VIBRATION ANALYSIS OPTICAL FIBER COUPLER · an optical fiber coupler with two spring supports, the bundle of optical fibers is modeled as a linear string, the substrate is rnodeled

VIBRATION ANALYSIS OF AN OPTICAL FIBER COUPLER

Changan Sun .

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Mechanicd and Industrial Engineering University of Toronto

@Copyright by Changan Sun 200 1

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Vibration Analysis of an Optical Fiber Coupler

Changan Sun

A thesis submitted in confomity with the requirements of the degree of Master of Applied Science

Department of Mechanical and Industrial Engineering University of Toronto

2001

Abstract

This thesis is focused on vibration analysis of an optical fiber coupler subjected [O a

shock motion. A linear dynamic mode1 is developed after proper assumptions are made. For

an optical fiber coupler with two spring supports, the bundle of optical fibers is modeled as a

linear string, the substrate is rnodeled as a Euler-Bemouli beam. and the silicon rubber pads

are modeled as two linear springs. For the optical fiber coupler with an elastic continuous

support, the bundle of optical fibers is modelect as a tinear string, the substrate is modeled as

a Euler-Bemouli beam, and the continuous silicon nibber pad is modeled as a linear elastic

foundation. In order to solve the equation of motion, a transformation of coordinate systern i s

introduced. Natural frequencies and normal modes of the beam and the string are obtained

using eigenvalue solutions. Furthemore, the mode superposition method is employed to

obtain the dynamic response of the optical fiber coupler.

The simulation software is written in C code. In order to verify the analysis and

solution, the finite element method using ANSYS is employed to explore the dynamic

response of the optical fiber coupler with two spring supports. Cornparison between the

results obtained in the simulation and those using ANSYS shows good agreement.

Finally, parameter studies are carried out to investigate the influence of differeni

parameten on the vibration of the coupler. Recommendations are made to improve the design

of the optic fiber coupler.

Acknowledgments

I would like to take this opponunity to thank those who assisted me during my Master's

candidacy. 1 would like to thank my supervisor, Prof. Jean W. Zu, for her guidance. support

and financial assistance. Thanks are due to my colleagues and friends especially Michüel

Cheng and Zichao Hou for their helpful suggestion and advice. Special thanks are also due to

Prof. W. L. Cleghom. whose insightful instruction in the vibration course strengthens my

understanding of the knowledge that is essential to this thesis. Most importantly, 1 would like

to thank my wife and daughter for their understanding and sacrifice.

iii

P.. u ............. . . . -

Contents

Absttact ................................................................................................... i

... Acknowledgements ...................................................................................... I I I

Nomenclature ............................................................................................ v i i

List of Table .............................................................................................. 1 x

List of Figures ............................................................................................ x

1 Introduction .......................................................................................... 1

1 . 1 Background .................................................................................... 1

1.2 Li terature Review ............................................................................. 3

1.3 Contributions of the Thesis .................................................................. 7

1.4 Organization of the Thesis .................................................................. 8

2 Dynamic Model ................................................................................... - 1 2

2.1 Introduction .................................................................................... 12

2.2 Model Consideration ........................................................................ -12

2.3 Equation of Motion of the Substrate under Two Spring Supports .................. 15

................... 2.4 Equation of Motion of the Substrate under an Elastic Foundation 18

2.5 Equation of Motion of the Optical Fiber ................................................ 20

........ 3 Dynamic Analysis of the Optical Fiber Coupler witb Two Spring Supports 23

3.1 Introduction ................................................................................. 33

3.2 Equations of Motion ........................................................................ 24

.................................................................. 3.3 Free Vibration Analysis 29

..................... 3.3.1 Natural Frequencies and Normal Modes of the Substrate 29

................ 3.3.2 Natural Frequencies and Normal Modes of the Optical Fiber 35

............................................................... 3.4 Forced Vibration Analysis 38

.................................... 3.4.1 Forced Vibration Analysis of the Substrate 39

............................................ 3.4.2 Forced Vibration of the Optical Fiber 44

..................................................................... 3.5 Numerical Simulation 60

.................................................................. 3.5.1 Simulation Scheme 60

............................................................... 3.5.2 Simulation Parameters 62

3.5.3 Cornparison with ANSYS ........................................................... 63

...................................................................... 3.5.4 Paramettic S tudy -65

.................................................................................. 3.6 Conclusions -68

4 Dynamic Analysis of the Optical Fiber Coupler with an Elastic

.............................................................................. Continuous Support 86

4.1 Introduction .................................................................................. 86

........................................................................ 4.2 Equations of Motion 87

.................................................................... 4.3 Free Vibration Analysis 90

4.3.1 Natural Frequencies and Normal Modes of the Substrate ...................... 90

4.3.2 Natural Frequencies and Normal Modes of the Optical Fiber .................. 92

................................................................. 4.4 Forced Vibration Analysis 93

................................................... 4.4.1 Forced Vibration of the Substrate 94

.............................................. 4.4.2 Forced Vibration of the Opticai Fiber 95

....................................................................... 4.5 Numerical Simulation 98

............................................................... 4.5.1 Simulation Parameters 98

4.5.2 Parametric Study ...................................................................... 99

4.6 Conctusims ................................................................................. 101

................................................................ S Conclusions and Future Work 106

5.1 Introduction ................................................................................. IO6

................................................................ 5.2 Summary and Conclusions IO6

................................................................................. 5.3 Future Work 108

References ............................................................................................. 110

........... Appendix Derivation of Natural Frequency of the Substrate under Two Springs 116

Nomenclature

Cross-sectional area of string

Cross-sectional area of beam

Young's modulus of beam

Amplitude of acceleration of shock motion

Stiffness constant of spring

Foundation modulus

Moment of inertia of beam cross section about z axis

Length of ith segment of beam

Length from the end of the string to the second bonding point

Length of beam

Bending moment of beam

Tension of string

Shear force

Generalized coordinate of beam

Generalized coordinate of string

Suppon shock motion

Displacement of the optical fiber r with resp O the steel tube

Displacement of the substrate with respect to the steel tube

vii

Transverse displacement of ith segment of the beam with respect to the

steel tube

Normal mode of ith segment of the beam with respect to the steel tube

Transverse displacement of the string with respect to the substrate

Transverse displacement of ith segment of the string with respect to the

substrate

Normal mode of hh segment of the string with respect to the substrate

Mass density of string

Mass density of beam

Frequency of shock motion

Angle between silicon pad and plumb line

Natural frequency of beam

Natural frequency of string

Angle the deflected string makes with the x axis

viii i

List of Tables

...................................................... Optical fi ber coupler parameters 63

.................................................................... Parameters for study 63

................................... Comparison of natural frequency of the substrate 74

.................................. Cornparison of natural frequency of the substrate 74

................................... Cornparison of natural frequency of the substrate 75

Cornparison of the dynamic response of the optical fiber coupler ................ 76

Cornparison of the dynamic response of the unsymmetrical optical coupler ... 77

Comparison of the dynamic response of the unsymmetrical optical coupler .... 78

Parameters for study ..................................................................... 98

Optical fi ber coupler parameters ....................................................... 98

List of Figures

.................. End view of the optical fiber coupler with two spring supports 9

...... A-A sectional view of the optical fiber coupler with two spring supports 9

............... End view of the optical fiber coupler with an elastic foundation 10

A-A sectional view of the optical fiber coupler with an elastic foundation ... 10

.......................................... Top view of a substrate and optical fibers I 1

......................................................................... Model diagram I 4

.............................................................................. Model diagr I 5

........................... Free-body diagram of an element of a beam vibration 16

........................... Free-body diagram of an element of a beam vibration 19

............................ Free-body diagram of an element of a string vibration 21

............................................. Coupled relationship between u and v 24

................................................... Coordinate system for the substrate 26

............................................. Coordinate system for the optical fibers 27

Combination of beam, string and springs ............................................ 48

........................................... Combination of beam, string and springs 50

.......................................... Combination of beam. string and springs 51


Combination of beam. string and springs ............................................ 55

........................................... Combination of beam. string and springs 56


.................................................................. Simulation flowchart 69

............... Fiowchart of subroutine for natural frequencies of the substrate 70

............ Flowchart of subroutine for the dynamic response of the substrate 71

......... FIowchart of subroutine for the dynamic response of the optical fi ber 72

............................................................... Flowchart for magnstr 1 73

Comparison of the dynamic response of the symmetrical optical

........................................................................... fi ber coupler 76

Comparison of the dynamic response of the unsymmetrical optical

........................................................................... fiber coupler 77

Comparison of the dynamic response of the unsymmetrical optical

........................................................................... fiber coupler 78

Amplitude of the optical fiber coupler with 1. as a parameter .................. 79

Amplitude of the optical fiber coupler with 1. as a parameter .................. 79

Maximum amplitude of the optical fiber coupler with respect

................................................................... to the substrate vs . 1. 80

Amplitude of the optical fïber coupler with P as a parameter ..................... 80

Amplitude of the optical fiber coupler with P as a parameter .................... 81

Maximum amplitude of the optical fiber with respect to

..................................................................... . the substrate vs P 81

Amplitude of the optical fiber coupler with a as a parameter .................. 83

Maximum amplitude of the optical fiber with respect to

..................................................................... the substrate vs a 82

3.27 Amplitude of the optical fiber coupler with k as a parameter ................... 83

3.28 Amplitude of the optical fiber coupler with k as a paramete .................... 83

3.29 Maximum amplitude of the optical fiber with respect to

.................................................................... the substrate vs k 84

3.30 Amplitude of the optical fiber coupler with 1, as a parameter .................... 84

3.3 1 Maximum amplitude of the optical fiber with respect to

...................................................................... the substrate vs 1. 85

4.1 Coordinate system for the optical fiber .............................................. 89

4.2 Amplitude of the optical fiber coupler with k f as a parameter .................. 102

4.3 Maximum amplitude of the optical fiber with respect

to the substrate vs kf ; ................................................................ IO2

4.4 Amplitude of the optical fiber coupler with Io as a parameter .................. 103

4.5 Maximum amplitude of the optical fiber coupler with

1. as a parameter ..................................................................... 103

4.6 Amplitude of the opticai fiber coupler with P as a parameter .................. 104

4.7 Amplitude of the optical fiber coupler with P as a parameter ................. 104

4.8 Maximum amplitude of the optical fîber with respect to

..................................................................... the substrate vs P 105

xii

Introduction

1.1 Background

An optical fiber coupler is a basic interconnection element for assembling a vüriety of

distribution networks that employ optical fibers. It combines optical signals from different

paths into one fiber, splits optical power in two or more output fibers, or performs both

functions. For a majority of communications networks, the performance of the coupling

elements, rather than the transmission characteristics of the fiber lines themselves, limits the

performance of the networks and detemines the optimum network configuration. Therefore.

the performance of optical fiber couplers is very important to optical fiber communications.

