LE

I . . P n T. 0' ,.
ll! 'r L .11. e.
. I: . . IL
. Iii-n? . , . ¢ . .W-Ihh. .w ..vrfl'b.3r:r
91.31?er rays...‘ h. f... 2 n, ..
$W'r .p is... flu. 7[.,I§1:9T§Umrcvlvr.tu. vvr
. 5...”.450. 1911'}! :.:v":.f:..o .JI
:. 1-; :wh o4~..0..§£.lfo2.u. .llluvv. .
.o - .tVi. IQ > '.-v."l,r‘f
011:5. ‘ I. . v, 1. f..L$
I you” .. Cusp-:5 It.)
. . o It... . . r .mfiru“u.n.nvl 7117...,
if c. . .4|.l.. .
‘ 11L...vltl¢49,.xcrtfl!_! 0...! «innsvv: E I.

uh
,
v
.

'.;,..

g
D

 

.A a?! 3. 1,10%..30‘ bu

:a‘qflfimmtfiflflrflzflmfl
. 35, “HI

. vb.

"t*'
l v.25

.5
- .Lv'a. cm? .
thin??? «in. £:
.t... . .9: v2.15:
"€I'¥.’\"fl.I-h ‘ib.lw ‘7
3...! is? t...

c Q "I. ”Ema“.

.. :2.
- sci bri‘zf

.
p!
u»! r .
a c...
. :KHMWK...
. I)

.7

..l' >
ya" 1

’1' .3
Eg‘lrl'
IXDVVLCOIIV.‘ . O

p] it‘ll‘jfif
. . .

g?!"

A
g.
'r

i

vl"l-I.

7: '- ~
ct»
. , -
5%?
. i. 2‘
5.5:?“
- viii, .
I {fin

n't

.z‘rli ...v
’33;;.’. 0’

.. cog
it. .9‘ no qo’.
.I ‘5. 4v .

.l'oll‘llin
. . .. ‘ . .. . .. h... t , 3.3!] filti‘illl
It. :15... 7. u ‘ . av. . .. . u ‘ ‘
is. .-. :19, .. ,. .

15...}...

u. . s 31:33.}..-

. . i..- 53..-.

.inriuls 1.12.31“. 5.4 v:
4...; .

 

. I'D-I‘llull..rhylrll 3
avLF-ofidfi.c‘l.l‘.l.llrpf
1-11.35)? I? Oil-T 9.1.).-
1.5. ‘ Iva. .31 5:. {to}! 1‘3;

.93.!

w,
I“

.
.

' t
'1‘”;

”WW.
. .5.
.‘h ,

1" ‘Fl‘. .
1"”.“1'
I »

' 'u;'
5:615.“
. ' W
'l

 

!
q:

a
H!

‘1‘
r

13""! I‘ll"! D.» . . . ‘ , . vltkb
vl I.1‘\I. Ila... v. 1‘ ‘ A ; l
..l .ul-.l|rk'rln.lot.|lr I ... . Al. .. y
‘3 ‘ .¢I|. vl.§l.r.\l.
‘ aol Ivb'..l.fu3liuvtn

l 0
‘
.
l
f
b

' m

 

:4

 

7:!

413

M

 

oh
WI
1

 

2:14!
‘ 1‘
WI

. v"
J
H

3.. ,........w...... -. _.

. V .
I‘Ili‘xtlaIOYS ‘ . I‘IINO .. A u p. 3:)
1,141.]! 73.123” «Inf... . .u . I .t? A

    

 

\l.»

 

1 I.-.£f.i!| k tl").a .‘.3uvrvl|sv.|li.l A“ I
‘ v .' V- I I it -O'-‘ ".V-.l-“vl.’ {‘1
. ‘ citslbv,‘.a00.vlil.lll ..v.|l.)l.l?"¢||lll .lln.u' t ‘
. . ‘ ‘ .III I. .a }.-‘Il-l'.of .- t ‘ I
o a nu . 1.1. .5.n.svfcu!..<a.l. . .l A :V o. v. .n . y . m
’ 131...: ‘ \ .JII V.)...¢x7 .«I..I.i|. ‘ ‘1 - tl . .p .33". 44¢ng
I . I will... ¢. I ..a vfua| o... a... . v . . .1 l1. ' l? .n’... |.. ..1. 9. . 0 .IVV |
. 15.: I . S K. = r. . r In:
kilo, 31-1. tn. é N v t. i ‘ . y . |

 

 

 

 

p

1E5

\lixtm

This is to certify that the

dissertation entitled

MULTI-OBJECTIVE OPTIMIZATION DESIGN STRATEGIES FOR
ARTICULATING MECHANISMS AND ROBOTIC MANIPULATORS
FABRICATED FROM ADVANCED POLYMERIC FIBER—REINFORCED

COMPOSITE MATERIALS

presented by

LU-PING CHAO

has been accepted towards fulfillment
of the requirements for

Pho Do degreein MoE.

