On the dynamics of an inclined stretched string

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On the Dynamics

of

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On the Dynamics

of

an Inclined Stretched String

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen

op donderdag 23 december 2004 om 10.30 uur

door CASWITA

Magister Sains Matematika, Universitas Gadjah Mada, Indonesi¨e,

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Toegevoegd promotor: Dr. ir. W.T. van Horssen Samenstelling promotiecommissie:

Rector magnificus,

Prof. dr. ir. A.W. Heemink Dr. ir. W.T. van Horssen Prof. dr. ir. A.J. Hermans

Prof. ir. A.C.W.M. Vrouwenvelder Prof. dr. ir. F. Verhulst

Prof. dr. A. Abramyan Prof. dr. I.V. Andrianov

voorzitter,

TU Delft, promotor

TU Delft, toegevoegd promotor TU Delft

TU Delft

Rijks Universiteit Utrecht Russia Academy of Sciencies, St. Petersburg

Pridneprovska State Academy, Ukraine

Caswita

On the Dynamics of an Inclined Stretched String. Thesis Delft University of Technology.

With summary in Dutch.

ISBN 90-8559-023-X

Copyright c_{2004 by Caswita}

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright holder. Printed by [OPTIMA] Grafische Communicatie, Rotterdam

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List of Figures

2.1 A simple vibrating system including parametrical and transversal ex-citation of a stretched inclined string. . . 4 2.2 The response curve R2 = ¯A22 of equation (2.3.4) for α = 0 and F = 6:

(a) Response curve as function of frequency of excitation; (b) Re-sponse curve as function of the angle of inclination on string: The stability only applies to the ( ¯A2, ¯B2)-plane. . . 11

2.3 The behaviour of critical points of equation (2.3.4) in the ( ¯A2, ¯B2)−plane

with α = 0, F = 6, and ϕ = 0.8. The horizontal and vertical axis are the ¯A2-axis and the ¯B2-axis, respectively. . . 12

2.4 Four domains with real solutions of system (2.3.7) for F=6: The curve P2P6 comes from ∆1 = 0 and the others come from Cond1 = 0, the

value of ∆ is negative in domains I, II, and III and positive in domain IV. . . 12 2.5 Eight domains with real solutions of type 1 and 2 of system (2.3.5)

for F=6: The curve P7P9 comes from the radicand of (2.3.8) ∆0 = 0. 13

2.6 Twelve domains with real solutions of type 1, 2, and 3 of (2.3.5) for F = 6: The new curves P6P11 and P5P10 are found from the radicand

of the first equation of (2.3.9) ∆2 =

h 6B 5π3 i2 +h₋ 8 15π3(η − β) i3 = 0 and Cond2 = 0, respectively. . . 14

2.7 Stability response-curves of the periodic solutions of (2.3.4) with re-spect to the detuning for F = 6, ϕ = 1₄π, and α = 0. The solid and dashed line represent stable and unstable solutions, respectively. . . . 16 2.8 Stability response-curves of the periodic solutions of (2.3.4) with

re-spect to the angle of excitation for F = 6, η = 7, and α = 0. The solid and dashed line represent stable and unstable solutions, respectively. . 16 2.9 The behaviour of critical points of (2.3.4) in the ( ¯A2, ¯B2)−plane with

α= 0.25, F = 6, and ϕ = 0.8 (no homoclinic orbit). The horizontal and vertical axis are the ¯A2-axis and the ¯B2-axis, respectively. . . 17

2.10 Eighteen domains with real solutions of types 1, 2, and 3 of (2.3.10) for α = 2π and F = 6. . . 19 2.11 Stability response-curves of the periodic solution of (2.3.4) with

re-spect to the detuning for F = 6, ϕ = 1

4π, and α = 2π. The solid and

dashed line represent stable and unstable solutions, respectively. . . . 21 iii

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2.12 Stability response-curves of the periodic solution of (2.3.4) with re-spect to the angle for F = 6, ϕ = 1₄π, and α = 2π. The solid and dashed line represent stable and unstable solutions, respectively. . . . 22 3.1 The dynamic state of the string suspended between a fixed support

at x = 0 and a vibrating support at x = l. . . 25 3.2 Four domains with real solutions of type 1 - 4 of system (3.4.5) in the

(¯η, ¯β)-plane for ¯α1 = ¯α2 = 0. . . 31

3.3 Projection of the curves S, U, and L for ¯α1 = ¯α2 = 0 and ¯η > ¯β

on: (a) the ( ¯As, ¯Bs) (( ¯Cs, ¯Ds)) -plane; (b) the ( ¯As, ¯Cs) (( ¯Bs, ¯Ds))

-plane; (c) the ( ¯As, ¯Ds) (( ¯Bs, ¯Cs)) -plane (with ρ−o =

q

2

3(2¯η− ¯β) and

¯

ρo =√2¯η). The solid and dashed line represent stable and unstable

solutions, respectively. . . 32 3.4 The stability response-curves r1 =

q ˜ A2

s+ ˜Bs2 with respect to ¯η for

system (3.4.2) with ¯α1 = ¯α2 = 0 and ¯β = 1.25: The curves for

CP-type 2 and 3 represent response-curves in Hkfor k = 1 and the curves

for CP-type 4 with k2 = −2k1, where k1 = ˜ As ˜ Cs and k2 = ˜ Bs ˜ Ds, represent response-curves of periodic solutions with all of the components non-zero. . . 34 3.5 Behaviour of the critical points of system (3.4.2) in the hyperplane

Hk for k = 1, ¯α1 = ¯α2 = 0, and ¯β = 1.25. The horizontal and the

vertical axis are the ¯As (or the ¯Cs) -axis and the ¯Bs (or the ¯Ds) -axis,

respectively. . . 35 3.6 The stability response-curves r1 =

q ˜ A2

s+ ˜Bs2 of system (3.4.2) with

respect to ¯β for ¯α1 = ¯α2 = 0 and ¯η = 0.75: the curves of CP-type

2 and 3 represent response-curves in H1, and the curves of CP-type

4 with k2 = −2k1, where k1 = ˜ As ˜ Cs and k2 = ˜ Bs ˜ Ds, represent response curves of the periodic solutions in case all of the components are non-zero. . . 36 3.7 The types of stable motion of the string for ¯α1 = ¯α2 = 0, ¯η= 1.6014,

and ¯β = 1.2159. The first three figures describe planar motions, while the other figures describe whirling (non-planar) motions. . . 37 3.8 The six domains describing the critical points of system (3.4.2) in the

(¯η, ¯β)-plane for ¯α1 = ¯α2 = α. . . 38

3.9 Projection of the curves Sα (= Uα) for ¯α1 = ¯α2 = ¯β and ¯η > 0 to:

(a) the ( ¯As, ¯Bs) (or ( ¯Cs, ¯Ds))-plane ; (b) the ( ¯Bs, ¯Ds)− plane. The

dashed line represents unstable solution. . . 40 3.10 The stability response-curves r1 =

q ˜ A2

s+ ˜Bs2 of system (3.4.2) as

function of ¯η in the hyperplane Hk for k = 1, with α = 0.75. . . 41

3.11 Behaviour of the solutions near the critical points of system (3.4.2) in the hyperplane Hk for k = 1, α = 0.75, and ¯β = 1.25. The horizontal

and vertical axis are the ¯As (or ¯Cs) -axis and the ¯Bs (or ¯Ds) -axis,

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LIST OF FIGURES v

3.12 Behaviour of the solutions near the critical points of system (3.4.2) in the hyperplane Hk for k = 1 and α = ¯β = 0.75. The horizontal and

the vertical axis are the ¯As (or ¯Cs) -axis and the ¯Bs (or ¯Ds) -axis,

respectively. . . 43 3.13 The stability response-curves r1 =

p_¯ A2

s+ ¯B2s of system (3.4.2) as

function of ¯β in the hyperplane Hk for k = 1, α = 0.75, and η = 0.75. 43

3.14 The stability diagram of the critical points of system (3.4.2) in the (¯η, ¯β)-plane for ¯α1 <α¯2. . . 44

3.15 Behaviour of the solutions near the critical points of system (3.4.2) projected to the (r1, r2)-plane; r1 =

p_¯ A2

s+ ¯Bs2 is the horizontal axis

and r2 =

p_¯ C2

s + ¯Ds2 is the the vertical axis. The first four figures

describe the case ¯α1 < ¯β < α¯2 (¯α1 = 0.65, ¯α2 = 0.90, and ¯β = 0.81)

and the other figures describe the case ¯α1 <α¯2 < ¯β (¯α1 = 0.65, ¯α2 =

0.77, and ¯β = 1.01). . . 47 4.1 The inclined string in the dynamic state including normal and

para-metrical excitation at x = 1. . . 54 4.2 Four domains with real solutions of the equations (4.3.7) and (4.3.8):

∆1 = (4₃β)2+ (−8₉η)¯ 3 and ∆2 = (3₂β)2+ (−2₃η)¯3. . . 60

4.3 The stability response-curves R3 = ˜A2m and R4 = ˜D2m of system

(4.3.3) with respect to ¯η for ¯α1 = ¯α2 = 0 and β = 2.0. The solid and

dashed line represent stable and unstable solutions, respectively. . . . 61 4.4 The stability response-curves R3 = ˜A2m and R4 = ˜D2m of system