There are many kinds of optical fiber couplers. Typically, an optical fiber coupler is

compoxd of a bundle of fused optical fibers, a substrate, and a steel tube. The bundle of

fused optical fikrs is the most important part of the optical fiber coupler, in which the

optical signals are transmitted, combined or split. The bundle of fused optical fibers is

bonded to the substrate using adhesive materials at four points as shown in Figure 1 S. The

substrate, which is made of fused silicon glass. provides the housing for the optical fiben. - - - - - - - -

The substrate and fibers are wrapped inside the steel tube.

Optical fibers in couplers rnay experience large vibration and occasional breaks under shock

and impact in communications lines. Such large vibration and occasional breaks severely

interfere with proper signal transmission. For example, the coupler will exhibit potential

modal distortion, dispersion and bandwidth-limiting effects. A major reason for the problem

is the lack of proper design of couplers to satisfy required dynamic characteristics. The

mechanical component of couplers presentl y avai lable on the market is designed main l y

based on experience and on trial and error. Thus. quantitative vibration anal ysis is imperüiive

to improve the design of the coupler.

To reduce the vibration of the optical fiber couplers, it is proposed that rubber pads be pliiced

between the steel tube and the substrate to help cushion the impact when the coupler is in

motion. There are two ways the rubber pads are placed in the coupler. One is the optical fiber

coupler with two spring supports, in which four rubber pads are placed between the substrate

and the steel tube as shown in Figure 1.1 to Figure 1.2. The other is the optical fiber coupler

with an elastic continuous support, in which a rubber pad is placed between the substrate and

the steel tube as shown in Figure 1.3 to Figure 1.4.

It is the objective of this thesis to perfonn vibration analysis of these two kinds of optical

fiber couplers. This work will provide a basis for the improved design of optical fiber

couplers

Literature Review

A literature survey is conducted to identify the existing research that can be related to the

topic of the vibration analysis of the optical fiber coupler. .

The term of optical fiber coupler appeared in the late of the 1970s after the glass fiber with

adequate performance was developed [ I l . Since then, many invesiigators have been working

on optical fiber couplers. Helemutf Wolf described the fundamental principle of the optical

fiber coupler from the point of view of optics il]. Joseph C. Palais gave the simple

construction of an optical fiber coupler [2]. The first fused-twin-biconical taper coupler

fabricated from multimode fi ber employing the fuse-pull-and-taper method was

demonstrated early in 1977 [3]. Since the 1980s, researchers' interests have been on the

optimum design and manufacturing methods of optical fiber couplers. Their work mainly

concentrated on the material structure of the bundle of optical fibers rather than the whole

structure of the optical fiber coupler. However, no literature related to vibration analysis of

opticai fiber couplers was found.

In order to explore the vibration characteristics of optical fiber couplen, the literature on the

vibration analysis of a structure similar to that of optical fiber couplers will be reviewed. It is

reasonable to mode1 the substrate as a beam and the bundle of fused optical fiben as a string.

Therefore, the following literature review is focused on vibration analysis of beams and

strings.

Many modem engineering structures often make use of one-dimensional continuous elements -

resistant to tension but not to bending. such as strings. cables, ropes and chains [4-IO]. A

string, king the simplest mode1 of a one-dimensional continuous system, has ken a subject

of great scientific interest for a long time. Classic theory for string vibrations using analytical

methods is discussed in a number of monographs by Bishop and Johnson [SI, Den Hartog

161, Fraba [7], Timoshenko[8]. Nowacki [9],etc. They explored the linear vibration of a string

based on the assumption that the change in the tension of the string during its vibration is

negligible as compared to its equilibrium tension.

While it is not difficult to satisfy ihis requirement when the string is executing free vibrations

by merely limiting the initial displacement to sufficiently small values, the tension will Vary

significantly when the string is set into forced vibration near any of its resonance frequencies.

Unless the string is heavily damped, as the driving frequency approaches a resonance

frequency, the amplitude, which is a dependent variable now, also increases. causing the

tension to Vary and making the motion of the string essentially nonlinear, however srnaIl the

driving force may be [IO]. Consequently, many researchers have been working on the

nonlinear vibration of a string [10- 151. For example. Murthy and Ramakrishna [IO] presented

an analysis that leads to a pair of coupled non-linear partial differential equations for the two

components of the transverse motion of the string. Approximate solutions of these equations

gave a very good description of the observed resonance response of a string, especially of the

jump and hysteresis phenornena and of the tubular motion. Miles [12] studied the stable

States and motion in forced oscillations, especially with reference to the occurrence of tubular

motion near resonance. He clearly set out the assumptions underlying the analysis, and

employed a formulation quite different from that of Munhy and Ramakrishna; but his basic a - - -

equations and his results for the critical frequency and amplitude are identical. Anand [I 11

studied the effect of introducing darnping into the system. On the other hand. for planar

motion, a different equation was proposed by Oplinger and the results of his analysis are in

good agreement with those from his own experiments in which the string was allowed only

to oscillate in a plane. Gottleb [15] explored the effect of nonlinearity due to the purely

geometrical property of curvature on the vibrations of a constant-tension string.

There are many publications pectaining to vibration analysis of beams [15-45). Classic

theories for beam vibrations using analytical method is discussed in a number of books by

Thomson [16], Singiresu S. Rao 1171, S. P. Timoshenko and D. H. Young [8]. H. Mc.

Cailion (181, ect. There are three beam theories available to mode1 beam vibration [ 181. The

simplest approach is based on the classical Euler-Bernouli beam theory, which is formed on

the assumption that both shear deformation and rotational inertia of the cross section are

negligible. Hence, the theory does no< suffice for the bending vibration of higher modes and

also in the lower modes for thick beams. Subsequently, Rayleigh improved the classical

theory by including the rotational inertial effects of the cross-sections of the beam. Further,

Timoshenko extended the theory by incorporating both the effects of rotational i nertia and

transverse shear defonnations. Timoshenko beam theory can be used for bending vibrations

of stubby as well as slender beams.

Many researchers have devoted themselves to the study of the vibration of a beam with

different boundary conditions [21-40). R. S. Ayer et e1.[21] used a simple graphical network

to determine the natural frequencies of flexural vibration of continuous beams having ariy --- - - - .- -- - - -

number of spans of uniform length even though its accuracy is lower. C. N. Bapat et a1.[22]

studied natural frequencies of a beam with non-classical boundary conditions and

concentrated masses. His approach was based on the transfer mntrix method, in which the

exact general solution for a uniform Euler beam was used, together with the continuity of

displacement and dope and the relationship between the shear force and bending moment ai

a support. J. S. Wu et d.[36] used both analytical methods and FEM to do free vibration

analysis of a cantilever beam canying any number of elastically mounted point masses.

P.L.Veniere de Irassar[38] explored the free vibration of a beam with an intemediate elastic

support using an approximate method. Chen Jin et al. [39] adopted analytical methods to

explore the free vibration and forced vibration of a beam with one clarnped end, a

concentrated mass on the other free end, and a simply support at the middle point of the

beam. In his dynamic model, the beam is discretized into two segments according to the

simply support so that the general analytical solution of homogeneous beams with uniform

cross-section can be used in each segment of the beam. In addition, Chen et al. used the mode

superposition rnethod to obtain the solution of the forced vibration. Some researchers [27-

29,371 have explored the vibration of a Timoshenko beam carrying various concentrated

elements, such as springs, using analytical methods.

The dynarnic response of a beam supported by an elastic foundation has also been studied by

many investigators [42-44]. Different foundation rnodels such as Win kler, Pasternak. Vlasov,

Filonemko-Borodich have been used in these studies [42]. The Winkler model, in which the

medium is taken as a system composed of infinitely close linear springs. is the simplest one

e i s often adopted. Timoshenko L81 ugd_@e gialytical method to obtain free vibration of P = - L & ...- = -- -.-A - - * - - - - a L - - - - --

the beam supported on a Winker foundation.

The conclusions drawn from the literature survey are as follows:

r There is a large amount of publications on vibration analysis of a beam and a string.

respective] y.

There is a lack of dynamic analysis on optical fiber couplers

r No vibration analysis has been done for a structure simiiar to that of the optical fiber

coupler.

1.3 Contributions of the Thesis

This thesis investigates the dynamic response of the optical coupler subjected to a shock

motion. The contributions of this thesis are outlined as follows:

Dynamic analysis of the optical fiber coupler subjected to a shock motion is performed

using the analytical method.

Simulation software is developed, which can be used to calculate natural frequencies and

dynamic response of similar structures.

Recommendations are made for improving the optical fiber coupler design

- - -

1.4 Organization of the Thesis

This thesis is composed of five chapters.

Chapter 1 introduces the background and objectives of the thesis, l i terature revie W.

and contributions of the thesis.

Chapter 2 presents the dynamic mode1 of the optical fiber coupler subjected to a

shock motion

Chapter 3 focuses on the vibration analysis of the

spring supports.

Chapter 4 concentrates on the vibration anal ysis of 1

elastic continuous support.

optical fiber coupler with

:he optical fiber coupler wii

Chapter 5 summaries the results and proposes the future work.

two

Steel tube

Substrate

Optical fibers

Figure 1.1 End view of the optical fiber coupler with two spring supports

Steel tubc

1 Optical fi bers

Substrate

Silicon rubber püd

Figure 1.2 A-A sectional view of the optical fiber coupler with two spring supports

Steel tube

Substrate

Optical fibers

-Silicon rubber pad

Figure 1.3 End view of the optical fiber coupler with two spring supports

Steel tube

t Optical fi bers

Substrats

Silicon rubber püd

Figure 1.4 A-A sectional view of the optical fiber couple with an elastic foundation

Substrate Bonding point

Figure 1.5 Top view of a substrate and optical fibers

-=- - -Châpter 2- - - - --

Dynarnic Model

2.1 Introduction

In this chapter, a dynamic model for the optical fiber coupler is developed and equations of

motion of the substrate and the optical fibers are established, respectively. For the optical

fiber coupler with two spnng supports, linear springs are used to model the silicon rubber

pads; the Euler-Bernoulli beam is used to model the substrate; and the linear string theory is

adopted to model the bundle of optical fibers. For the optical fiber coupler with an elastic

continuous support, an elastic foundation is used to model the silicon rubber pad.