 

 

Cf V ngfloU/t.

Major professor

Date 73/26 1/?0

MSU Lt an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

 

r LIBRARY
Michigan State

University J

_

 

 

is

PLACE IN RETURN BOX to remove this checkout from your record. »
TO AVOID FINES return on or before date due.

N'
DATE DUE DATE DUE DATE DUE i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘ WWI;

 

 

MSU Is An Affirmetive Action/Equal Opportunity Institution
emeritus

 

 

 

MUETI-OBJECTIVE OPTIMIZATION DESIGN STRATEGIES FOR
ARTICULATINC MECHANISMS AND ROBOTIC MANIPULATORS FABRICATED FROM
ADVANCED POLXMERIC FIBERrREINFORCHI COMPOSITE MATERIALS

by

Lu-Ping Chao

A DISSERTATION

submitted to
Miehigan State university
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Depart-cut of Mechanical Engineering

1990

ABSTRACT

MUETI—OBJECTIVE OPTIMIZATION DESIGN STRATEGIES FOR
ARTICULATINC MECHANISMS AND ROBOTIC MANIPULATORS EABRICATHI FROM
ADVANCED POLTMERIC FIBERrREINFORCHI COMPOSITE MATERIALS

by

Lu-Ping Chao

A theoretical and experimental study is undertaken to investigate
the performance of articulating linkage mechanisms incorporating
symmetric composite laminated links operating in various hygrothermal
environments. The theoretical investigation is focused upon developing a
finite element model to predict the hygrothermoelastodynamic response of
four-bar linkage mechanisms. In a complementary investigation, an
experimental program focuses on the elastodynamic response of a four—bar
mechanism with composite laminated coupler links fabricated from various
configurations of Hercules ASA/3SOl—6 prepreg and subjected to a variety
of hygrothermal conditions.

The single-objective optimization design methodology is developed
for robotic manipulators fabricated from fiberous composite materials. A
numerical simulation demonstrates that the optimal design methodology has
a efficient design process when compared with arbitrarily intuitive

design.

A strategy for the application of fuzzy set theory to multi-
objective optimal design requirements of a composite-based robot arm is
developed to choose the compromise solution. The formulation of the
fuzzy multi-objective optimization design problem is accomplished by
integrating design variables and by defining the fuzzy domains
corresponding to the objective functions and the constraints. The fuzzy
multi-objective optimization problem is solved by using conventional
single-objective function programming techniques.

Design for manufacturing strategies for incorporating manufacturing
uncertainties in the robust design of composite structures are proposed.
A methodology for the application of chance constrained programming to
stochastic design optimization of composite-based structural is
presented. A sensitivity analysis of the dynamic response of composite
structures is developed as a pre-requisite to obtaining the optimal
solution. This formulation is then employed to study the transient
response of cantilevered beams in order to demonstrate the significance
of incorporating random parameters into traditional deterministic

optimization design methodologies.

To my parents
and
to my wife

Chung-Shu

iv

ACKNOWLEDGMENTS

The author wishes to express sincere gratitude to his co-advisors,
Dr. Mukesh V. Gandhi and Dr. Brian S. Thompson, for constant guidance,
support, and encouragement during the course of this research program.
The author is grateful to the guidance committee, Dr. Byron Drachman and
Dr. Alan G. Haddow, for their constructive comments, insight, and
suggestions.

The author would like to extend thanks to his friends at Michigan
State University for being excellent sources of discussion and
suggestions. Also, the author would like to thank his colleagues in
Machinery Elastodynamics Laboratory and Intelligent Materials and
Structures Laboratory for their valuable suggestions and assistance.

Particular gratitude goes to the author's lovely wife, his two sons
for being a wonderful family with patience and understanding.

The author wishes to express sincere gratitude to his parents,
parents-in-law, sisters and younger brother for their encouragement and
support.

Finally, the author wishes to gratefully acknowledge the financial
support of this research program by the Defense Advanced Research
Projects Agency, under contract DAALO3-87-K—0018, the U. S. Army Research
Office, under contract DAAL03-88-K-0022, the National Science Foundation,
under Grant No. MSM-8514087, and also the state of Michigan, Department

of Commerce, Research Excellence and Economic Development Fund.

TABLE OF CONTENTS

LIST OF TABLES ....................................................

LIST OF FIGURES ...................................................

CHAPTER I. INTRODUCTION ..........................................

1.1 Motivation ................................................

l.
l.
1.

l.
l.