(4.3.3) with respect to β for ¯α1 = ¯α2 = 0 and ¯η= 2.0. The solid and

dashed line represent stable and unstable solutions, respectively. . . . 62 4.5 The stable motions of the string in the 1 : 1 resonance case for the first

mode, and = 0.0025, ¯α1 = ¯α2 = 0, ¯η = 0.6450, and β = 0.4561. In

(i)-(iii) planar motion is presented, and in (iv)-(vi) non-planar motion is given. . . 63 4.6 The domains with real solutions of (4.3.13) in which the conditions

2-4 are satisfied for ¯α1 = ¯α2 = 0.50: ∆12= (1₂δ12)2+ (1₃κ12)3. . . 65

4.7 The domains with real solutions of system (4.3.9) for ¯α1 = ¯α2 = 0.50:

∆11 = (1₂δ11)2+ (1₃κ11)3 and ∆12= (1₂δ12)2+ (1₃κ12)3. . . 65

4.8 The domains with real solutions of system (4.3.9) as illustration for the case ¯α1 6= ¯α2: ∆13 = (1₂δ13)2+ (1₃κ13)3. . . 66

4.9 The stability response-curves of system (4.3.3) with respect to the detuning parameter ¯η for ¯α1 = ¯α2 = 0.50 and β = 2.0. The solid and

dashed line represent stable and unstable solutions, respectively. . . . 68 4.10 The stability response-curves of system (4.3.3) with respect to the

excitation-amplitude β for ¯α1 = ¯α2 = 0.50 and ¯η = 2.0. The solid

and dashed line represent stable and unstable solutions, respectively. . 68 4.11 The stability response-curves of system (4.3.3) with respect to the

detuning parameter ¯η for β = 2 and ¯α1 6= ¯α2. In the upper figures

¯

α1 = 0.25 and ¯α2 = 0.75 (¯α1 <α¯2) and in the lower figure ¯α1 = 0.75

and ¯α2 = 0.25 (¯α1 >α¯2). The solid and dashed line represent stable

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4.12 The stability response-curves of system (4.3.3) with respect to the excitation-amplitude β for ¯α1 6= ¯α2 and ¯η = 2.0. In the upper figures

¯

α1 = 0.25 and ¯α2 = 0.75 (¯α1 <α¯2) and in the lower figures ¯α1 = 0.75

and ¯α2 = 0.25 (¯α1 >α¯2). The solid and dashed line represent stable

and unstable solutions, respectively. . . 70 4.13 The stable motions of the string in the 1:1 resonance case for the first

mode with ¯ = 0.0025, ¯α1 = ¯α2 = 0.2580, β = 0.4561, and ¯η= 0.3225.

The upper figure presents planar motion and the lower figure presents non-planar motion. . . 71 4.14 Twenty-two domains with real solutions of system (4.4.8) for ¯α > 0

and ˜σ = 1: ∆3 = 1₄δ212 + 271κ213 and ∆4 = 1₂δ222 + 271κ322. The vertical

axis is the ˜β axis and the horizontal axis is the ˜η axis. . . 76 4.15 The stability response-curves for system (4.4.7) for ¯α > 0, ˜σ = 1,

and ˜β = 2. The solid and dashed line represent stable and unstable solutions, respectively. . . 78 4.16 Stable motions of the string for ¯α >0, ˜σ = 1, and ˜β = 2. Middle and

right part of the upper figure are motions corresponding to CP-type 2 and CP-type 3, respectively, and in the lower figure the motions correspond to CP-type 4. . . 79 5.1 The inclined stretched string in static state due to gravity. . . 88 6.1 The inclined string in the dynamic state with a parametrical and

transversal excitation at ¯X = ¯L. . . 96 6.2 The two domains of critical points of (6.4.2); ∆0 = 1₄F¯02− 271 Ω¯

3 0 and

∆α = ₂₉₁₆1 (2 ¯Ω3α+ 18 ¯Ωα− 27 ¯Fα2)2+ ₂₇1(1 − 1₃Ω¯2α)3. . . 104

6.3 Phase portraits of system (6.4.2) for different values of ¯Ωα and ¯Fα,

where the horizontal and the vertical axis are the ˜An-axis and the

˜

Bn-axis respectively. In (i)-(iii): α = 0, and in (iv)-(vi): α > 0. . . 105

6.4 The amplitude response-curves ¯Rn = ˜A2n+ ˜Bn2 are given as function

of the parameter ¯Ωα (with ¯Fα = 2) and as function of ¯Fα(with ¯Ωα = 3).107

6.5 The bifurcation diagram of the critical points of system (6.4.11) for ¯

α= 0.25 and for several values of ¯βs≥ 0 (y and z are the real solutions

of (6.4.16) for j = 1 and j = 2, respectively). . . 111 6.6 The bifurcation diagram of the critical points of system (6.4.11) for

¯

α= 0.90 and for several values of ¯βs≥ 0 (y and z are the real solutions

of (6.4.16) for j = 1 and j = 2, respectively). . . 112 6.7 The stability response-curves ¯Rn = ˜A2n + ˜B2n and ¯Rs = ˜A2s + ˜Bs2 of

system (6.4.11) for ˜α= 0.90 and ¯M = 2. The dashed line represents an unstable solution and the solid line represents a stable solution. . . 114

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List of Tables

2.1 The critical values of p2 _{and λ . . . .} ₉

2.2 The number of critical points of (2.3.4) as described in Fig. 2.6. . . . 15 2.3 The number of critical points of (2.3.4) describing Fig. 2.10. . . 20 3.1 The critical points of system (3.4.2) and their stability for ¯α1 = ¯α2 =

0. The stability is determined by using the linearisation method. . . . 32 3.2 The critical points of system (3.4.2) and their stability in the

hy-perplane G0 for ¯α1 = ¯α2 = α and ˜βα =

q_¯

β−α ¯

β+α. The stability is

determined by using the linearisation method. . . 39 3.3 The critical points of system (3.4.2) and their stability in the ( ¯As, ¯Bs, ¯Cs, ¯Ds

)-space for 0 < ¯α1 <α¯2 and ˜βαi = q_¯

β−¯αi

¯

β+ ¯αi,i = 1 or 2. The stability is determined by using the linearisation method. . . 45 4.1 The number of critical points of system (4.3.3) as described in Fig. 4.2. 61 4.2 The number of critical points of system (4.3.3) as described in Fig. 4.7. 66 4.3 The number and the type of critical points of system (4.3.3) as

de-scribed in Fig. 4.8(ii), with ¯α1 = 0.75 and ¯α2 = 0.25. . . 67

4.4 The number of critical points of system (4.4.7) for ¯α > 0 and ˜σ = 1 as described in Fig. 4.14. . . 77 4.5 The three cases which arise from the relation between the solutions

of (B.14) and (B.16). . . 84 6.1 The number of critical points of (6.4.11) as described in the Figures

6.5 and 6.6. . . 113

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Chapter 1 Introduction

The aim of this thesis is to gain a better insight into the dynamics of inclined stretched strings which are suspended between a fixed support and a vibrating sup-port. Some interesting problems, which can be modelled by these string-problems, are for instance the oscillations of cables in suspension bridges, the oscillations of cables in cable-stayed-bridges, and the oscillations of overhead power transmission lines. For a cable-stayed bridge the oscillation of the deck or of the pylon tower (from which the cable is suspended) can act as a vibrating support for the cable.

The oscillations of a stretched string can consist of two types of vibrations: lon-gitudinal vibrations and transversal (in- and out-of-plane) vibrations. Also the force applied at the vibrating support can act in the longitudinal direction and/or in the transversal directions. A well-known experiment done by Melde [25] in 1860 showed that a parametrical excitation (that is, an excitation in longitudinal direction) at one of the ends of string can lead to a transversal response of the string. In this thesis support-excitations in the longitudinal (or parametrical) direction and in the in-plane transversal direction are considered both. For certain excitation-frequencies it will turn out that complicated internal resonances can occur, and for those fre-quencies the usually small excitation-forces can lead to large amplitude-responses of the string. To identify the parameters (that is, the properties of the string) that give rise to these resonances is one of the aims of this thesis.

In order to derive a mathematical model describing the dynamics of the string certain assumptions have to be made. The most important assumptions are that:

• The string is elastic, and the bending stiffness of the string can be neglected, • The excitation force applied at one end of the string only acts in the plane of

the string,

• The tension in the string is sufficiently large such that the sag of the string due to gravity is small, and

• damping forces are small, and are proportional to the velocity of the string. 1

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Based on these assumptions the equations of motion (describing the longitudinal and transversal displacements) of the string can be derived by using a variational principle (see for instance in [12]). By using Kirchhoff’s approach the so-obtained system of partial differential equations (PDEs) can be reduced to a system of PDEs describing the transversal displacements of the string. The existence and the stability of (almost) time-periodic solutions are obtained analytically by using the averaging method, the Galerkin truncation method, and the linearisation method.