2.2 Model Consideration

As shown in Figure 1.1 to 1.4, an optical fiber coupler is composed of a bundle of opticül

fibers, a substrate, a steel tube, four silicon mbber pads or a continuous silicone rubber pad

between the steel tube and the substrate. A tension is applied to the bundle of fibers before

the fibers are bonded to the substrate. Thenfore, when the optical fiber coupler vibrates.

there is. prestress- in the optical fiber.- The following assumptions are made in the

establishment of a dynamic model:

(i) While the collision between the optical fiben and the substrate may occur and

rnay be one reason for the fracture of the optical fibers, only the effect of the vibration of the

optical fiberis considered. It is assumed that the optical fiber can vibrate beIow the substrate

and there is no collision between the optical fiber and the substrate. This makes the

analytical analysis of vibration much easier.

(ii) The effect of the pretension of the optical fiber on the substrate is neglected

because the pretension is no more than 0.6N so that the axial deflection of the substrate

caused by this force is small.

(iii) During the optical fiber vibration. if the amplitude is large, the tension will

change, especially when the frequency of the excited force is at the resonance frequency of

the optical fibers. It is assumed that the amplitude is not large and the tension is a constant.

i.e., the linear vibration of the optical fibers is considered.

(iv) The damping and the mass of silicon rubber pads are not considered.

(v) The steel tube is assumed as a rigid body because of its stiffness and dimensions

(vi) A half sine shock is approximated by a sine shock.

Based on the above assumptions, we can set up a linear dynamic model for the optical fiber

coupler as follows.

For the optical fiber coupler with two spring supports, the substrate is modeled as an Euler-

Bernoulli beam; the bundle of the optical fibers is modeled as a linear string and the four

silicone rubber pads are modeled as two linear springs. Thereafter the beam represents the - - U T - L - - L

substrate; the string represents the optical fibers; and the springs represen t the si l icone

mbber pads. Finally the dynamic model is developed as shown in Figure 2.1.

String Bearn

-*-.---.-..L Shock Motion

Fig 2.1 Model Diagram

For the optical fiber coupler with an elastic continuous support, an Euler-Bernoulli beam is

employed to model the substrate; a linear string is employed to model the optical fibers; and

an linear elastic foundation is employed to model the silicon rubber pad. Finally. the

dynamic rnodel is developed as shown in Figure 2.2.

Beam

X

Elastic founda

Steel tube

_._-_.- Shock Motion

Figure 2.2 Dynamic mode1

In Figure 2.1 and Figure 2.2, it is defined that s denotes the transverse support motion. if

denotes the transverse displacement of the beam with respect to the steel tube, and rc denotes

the transverse displacement of the stiing with respect to the steel tube.

2.3 Equation of Motion of the Substrate under Two Spring

Supports

This section is focused on derivation of equation of motion of the substrate under two springs

subjected to a sine shock. Consider the free-body diagram of an element of the beam shown

in Figure 2.3, where M(x,t) is the bending moment, and Q(x, t) is the shear force.

Figure 2.3 Free-body diagram of an element of a beam vibration

The inenia force acting on the element of the beam is

where p, is the mass density and A#) is the cross-sectional area of the beam. The first

term of the above expression is caused by the shock motion. From Newton second law. we

obtain

By writing

and

the equilibrium of the moment leads to

Disregarding tenns involving second powen of dr. Eq. (2.1 ) and Eq.(2.2) can be written as

aM -- ax - Q

By substituting Eq. (2.4) into Eq. (2.3). Eq.(2.3) becomes

a Z y -= a= ax2 - -p2~?dxl (s(x , t ) + v (x , t ) )

at -

From the thin beam theory, the relationship between bending moment and deflection c m be

exprcssed as

where E is Young's modulus and I is the moment of inertia of the beam cross section about --,--. - 2.- ---=--- . . - - ... . - - .a- --p. -

the z axis. Inserting Eq.(2.6) into Eq.(2.5), we obtain the equation of motion for the lateral

vibration of the beam subjected to a shock as follows:

For a sine shock

a 2 s - = F sin Qr at

the final vibration equation of the beam is

where F is the amplitude of the acceleration of the shock motion and R is the angular

frequency of the shock.

2.4 Equation of Motion of the Substrate under an Elastic

Foundation

This section is focused on derivation of the equation of motion for the substrate under an

elastic foundation support. Consider the free-body diagram of an element of a beam shown

in Figure 2.4, where M(x,t) is the bending moment about z mis, Q(x, t ) is the shear force,

and k, is the foundation modulus as the load per unit length of the beam necessary to ---y ---- -- .- - - &

produce a unit displacement of the foundation.

Figure 2.4 Free-body diagram of an element of a beam vibration

From the Newton's second law, we can obtain:

From the moment equilibrium, we can obtain

Since

dx'

& - - su6stïtuting the above expressions TntoEQ (2.9) andEq(2.10) , we can obiain

El a 4 v ( x J ) kf - +- a2v(x , t ) a ' s = -- p 2 4 ax4 p2A2 at at

For a sine shock

we obtain the vibration equation of a beam on an elastic foundation subjected to a sine shock

motion as follows:

EI a 4 ~ ( ~ , t ) +- kf v(x, t ) + il ?v(x. t ) = -F sin fit P?A? axJ P î A 2 at

where F is the amplirude of the acceleration of the shock motion and l2 is the angular

frequency of the shock.

2.5 Equation of Motion of the Optical Fiber

This section is focused on derivation of the equation of motion of the optical fiber. As we

discussed previously, the fiber is modeled as a linear string. Consider the free-body diagram

of an element of a string as shown in Figure 2.5.

Fig 2.3 Free-body diagram of an element of a string vibration

The equation of motion in y direction gives

# ( S ( X , I ) + u ( x , ~ ) ) (P + d ~ ) sin@ + dû) - &in û = p, A,& dr '

where P is the tension, p, is the mass per unit length, A, is the cross section area and 8 is

the angle which the deffected string makes with the x axis. For an elemental length ds. we

have

and

Hence the vibration equation of the string Eq.(2.13) can be simplified to - %. - - . - . . - - -:- --

Since the stnng is uniform and we assume that the tension is constant. Eq. (2.17) becomes

After simplification, it becomes

For a sine shock

the vibration equation of the string is

where F is the amplitude of the acceleration of the shock and R is the angular frequency of

the shock.

Dynamic Analysis of the Optical Fiber Coupler

with Two Spring Supports

3.1 Introduction

This chapter focuses on vibration analysis of the optical fiber coupler with two spring

supports. In Chapter 2, a linear dynamic model for this structure has been developed as

shown in Figure 2.1. In this model, the springs can be anywhere between the steel tube and

the substrate. In this chapter, the equations of motion of the system and their corresponding

boundary conditions are presented. Then the naturai frequency and normal mode are obtained

from free vibration analysis. Furthemore, forced vibration analysis is performed using mode

superposition methods. Numerical simulations are carried out and the results obtained in the

analysis are compared with those using ANSYS. Finally, the conclusions and

recommendations are presented.

3.2 Equations of Motion

From Eq. (2.8) and Eq. (2.20). we know the equation of motion of the optical fiber coupler as

follows:

It is obvious that u and vare coupled at the four bonding points O. A. B. C as shown in

Figure 3.1

Figure 3.1 Coupled relationship between u and v

We have

In order to solve the coupled Eq. (3.1) and Eq. (3.2), a transformation of the coordinate

system is introduced to decouple these two equations. We introduce

2=11-v (3.3)

where z represents of the motion of the fiber relative to the motion of the substrate.

From Eq. (3.3). we have

u=z+v

Substituting the above expression into Eq (3.2). we obtain the vibration equation of the

optical fibers subjected to a sine shock as

where

Finally we obtain the equations of motion of the optical fiber coupler with two spring

supports as follows:

In order to make the cdculation easy, the substrate can be treated as three segments

according to the support points of the springs. Correspondingly. the coordinate systems are

set up as shown in Figure 3.2. The symbols V I , vz and v3 are used io represent v for the three

segments of the beam, respectively. Similarly, the optical fiber can also be treated as three

segments according to the bonding points of the optical fiber to the substrate. The coordinate

system for the string is show in Figure 3.3, where ri, 22 and 2, denote 2 for different r range

[O, hl* [ld-!,II and [L-lu, LI*

Figure 3.2 Coordinate sysiem for the substrate

Figure 3.3 Coordinate system for the optical fibers

The boundary conditions and compatibility conditions can be written as:

Shear force and moment free condition of the substrate at x, = O gives,

Displacement and slope continuity of the substrate ai x, = 1, gives,

a Equilibrium of shear force and spring force of the substrate at x, = 1, gives.

-m. -.- -. . @ Equilibrium . - - of the moment of the substrate at x, = 1, gives

@ Displacement and slope continuity of the substrate at x, = 1, gives,

Equilibrium of shear force and spring force of the substrate at x, = l 2 gives.

* Equilibrium of the moment of the substrate at x2 = I I gives

Shear force and moment free condition of the substrate at .r, = I , gives,

Same displacement of the optical fiber and substrate at the adhesive points gives,

From the above boundary conditions and compatibility conditions as well as Eq. (3.6) and

Eq.(3.7), we can find v is independent from z . Therefore, Eq. (3.6) can be solved first.

Then the solution of v can be substituted into Eq.(3.7) and thus the Eq.(3.7) can be solved .

3.3 Free Vibration Analysis

In this section, free vibration analysis is performed to obtain the natural frequencies and

normal modes of the system.

3.3. Naturat Frequencies and Nornal Mdesof the SnbJtrate

For free vibration of the substrate, Eq.(3.6) becomes

By using the method of separation of variables. it is assumed

---- V ( X , t ) = V(x)T ( t )

Substituting Eq.(3.27) into Eq.(3.26) leads to

where tu, is a positive constant.

Eq. (3.28) can be written as two equations:

w here

The solution of Eq. (3 -30) can be expressed as

T( t ) = T sin(o,t + cp)

where T and cp are constants that cm be found from the initiai conditions.

The solution of Eq.(3.29) can be found

V ( x ) = Acos Br + Bsin Br + Ccosh fi + Dsinh Px (3.33)

where A, B, C, D are the constants that can be found from the boundary conditions.

For dl three segments of the bearn as shown in Figure 3.2, Eq.(3.33) can be written as

Combining Eq.(3.34) . Eq.(3.35), Eq.(3.36) and boundary conditions (Detailed deri vation is

given in Appendix 1 ) gives.