2.1 Dynamic Modeling ......................................
2.2 Dynamic Response Characteristics ......................
2.3 The Single-objective Optimal Design of

Isotropic Elastic Structures ..........................

.2.4 The Single-objective Optimal design of

Fiberous Composite Materials ..........................
2.5 Fuzzy Sets and Multi-objective Optimal Design .........
2.6 Reliablility-based Optimal Design .....................

1.3 Thesis Organization .......................................

1.4 Principal Contributions ...................................

CHAPTER II. MODELING OF FLEXIBLE STRUCTURES AND MECHANISMS

NNNNNN
O‘U'ibWNH

FABRICATED FROM FIBER-REINFORCED COMPOSITE MATERIALS

Governing Equations of Motion .............................
Finite Element Formulation ................................

Summary of the Chapter ....................................

vi

CHAPTER III. AN EXPERIMENTAL INVESTIGATION OF THE
HYGROTHERMOELASTODYNAMIC RESPONSE OF LINKAGE
MECHANISMS FABRICATED FROM POLYMERIC

COMPOSITE MATERIALS ................................. 54

3.1 Introduction .............................................. 54
3.2 Surface-fit Functions of Ei and $1 ........................ 55
3.3 Experimental Apparatus and Procedures ..................... 60
3.3.1 Fiberous Polymeric Composite Materials ................ 60
3.3.2 Four-bar Linkage and Hygrothermal Chamber ............. 61
3.3.3 Strain Gauges ......................................... 67
3.3.4 Moisture Raising of Coupler Links ..................... 72
3.4 Results and Discussions ................................... 75
3.5 Summary of the Chapter .................................... 81
CHAPTER IV. SINGLE-OBJECTIVE OPTIMIZATION DESIGN METHODOLOGY ..... 83
4.1 Introduction .............................................. 83
4.2 Formulation of Single-objective Optimization Design ....... 84
4.3 Generalized Reduced Gradient Method ....................... 89
4.4 Illustrative Example ...................................... 96
4.5 Summary of the Chapter .................................... 105

CHAPTER V. A FUZZY SET AND MULTI-OBJECTIVE OPTIMIZATION DESIGN

METHODOLOGY ........................................... 106
5.1 Introduction .............................................. 106
5.2 Fuzzy Set and Multi-objective Optimization Problem ........ 107
5.3 Computational Strategy .................................... 112
5.4 Numerical Example ......................................... 120
5.5 Summary of the Chapter .................................... 125

CHAPTER VI. DESIGN-FOR-MANUFACTURING STRATEGIES FOR
INCORPORATING MANUFACTURING UNCERTAINTIES

IN THE ROBUST DESIGN OF COMPOSITE STRUCTURES ......... 126

6.1 Introduction .............................................. 126
6.2 Formulation of the Stochastic Optimization Problem ........ 128
6.3 Design Sensitivity Analysis of Dynamic Problems ........... 135
6.3.1 Dynamic Response Sensitivities ........................ 136

vii

6.3.2 Natural Frequency Sensitivities ....................... 141

6.4 Numerical Example ......................................... 143
6.5 Summary of the Chapter .................................... 149
CHAPTER VII. CONCLUSIONS AND RECOMMENDATIONS ..................... 153
7.1 Conclusions of the Thesis ................................. 153
7.2 Recommendations for Future Work ........................... 156
APPENDIX A ........................................................ 158
The Modified Rule-of-Mixtures Approach .................... 158
BIBLIOGRAPHY ...................................................... l6 2

viii

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

LIST OF TABLES

Geometrical and Material Properties of Flexible

Coupler ...........................................

Geometrical and Material Properties of

Robotic Arm .......................................

Single-objective Optimization Results of

Robotic Arm .......................................

Geometrical and Material Properties of Robotic Arm

Multi-objective Optimization Results of Robotic Arm

Standard Normal Distribution Table ................

Geometrical and Material Properties of

Beam-like Structure ...............................

Stochastic Optimization Results of a

Cantilevered Beam .................................

First Order Derivatives of Objective Function and
Constraints with respect to Random Variables

when P-0.5 ........................................

First Order Derivatives of Objective Function and
Constraints with respect Random Variables

when P-0.9999997 ..................................

ix

Page

Figure

Figure

Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.10

.11

.12

.13

LIST OF FIGURES

Programming Methods in Structural Design .............
Design-for-Manufacture Strategy for Components
Fabricated in Composite Materials

Configuration of Composite Laminate

Configuration of a Composite Laminate

The Nodal Displacements of One Element

The Surface-fit Function of Young's Modulus E
The Surface—fit Function of Young's Modulus E

The Surface-fit Function of Shear Modulus G12 ........