This thesis is organized as follows. In the chapters 2, 3, and 4 it is assumed that the tension in the string is so large that gravity-effects (such as the sag of the string) can be neglected. In chapter 2 only the in-plane transversal vibrations of the string due to a parametrical and a transversal excitation at one of the ends of the string will be studied. In the chapter 3 and 4 the in-plane and the out-of-plane transversal vibrations of the string are studied. In the third chapter the analysis is restricted to a parametrical excitation at one of the ends of the string, but in the fourth chapter both a parametrical and a transversal excitation are considered. Effects due to gravity are considered in the chapters 5 and 6. First the static state due to gravity is studied in chapter 5, and the in-plane transversal motion of the string due to a parametrical and a transversal excitation at one of the ends of the string is studied in chapter 6.

In the analysis of the model equations in the chapters 2, 3, 4, and 6, many bifurcation diagrams and pictures are given. The bifurcation diagrams and the response curves are depicted by using the analytical software package Maple, whereas the phase plane figures and the stabilities of the motions are determined by using the numerical software package Matlab. In the bifurcation diagrams, for instance Figs. 2.10, 4.14, and 6.6, bifurcation curves are crossing. The intersection points correspond to higher order bifurcations. Hence, for parameters in a neighbourhood of these intersection points the analysis should be extended to higher order terms. However, these problems are not considered in this thesis.

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Chapter 2 Combined parametrical and

transversal in-plane harmonic

response of a stretched string

‡

Abstract. In this chapter, a two-point boundary value problem for an integro-differential equation is studied. This equation describes the dynamics of an inclined stretched string suspended between a fixed support and a vibrating support. Due to the inclination, the string will vibrate under combined parametrical and transversal excitation. The attention will be focused on time-periodic solutions consisting of one mode in transversal direc-tion (semi-trivial soludirec-tion) generated by transversal (external) excitadirec-tion and two modes in transversal direction (non-trivial solution) generated by combined parametrical and transversal excitation. When the parametrical and transversal excitation amplitudes are equal, the modes generated by parametric excitation may easily have amplitudes ten times larger than the amplitudes of the transversally excited modes.

2.1 Introduction

Consider a perfectly flexible string in a stretched situation. At the end x = 0 the string is attached to a horizontal plane and at x = 1 the other end is fixed to a vertical rigid bar, which is excited in horizontal direction. A sketch of the vibrating system is given by Figure 2.1. The case is studied that the system is embedded in an elastic medium where Hooke’s law applies. In the literature [28, 36, 41] one usually studies vibrating strings positioned along a horizontal plane and excited either vertically or horizontally. Belhaq and Houssni [3] studied the dynamic response of a one-degree-of-freedom system with quadratic nonlinearities and subjected to combined parametric and external excitations. The system can serve as a model for the

one-‡_{A slightly revised version of this chapter has been published in Journal of Sound and Vibration}

([5]), Combined parametrical and transversal in-plane harmonic response of an inclined stretched string.

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mode vibration of a heavy elastic structure suspended between two fixed supports at the same level and excited by a quasi-periodic forcing. Perkins [31] studied modal interactions in the non-linear response of suspended elastic cables under parametrical and external excitations with horizontal position of the cable. Zhang and Tang [42] investigated analytically the global dynamic behaviour of an elastic cable under combined parametrical and external excitations. The system describes a model for the coupling between in-plane and out-of-plane modes of a suspended elastic cable between a fixed support and a vibrating support. Lilien and Pinto da Costa [21] studied pure parametric excitation of an inclined elastic cable with small sag. He at al. [15] studied the control of seismic excitation of a cable-stayed bridge by means of special dampers.

X = 0

V(X , t)

X − axis

ϕ

Figure 2.1: A simple vibrating system including parametrical and transversal excitation

of a stretched inclined string.

In a recent paper by Nielsen and Kirkegaard [30] the in- and out-of-plane ex-citation of an inclined elastic cable with small sag is investigated. In contrast to the present paper primary parametric excitation due to longitudinal excitation is not considered by Nielsen and Kirkegaard. As is known from experimental work by e.g. Melde [25] this primary parametric excitation in stretched strings may lead to a transverse response. In this chapter a system with combined transversal (i.e. perpendicular to the X-axis as indicated in Fig. 2.1 and longitudinal (i.e. in the di-rection of X-axis) excitation due to the angle ϕ between the string and the horizontal plane will be investigated.

The mathematical model for this system is an extension of the model as given in [4, 36]. The extension concerns the term p2_V_¯ _{describing the elasticity of the medium}

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2.2 Analysis of the model equation 5

The mathematical model obtained in this way is as follows: ¯ Vτ τ(x, τ ) − ¯Vxx(x, τ ) + p2V¯(x, τ ) = ¯Vxx(x, τ ) h1 2 Z 1 0 ¯ Vx2(x, τ )dx + Asin(λτ )i_{− α ¯}Vτ(x, τ ), 0 < x < 1, τ > 0, ¯ V(0, τ ) = 0, V¯(1, τ ) = B sin(λτ ), τ >0, ¯ V(x, 0) = ¯V0(x), V¯t(x, 0) = ¯V1(x), 0 ≤ x ≤ 1, (2.1.1)

where 0 < << 1, α ≥ 0, p2 _{≥ 0, λ > 0, A = F cos(ϕ), B = F sin(ϕ), F 6= 0 is}

_{−independent, 0 < ϕ <} 1₂π _{with ϕ is −independent, and ¯}V(x, τ ) is the transversal displacement. The magnitudes of A and B are of the same order implying that the angle ϕ between the string and the horizontal plane is of order 1. In model (2.1.1) gravitation is not considered, implying that there is no sag. Hence, the parametric excitation only applies to elastic elongations. Additionally elongation due to the presence of the sag due to gravity will be studied in a subsequent chapter. This type of elongation is also relevant in a practical situation. Model equation (2.1.1) is of particular relevance for shorter stays in cable-stayed bridges. In this chapter formal approximations to the solutions of the boundary value problem (2.1.1) will be constructed by using a Fourier mode expansion. Subsequently, the averaging method [39] will be applied and for special combinations of λ and p2_{, periodic solutions will}

be studied. Values of λ and p2 _{for which a periodic solution consisting of two}

modes exists, are determined. Those values will give rise to mode interaction. The interesting cases p2 _{= 0 and λ near 2π are considered. Especially the interaction}

between the first and the second mode is studied.

2.2 Analysis of the model equation

Consider the two-point boundary value problem for the integro-differential equa-tion (2.1.1) with a small parameter . To reduce the problem to a problem with homogeneous boundary values, the following transformation is used:

¯

V(x, τ ) = Bx sin(λτ ) + v(x, τ ). (2.2.1)

Substitution of (2.2.1) into (2.1.1) yields

vτ τ(x, τ ) − vxx(x, τ ) + p2v(x, τ ) = vxx(x, τ ) h1 2 Z 1 0 v_x2(x, τ )dx + A sin(λτ )i +(λ2_{− p}2_{)Bx sin(λτ ) − αv}τ(x, τ ) + O(2), 0 < x < 1, τ > 0, v(0, τ ) = v(1, τ ) = 0, τ >0, v(x, 0) = ¯v0(x), vτ(x, 0) = ¯v1(x), 0 ≤ x ≤ 1, (2.2.2)

where ¯v0(x) = ¯V0(x) and ¯v1(x) = ¯V1(x) − Bλx. The eigenfunctions and the

eigen-values of the Sturm-Liouville problem: d2vn(x)

dx2 − p 2_v

n(x) = −µ2nvn(x),

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related to the homogeneous unperturbed system (2.2.2) (i.e. = 0) are vn(x) =

qnsin(nπx) and µ2n = p2+ n2π2, respectively, where qn are constants, n = 1, 2, 3, ....

The O(2_{) terms are neglected; exact solutions v(x, τ ) of (2.2.2) exist in the form of}

a Fourier sine-series (eigenfunction-series): v(x, τ ) =

∞

X

n=1

qn(τ ) sin(nπx). (2.2.4)

By substituting (2.2.4) into (2.2.2) and using the orthogonality properties of the eigenfunctions, one obtains an infinite dimensional system for qn(τ ):

¨ qn(τ ) + µ2nqn(τ ) = − h n2π2qn(τ ) _X∞ k=1 1 4k 2_π2_q2 k(τ ) + A sin(λτ ) + α ˙qn(τ ) − (λ2 _{− p}2)Bcnsin(λτ ) i , q(0) = 2 Z 1 0 [¯v0(x) sin(nπx)]dx, ˙q(0) = 2 Z 1 0 [¯v1(x) sin(nπx)]dx, (2.2.5)

where cn = (−1)(n+1) 2_nπ, n = 1, 2, . . ., and the dot represents differentiation with

respect to τ .

Notice that system (2.2.5) can be expected to have solutions whenever the series in the right hand side converges. It is assumed that v(x, τ ) is a twice continuously differentiable function with respect to x on the open interval (0, 1). It is reasonable to choose appropriate initial conditions, for instance ¯v0(x) ∈ C4(0, 1) and ¯v1(x) ∈

C3_{(0, 1). By using integration by parts it will follow that |q}

n(τ )| ≤ constant_n2 . It means that the series in the right hand side of (2.2.5) converges.