K',, A, + K, B, = O K A,+K,B,=O

w here

(COS pl, COS Pl2 COS Pl3 + COS pl, COS Pl2 cash Plz ) K, = cos pl, sin p(1, + 4 ) - - 2 EIP ?

k (a, sin pl2 cos PI, +a, sin pz c o ~ h PI, ) - - -a, cosD(l2 +If - -

2Eg3 2 E@ '

- (sinh cos #%, tsinh COS^ &) 2 EIp '

--. -- -(&ptl cos@? cos/~- +sin pf, cos pz cosh P, Y- a, cosfi(f2 + 1, - KG = sSiir&-~in$(b~ + r3 )- -

2 ~ 1 p

-- ka, (sin pl2 cos pl3 +sin & cosh &) + sinh pl, cosh plz sinh pl, + sinh Pl, s inh pl, cosh pl, 2EIp3

-- (sinh pl, cosh PL2 cos p, + sinh pl, cosh plz COS^ plj ) + ce4 (sinh pl2 sinh Pl, 2 E I ~

(sinh pl2 COS pl3 + sinh & cash ) + cosh pl2 COS^ pl3 ) - - 2 E I p 3

(COS pll COS pl2 sin pl, +cos pl, cos Pl, sinh Pl,) Ku? =-cos@, cosB(lz +13) - - 2 E g 3

(sin pl, sin pl, + sin pl? sinh pl,) + cosh pl, cosh pl, cosh pl, -al sin P(l2 + l , ) - - 2EIp '

(cosh Pl, cosh & sin PI, + cosh Pl, cosh pl2 sinh pl, + cosh pl, sinh @, sinh Pl, - - 2 EIP '

(sinh Pl2 sinh pl, + sinh Pl, sinh Pl i 1 + a , (sinh Pl, cosh & + cosh Pl2 sinh pl,) - - 2 E v '

(sin pl cos sin 131, +sin Pl, cos pl, sinh pl, > K,: = -sin PI, cos P(1, + 1, ) - - 2 EIP '

(sin B, sin pl, +sin pl, sinh pl,) + sinh pl, cosh pl, cosh pl, - a2 sin P(1, + 1, ) - - 2 EIP '

(sinh pl, cosh sin fl , + sinh Pl, cosh pl, sinh Pl, ) + sinh pl, sinh @, sinh & - - 2m3

(sinh pl2 sinh + sinh pl: sinh p/; ) + a , (sinh p, cosh fl, + cosh pl2 sinh 01, ) - - 2EIP '

(cos Sl, + cosh a, ) a, = -sin pi +- 2 ~ 1 p '

(sin pl + sinh ) a: = COS pl + - 2 E I ~

In order to obtain non-zero solution of Eq (3.37), we must have

Thus we obtain

Using the appropriate numerical methods, we can obtain the solution of Eq(3.38) and ihus we

gei the natural frequencies of the substrate.

Afier the natural frequency is obtained. we can easily obtain

sin pl Al = A, (COS & - - K,

-..-AI_. K., B2 = AL (a, - - al

K4

Ku ' sinh Pl,) C2 = Al COS^ pl -- K,

a,)sinh pl, sinh P,)sinh& +(a, -- ( C O S ~ pl, - - K, K , 1

a,)cosh pl, Ku ' sinh pl, ) sinh pll + (a, - - COS^ plI - - K, K,

Ku' or,)sinh pl2 Ku ' sinh pll )cosh PZ + (a, - - 2 E I ~ " 6 1

Therefore normal modes can be expressed:

Segment 1: VI (x,) = Al cos /k, + BI sin & + Cl cash fllx + Dl sinh &r,

34

----z - - -

Segment II: V2 (xz ) = A2 COS &i2 + B2 sin & + C2 cash PIx2 + D? sinh fi,

(3 .$O)

Segment III: V,(x,) = A, cos & + B3 sin fi3 + C, cosh & + 4 sinh fi,

(3.41 )

In the xoy coordinate system as shown in Figure 3.2, the above the normal modes cm be

expressed as follows:

V ( x ) = A, cos p,x + B, sin p,r + Cl cosh P,, x + Di sinh Pl, r O < . r c I ,

V ( X ) = A3 cos p , (X - fI - l2 ) + B3 sin p, (x - 1, - i l ) + c3 cash p,, (x - 1 , - l2 ) -

+ D, sinh p,, ( x - 1, - 1,) 1, + 1 2 i x 6 l1 + 1 2 + l Z

3.3.2 Natural Frequencies and Normal Modes of the Optical Fiber

For free vibration of the optical fiber, from Eq (3.7), we can have

Eq.(3.45) can be written as

Eq. (3.46) can be solved by the method of sepmation of variables. We assume that

z(x, t ) = Z(x)T( t )

Substituting Eq. (3.47) into Eq. (3.46), we obtain

where a, is a positive constant. The equations implied in Eq. (3.48) can be written as

These solutions of these equations are given by

. -- - ..- - -

where A, B , T , and q are constants that can be evaluated from the boundary conditions and

initial conditions.

For the three segments of a uniform string as shown in Figure 3.3, Eq. (3.5 1 ) cm be wriiten

as

u s 0, 2, (x) = Ai cos - x + BI sin -x (3.53) C C

W 0 . r Z2 (x) = A2 COSIX + B2 sin -

C C

where A,, B, , A , , - B,, - .4, and 8, are the constants that can be evaluated from the boundüry

conditions.

After applying boundary conditions, we obtain the natural frequencies and normal modes as

follows:

For O ' , x d , .

Natural frequencies:

Normal modes:

n7t Z, (x) = Bi sin -x 4.l

For 1 , I x l L - l , ,


Nomal modes:

n~ nK nR Z ? ( X ) = B @ n - x - tan- 1, cos - x L - 21, L - 2 4 L - 21(]

ForL-l , ,SxlL,

Natural frequencies

Normal modes

nR nlC nK 2, = B, (sin - x - tan - L cos - .Y) n = 1,2,3 ...

10 10 1,

3.4 Forced Vibration Analysis

In section 3.2, the equations of motion of the optical fiber coupler subjected to a sine shock

are derived. The mode superposition method will be used to solve these equations for forced

vibration.

-- 2% -- -- - - -

3.4.1 Forced Vibration Analysis of the Substrate

In order to use the mode superposition method, the orthogonality of the normal mode will be

derived first.

From free vibration of the optical coupler, Eq.(3.28) gives

Substituting the normal mode V, (x) into Eq (3.62). we have

Since fl: = p z ~ p i n / E l , we have the nth normal mode equation

d4v, (XI - &yn (x) = O dx4

Similarly, we have the mth normal mode equation

p..--- L. ... . 2hltiplyifig Eq (3.63) by V,,, (x) -an&-Eq (3.64) by V* (x) and subnaaing the resu lting

equation one from the other, we can obtain

v,, ( x ) 4vm - 4Vn(X) vm (x) = (p; - p : ) ~ , ( x ) ~ , (x) dr4 dx4

In tegrating Eq (3.65) from O to 1,. . it gives

where i = 1.2,3. corresponding to the xiyl coordinate system, the xrv2 coordinate sysiem and

the x3y3 coordinate system, respectively.

Let

Y, = V,, (x)V,;(x) -4, (xYi;, ( ~ 0 + v;n::,(x)v,., (XI - y;, (.,y., (-y>

We can obtain

Substituting the boundary conditions and the compatibility conditions yields

btv, (x)V, (x)& = O

-.Lm- -

Therefore we have proved that the normal modes are orthogonal. We can use the normal

mode superposition method to solve the forced vibration equation.

Let

where

VI, (+) = A, cos &+ + B, sin &x + C, cosh &x + D, sinh P,,-r

Substituting Eq. (3.69) into Eq. (3.6), we obtain

P 4 ~ 1 Since A = a;, and o,, is the natural frequençies of the beam, Eq (3.70) is sirnpl ified

PA

as follows

(q: ( t ) + abn )Vin (x) = - F sin Qt n=l

Muitiplying i(,, (x) to the both sides of Eq (3.7 1 ) and integrating from O to L. we obtüin

where

Let

the steady state solution of Eq (3.7 1) is

where

Therefore the dynamic response of the beam under a sine shock is

0

V ( X , t ) = V,,, (x)q,,,, sin Rt 11=1

where i = 1,2,3.

Eq43.78) can be solved by an appropriate numerical method.

3.4.2 Forced Vibration of the Optical Fiber --: - - -

From Eq (3.7), we know that the equation of motion of the optical fiber with respect to the

substrate when subjected to a sine shock motion is

Substituting Eq. (3.78) into the above expression yields

Pz(x,~) c - - = f (x) sin Rr ax at -

w here

2 f(x- = F - 2 $'tlc ( - A ~ c d n X , - B~ sin ~ , , x , + C, cosh ~ , , x , + D, sinh P,,+,

,l=ioin -a2 = cna2

- x 9 (A , cosfl,x, + Bi sin &xi + Ci cosh P,#-r, + Dl sinh &-r, .=i a;, - n2

(3.SO)

in which i = 1,2,3 and

- ...

Let

where Zi, (x) is the nth nomal mode of the ith segment of string. Substituting Eq (3.8 1 ) into

Eq (3.80), we can obtain

It is easy to prove

Multipl ying Eq(3.83) by 2, (x) and integrating, we obtain

For i = 1

The solution is

For i = 2,

The solution is

x-tan- COS

w here

For i = 3

Q.'(t) + ( t ) = - f3 (x) sin Rt

The solution is

Substituting v and z,, into Eq(3.3). we can obtain the displacement of the optical fiber. f i .

Denoting

g, (x ) = F - 2 "&c2 ( - A , cos &,x - 8, sin p,,x + C, cosh f l , t . v + D, sinh 311.v) ,,=Io,, - '

œ

- z ( A , cos &x + B, sin P,x + C, cosh pf,x + Dl sinh &r) n = i o & -n2

- &,2cnc' g ? ( x ) = F - I (-A2 COS 8, (X - 1, ) - B2 sin p,(x - f 2 ) + C2 cosh PI, (X - il ) "=lah - n2

the expression for u will be simplified and concise.

