles, and then the residues, and then use the convolution integral theorem to find the inverse Laplace transform, we obtain
v1(x, t0, t1) = k∗ X n=1 hn1 4(Ψ[1]n+ eΨ[2]n) + 1 8(Ψ [1] [2]n+ Ψ [3] [1]n) o (t0sin(ωnt0)) + n1 4 Ψe[1]n− Ψ[2]n + 1 8 Ψ [1] [1]n− Ψ [3] [2]n o (t0cos(ωnt0)) i + ∞ X n=k∗+1 hn1 4 Ψ[1]n+ eΨ[2]n + 1 8 Ψ [1] [2]n+ Ψ [3] [1]n+ Ψ [2] [2]n+ Ψ [4] [1]n o (t0sin(ωnt0)) + n1 4 Ψe[1]n− Ψ[2]n + 1 8 Ψ [1] [1]n− Ψ [3] [2]n+ Ψ [2] [1]n− Ψ [4] [2]n o (t0cos(ωnt0)) i + k∗−1 X n=1 h1 8 n − Ψ[5][1]n+ Ψ[6][2]no(t0sin(ωnt0)) + 1 8 n Ψ[6][1]n+ Ψ[5][2]no(t0cos(ωnt0)) i + terms with non-secular behavior,
(2.58) where Ψ[1]n(x, t1) = wnF[1]n+ pnF[2]n, Ψ[2]n(x, t1) = wnF[2]n− pnF[1]n, e Ψ[1]n(x, t1) =wenF[1]n+epnF[2]n, Ψe[2]n(x, t1) =wenF[2]n−penF[1]n, Ψl[1]n(x, t1) = wlnF[1]n+ plnF[2]n, Ψl[2]n(x, t1) = wnlF[2]n− plnF[1]n, (2.59)
with l = 1, 2, 3, 4, 5, 6, and w[1]n = 1 2 Z 1 0 ϕn+k∗+ eζn+k∗ πn (−F[2]n)dx, p [1] n = −1 2 Z 1 0 ϕn+k∗+ eζn+k∗ πn (−F[1]n)dx, wn[2]= 1 2 Z 1 0 −ϕn−k∗+ eζn−k∗ πn (−F[2]n)dx, p [2] n = −1 2 Z 1 0 −ϕn−k∗+ eζn−k∗ πn (−F[1]n)dx, wn[3]= 1 2 Z 1 0 −ϕen+k∗+ ζn+k∗ πn (−F[2]n)dx, p [3] n = −1 2 Z 1 0 −ϕen+k∗+ ζn+k∗ πn (−F[1]n)dx, w[4]n = 1 2 Z 1 0 e ϕn−k∗+ ζn−k∗ πn (−F[2]n)dx, p [4] n = −1 2 Z 1 0 e ϕn−k∗+ ζn−k∗ πn (−F[1]n)dx, w[5]n = 1 2 Z 1 0 −ϕek∗−n+ ζk∗−n πn (−F[2]n)dx, p [5] n = −1 2 Z 1 0 −ϕek∗−n+ ζk∗−n πn (−F[1]n)dx w[6]n = 1 2 Z 1 0 ϕk∗−n+ eζk∗−n πn (−F[2]n)dx, p [6] n = −1 2 Z 1 0 ϕk∗−n+ eζk∗−n πn (−F[1]n)dx. wn(x, t1) = 1 2 Z 1 0 Θn(x, t1) πn (−F[2]n)dx, pn(x, t1) = −1 2 Z 1 0 Θn(x, t1) πn (−F[1]n)dx, e wn(x, t1) = 1 2 Z 1 0 e Θn(x, t1) πn (−F[2]n)dx, pen(x, t1) = −1 2 Z 1 0 e Θn(x, t1) πn (−F[1]n)dx. (2.60)
It follows from Eq. (2.58) that the solution of the O(ε)-problem does not contain secular terms if and only if Ψ[1]n(x, t1) + eΨ[2]n(x, t1) +12 Ψ[1][2]n(x, t1) + Ψ[3][1]n(x, t1) − Ψ[5][1]n(x, t1) + Ψ[6][2]n(x, t1) = 0, e Ψ[1]n(x, t1) − Ψ[2]n(x, t1) +12 Ψ[1][1]n(x, t1) − Ψ[3][2]n(x, t1) + Ψ[6][1]n(x, t1) + Ψ[5][2]n(x, t1) = 0, (2.61) for n = 1, 2, . . . k∗− 1, and Ψ[1]n(x, t1) + eΨ[2]n(x, t1) +12 Ψ[1][2]n(x, t1) + Ψ[3][1]n(x, t1) + Ψ[2][2]n(x, t1) + Ψ[4][1]n(x, t1) = 0, e Ψ[1]n(x, t1) − Ψ[2]n(x, t1) +12 Ψ[1][1]n(x, t1) − Ψ[3][2]n(x, t1) + Ψ[2][1]n(x, t1) − Ψ[4][2]n(x, t1) = 0, (2.62) for n = k∗, k∗+ 1, . . . .