In system (2.2.5) the frequency of the vertical (that is, perpendicular to the X-axis) and parametrical excitation is λ. To have the larger effect of excitation there are two possibilities for the values of λ which are of interest for the study of periodic solutions:

(a) λ = µm ; (b) λ = 2µs,

where m and s are certain values of n. As is well-known (a) corresponds to elementary resonance whereas (b) corresponds to parametrical resonance. If no positive integers m and s exist such that (a) and/or (b) hold then the effects of the excitation on the solutions are of higher order i.e. of order 2_{. Both types of resonance can be}

expected if µm = 2µs i.e.

p2+ m2π2 = 4p2+ 4s2π2 or 3p2 = (m2_{− 4s}2)π2. (2.2.6) In other words by choosing the integers m and s such that m2_{− 4s}2 _{≥ 0 and p such}

that (2.2.6) holds, then λ is defined by λ = µm = 2µs and two types of resonance

may be expected. Introduce the transformation (qn(τ ), ˙qn(τ )) → (An(τ ), Bn(τ )) as

follows:

qn(τ ) = An(τ ) sin(µnτ) + Bn(τ ) cos(µnτ),

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2.2 Analysis of the model equation 7

The transformation (2.2.7) implies that ˙An(τ ) sin(µnτ) + ˙Bn(τ ) cos(µnτ) = 0.

Sub-stitute (2.2.7) into (2.2.5) and solve the equations for ˙An(τ ) and ˙Bn(τ ), giving

˙ An(τ ) = −Fn(A, B; ϕ, α, τ ) cos(µnτ), ˙ Bn(τ ) = Fn(A, B; ϕ, α, τ ) sin(µnτ), (2.2.8) where A = (A1(τ ), A2(τ ), . . . , An(τ ), . . .), B = (B1(τ ), B2(τ ),. . . , Bn(τ ), . . .), and Fn = 1 µn n An(τ ) sin(µnτ) + Bn(τ ) cos(µnτ) h_X∞ k=1 1 8n 2_π4_k2_(A2 k(τ ) + B_k2(τ )) + (B_k2_{(τ ) − A}2_k(τ )) cos(2µkτ) + 2Ak(τ )Bk(τ ) sin(2µkτ) + n2π2Asin(λτ )i+ αµn An(τ ) cos(µnτ) − Bn(τ ) sin(µnτ) − (λ2_{− p}2)Bcnsin(λτ ) o .

As is known from the theory of averaging, the averaged equations have solutions ¯

An(τ ) and ¯Bn(τ ) which are O() approximations to An(τ ) and Bn(τ ) respectively

on a long 1

time-scale. Isolated stable critical points of the averaged system

corre-spond with stable (quasi) periodic solutions of (2.2.8) in case that the systems are finite dimensional. Here, however, (2.2.8) is an infinite dimensional system. To the knowledge of the author it seems not to be known whether these stability properties of the averaged system correspond with stability properties of (2.2.8). However, it is assumed that these results for finite dimensional systems also hold for infinite dimensional systems.

The terms on the right hand side of (2.2.8) are periodic functions with respect to τ . This means that one can approximate the functions An(τ ) and Bn(τ ), for all

n, by using the averaging method. In order to apply this method to system (2.2.8), the value of λ must be determined. If λ is not O() close to µk and 2µk for all

k = 1, 2, 3, . . ., the averaged equations of (2.2.8) are as follows: ˙¯ An(τ ) = − 1 2 α ¯An+ n2π4 8µn ¯ Bn h_X∞ k=1 k2( ¯A2_k+ ¯B_k2) + 1 2n 2_{( ¯}_A2 n+ ¯Bn2) i , ˙¯ Bn(τ ) = − 1 2 α ¯Bn− n2_π4 8µn ¯ An h_X∞ k=1 k2( ¯A2_k+ ¯B_k2) + 1 2n 2_{( ¯}_A2 n+ ¯Bn2) i , (2.2.9)

for n = 1, 2, 3, . . . . From equation (2.2.9) it follows that if ¯An(0) = ¯Bn(0) = 0, then

¯

An(τ ) ≡ ¯Bn(τ ) ≡ 0 for ∀τ > 0. It means that if there is no initial energy in the

nth-mode, there will be no energy present up to O() on a time-scale of order −1_.

This allows us to truncate to those modes that have non-zero initial energy. If λ is O() close to µm but not to 2µs, where m and s are certain values of n, extra terms

probably occur in the equation for ¯Am(τ ) and ¯Bm(τ ). Whereas if λ is O() close

to 2µs but not to µm, extra terms multiplying ¯As(τ ) and ¯Bs(τ ) in the equations

for ¯As(τ ) and ¯Bs(τ ) probably occur. Thus, if λ = µm = 2µs (full resonance) so

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¯

Am(τ ), and ¯Bm(τ ). After some calculations, the obtained averaged equations for

¯

An(τ ) and ¯Bn(τ ), n = s, m, are as follows:

˙¯ As = − 1 2 (α +s 2_π2_A 2µs ) ¯As+ s2_π4 8µs ¯ Bs hX∞ k=1 k2( ¯A2_k+ ¯B_k2) + s 2 2( ¯A 2 s+ ¯Bs2) i , ˙¯ Bs = − 1 2 (α − s 2_π2_A 2µs ) ¯Bs− s2π4 8µs ¯ As h_X∞ k=1 k2( ¯A2k+ ¯Bk2) + s2 2( ¯A 2 s+ ¯Bs2) i , ˙¯ Am = − 1 2 α ¯Am+ m2π4 8µm ¯ Bm h_X∞ k=1 k2( ¯A2_k+ ¯B_k2) + m 2 2 ( ¯A 2 m+ ¯B 2 m) i , ˙¯ Bm = − 1 2 α ¯Bm− m2π4 8µm ¯ Am h_X∞ k=1 k2( ¯A2_k+ ¯B_k2) + m 2 2 ( ¯A 2 m+ ¯Bm2) i + (−1)m+1B2mπ µm . (2.2.10)

For n 6= s and n 6= m the equations for ¯An(τ ) and ¯Bn(τ ) are given by equation

(2.2.9). Supposing that ¯An(0) = ¯Bn(0) = 0, n 6= m, it follows that ¯An(τ ) ≡ 0 and

¯

Bn(τ ) ≡ 0 for τ > 0. It is supposed that ¯An(0) = ¯Bn(0) = 0 for n 6= s and n 6= m.

It then follows from (2.2.9)-(2.2.10) that the following system holds: ˙¯ As = − 1 2 h (α +s 2_π2_A 2µs ) ¯As+ s2π4 16µs ¯ Bs 3s2( ¯A2_s+ ¯B_s2) + 2m2( ¯A2_m+ ¯B_m2)i, ˙¯ Bs = − 1 2 h (α − s 2_π2_A 2µs ) ¯Bs− s2π4 16µs ¯ As 3s2( ¯A2_s+ ¯B_s2) + 2m2( ¯A2_m+ ¯B_m2)i, ˙¯ Am = − 1 2 h α ¯Am+ m2π4 16µm ¯ Bm 2s2_{( ¯}_A2 s+ ¯Bs2) + 3m2( ¯A2m+ ¯B2m) i , ˙¯ Bm = − 1 2 h α ¯Bm− m2π4 16µm ¯ Am(2s2( ¯A2s+ ¯Bs2) + 3m2( ¯A2m+ ¯Bm2) + (−1)m+1B2mπ µm i . (2.2.11)

It may be clear that there is a coupling between the modes s and m and if one wants to truncate the series (2.2.4) for the study of resonance one has to take into account at least m modes, where m is determined by the values of p, λ, and the initial values. Based on the above results it follows that only specific combinations of λ and p2 _may

give periodic solutions. The values of p2 _{and λ for which periodic solutions consisting}

of two modes are found are called critical values and can easily be determined. For these critical values mode interaction will occur. The critical values of p2 _{with the}

corresponding values of λ are given in Table 2.1.

2.3 Modal interaction for the specific

combina-tions of values of p

2

and λ

As stated above only the equations for n = s and n = m are considered. An inter-esting case is p = 0 corresponding to m = 2s and λ = 2sπ for a fixed s. The model

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₉

Table 2.1: The critical values of p2 and λ

p2 ₌ _λ2 ₌ _{critical points}

1 3π

2_(m2_{− 4s}2₎ 4 3π

2_(m2_{− s}2₎ _{one mode : two modes}

4π2 _(m=2) _{(0, 0, ¯}_A 20, ¯B20) : ( ¯A12, ¯B12, ¯A21, ¯B21) 16π2 _(m=4) _{(0, 0, ¯}_A 40, ¯B40) : ( ¯A24, ¯B24, ¯A42, ¯B42) 0(m = 2s) 36π2_(m=6) _{(0, 0, ¯}_A 60, ¯B60) : ( ¯A36, ¯B36, ¯A63, ¯B63) ... ... λ2 _{= 4s}2_π2 _{(0, 0, ¯}_A m0, ¯Bm0) : ( ¯Asm, ¯Bsm, ¯Ams, ¯Bms) 5 3π 2_{(s=1, m=3)} 32 3π 2 _{(0, 0, ¯}_A 30, ¯B30) : ( ¯A13, ¯B13, ¯A31, ¯B31) 4π2 _(s=1,m=4) _20π2 _{(0, 0, ¯}_A 40, ¯B40) : ( ¯A14, ¯B14, ¯A41, ¯B41) 3π2 _(s=2,m=5) _28π2 _{(0, 0, ¯}_A 50, ¯B50) : ( ¯A25, ¯B25, ¯A52, ¯B52) ... ... ...