According to different locations of the springs and bonding points dong the beam, the

solution of u(x , t ) is given as follows:

( 1 ) The combination satisfies 1, = 1, = 1, as shown in Figure 3.4

Figure 3.4 Combination of beam, string and springs

- L . -- - - - . - -

where

For Io S x < L-1,

w here

For L-1, 5 x 5 L

00

L ~ ( x , I ) = f ~ ~ , ( x ) q ~ , sin f i t + ZZ,,,W f d ~ ) si"Rl n=l n=l R' - o,?,

w here

(2) This combination satisfies 1, < 1, < 1, + 1, and L - 1 < 1, + l2 as shown in Figure 3.5


l E y,l (x)qo,, sin + i z , ~ 0) f ' ( X ) sinRi O < x < l , n=l n-I $2' -

u ( x , ~ ) = g v,, (x)qo, sin ~t + g z,,, O) s i n 0 1 , 4 x S l ,

n=l n=l a' -

where

w here

For L-1 , 5x5 L

i vZn (x)qOn sin ~t + E z3, ( x ) n=I n=l

f 3 ( x ) sin Rt az - w; u(x, t ) =

f v3, (x)qon sin + Z z3, 0) n=l n=l

f 3 ( X ) , sin Ri R? -a,;,

w here

(3) This combination satisfies 1, c f , and f , + 1, c L- Io as shown in Figure 3.6.


u ( x , ~ ) = ~ v ~ ~ ( x ) ~ ~ ~ sin Rt + Z Z , , ( X ) n=l n=I

f l ( X ) sinnt n - a.:,

where

For 1, 5 x 5 L - Io

5 v,, (x)q, , sin Qt + I qn 0) f 2 ( X ) , sin ~t i l + I? < x I L - I,, n=I 11=l n2 - O ,

w here

For L - 4 I x S L


w here

(4) This combination satisfies 1, + 1, < 1, as shown in Figure 3.7

i V I , (xlq,, sin m+ E z,, (x) n=l

l sin nt n2

u(x, t ) = { s v2. mon sin f i t + i 2,. n=l n=L

f i sin f i t n2 -4,

Z v3, (x)qo, sin nt + f, z,, (x) f i (X ' sinwt I , + ~ , s . v < L - 1 , n=l n2 CO:^

For 1, $ x < L - I o

where

For L-1, < x < L

w

U ( X , t ) = x Vjn (x )qh sin ~t + T z,, (x) n=l n=l R' f3(X), - w , sin RI

where

(5) This combination satisfies 1, < 4, 4 < 1, + 1, and 1, + 1, c L - I o as shown in Fig. 3.8


w here

For 1, I x 5 L-1,

For L-1, a x s L

U(X, f ) = t v3n (x)qon sin + ? z3,, (x) n=l n=l

f3(X) sin Gr s22 - m.;i

where

(6) This combination satisfies 1, > I o , 1, < L - Io and 1, + l2 z 1, + Z2 - 1, as shown in

Figure 3.9.


For O l x S 1,: - ---.Y- 8 - - --

u ( x , ~ ) = E vin ( X I ~ ~ ~ sin ~i + Z z,, ( x ) n=l n=l

f l ( X ) s i n o f n2 - a.;n

w here

For Io 5 x l L-1,

0 i v,, (x lq , sin Qf + Z Z2. ( x ) f d ~ ) sin** 1, < x q n=l n=l n2 -u:,

u ( x , t ) = 0 ? v ~ , (XI%, sin ~r + 1 z?. (x) f d x ) sin*I 1, c x l L - f , ,

n=l ri=, R? -o.:,

w here

For L-Io I x S L

sin Or

s i n Qt

- A

where

(7) This combination satisfies 1, > L - 1, as shown in Figure 3.10


For O S x l l , :

U ( X . t ) = (x)%,, sin nt + Z z,,, (x) n=L n=l

f ' (X) sinRt R? - o.:,

where

01

u (x, t ) = f v,,~ (x)qOn sin ni + I: Zzn (XI f d ~ ) n=l n =I a2 - ~~i~

w here

For L-1, I x S L

0 Ev,, (x)q,,, sin nt + T 23, (XI n=I n=l

f 3 ( x ) sin Rt R? -a:,

fv2, ( x ) ~ ~ , ~ sin nt + i~ , , (x) - t ' ( e t ) , sin*I )I=I n=l R - a;,

2 v,, (xIq,, sin + ZZ,, ( x ) 1 n=i n=l f 3 ( x ) sin RI

R~ -o.;

w here

PA- -- 3.5 Numerical Simulation-

Numerical simulations are conducted in this section to show the dynamic characteristics

of optical fiber couplers. First the simulation scheme is introduced. Then the results

obtained by simulation are compared with those obtained using ANSYS. Finally, the

effects of the parameters on the dynamic characteristics are thoroughly investigated.

3.5.1 Simulation Scheme

A simulation program written in C code is developed. The simulation program which

consists of three subroutines is outlined in Figure 3.1 1.

The first subroutine is to calculate the natural frequency of the substrate. In section 3.3,

we obtain the characteristic equation for the natural frequencies of the substrate as ,

fol iows:

K , , K , - K d I K 4 = O (3.1 40)

This equation is a nonlinear equation. Due to its complexity, a numerical scheme is

proposed.

This method begins by defining a function

--.-. ~ _ - --The function g is a function of the natural frepuency e,. Fi-, an initial guess is taken

and substituted into Eq.(3.141). The function g is calculated and its value is given to a

variable gdd. If guld does not equal zero, the g is recalculated by adjusting the initial guess

with a fixed increment. If the product of g and g,,d is more than zero. the value of g is

denoted as gdd and the above procedure is repeated. If the product of g and g,ld is no more

than zero, the corresponding to g is one root of Eq. (3.140). In order to obtain more roots,

the above procedure will continue until al1 the roots required are obtained.

The above method has some shortcomings. If the increment is small. the calculation

precision will be higher but the speed will be slower. In order to improve the calculation

speed and maintain the calculation precision, the above method must be modified.

First, the fixed increment is set larger. The larger increment is used to find the range of

the root of Eq. (3.140). After the range of the root in Eq. (3.140) is found. the fixed

increment is set smaller. Within the range of the root, the above procedure is repeated

using the smaller fixed increment until the high precision root of Eq.(3.140) is obtained.

Using the modified method, the calculation speed is improved significantly. The

fiowchart of the above numerical solution method is outlined in Figure 3.12.

The second subroutine is the one for dynamic response of the substrate. From section 3.4,

the dynamic response of the substrate is obtained by the mode superposition method as

follows

The flowchart is outlined in Figure 3.13. ------- -- - - *--- - -

The third subroutine is the one for the dynamic response of the optical fiber. Its flowchart

is shown in Figure 3.14 and Figure 3.15. In Figure 3.14. the general structure of the

subroutine for the vibration amplitude of the string is illustrated. The functions magnstl

to magnst7 deal with the dynamic response of the optical fiber with different locations of

the springs and bonding points of the string. Figure 3.15 outlines how to implement the

function magnstr 1. Since the principle of function magnstr2 to function magnstr7 is the

same as that of function magnstrl, only the flowchart of function magnstrl is given here.

In calculating the dynamic response, only the first five modes are considered because the

higher order mode has little effect on the dynamic response. In fact. the fifth mode

contributes almost zero to the magnitude of the vibration.

3.5.2 Simulation Parameters

The parameters of a typical optical fiber coupler are provided by the Resonance

Photonics Inc. These parameters are given in Table 3.1. The range of values used for

parametric study is given in Table 3.2.

Table 3.1 Optic@ fibgr supler parameters

1 Young's modulus of the beam E 1 7 .24~ 10"~a 1

Length of the beam L Cross-sectional area of the beam A2 Mass density of the beam p,

w

1 Moment of inectia of the beam cross section I 1

I 4 .34~ 1 0 - 1 ~ ~ ~ I

0.04m 6.6 1 x 1 oe6rn2 2200kg/m3

1 Cross-sectional area of the string Ai 3. 1xl0'hm' 1 1 ~ a s s density of the string p, 1 2200kg/mJ 1

Table 3.2 Parameters for study

- - . Acceleration of the shock motion F Circular frequency of the shock motion @

9800mlsL 2m 1 03radls

35.3 Cornparison with ANSYS

Tension of string P Spring stiffness k Position of bounding points Angle between the silicon pad and plumb line a

. Position of the two springs f i , 12, 1,

In order to verify whether the above numencal solution to the optical fiber coupler is

nght or not, the sarne problem was investigated by ANSYS. A subroutine in ANSYS was

developed to calculate the natural frequencies of the substrate. In this subroutine, beam

elements and spring elements have been used to calculate the natural frequencies of the

substrate. Tables 3.3 to Table 3.5 give thne sets of cornparison data. From these tables.

we can find that the largest difference for the 1" natural frequency is 0.2596, the 2"d

natural frequency is 0.0646, and the 3" natural frequency is 1.02%.

0.0 1 -0.60N 1000-4ûûûûN/m 0.005-0.0 t7Sm

IdI2-51~/12 0-0.04m

-- a

Another subroutine was developed in ANSYS to calculate the dynamic response of the

optical fiber coupler. In this ANSYS subroutine. LINKl elements are used to

approximate the optical fiber, beam elements are used to approximate the substrate and

spring elements are used to approximate the two springs. The effect of the string prestress

on the beam was considered. Three sets of comparison data are given in Table 3.6-3.8

and Figure 3.16-3.18, respective1 y.

The structure of the optical fiber coupler given in Table 3.6 is symmetric. with two elastic

pads placed under the substrate symmetrically. Corresponding to this structure. the

comparison of the results obtained by ANSYS and the analytical method is chown in

Figure 3.16. The structures of the optical fiber coupler given in Table 3.7 and Table 3.8

are unsymmetrical, with two elastic pads placed under the substrate unsymmetrically.

Corresponding to these structures, the comparison of the results is shown in Figure 3.17

and Figure 3.18, respectively. From the comparison of the results, it is shown that the

largest differences of the amplitude of the substrate and optical fibers are just 0.85% and

2.23%. respectively. Therefore the analytical method developed is right.

When we developed the dynarnic model, the effect of the prestress on the beam was

ignored. However, this effect was considered in the ANSYS subroutine and since the

results obtained by the two methods are in good agreement, it is proved thai i t is

reasonable that we ignore this effect in the dynamic model.

The parameten in Table 3.2 are used to explore their effects on the vibration of the

Jb

optical fiber couple. First. the five parameters are chosen as P = 0.6N. a = -. 4

1, = 0.01m, k, = 50ûûN/m, 1, = !, = O . Since the equations of motion of the optical

fiber coupler are linear, one parameter is changed once a time so that its effect on the

vibration amplitude of the optical fiber coupler can be explored.

( 1 ) Position of the Two Springs

Let us consider a symmetrical optical fiber coupler structure, in which the two springs

are placed syrnmetrically, Le., 1, = 1,. The value of 1, and 1, varies from O to 0.0 18m. The

amplitude of vibration of the optical coupler with f i as a parameter is shown in Figures

3.19 and 3.20. It is easily found that the position of the maximum amplitude of the

optical fiber is at the midpoint. The maximum amplitude of the optical fiber wiih respect

to the substrate versus Ii is drawn in Figure 3.21. It is shown from Figure 3.2 1 that wiih

the increase of 11, the maximum amplitude of the optical fiber with respect to the

substrate increases monotonically. When I , = O , the smallest amplitude of the optical

fiber with respect to the substrate occurs. It is recommended that i, be zero. The two

springs (elastic rubber pads) should k placed at the two ends of the substrate.