By defining A00≡ 0 and B00≡ 0, it follows from Eqs. (2.61) and (2.62) that for all n = 1, 2, . . . ,
Ψ[1]n(x, t1) + eΨ[2]n(x, t1) +12 Ψ[1][2]n(x, t1) + Ψ[3][1]n(x, t1) + Ψ[2][2]n(x, t1) + Ψ[4][1]n(x, t1) −Ψ[5][1]n(x, t1) + Ψ[6][2]n(x, t1) = 0, e Ψ[1]n(x, t1) − Ψ[2]n(x, t1) +12 Ψ[1][1]n(x, t1) − Ψ[3][2]n(x, t1) + Ψ[2][1]n(x, t1) − Ψ[4][2]n(x, t1) +Ψ[5][2]n(x, t1) + Ψ[6][1]n(x, t1) = 0. (2.63)
We rewrite Eq. (2.63) in a more simple form as (
F[1]n(x)Φ[1]n(t1) + F[2]n(x)Φ[2]n(t1) = 0,
F[1]n(x)Φ[2]n(t1) − F[2]n(x)Φ[1]n(t1) = 0,
where Φ[1]n(t1) = wn(t1) −pen(t1) + 1 2 w[3]n (t1) − p[1]n (t1) + w[4]n (t1) − p[2]n (t1) − wn[5](t1) − p[6]n (t1) , Φ[2]n(t1) =wen(t1) + pn(t1) + 1 2 w[1]n (t1) + p[3]n (t1) + w[2]n (t1) + p[4]n (t1) + wn[6](t1) − p[5]n (t1) . (2.65) System (2.64) can be viewed as a system for two unknowns Φ[1]n(t1) and Φ[2]n(t1). It should be
observed from Eq. (2.64) that the determinant of this system is equal to (F[1]n)2+ (F[2]n)2 6= 0,
for all x ∈ (0, 1). It then follows that Φ[1]n(t1) = 0 and Φ[2]n(t1) = 0, or equivalently,
wn(t1) −pen(t1) + 1 2 wn[3](t1) − p[1]n (t1) + wn[4](t1) − p[2]n (t1) − w[5]n (t1) − p[6]n (t1) = 0, e wn(t1) + pn(t1) +12 wn[1](t1) + p[3]n (t1) + wn[2](t1) + p[4]n (t1) + w[6]n (t1) − p[5]n (t1) = 0. (2.66) It can be seen that the system (2.66) involves the functions dAn0
dt1 ,
dBn0
dt1 , An0(t1) and Bn0(t1). After
some lengthy, but elementary calculations, the following system of ODEs for An0(t1) and Bn0(t1)
can be obtained from (2.66): dAn0 d ¯t1 = (n + k∗)hµk∗A(n+k∗)0+ ηk∗B(n+k∗)0 i − (n − k∗)hµk∗A(n−k∗)0− ηk∗B(n−k∗)0 i + (k∗− n)hµk∗A(k∗−n)0+ ηk∗B(k∗−n)0 i , dBn0 d ¯t1 = (n + k∗) h − ηk∗A(n+k∗)0+ µk∗B(n+k∗)0 i − (n − k∗) h ηk∗A(n−k∗)0+ µk∗B(n−k∗)0 i + (k∗− n)hηk∗A(k∗−n)0− µk∗B(k∗−n)0 i , (2.67)
for n = 1, 2, 3, ..., and the functions An0 and Bn0 are zero for n ≤ 0, where
¯ t1 = αt1 2k∗, µk∗ = (−1) k∗+1sin(k∗ πV0), ηk∗ = (−1)k ∗ cos(k∗πV0) − 1. (2.68)
For convenience, we will drop the bar from ¯t1. System (2.67) is an infinite dimensional system
of coupled ordinary differential equations (ODEs), which exhibits infinitely many interactions between the vibration modes for Ω = ωk∗. The solution of this coupled ODEs for the functions
An0(t1) and Bn0(t1) will yield the amplitude-response and the energy of the system. As described,
there are infinitely many interactions between the vibration modes, the closed-form solution of the coupled ODEs is not easily attainable. In following subsections we deal with investigating the system (2.67) by truncating the infinite dimensional system to finite dimension one and by computing the energy of the belt system in terms of the infinite dimensional system.
2.4.3 Application of the truncation method
In this subsection, the infinite dimensional system (2.67) will be truncated to finite one. To do so, we will use some first few modes and neglect the higher order modes. We truncate the system for k∗ = 2, k∗ = 3, k∗ = 4, k∗ = 5 upto 10 modes by using the computer software package Maple. The truncation of the system (2.67) for k∗ = 1 up to 10 modes is given in Ref.[49]. It is shown that the system yields either the zero or purely imaginary eigenvalues in the first resonance case. From Table2.2, Table2.3 and Table 2.4, it can be seen that the eigenvalues of the truncated system are always either real, complex or purely imaginary. It is well known in mathematics that in this case no conclusion can be drawn for the infinite dimensional system. Moreover, it can