equation as presented here can be used for the study of the dynamics of inclined stay-cables connecting the bridge deck and pylon of a cable-stayed bridge by assum-ing that the motion of the deck can be ignored. In terms of accuracy of the model equation one could say that the motion of the bridge deck at the endpoint of the stay-cable is assumed to be of O(2_{). Substitution of q}

n(τ ) = 0, n 6= s and n 6= m,

into equation (2.2.5), gives the two coupled second order equations: ¨ qm(τ ) + µ2mqm(τ ) = − m2π2qm(τ ) hs2π2 4 q 2 s(τ ) + m2π2 4 q 2 m(τ ) + A sin(λτ ) i +α ˙qm(τ ) − (λ2− p2)Bcmsin(λτ ) , ¨ qs(τ ) + µ2sqs(τ ) = − s2π2qs(τ ) hs2π2 4 q 2 s(τ ) + m2_π2 4 q 2 m(τ ) + A sin(λτ ) i +α ˙qs(τ ) − (λ2− p2)Bcssin(λτ ) . (2.3.1)

It may be clear that if λ 6= µm + O() and λ 6= 2µs + O() and then by using

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equations with a structure similar to (2.2.9), has as critical point (0, 0, 0, 0) which is stable for α > 0. If λ = µm+ O() but λ 6= 2µs+ O() then (0, 0, 0, 0) is not a critical

point unless ϕ = 0 (or O()), corresponding with B = 0. In this case the stable critical point (0, 0, ¯Am, ¯Bm) is found. If λ 6= µm+ O() but λ = 2µs+ O() then in

the (ϕ − α) parameter plane there is a domain defined by (cos(ϕ)-2 αµs

Cs2_π2) > 0 where there are two critical points: one, the origin, is unstable, the other, ( ¯As, ¯Bs,0, 0), is

stable. In the complement of this domain the only critical point is the stable origin. All cases discussed above concern systems with one stable critical point and hence one mode, implying that there is no interaction between modes s and m. Therefore in what follows only the case λ = 2µs + O() and µm = 2µs will be studied. By

setting λτ = 2t, where λ = 2(µs+ η), system (2.3.1) becomes:

q_m00(t) + 4qm(t) = − µs hm2π2 µs qm(t) s2π2 4 q 2 s(t) + m2_π2 4 q 2 m(t) + A sin(2t) + αq0m(t) − βmsin(2t) − 8ηqm(t) i + O(2), q00s(t) + qs(t) = − µs hs2π2 µs qs(t) s2π2 4 q 2 s(t) + m2π2 4 q 2 m(t) + A sin(2t) + αq0_s_{(t) − β}ssin(2t) − 2ηqs(t) i + O(2_), _(2.3.2) where βm = (−1)m+12mπ_µ_s B, βs = (−1)s+12m 2 π

sµs B, and η is the detuning coefficient of the frequency of excitation. A prime denotes differentiation with respect to t. For the sake of simplicity, the case λ near 2π is considered. This value implies a system describing the interaction between first (s=1) and second (m=2) mode:

q₂00(t) + 4q2(t) = − π h 4πq2(t) 1 4π 2_q2 1(t) + π2q22(t) + A sin(2t) + αq0 2(t) + 4B sin(2t) − 8ηq2(t) i , q00₁(t) + q1(t) = − π h πq1(t) 1 4π 2_q2 1(t) + π2q22(t) + A sin(2t) + αq₁0_{(t) −} 8B sin(2t) − 2ηq1(t) i . (2.3.3)

In the first equation of (2.3.3) the excitation term 4B sin(2t) is relevant for having an O(1) amplitude response, whereas in the second equation πq1Asin(2t) is the

relevant excitation term. Clearly, the first term describes ordinary and the second one, parametric resonance. System (2.3.3) can be used for the study of rotor-bearing system as well [14].

By using transformation (2.2.7) for n=1 and n=2 the following averaged system is obtained: ¯ A0₂_{(t) = −} 2π α ¯A2+ ¯B2 h γ2(( ¯A21+ ¯B12) + 6( ¯A22+ ¯B22)) − 4η i , ¯ B0₂_{(t) = −} 2π α ¯B2 − ¯A2 h γ2(( ¯A21+ ¯B12) + 6( ¯A22 + ¯B22)) − 4η] − 2B , ¯ A0₁_{(t) = −} 2π β+A¯1+ ¯B1 h γ1(3( ¯A21+ ¯B12) + 8( ¯A22+ ¯B22)) − 2η i , ¯ B0₁_{(t) = −} 2π β₋B¯1− ¯A1 h γ1(3( ¯A21+ ¯B12) + 8( ¯A22+ ¯B22)) − 2η i , (2.3.4)

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₁₁

where γ1 = ₁₆1 π3, γ2 = 1₄π3, and β± = α ±21πA.

In what follows the critical points and their dependence on the parameters α, η, A and B will be investigated. Recall that these parameters are supposed to be O(1) and that A and B are defined by: A = F cos(ϕ) and B = F sin(ϕ). The following special cases will be studied: α = 0 and α > 0.

0 0.5 1.5 1.0 −5 5 10 15 20 η 0 0.2 0.6 1.0 1.4 0.2 0.4 0.6 2 R 0.8 (a) ϕ= 1 4π (b) η= 7

Figure 2.2: The response curve R2 = ¯A22 of equation (2.3.4) for α = 0 and F = 6: (a)

Response curve as function of frequency of excitation; (b) Response curve as function of the angle of inclination on string: The stability only applies to

the ( ¯A2, ¯B2)-plane.

2.3.1 The case without damping,

i.e.

α

= 0

Because of B 6= 0, it follows from the first two equations of (2.3.4) that ¯B2 = 0. As

a consequence the following system of algebraic equations is obtained: ¯ A2 h γ2 ( ¯A2₁+ ¯B₁2) + 6 ¯A2₂_{− 4η] + 2B = 0,} β ¯A1+ ¯B1 h γ1 3( ¯A2₁+ ¯B₁2) + 8 ¯A2₂_{− 2η}i = 0, β ¯B1+ ¯A1 h γ1 3( ¯A2₁+ ¯B₁2) + 8 ¯A2₂_{− 2η}i = 0, (2.3.5) where β = 1₂πA. For nontrivial solutions of the last two equations of (2.3.5) it follows that ¯A1 = ± ¯B1 under the condition that β2 − [γ1(3( ¯A21 + ¯B12) + 8 ¯A22) − 2η]2 = 0.

Clearly the following type of critical points are found: CP-type 1: ( ¯A1, ¯B1, ¯A2, ¯B2) = (0, 0, ˜A2,0),

CP-type 2: ( ¯A1, ¯B1, ¯A2, ¯B2) = ( ˜B1, ˜B1, ˜A2,0),

CP-type 3: ( ¯A1, ¯B1, ¯A2, ¯B2) = (− ˜B1, ˜B1, ˜A2,0), (2.3.6)

where ˜B1 and ˜A2 satisfy (2.3.5). The first type of critical points are on the ¯A2-axis

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on η is well-known and is depicted in Fig. 2.2 and the behaviour of it is presented in Fig. 2.3. 0 −0.4 −0.8 −1.2 0.4 0 −0.4 −0.8 0.4 0.8 0.8 0.4 0 0.4 0.8 1.2 0.8 0.4 0 0.4 1.2 0.8 1.2 0.8 0.4 0 0.4 1.2 0.8 0.4 0 0.4 (a) η= 7.0 (b) η= 7.1396 (c) η= 8.0

Figure 2.3: The behaviour of critical points of equation (2.3.4) in the ( ¯A₂, ¯B₂_)−plane

with α = 0, F = 6, and ϕ = 0.8. The horizontal and vertical axis are the ¯

A2-axis and the ¯B2-axis, respectively.

Figure 2.4: Four domains with real solutions of system (2.3.7) for F=6: The curve P2P6

comes from ∆1 = 0 and the others come from Cond1 = 0, the value of ∆ is

negative in domains I, II, and III and positive in domain IV.

In what follows a (ϕ, η)-diagram will be constructed which gives an overview of all possible critical points for F fixed. Starting with ¯A1 = ¯B1, system (2.3.5) can be

rewritten as: ˜ A3₂₋ 8 5π3(η + β) ˜A2+ 12B 5π3 = 0, Cond1 = (4η − 2β) − π3A˜22 >0, ˜ B₁2 = 8 3π3(2η − β) − 4 3A˜ 2 2. (2.3.7)

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₁₃

The case 4η − 2β − π3_A_˜2

2 = 0 corresponds with the first type of critical points in

(2.3.6). System (2.3.7) defines domains in the (η − ϕ)-plane where there exists one, two, or three real non-zero solutions ( ˜B1, ˜B1, ˜A2,0). These domains are found by

determining the boundary curves which follow from ∆1 = 0, where ∆1 =

h 6 5π3B i2 + h 8 15π3(η + β) i3

is the radicand of the cubic equation in standard form in (2.3.7) and the equality (4η − 2β) − π3_A_˜2

2 = 0. In this equality ˜A2 as solution of the first

equation of (2.3.4) (obtained from the Cardano formulas) depending on η and ϕ is substituted. As a result one obtains Fig. 2.4. In this figure there are four domains I -IV with one, two, or three critical points: I-1, II-2, III-3, -IV-1. Here II-2 means that in domain II there exist 2 critical points of the type ( ˜B1, ˜B1, ˜A2,0). When one looks

separately at the case ¯A1 = ¯B1 = 0 corresponding with the so-called semi-trivial

solution one obtains the following cubical equation for ˜A2:

3 2π

3_A_˜3

2− 4η ˜A2+ 2B = 0. (2.3.8)

This equation differs from the first equation in (2.3.7). By setting the radicand ∆0 = [_3π23B]

2_{+ [−} 8 9π3η]

3 _{= 0 one obtains a curve indicated by curve P}

7P9 in Fig. 2.5

on which there are two critical points of the type (0, 0, ˜A2,0). On the left hand side

there is one critical point and on the right hand side there are three critical points. In the domains in Fig. 2.5 the notation (n, m) means that there are n critical points of type (0, 0, ˜A2,0) and m of type ( ˜B1, ˜B1, ˜A2,0).