(2) Pretension of the Optical Fiber

Whea the pretension P varies f q 0.01 - to 0.6N - the amplitude of vibration of the

optical fiber coupler is shown in Figures 3.22 and 3.23. It is easily found that the position

of the maximum amplitude of the optical fiber is not always at the midpoint. When

P = O.OlN , the position is at x = 0.01 Sm and x = 0.025m . The maximum amplitude of

the optical fiber with respect to the substrate is shown in Figure 3.24. It is easily found

that the effect of the pretension is not linear. There are two peaks of the amplitude. When

P = O. 1 IN , the largest amplitude occurs, where z = 0.0043rn. When P = 0.03N . the

second largest amplitude occurs, where z = O.022m . From P = O.2N to P = 0.6N , the

amplitude of the midpoint decreases monotonically with the increase of P . When

P = 0.6N, the amplitude is the smallest, where z = 0.00005m. Therefore it is

recommended that P be 0.6N in order to make the amplitude of the optical fiber

smal lest.

(3) Angle a

When a varies from O. ln to 0 . 4 ~ . the amplitude of vibration of the optical coupler with

respect to a is shown in Figure 3.25. It is easily found that the position of the maximum

amplitude of the optical fiber is at the midpoint. The maximum amplitude of the optical

fiber with respect to the substrate versus a is shown in Figure 3.26. It is found that the

larger the a is, the smaller the maximum amplitude of the optical fiber relative to the

substrate is.

(4) Spring Constant

-T.ht_springconstmt k varies from l OOJ JO- 4ooqp-Nlm . - - amplitude - of vibration of the

optical fiber coupler is shown in Fipre 3.27 and Figure 3.28. It is easily found that the

position of the maximum amplitude of the optical fiber is at the midpoint. The maximum

amplitude of the midpoint of the optical fiber with respect to the substrate is shown in

Figure 3.29. From k = 5000 Nlrn to k = 1 1000 Nlrn . the amplitude increases very

slowly. There is a largest amplitude peak between k = 1 1 ûûû Nlm and k = 14ûûONlm .

When k = 12OOONlm , the amplitude reaches the largest value 2 = 0.001 2m . From

k = l4OOO Nlrn to k = 40000 Nlm , the amplitude decreases slow1 y. When

k = 4 O N l m , the smallest amplitude occurs, where 2 = 0.000096 m . Therefore it is

reconimended that k be 4ûûûûNlm and k = 12000Nlm be avoided.

(5) Position of the bonding point

The position of the bonding point 1, varies from 0.005 to 0.018m. The amplitude of

vibration of the optical fiber coupler is shown in Figure 3.30. It is shown that as 1,

increases. the amplitude of the middle segment of the optical fiber decreases; however.

the amplitude of the other two segments increases. So the maximum amplitude does noi

always occur at the middle segment of the optical fiber. For 1, greater than a certain

value, the maximum amplitude will occur at the two end segments of the optical fïber.

The maximum amplitude of the optical fiber with respect to the substrate i s shown in

Fipre 3.3 1. From Fipre 3.3 1, it can be found that from 1, = 0.001 m to Io = 0.0 Mm,

the maximum amplitude decreases monotonically. When 1, = 0.014 m , the smallest

.--- - - amplitude occ-urs. For 1, > 0.014 m , - the -- amplitude - increases - with the increase of 1,. It is

recommended that 1, be 0.014 m.

3.6 Conclusions

In this chapter, an analytical approach is employed to perform the free and forced

vibration analysis of the optical fiber coupler with two spring supports. First. the

equations of motion of the optical fiber coupler are developed. Tlien the free vibration is

analyzed. In order to obtain the solution of the forced vibration equation. the

orthoganality of the normal mode is derived. The mode superposition method is used to

obtain the dynamic response of the optical fiber coupler. A simulation program written in

C is developed. To verify validity of the solution, the same problem i s investigated by

ANSYS. It is shown that the results obtained by the cutrent approach and ANSYS are

almost the same. Thesefore the anal ytical method is right. From the simulation resu lts.

we can draw conclusions as follows:

The two springs (elastic mbber pads) should be placed ai the two ends of the

substrate.

It is recommended that P be O.6N.

The larger a is, the smaller the amplitude will be.

It is recommended that k be 400ûûNlm.

It is recommended that 4 be 0.0 14 m.

SUBROUTXNE FOR NATURAL FREQUENCIES

OF SUBSTRATE

SUBROUTINE FOR THE DYNAMIC RESPONSE OF

THE SUBSTRATE

DYNAMIC RESPONSE OF THE OPTICAL FIBER

Figure 3.1 1 Simulation flowchart

1 START I INITIAL ESTIMATE FOR

CACULATE g gdd =g

FWED INCREMENT

RECALCULATE g u

1 ADJUST ah. BY A SMALLER 1

CALCULATE

YES

Figure 3.12 Flowchart of subroutine for natural frequencies of the substrûte

NATURAL FREQUENCY OF THE SUBSTRATE

CALCULATE TNE NORMAL MODE OF THE SUBSTRATE

CALCULATE a, BY CALLING THE INTEGRATION

FUNCïiON AM) NORMAL r MODE FUNCTION

THE INTEGRATION FUNCTION AND NORMAL

MODE FUNCTION

I CALCULATE C,

CALCULATE V ( x )

Figure 3.13 Flowchart of subroutine for the dynamic response of the substrate

Fig. 3.14 Flowchart of subroutine for the dynamic response of the optical Fiber

CALLING gi FUNCTION AND THE STRING NORMAL MODE z FUNCTION

--

CALCULATE THE INTEGRATION OF THE SQUARE OF THE NORMAL MODE

FCMCTION

CALCULATE THE SECOND TERM OF EQ(3.98)

CALCULATE THE FIRST TERM OF EQ.(3.98)

CALCULATE THE STRING VIBRATION MAGNITUDE V ( x )

OUTPUT THE DATA I

Figure 3.15 Flowchart for magnstr l

ANSYS

Simulation

Difference

ANSYS

SimuTation

Difference

Table 3.3 Comparison of natural frequency of the substrate

Table 3.4 Comparison of natural frequency of the substrate

--.-- - A .. .- . Table 3.5 Compyison-ofnatural frequency of - the . substrate

Di fference l

Table 3.6 Cornparison of the dyaamic response of the optical fiber coupler

lo 11 l2 h Amplitude at the Amplitude at the (m) (m) (m) (m) rnidpoint ofthe midpoint of the

optical fiber substrate (m) (m)

ANSYS

Figure 3.16 Cornparison of the dynamic response of the symmetrical optical fiber coupler

0.01 0.01 0.02 0.01 0.00048988 0,0004375 1

Simulation 0.01 0.01 0.02 0.01

Table 3.7 Cornparison of the dynamic fesponse of the unsymmeûical optical coupler

[O II f2 li Amplitude at the Amplitude at the (m) (m) (m) (m) midpoint of the midpoint of the

optica! fiber substrate (m) (m )

ANSYS 1 0.01 0.008 0.004 0.028 0.0008 1948 0.0006958

Difference 1 2.23% 0.85%

Simulation

Figure 3.17 Cornparison of the dynamic response of the unsymmetrical opticai fiber coupler

77

0.01 0.008 0.004 0.028

Table 3.8 Cornparison of the dynarnic response of the unsymmetncal optical fiber coupler

10 4 l2 ii Amplitude at the Amplitude at the (ml (m) (m) (m) midpoint of the midpoint of the

string beam (m) (m)

ANSYS 1 0.01 0.02 0.0052 0.0148 0.0004975 0.00044148

Difference

Simulation

- ANSYS - Simulation

0.01 0.02 0.0052 0.0148 0.0005009 0.00044339

Figure 3.18 Cornparison of the dynamic response of the unsymmetrical optical fiber coupler

..-.*.--.. beam string I,=O.O l,=0.004 l,=0.008 /,=O .O 1

Figure 3.19 Amplitude of the optical fiber coupler with f as a parameter

. - - . . - . . . . beam string

1,=0.012 -- i,=0.016

I,=0.018

Figure 3.20 Amplitude of tbe optical fiber wuplers with Il as a parameter

Figure 3.21 Maximum amplitude of the optical fiber with respect to the substrate vs. I I

/' ..*.-.-... beam /

/ string ,i wo.0 1

-- P0.05 &O. 1

Figure 3.22 Amplitude of the optical fiber coupler with P as a parameter

..-...*-.- beam string -0.25

. - -- -0.30 F 0 . 3 5 P=0.40

-- P=0.50 T0=0.60

Figure 3.23 Amplitude of the optical fiber coupler with P as a parimeter

Figure 3.24 Maximum amplitude of the optical fiber with respect to the substrate vs. P

Figure 3.25 Amplitude of the optical fiber coupler with a as a parameter

Figure 3.26 Mavimum amplitude of the optical fiber with respect to the substtate vs. a

beam ................................ 0.00070 ............

......... IF1000 0.-

Figure 3.27 Amplitude of the optical fiber coupler with k as a parameter

......... beam

Figure 3.28 Amplitude of the optical fiber coupler with kas a parameter

-0.002 1 . , . , . , . , . , . , . , . , . i O 5000 10000 15000 20000 25000 30000 35000 40000 45000

k (Nlm)

Figure 3.29 Maximum amplitude of the optical fiber with respect to the substrate vs. k

..... * ..... beam string lo=0.008 Io= O .O 1 ro=o.ol 4 io=O.Of 8

Figure 3.30 Amplitude of the optical fiber coupler with fo as a parameter

Figure 3.3 1 Maximum ampiitude of the optical fiber with respect to the substrate vs. 10

Dynamic Analysis of the Optical Fiber

Coupler with an Elastic Continuous Support

4.1 Introduction

This chapter is focused on vibration analysis of the optical fiber coupler with an elastic

continuous support, the structure of which is shown in Figure 1.3 to 1.4. In this system. a

continuous elastic pad is placed between the substrate and the steel tube to cushion the

impact when the coupler is in motion. In chapter 2, the dynamic mode1 has been

developed as shown in Figure 2.2. In this chapter, the equations of motion of the opticaf

coupler and their comsponding boundary conditions are presented. Then the naturül

frequencies and normal modes are obtained from free vibration anülysis. Furthemore.

forced vibration analysis is perfonned using the mode superposition method. Numerical

simulations are carried out. Finally, the conclusions are drawn.

