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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 53
Combination of beam. string and springs ............................................ 55
........................................... Combination of beam. string and springs 56
.......................................... Combination of beam. string and springs 58
.................................................................. 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 , ,
Natural frequencies:
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
Figure 3.5 Combination of beam, string and springs
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.
Figure 3.6 Combination of beam, string and springs
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
Figure 3.7 Combination of beam, string and springs
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
Figure 3.8 Combination of beam, string and springs
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.
Figure 3.9 Combination of beam, string and springs
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
Figure 3.10 Combination of beam, string and springs
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 , ,
Natural frequencies:
Normal moâes:
n z 2, (x) = BI sin - x
10
Natural frequencies:
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
Figure 4.6 Amplitude of the optical fiber coupler with P as a parameter
. . . * . . . - . . beam string pr0.25
--- m.3 m . 3 5 P0.4 m . 5 Pû.6
Figure 4.7 Amplitude of the optical fiber coupler with P as a parameter
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.