Figure 2.5: Eight domains with real solutions of type 1 and 2 of system (2.3.5) for F=6:

The curve P7P9 comes from the radicand of (2.3.8) ∆0 = 0.

In a similar way the critical points of the type (− ˜B1, ˜B1, ˜A2,0) are analyzed.

They are found from the system: ˜

A3₂₋ 8

5π3(η − β) ˜A2+

12B 5π3 = 0,

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Cond2 = (4η + 2β) − π3A˜22 >0, ˜ B₁2 = 8 3π3(2η + β) − 4 3A˜ 2 2. (2.3.9)

The resulting boundary curves are additionally presented in Fig. 2.6 which follows from Fig. 2.5. Clearly, new curves P10P5 and P11P6 are found defined in the 12

domains. The type and the number of critical points in these domains and on the boundary curves are given in Table 2.2.

Figure 2.6: Twelve domains with real solutions of type 1, 2, and 3 of (2.3.5) for F = 6:

The new curves P6P11 and P5P10 are found from the radicand of the first

equation of (2.3.9) ∆2 = h 6B 5π3 i2 +h₋ _15π83(η − β) i3 = 0 and Cond2 = 0, respectively.

It is of interest to look what happens if one chooses ϕ fixed for instance ϕ = 1₄π and starts in domain I and then increases η. It is aimed to know the effect of the frequency of excitation to the periodic solutions. In particular it is of interest to compute explicitly the amplitude of the periodic solutions R1 = ˜A21+ ˜B12and R2 = ˜A22

as a function of η. The results are presented in Fig. 2.7. Clearly, in domain I there is one real solution R2 as sketched in Fig. 2.6. In Q1 this solution bifurcates in a

stable (in the sense of Lyapunov) and an unstable one. When one arrives in Q2 two

new solutions appear, a stable and an unstable one. Analogously in Q3 two new

unstable ones appear whereas in Q4 one new unstable and in Q5 one stable and one

unstable solution appear leading totally to nine critical points in domain XII. All bifurcation points Qi, i = 1, 2, . . . , 5 are indicated in Fig. 2.6 and in Fig. 2.7.

In Fig. 2.8 the amplitude of periodic solutions are depicted as function of the angle (ϕ) for η and F are given. In this figure one can see clearly that the critical points type 2 are always unstable solutions whereas type 3 are always stable solution. The critical points type 1 will be a stable solution if ϕ > ϕ4 whereas for 0 < ϕ < ϕ3

they will be stable solutions if the amplitude of them is smaller.

The semi trivial solution as indicated in Fig. 2.7(a) and in Fig. 2.8(a), respec-tively correspond with the response curve in Fig. 2.2(a) and in Fig. 2.2(b). The

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₁₅

Table 2.2: The number of critical points of (2.3.4) as described in Fig. 2.6.

Domains/ The number of critical points of (2.3.4)

Curves/Points CP-type 1 CP-type 2 CP-type 3 Total

I 1 0 0 1 II 1 0 1 2 III 1 1 1 3 IV and V 3 0 1 4 VI and VII 3 1 1 5 VIII 3 2 1 6 IX 3 1 3 7 X 3 3 1 7 XI 3 2 3 8 XII 3 3 3 9 P5P10 1 0 0 1 P5P8 1 0 1 2 P7P8 2 0 1 3 P1P2 and P4P8 3 0 1 4 P8P9 2 1 1 4 P2P4 and P2P12 3 1 1 5 P4P6 and P4P13 3 2 1 6 P11P12 3 1 2 6 P3P12 3 1 3 7 P12P13 3 2 2 7 P3P13 3 2 3 8 P6P13 3 3 2 8 P5 1 0 0 1 P2 and P8 2 0 1 3 P9 2 1 1 4 P4 3 1 1 5 P12 3 1 2 6 P6 and P13 3 2 2 7

difference, however, is that in Fig. 2.7(a) and in Fig. 2.8(a) parts of the curves with stable solutions have become unstable due to the interaction with the first mode. Apparently these parts are unstable in the four dimensional phase space. The most interesting solutions are the non-trivial ones in the four dimensional phase space of type 3 in (2.3.6). Considering the amplitude as function of η and starting from a large value of η < 36 and subsequently decreasing η one observes at Q5 two jumps,

R1 decreases whereas R2 increases. A remarkable result is that the amplitude of the

parametrical induced mode at Q5 before the jump is much larger (i.e. a factor 30)

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Q Q _Q Q Q Q Q Q Q Q Q 1 2 3 4 5 6 1 2 3 4 5 6 ’ ’ Q’ ’ ’ ’ CP’s−type 1 CP’s−type 2 CP’s−type 3 Q Q Q Q Q Q Q Q 1 2 3 4 ₅ ₆ Q3 5 6 ’ ’ ’ CP’stype 3 CP’stype 2 CP’stype 1

(a) second mode (b) first mode

Figure 2.7: Stability response-curves of the periodic solutions of (2.3.4) with respect to

the detuning for F = 6, ϕ = 1₄π, and α = 0. The solid and dashed line

represent stable and unstable solutions, respectively.

CP’stype 1 CP’stype 3 CP’stype 2 ϕ ϕ ϕ ϕ ϕ 1 2 3 4 5 CP’stype 3 CP’stype 2 CP’stype 1 R1 ϕ ϕ ϕ ϕ ϕ ϕ 1 3 4 5 2

Figure 2.8: Stability response-curves of the periodic solutions of (2.3.4) with respect to

the angle of excitation for F = 6, η = 7, and α = 0. The solid and dashed line represent stable and unstable solutions, respectively.

2.3.2 The case with (positive) damping,

i.e.

α >

0

When there is no damping system (2.3.4) has both unstable and stable critical points of type 1. The eigenvalues of the stable critical points of type 1 have zero real part implying that no conclusions can be drawn about the stability of the corresponding periodic solutions of the original system. The curves on which these critical points are located are given in Fig. 2.2, in Fig. 2.7(a), and in Fig. 2.8(b) indicated with

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₁₇

semi trivial solutions.

(a) η= 7.0 (b) η= 7.1392 (c) η= 8.0

Figure 2.9: The behaviour of critical points of (2.3.4) in the ( ¯A2, ¯B2)−plane with α =

0.25, F = 6, and ϕ = 0.8 (no homoclinic orbit). The horizontal and vertical

axis are the ¯A2-axis and the ¯B2-axis, respectively.

When one considers, however, small positive damping the centre points in the ( ¯A2, ¯B2)-plane as indicated in Fig. 2.2 now become positive attractors as depicted in

Fig. 2.9. In what follows the number of critical points of system (2.3.4) with α > 0 and their stability will be studied in more detail. The critical points of (2.3.4) are solutions of: α ¯A2+ ¯B2 h γ2 ( ¯A2₁+ ¯B₁2) + 6( ¯A2₂+ ¯B₂2)_{− 4η}i= 0, α ¯B2− ¯A2 h γ2 ( ¯A2₁+ ¯B₁2) + 6( ¯A2₂+ ¯B₂2)_{− 4η}i_{− 2B = 0,} β+A¯1+ ¯B1 h γ1 3( ¯A2₁+ ¯B₁2) + 8( ¯A2₂+ ¯B₂2)_{− 2η}i= 0, β₋B¯1 − ¯A1 h γ1 3( ¯A2₁+ ¯B₁2) + 8( ¯A₂2+ ¯B₂2)_{− 2η}i= 0. (2.3.10) It is obvious that ¯A1 = ¯B1 = 0 is a solution of (2.3.10) with ¯A2 = −R_B(3γ2R− 2η)

and ¯B2 = _2Bα R, where R are positive real solutions of the following equation:

R3₋ 4η 3γ2 R2+ 1 36γ2 2 (α2 + 16η2_{)R −} B 2 9γ2 2 = 0. (2.3.11)

In order to have non trivial solutions for ¯A1 and ¯B1 from the last two equations

of (2.3.10) it follows that the condition A2 _{− 4}α2

π2 ≥ 0 should hold. The last two equations of (2.3.10) can be reduced to:

γ1(3( ¯A21+ ¯B12) + 8( ¯A22+ ¯B22)) − 2η

2

= β2_{− α}2. (2.3.12)

In the case β = α (A − 2α

π = 0) the value of ¯A1 in equation (2.3.12) should be zero

but ¯B1 6= 0. The equation (2.3.12) implies the possibilities:

(i) γ1 3( ¯A2₁ + ¯B2₁) + 8( ¯A2₂+ ¯B₂2)_{− 2η = −}pβ2_{− α}2_, (ii) γ1 3( ¯A2₁ + ¯B2₁) + 8( ¯A2₂+ ¯B₂2)_{− 2η =}pβ2_{− α}2_. _(2.3.13)

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From the last two possibilities and the case ¯A1 = ¯B1 = 0 it follows that the

coordi-nates of the critical points can be classified in three types: CP-type 1: ( ¯A1, ¯B1, ¯A2, ¯B2) = (0, 0, Ã2, ˜B2), CP-type 2: ( ¯A1, ¯B1, ¯A2, ¯B2) = ( s |β−| β+ ˜ B1, ˜B1, Ã2, ˜B2), CP-type 3: ( ¯A1, ¯B1, ¯A2, ¯B2) = (− s |β−| β+ ˜ B1, ˜B1, Ã2, ˜B2), (2.3.14)

where ˜Aj and ˜Bj, j = 1, 2, satisfy equation (2.3.10). Substitution of the first equation

(2.3.13) in the first and second equations of (2.3.10) gives: R2[γ2(R1+ 6R2) − 4η]2 + α2R2− 4B2 = 0, Cond3 = (2η − p β2 _{− α}2_{) − 8γ} 1R2 >0, R1 = 1 3γ1(2η − p β2_{− α}2_{) −}8 3R2, (2.3.15)

where R1 = ¯A21 + ¯B12 and R2 = ¯A22 + ¯B22. Now substitution of the third into the

first equation of (2.3.15) and then introducing the new variable R2 = Y + _15γ42(η + pβ2_{− α}2_{), yields:} Y3+ κ11Y + δ11= 0, Cond30= 2 15(22η − 23 p β2_{− α}2_{) − π}3_{Y >}_0, _(2.3.16) where: κ11 = − 4 25π6 h16 3 (η + p β2_{− α}2₎2 − 9α2i, δ11 = − 16 25π6 h 9B2₋ 4 5π3(η + p β2_{− α}2₎16 27(η + p β2_{− α}2₎2_{+ 3α}2i_.

In a similar way, but by using now the second equation of (2.3.13) one obtains a second system of equations with solutions of type 3:

Z3+ κ12Z+ δ12 = 0, Cond4 = 2 15(22η + 23 p β2_{− α}2_{) − π}3_{Z >}_0, _(2.3.17) where κ12 = − 4 25π6 h16 3 (η − p β2_{− α}2₎2_{− 9α}2i_, δ12 = − 16 25π6 h 9B2₋ 4 5π3(η − p β2_{− α}2₎16 27(η − p β2_{− α}2₎2_{+ 3α}2i_, R2 = Z + 16 15π3(η − p β2_{− α}2_), R1 = 1 3γ1 (2η +pβ2_{− α}2_{) −}8 3R2.

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2.3 Modal interaction for the specific combinations of values of p2 _{and λ} ₁₉

As indicated in section 2.3.1 an overview of the number of real solutions of type 1, 2, and 3 of (2.3.10) and their dependence on η and ϕ for certain values of α and F can be given in the diagram in Fig. 2.10. Apparently in this figure 18 domains can be distinguished. The boundary curves separating the domains are defined by using equations (2.3.11), (2.3.16), and (2.3.17).

Figure 2.10: Eighteen domains with real solutions of types 1, 2, and 3 of (2.3.10) for

α= 2π and F = 6.

Let us suppose that η is increased while ϕ is held constant. This process is represented by the line through the points Q1, Q2, . . ., Q8 in Fig. 2.10. For η < Q1

only CP-type 1 exists. Between Q1 and Q2 there are CPs of type 1 and CPs of type

3. When η exactly reaches Q2 the critical point type 1 bifurcates to two, whereas

the critical point of type 3 still remains, etc. An overview of the domains and the number of critical points is given in Table 2.3.

In Fig. 2.11 R2 and R1 are plotted as a function of η with α = 2π, ϕ = 1₄π,

and F = 6. The presence of jump phenomena can be observed in these figures. These phenomena are due to the nonlinearities and excitations. To explain this one starts in domain I in Fig. 2.10 and follows the line indicated and parallel to the η-axis. At the point Q1 one enters domain II, at Q2 one enters domain VIII etc. All

points Q1, Q2, . . ., Q8 are also indicated in Fig. 2.11. In this figure, however, the

η-coordinate of Q0

i is indicated with Qi, i = 1, 2, . . . , 8. At points Q1 in Fig. 2.11(a)

one clearly leaves the CP-type 1 (corresponding to the transversally excited mode) because the parametrically excited mode comes in. By increasing η one arrives at Q8 and a jump occurs. Then by decreasing η again a jump occurs at Q5 where now

the amplitude increases while simultaneously in Fig. 2.11(b) at Q5 the amplitude

decreases with a jump. It is of interest to remark that additionally two types of stable critical points are present: the one on the upper Q0

3Q06 curve and the ones on

the lower curve starting at Q0

2 all belonging to the CP-type 1. When one follows

the solutions along these curves apparently at the ends only a jump downward in Fig. 2.11(a), and a jump upward in Fig. 2.11(b) are possible. At the left end of the

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Table 2.3: The number of critical points of (2.3.4) describing Fig. 2.10.

Domains/ Number of critical points of (2.3.4)

Curves/Points CP-type 1 CP-type 2 CP-type 3 Total

I and XVIII 1 0 0 1

II 1 0 1 2

IV , V and XV 1 1 1 3

VI 3 0 0 3

III and VIII 3 0 1 4

VII , X and XI 3 1 1 5 XIV 1 3 1 5 XVI 1 1 3 5 IX 3 2 1 6 XII 3 3 1 7 XVII 1 3 3 7 XIII 3 3 3 9 P1P2, P2P4, and P17P19 1 0 0 1 P2P5, P6P7, and P9P10 1 0 1 2 P3P4 and P17P18 2 0 0 2 P4P17 3 0 0 3 P5P6, P7P8, and P8P10 2 0 1 3 P4P5 and P10P11 2 1 1 4 P5P12 and P7P10 3 0 1 4 P11P15 1 2 1 4 P15P16 and P16P19 1 1 2 4 P6P11, P6P12, and P7P11 3 1 1 5 P11P12 and P12P13 3 2 1 6 P11P14 2 3 1 6 P14P15 1 3 2 6 P15P19 1 2 3 6 P13P14 3 3 2 8 P14P17 2 3 3 8 P2 and P19 1 0 0 1 P3, P4, P17, and P18 2 0 0 2 P8 1 0 1 2 P5, P6, P7, and P10 2 0 1 3 P13 3 0 0 3 P11 2 1 1 4 P16 1 1 2 4 P12 3 1 1 5 P15 1 2 2 5 P14 2 3 2 7

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2.4 Conclusion 21 CPtype 1 CPtype 3 CPtype 2 Q Q Q Q Q Q Q Q Q Q Q 1 1 2 2 2 3 3 4 4 5 5 6 6 7 7 8 8 Q’ ’ ’ Q’ ’ Q’ Q’ Q’ Q1 Q Q2 3 Q4 Q5 Q6 Q7 Q8 4 5 7 8 Q’ Q’ Q Q’ CPtype 1 CPtype 2 CPtype 3

Figure 2.11: Stability response-curves of the periodic solution of (2.3.4) with respect to

the detuning for F = 6, ϕ = 1₄π, and α = 2π. The solid and dashed line

Q3Q6 curve corresponding to the CP-type 1 the parametric excited mode comes in;

hence, a jump to the lower ST curve, starting at Q0

2 is not possible and the jump

should end at the Q0

1Q08 curve. At the right end of the Q03Q06 curve, no parametric

excitation comes in and hence a jump to the lower curve of critical points of type 1 takes place. As in the case without damping, the parametrically excited mode can easily be a factor 10 greater than the amplitude of the transversally excited mode. For ϕ = 1₄π the excitation amplitudes A and B are equal. This does not, however, imply that the excitation energy in both directions is equal. On the other hand, there is a non-linear interaction between the modes implying that energy transfer between the two modes is possible.

In Fig. 2.12 R2 and R1 are plotted as function of ϕ with α = 2π, η = 7, and

F = 6. The purpose is to know the effect of inclination of the string. When ϕ > ϕ6

the parametric excitation does not give any influence to the solution so that the system (2.3.4) only has one stable periodic solution corresponding with CP-type 1. It means that the string only moves in the plane. However, for 0 < ϕ < ϕ6 the

periodic solution corresponding to CP-type 3 exists and is always a stable solution. Whereas the periodic solution corresponding to type 2 will exist for 0 < ϕ < ϕ3 or

ϕ5 < ϕ < ϕ6 and is always unstable. In case a periodic solution of type 2 exists one

can expect an in plane stable motion of the string.