In section 2.4, the equation of motion of the optical fiber subjected to a sine shock has

been derived in Eq. (2.20), and the equation of motion of the substrate under an elastic

foundation has been derived in Eq.(2.12). Therefore we obtain the equation of motions

of the optical fiber coupler with an elastic continuous foundation as follows:

+- a % ( ~ , t ) = -F sin Rr at

Comparing these two equations with the equations of motion of the optical fiber coupler

- v on the left with two spring supports, it is easily found that there is an extra terni - ~ 2 4

side of Eq. (4.1) compared with Eq(3.1) while Eq. (3.2) is exactly the same as Eq.(4.2).

Using the same method as that used in Chapter 3, Eqs. (4.1) and (4.3) can be solved.

In ordet to solve these equations, a transformation of the coordination system i s

introduced. We define

From Eq.(4.3), we can derive

.- --- - - a Substiwting the ab- expression into Eq- (A?), we obtain - . the vibration equation of the

optical fiber subjected to a sine shock

where

Finally we obtain the equations of motion of the optical fiber coupler under elastic

foundation as follows:

In order to make the calculation easy, the optical Aber is discretized into three segments

according to the four adhesive points and the coordinate systems for the optical fiber are

shown in Figure 4.3. The symbols 2,. z2 and z , are used to represent z for different x

range [O . Io ] , [ l , ,L-Io] and [ L - l o , L ] .

Figure 4.1 Coordinate system for the optical fibers

Boundary conditions and compatibility conditions can be written as:

Shear force and moment free conditions of the substrate give.

v'(0, t ) = O

va(o, t ) = 0

vN(L,t) = O

V ~ ( L , I ) = 0

Same displacement of the optical fiber and the substrate at the adhesive points gives.

- - -

-- -- . - z3 (L, 1) = O (4.17)

From the above boundary conditions and compatibility conditions as well as Eq. (4.6)

and Eq.(3.7), we can find v is independent from z . Therefore. Eq. (4.6) can be solved

first. Then the solution of v is substituted into Eq.(4.7) and thus Eq.(4.7) can be solved .

4.3 Free Vibration Analysis

Free vibration analysis is performed for equations (4.6)-(4.17) to obtain the natural

frequencies and normal modes of the system.

4.3.1 Natural Frequencies and Normal Modes of the Substrate

For free vibration, from Eq.(4.6), we have

Assuming v(x,t) = V(x)q(t) and substituting it into Eq (4. la), we obtain

where o, is a positive constant.

Eq (4.19) can be written as -- - L A - - -

2

Denoting p4 = P2A2a)b -k' , we cari &tain the solution of Eq. (4.20) EI

The solution of Eq.(4.21) is

where A, B. C . D , qo and %are the constants that can be found from the boundary and

initial conditions. Applying the boundary conditions, we can obtain the equation of the

natural frequencies of the bearn on an elastic foundation

cos P, Lcosh /3, L = 1 (4.24)

Using the appropriate numerical method, we can solve Eq(4.24) and obtain the natural

frequencies of the substrate.

The normal mode of the substrate is

sin p,, x + sinh Brix + sin Bnr -sinh "' (cos B,r + cosh P,x) (4.25) COS^ j,~ -COS p,r

where n = l,2,3 ...

4.3.2 Natural Frequencies and Normal Modes of the Optical Fiber

For free vibration, it is easily found that the equation of motion agd boundary condition

of the optical fiber under the condition of an elastic foundation is the same as that under

the condition of two springs. So the results for the optical fiber derived in Chapter 3 can

be used directly as follows:

For O S x d , ,


Normal moâes:

n z 2, (x) = BI sin - x

10


Normal modes:

n~ nl t nlt Z, (x) = B2 (sin x - tan Io cos

L - 21, L - 21, X)

L - 2 4

For L-1, I x l L ,

Natural frequencies

Normal modes

4.4 Forced Vibration Analysis

Since the dynamic mode1 is linear, we can still use the mode superposition method to

derive the dynamic response of the optical fiber coupler

4.4.1 Forced Vibration of the Substrate -.---A--=- - - - .& -*... . - - .

Substituting Eq (4.32) into Eq (4.6), we obtain

It is easily shown that the normal modes are orthogonal as follows:

Multiplying Eq (4.33) by V,,, (x) and integrating from O to L, we can obtain

where L is the length of the bearn and

--.----- 3 - - .. - - - -

The solution of Eq (4.35) is

qn (1) = Cn sin ~t a;, - n

Thus the solution of Eq. (4.6) is

4.4.2 Forced Vibration of the Optical Fiber

0

Since v = TV,,(x) sin S2t , substituting it into Eq(4.7) yields n = I min -aZ

w here

- cnc'P' (-sin B,x + sinh /$x + f ( x ) = F - Z s i n P n L - s i n h B n L ( -EOS/~ , ,X+COS~B,~X) ) A O ~ ~ -fi2 COS^ bn L - cos p,, L .. c,n2 - = 2

(sin /3,x + sinh f lnx + sin BnL-sinh (COS &x + cosh P,p) ) n=i% -fi2 cash P, L - COS P, L

_ _ _ - - -JJsingthe same method as solving the forced vibration of the optical fiber in Chapter 3.

we can easily obtain the forced vibration of the optical fiber as follows.

For O S x 4 1 ,

where

w here

For L-1, S x $ L

w here

4.5 Numerical Simulation

Numerical simulations are conducted in this section to show the dynamic characteristics of

optical fiber couplers with continuous elastic support. A simulation program written in C

code is developed. Since the simulation scherne is alrnost the s a m e as that of the optical

fiber coupler with two spring supports, we do not discuss it again here. The effects of the

parameten on the dynamic characteristics are thoroughly investigated.

4.5.1 Simulation Parameters

The parameters for study are given in Table 4.1. The other parameters are given in Table

4.2.

Table 4.1 Parameters for study

Table 4.2 Opticaî fiber coupler parameters

Tension of string P Foundation modu tus kf Position of bounding points 10

1 Length of the beam L 1 0.04 m 1

0.0 1 -0.6ON 50000- 10000000 ~/m'

0.005-0.0 1 75m

Cross-sectional area of the substrate A2 I

1 6 . 6 1 ~ 1 0 ~ m2 1 Mass density of the beam p, 1 2200 kg/mJ

1 Cross-sectional ana of the optical fibers Al 1 3.1~10" rn2 1

Young's mdulus of the beam E Moment of inertia of the bearn cross section I

Mass density of the optical fibers p, 1 2200 kg/m3

7 . 2 4 ~ 1 0 ' ~ pa 4 . 3 4 ~ 1 O-'* m4

Acceleration of the shock motion F Circular freauencv of the shock motion

9800 m/s2 2lot 1 o3 rack

-.- - - - - - . - -

4.53 Pammetric Study

The parameters in Table 4.1 are used to explore their effects on the vibration of the optical

fiber couple. These parameters are initialized as P=0 .6N, 1, =0.01m, and

k, = 50000~lm~. Since the equations of motion of the optical fiber coupler are linear. one

parameter is changed so that its effect on the vibration amplitude of the optical fiber coupler

can be explored.

( 1 ) Foundation Modulus

When the elastic foundation modulus kf varies from 50000 to Iûûûûûûû ~ / m ' . the

amplitude of vibration of the optical fiber coupler is shown in Figure 4.2. It is observed that

the amplitudes of the optical fiber coupler with different kf are overlapped because there are

very little difference between these amplitudes. The maximum amplitude of the optical fiber

with respect to the substrate is shown in Figure 4.3. It is easily found that the amplitude of

the optical fiber coupler is very small and almost does not change with kt: Therefore the

amplitude of the optical fiber is robust to kl.

(2) Position of the Optical Fiber Bonded to the Substrate

The position of the optical fiber bonded to the substrate Io varies from 0.005 to 0.0175rn.

The vibration of the optical fiber coupler is shown in Figure 4.4. It is shown that as lo

incnases, the amplitude of the middle segment of the optical fiber decreases; however, the

- amplitude of the other two segments incpmes. For 1, greater than a certain value, the

maximum amplitude will occur at the two end segments of the optical fiber. The maximum

amplitude of the optical fiber with respect to the substrate is shown in Figure 4.5. It can be

observed that from Io = 0 .0 1 m to 1, = 0.014m , the maximum amplitude decreases

monotonically. When 1, = 0.0 14 m , the smallest amplitude occurs. For 1, > 0.0 14 m . the

amplitude increases with the 1, increase. It is recommended that 1, be 0.0 14 m.

(3) Pretension of the Optical Fiber

The pretension P of the optical fiber varies from 0.01 to 0.6N. The vibration of the optical

fiber coupler is shown in Figure 4.6 and Figure 4.7. It is shown that the position of the

maximum amplitude of the optical fiber is not always at the rnidpoint. When P = 0.01N . the position is at x = 0.015m and x = 0.025m. The maximum amplitude of the optical fiber

with respect to the substrate is shown in Figure 4.8. It is easily found that the effect of the

pretension is not linear. There are two peaks of the maximum amplitude. When P = 0.1 IN.

the largest amplitude occurs, where z = 0.0034m. When P = 0.03N , the second largest

amplitude occurs, where z = 0.00T25m . From P = O.2N to P = 0.6N , the amplitude

decreases slowly with increase of P. When P = 0.6N, the amplitude is the smallest, where

2 = 0.00007m. Thecefore ii is recommended that P be 0.6N in order to make the amplitude

of the optical fiber smallest.

Conclusions

In this chapter, an analytical method is employed to perform both free and forced vibration

analyses of the optical fiber coupler with an elastic continuous support. First. the equations

of motion of the optical fiber coupler are developed. In order to solve the equations. a

coordinate system is introduced. Then the free vibration is analyzed and the mode

superposition method is used to obtain the dynamic response of the optical fiber coupler. A

simulation program written in C is developed and the effects of some parameters on the

dynamic characteristics have been investigated and the results are drawn as follows:

( 1 ) The amplitude of the optical fiber is robust to kj. The amplitude of the optical fiber

is very small and changes little with increase of kF

(2) It is recommended that 1, be 0.0 14 m.

(3) It is recommended that P be 0.6N.

. a . a string kf=105 a

a a

t f i

string IO^ t ¶

4 string 4=107 8

Figure 4.2 Amplitude of the optical fiber coupler with kl as a parametet

Figure 4.3 Maximum amplitude of the optical fiber with respect to the substrate vs kf

........... 1 beam

Figure 4.4 Amplitude of the optical fiber coupler with Io as a parameter

Figure 4.5 Maximum amplitude of the optical fiber coupler with 10 as a parameter

.--....---- beam string -0.01

-- -0.05 -0.1 '

- - - -0.15 P O . 2


. . . * . . . - . . beam string pr0.25

--- m.3 m . 3 5 P0.4 m . 5 Pû.6


Figure 4.8 Maximum amplitude of the optical fiber with respect to the substrate vs. P

Chapter 5

Conclusions and Future Work

5.1 Introduction

This chapter presents the sumrnary and the conclusions of this thesis. In addition. it also

addresses the work to be done in the future.