2.4 Conclusion

In this chapter the simultaneous small amplitude excitation in horizontal and ver-tical direction of an end point of a inclined stretched string is studied. As the attention is focused to transversal standing wave modes a modified Kirchoff model

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CPtype 1 CPtype 3 CPtype 2 ϕ ϕ ϕ₁ _{2 3} ϕ4 ϕ5ϕ₆ _ϕ _ϕ ϕ ϕ ϕ _ϕ CPtype 3 CPtype 2 CPtype 1 1 2 3 4 5 6 0.8

Figure 2.12: Stability response-curves of the periodic solution of (2.3.4) with respect to

the angle for F = 6, ϕ = 1₄π, and α = 2π. The solid and dashed line

is used implying that acceleration of horizontal elements of the string are neglected. The mechanisms of mode generation are classical resonance combined with paramet-ric resonance. The conditions to have this combination of resonances are given by p2 = 1

3(m

2_{− 4s}2_)π2 _{and λ}2 ₌ 4 3(m

2_{− s}2_)π2 _{where p ≥ 0 describes the linear elastic}

behaviour of the medium in which the string is embedded, p = 0 corresponds with a model for a vibrating string in air under normal conditions, λ is the excitation frequency and m and s are integers representing mode numbers. An interesting case is p = 0, λ near 2π, and m = 2 and s = 1, describing the interaction of the first and second modes. Three parameters i.e. the damping coefficient (α), the angle of incli-nation (ϕ),and the detuning coefficient (η) are relevant to describe this interaction. Equations are derived for the time-varying behaviour of the mode amplitudes. Fixed points of these equations are analyzed corresponding with time-periodic solutions i.e. modes with constant amplitudes. A classification of all critical points and their stability is given. Of particular interest are the critical points in R4_{, corresponding}

with non-trivial solutions (CP’s-types 2 and 3) and describing mode interaction. A number of amplitude jumps are found with saddle-node bifurcation as underlying mechanism. A remarkable result is that when the horizontal and vertical excitation amplitudes are equal (corresponding with an inclination of 1₄π) the order of magni-tude of the mode response may be quite different: modes generated by parametric excitation may easily have amplitudes ten times larger than the amplitudes of the transversally excited modes. The model equation as presented here can be used for the study of the dynamics of inclined stay-cables connecting the bridge deck and pylon of a cable-stayed bridge. Due to the inclination of the stay cables an aerody-namically unstable pylon will simultaneously induce horizontal and vertical motion of the cable.

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Chapter 3 On the planar and whirling

motion of a stretched string due to

a parametric harmonic excitation

Abstract. In this chapter a model of the dynamics of a stretched string is derived. The sag of the string due to gravity is neglected. The string is suspended between a fixed support and a vibrating support. Due to the vibrating support the oscillation of the string in vertical direction is influenced by a parametrical excitation. The parametric term originates from a longitudinal vibration caused by an elastic elongation and then influences the transversal vibrations in- and out-of-plane. The study will be focused on the existence and stability of time-periodic solutions in transversal direction. The stability is analyzed by using a linearisation method. In addition the different types of periodic motions of the string will be determined.

3.1 Introduction

The vibrations of stretched strings have been investigated by many researchers, because a variety of physical systems can be described by stretched strings, e.g., the stay cable of a cable-stayed-bridge, an overhead power transmission line, and so on. Most of the studies of oscillations of stretched strings has mainly focused on the transverse displacements. The earliest experiment was successfully done by Melde in [25]. He observed that the string can oscillate transversally with an amplitude of about 4% of the length of string, although the excitation force is purely longitudinal. A number of papers about this subject has been published, for instance in [27, 29, 37]. The oscillations of the strings can be caused by many factors [19, 22]. Lilien and Pinto da Costa [21] studied the vibrations caused by a purely parametrical excitation of inclined cables of a cable-stayed-bridge. Pinto da Costa et al. [10] also studied the steady-state response of inclined cables when the ratio between the excitation frequency and the first natural frequency of the cables is close to two. The dominant

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phenomenon in that case is a parametric excitation. It has been shown in [5] that in this case the mode generated by a parametric excitation may easily have amplitudes ten times larger than the amplitudes of the transversally excited modes.

In [5, 10, 21] the vibrations of the strings are only studied in the plane. On the other hand if the frequency of excitation falls in a certain resonance range, the string movement in the plane becomes unstable, and leads to out of plane vibrations ( see also [26, 31, 32, 40]). An experiment to show this phenomenon has been done by Matsumoto et al. [23].

In a recent paper by Zhao et al. [43] the in- and out-of-plane excitation of an inclined elastic cable is investigated (without considering the primary parametric excitation in longitudinal direction). Lee and Renshaw [20] studied the stability of parametrically excited systems using a spectral collocation method. In this chapter, the model of an stretched string motion will be derived by neglecting the sag of the string due to gravity. The periodic solutions will be studied by using the av-eraging method, whereas their stability will be studied by linearizing the averaged equations.

3.2 The derivation of the model equation

We consider a perfectly flexible, elastic, unstretched string with length l < 1. Let (X,0,0) be the coordinate of each material point P of the string with X ∈ [0, l]. The string is stretched uniformly so that the stretched length is 1. In the stretched state the point P will have the coordinates ((1 + ωo)X, 0, 0), where ωo = 1_l− 1 is the initial

strain. Denote the dynamic displacement of the point P by U (X, ¯τ)i, V (X, ¯τ)j and W(X, ¯τ)k, where i, j and k are the unit vectors along the axes of the Cartesian coordinate system and ¯τ is time. U and V are the displacements in horizontal and vertical direction, respectively, whereas W represents the displacement perpendicular to the picture as indicated in Fig. 3.1. So the vector position R(X, ¯τ) of the point P in the dynamic state can be written as:

R(X, ¯τ) = [(1 + ωo)X + U (X, ¯τ)]i + V (X, ¯τ)j + W (X, ¯τ)k. (3.2.1)

The relative strain per unit length of the stretched string is: | _∂X∂ R(X, ¯τ_{) | −1 =}

q

[(1 + ωo) + UX]2+ VX2 + WX2 − 1, (3.2.2)

where UX, VX, and WX represent the derivative of U (X, ¯τ), V (X, ¯τ), and W (X, ¯τ)

with respect to X, respectively. Introduce the new coordinate x by:

x= (1 + ωo)X, (3.2.3)

implying that x ∈ [0, 1] and set: U(X, ¯τ) = U ( x (1 + ωo) ,τ) = ˜¯ u(x, ¯τ), V(X, ¯τ) = V ( x (1 + ωo) ,τ¯) = ˜v(x, ¯τ), W(X, ¯τ) = W ( x (1 + ωo) ,τ¯) = ˜w(x, ¯τ),

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3.2 The derivation of the model equation 25

l

X = l

X=0

f( )

τ

Figure 3.1: The dynamic state of the string suspended between a fixed support at x = 0

and a vibrating support at x = l.

then the relative strain r(x, ¯τ) per unit length becomes:

r(x, ¯τ) = (1 + ωo)p1 + (2˜ux+ ˜u2x+ ˜vx2+ ˜wx2) − 1. (3.2.4)

The kinetic and the potential energy densities of the system are given by: K = 1 2ρ(˜u 2 ¯ τ + ˜vτ2¯+ ˜wτ2¯) and P = 1 2Er 2_{(x, ¯}_τ_), _(3.2.5)

respectively, where ρ is the mass of the string per unit length and E is Young’s modulus. By assuming that | ˜ux |, | ˜vx |, and | ˜wx | are small with respect to 1, the

potential energy may be approximated by its Taylor expansion P6 up to terms of

the sixth degree: P6 = 1 2E h ω_o2+ 2ωo(1 + ωo)˜ux+ (1 + ωo)2u˜2x+ ωo(1 + ωo)(˜vx2+ ˜w2x) + (1 + ωo)˜ux(˜vx2+ ˜wx2) − (1 + ωo)˜u2x(˜vx2+ ˜wx2) + 1 4(1 + ωo)(˜v 2 x+ ˜wx2)2− 3 4(1 + ωo)˜ux(˜v 2 x+ ˜w2x)2 + (1 + ωo)˜u3x(˜vx2+ ˜wx2) + 3 2(1 + ωo)˜u 2 x(˜vx2+ ˜ w2_x)2 _{− (1 + ω}o)˜ux4(˜v2x+ ˜w2x) − 1 8(1 + ωo)(˜v 2 x+ ˜w2x)3 i . (3.2.6)

The Lagrangian density D = K − P6 is used in a variational principle [12] to obtain

the following equations of motion: ρu˜¯τ¯τ− E(1 + ωo)2u˜xx = 1 2E(1 + ωo) ∂ ∂x h (˜v2_x+ ˜w2_x_{) − 2˜u}x(˜vx2+ ˜wx2) − 3 4(˜v 2 x + ˜w2_x)2+ 3˜u2_x(˜v_x2+ ˜w_x2) + 3˜ux(˜v2x+ ˜w 2 x) 2 − 4˜u3x(˜v 2 x+ ˜w 2 x) i ,

On the dynamics of an inclined stretched string

On the Dynamics

of

On the Dynamics

of

an Inclined Stretched String

Contents

List of Figures

List of Tables

Chapter 1

Introduction

Chapter 2

Combined parametrical and

transversal in-plane harmonic

response of a stretched string

‡

2.1

Introduction

2.2

Analysis of the model equation

2.3

Modal interaction for the specific

combina-tions of values of p

and λ

2.3.1

The case without damping,

α

= 0

2.3.2

The case with (positive) damping,

α >

0

2.4

Conclusion

Chapter 3

On the planar and whirling

motion of a stretched string due to

a parametric harmonic excitation

3.1

Introduction

3.2

The derivation of the model equation

l

X = l

X=0

f( )

τ