5.2 Summary and Conclusions

In this thesis, the vibration analysis of an optical fiber coupler has been explored by the

anal ytical method. In order to obtain the dynamic characteristics, the optical fiber coupler is

simplified and a linear dynamic mode1 is developed in Chapter 2. For the optical fiber

coupler with two spring supports. the substrate has been modeled as a Euler-Bemouli beam:

the bundle of optical fibers has been modeled as a string; and the four silicon mbber pads

have been rnodeled as two linear springs. Similarly, for the optical fiber coupler with an

elastic continuous support, the substrate has been modeled as a Euler-Bemouli beam; the

bundle of optical fibers has been modeled as a string; and the silicon nibber pad has been

modeled as a linear elastic foundation.

In chapter 3, the analytical method is employed to explore the dynamic response of the

optical fiber coupler with two spring supports. Fint, the equations of motion of the opticül

fiber coupler along with the proper boundary conditions are developed. Then the free

vibration is analyzed. In order to obtain the solution of the forced vibration equations. the

orthoganality of the normal mode M proved and the mode superposition method is used to

obtain the dynamic response of the optical fiber coupler. A simulation program written in C

is developed. In order to verify the analytical method, the same problern was investigated by

ANSYS. It is found that the results obtained by the two methods are in good agreement.

From the simulation results, the following conclusions are drawn:

The two springs (elastic mbber pads) should be placed at the two ends of the

substrate.

a It is recommended that P be 0.6N.

a The smaller a is, the smaller the amplitude will be.

a It is recommended that k be 4ûûûûNlm.

a It is recommended that 1, be 0.0 14 m.

In chapter 4, the analytical method is employed to explore the dynamic response of the

optical fiber coupler. First, the equation of motion of the optical fiber coupler is developed.

Then the free vibration is analyzed and the mode superposition method is used to obtain the

dynamic response of the optical fiber coupler. Numerical simulations were carried out and

the following conclusions are drawn:

The amplitude of the optical fiber is robust to kl. The amplitude of the optical fiber

is very srna11 and changes little with increase of kF

It is recommended that 1, be 0.014 m.

It is recommended that P be 0.6N.

From the simulation results, it can be found that the amplitude of the forced vibration of the

optical fiber coupler with an elastic continuous support is less than that of the optical fiber

coupler with two spring supports. Therefore, from the point of view of vibration, the optical

fiber coupler with an elastic continuous support is recommended.

Future Work

In our dynamic model of the optical fiber coupler, it is assumed that the tension of the

optical fiber is a constant during vibration. This just applies to the srna11 amplitude of the

string. If the amplitude is larger, the tension must be considered to change with time.

Therefore, a nonlinear dynamic model should be developed in the future work.

In addition, damping is neglected in our dynamic mode]. Since damping affects the

amplitude, especially when the frequency of excitation is at or near one of the natural

frequencies of the system, damping is of primary importance. Therefore damping should be

considered in the future work.

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--

-a=-- --- -Appendh - - -

Derivation of Natural Frequency of the

Substrate under Two Springs

For free vibration of the substrate, Eq.(3.1) becomes

By using the method of separation of variables, it is assumed

v(x , t ) = V(x)T( t )

Substituting Eq.(A.Z) into Eq.(A.l) ; it leads to

where 0, is a positive constant.

Eq. (A.3) can be written as two equations:

w here

The solution of Eq. (AS) can be expressed as

T(t ) = T sin(^ + p)

where T and cp are constants that can be found from the initial conditions.

The solution of Eq.(A.4) can be found

V ( x ) = ACOS&+ Bsinfi+Ccoshfi+Dsinhfi (A.@

where A. B, C, D are the constants that can be found from the boundary conditions.

For three segments of a beam as shown in Figure 3.1, Eq.(A.8) can be written as

Segment 1: V, ( x , ) = A, cos fil + BI sin fi , + Cl COS^ p Ix + Dl sinh pl (A.%

Segment III: V,(x , ) = A, cos &r, + B, sin /3.+ + C, cosh fi3 + D, sinh Br, (A. l 1 )

The boundary condition has been given in Chapter 3 as follows:

(A. 12)

(A. 1 3)

(A. 14)

(A. 16)

(A. 17)

(A. 18)

(A. 19)

(A.20)

Combining the solution of the three separate beams and their boundary and compatibility

conditions, the natural frequencies of the substrate can be obtained.

Substituting Eq (A.9) into Eq(A. 12) and Eq(A. 13), we obtain

--

pu -PL.

Substituting the above results into Eq(A.9). we have

V, = A, (cos fi, + cosh fi, ) + BI (sin bI + sinh @, )

Substituting Eq(A.20) and Eq.(A. IO) into Eq(A. 14). we have

A, (COS Pl, + cosh fil, ) + B, (sin pl, + sinh pl, ) - A, - Cz = O

Substituting Eq(A. IO) and Eq.(A.20) into Eq (A. 15), We have

A, (-sin j31, + sinh pl, ) + BI (cos pl, + COS^ Pl ) - B, - Dz = 0

Substituting Eq.(A. IO) and Eqm(A.24)into Eq(A. 16) and Eq(A. 17), we have

EIP ' (-B2 + D,) - EIP 'A, (sin pl + sinh pl,) - B, ~ l P ' ( - c o s Pl, + cosh pi, ) (A.27) + &[A, (cos pl1 + cosh pl, ) + B, (sin pl + sinh pl, )] = O

Substituting Eq(A. 10) and Eq(A. 1 1) into Eq(A. 18) and (A. 19), we obtain

A, + C, - A, cos 81, - B, sin @, - C, cosh -O, sinh Pl2 = O

119

Substituting Eq(A. 10) and Eq (A. 1 1)into Eq(A.20) and Eq(A.2 1). we obtain

Elp ( -B , + 4,) - E@(A, sin pl2 - B2 COS& + C1 sinh pl, + D, cosh pl,) (A.3 1 )

+&(A, +C,) = O

A? COS Pl2 - B2 sin P, - C, cosh b, - D, sinh pl2 - A, + C, = O

Substituting Eq(A. 10) and Eq(A.11) into Eq(A.22) and (A.23), we obtain

A, sin fl, - B, cos pl, + C, sinh Pi, + D, cosh Pl, = O

- A, cos pl3 - B, sin pl3 + C, cosh p3 + 4 sinh = O

The combination of Eq (A.25) and Eq(A.28) produces

C2 = Al cosh 8, + BI sinh pl

A, = Al cos pl, + BI sin pl

The combination of Eq(A.26) and Eq(A.27) produces _YYv--_-- -2 - IC - - . - - - - -. - - - - - - -

k sinh Pl - -- (cos + cosh pll ) - (sinhfl, +s in&) 2 E Z ~ ' 1

k -sin&+- (cos PI, + cosh /?LI ) (sin pl, + sinh pl ) 2 E I ~ ' 1

Let

W here

(COS 81, + cosh 81, ) a, = -sin Pl, +- 2 EIB

a, =cos& +- (sin Pl + sinh pl, ) 2 ~ 1 p

k a, = sinh pl, -- (cos BI, + cosh P, ) 2 ~ 1 p

Thus we have

fsinbp, +sin pl, ) a, =costt& -- 2 EIP '

A, = Al cos pl + B I sin Pl, B2 = A l a l + B p , C, = A, cosh P, + B, sinh pl, D2 = A , a , + B,a,

The combination of Eq (A.30) and Eq (A.3 1 ) yields

k A, sin -B2cos& + Bj --

2 EIB ' (A, +C,) = O

From Eq(A.29), we can obtain

A, + C, = A, cos& + B, sin fl , + C2 cash pl2 + D2 sinh &

Substituting Eq (A.46) into Eq (A.47), we obtain

k B, = -A, sin pl, + B, cos p, + - (A, cosp, + B, sin pl, + Cl cosh & + D? sinh Pl, )

2 E I ~

The combination of Eq (A.30) and Eq(A.31) also gives

D, = C, sinh & + DZ COS^ pl2 -- (A3+C3) 2 EIP '

---- - -. -- -

Substitution of Eq (A.47) and Eq(A.49) gives

k D, = Cl sinh Pl, + D, coshp, -- ((A, cos pl2 + B, sin pl2 + C l cosh Pl? + D1 sinh pl2 )

2 EIfl

The combination of Eq (A.29) and Eq(A.30) gives

A, = A, cos + B2 sin pl,

C, = C2 cosh Pl1 + D2 sinh pl2

Thus we have

( A, = A, COS & + B2 sin fl , k

B, = -A, sin /il, + B, cos & + - (A, cos Dl? + B2 sin pl, + C, cosh Pl, + D2 sinh pl2 ) 2 E V ' -

C, = C, cosh + D2 sinh pz k

0, = C, sinh + Di cosh p,, - - (A, cos& + B2 sin pl, + C, cosh Pl, + D2 sinh i 2EIp ' *

Substitution of Eq (A.53) into Eq (A.33) and Eq(A.34) , after simplification . we can

obtain

where

k -a, c o s ~ ( l z + f i ) -- (a, sin pl, cos 8, +a, sin pl, cosh pi ) - - ( E O S ~ pli COS^ COS PI: 2 ~ 1 ~ 2 EIp '

(sinh pl, cos pl, + sinh pl, cosh pl, ) + cosh pl2 cosh pl, ) - - 2 EZP '

k K LI? = - c o s ~ l , c o s ~ ( l z + 1 3 ) - - (COS pl, COS Pl2 sin pl, + cos Pl, cos Pl2 sinh Pl1 )

2 EIP '

(sin pl, sin pl, +sin fil2 sinh pl,) + cosh Pli cosh Pl2 cosh Pl, -a, sin 0(12 + 1 3 ) - - 2 EIP '

(cosh P, cosh P, sin PZ, +cash PI, cash PI? sinh pll + cosh p, sinh pl, sinh pl3 -- 2 Elp '

(sinh pl, sinh Pl, + sinh BI2 sinh pl, ) +a, (sinh & cosh BIj + cosh pl2 sinh & ) - - 2 EIp '

In order to obtain non-zero solution of Eq (A.54), we must have

Thus we obtain

Using the appropriate numerical rnethods, we can obtain the solution of Eq(A.55) and

thus we get the natural frequencies of the substrate.

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