The Surface-fit Function of Specific Damping
Capacity $1

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

The Surface—fit Function of Specific Damping
Capacity $2

The Surface-fit Function of Specific Damping
Capacity $12

Curing Cycle for AS4/3501-6 Graphite—Epoxy
Prepreg

0000000000000000000000000000000000000000000000

Photograph of the Experimental Four-bar Linkage ......
Schematic Diagram of the Experimental

Four-bar Linkage .....................................
Photograph of the Hygrothermal Chamber and Associated
Instrumentation ......................................
Schematic Diagram of the Experimental Apparatus and
Associated Instrumentation

The Detail-design Features of Coupler Link

Photograph of the Calibration-fixture

X

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.14

.15

.16

.17

.18

.19

.20

.21

.22

.23

.24

.25

.26

.27

Photograph of the Calibration-fixture with Coupler
Link

Photograph of the Calibration System .................
Calibration Relationship of the Amplified Voltage
and the Mid-span Deflection of Coupler

Specimen Preparation: Environmental Chamber

Photograph of Specimen Preparation Environmental
Chamber ..............................................
The Experimental Transverse Acceleration
Characteristics of the Coupler for Four-bar

Mechanism Operating Speed 130 rpm ....................
The Numerical Transverse Acceleration Characteristics
of the Coupler for Four-bar Mechanism Operating

Speed 130 rpm ........................................
The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/45/—45)2]S

Coupler Link: Mechanism Operating Speed 130 rpm

......

The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/30/-30)2]S

Coupler Link: Temperature of 80 F

The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/30/-30)2]S

Coupler Link: Temperature of 750 F

OOOOOOOOOOOOOOOOOOO

The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/30/-30)2]S

Coupler Link: Temperature of 2000 F

Experimental Results for the Mid-span Transverse
Deflection of the [(0/30/-30)2]S Coupler Link:

Mechanism Operating Speed 130 rpm

OOOOOOOOOOOOOOOOOOO

Experimental Results for the Mid-span Transverse
Deflection of the [(0/0/0)2]S Coupler Link: Mechanism

Operating Speed 130 rpm

The Experimental Results of the Effects of
Layup of the Laminae upon the Elastodynamic
Response Profile

xi

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure

Figure

Algorithm for Generalized Reduced Gradient
Method ...............................................
Schematic Diagram of a Single-link Flexible

Robotic Arm ..........................................
Methodology for Optimal Structure Design of a

Flexible Single-link robotic Arm Fabricated

from Composite Laminate ..............................
Test Input Torque for Evaluation of the

Objective Function ...................................
Tip Deflection Between Initial and Optimal

Design Due to Test Input Torque

Tip Position for Optimal Design

OOOOOOOOOOOOOOOOOOOOOO

Tip Position for Initial Design

Diagram of Membership Function For Fuzzy Set .........
Methodology for the Fuzzy Multi-Objective
Optimization Problem

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

OOOOOOOOOOOO

Methodology for Optimal Structural Design

Configuration of the Planar Motion of a
Single-Link Robotic Arm

Input Torque for Operating the Robotic Arm ...........
Iteration History of Individual Membership
Function

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Standard Normal Distribution

OOOOOOOOOOOOOOOOOOOOOOOOO

Transient Response of a Cantilevered Beam

The Comparison of Dynamic Responses for P=0.5 and
P-0.9999997

xii

CHAPTER I

INTRODUCTION

1.1 Motivation

The extensive utilization of fiber reinforced composite materials in
the defense, aerospace and manufacturing industries mandates that the
dynamic response characteristics of these advanced materials be carefully
studied under severe hygrothermal environments in order that reliable
design guidelines be distilled for the advanced optimal design of
composite parts. Since structures fabricated from fiberous polymeric
composite materials are involving more design parameters, more
complicated manufacturing processes, and more sensitive to hygrothermal
environments than those associated with the fabrication of the commercial
metals, it is imperative that the hygrothermoelastodynamic
characteristics of these materials be carefully investigated and
incorporated into design methodologies.

Traditionally, the design of articulating mechanisms has been based
on rigid-body analyses in which all mechanism members are treated as
rigid bodies [R1,Sl]. Stresses in the mechanism members due to inertia
forces and external loading are then calculated in a sequential manner
using quasi-static analyses. In the past two decades, mechanisms have
been designed to operate at higher and higher speeds in order to increase
industrial productivity. A consequence of this trend is that the members

of these systems can no longer be treated as rigid bodies since inertial

2
loads are dominant at high speeds [E3,L1,L2,T1]. In order to reduce the
inertial loading imposed on the machine, the weight of mechanism members
need to be made as light as possible without sacrificing structural
integrity. However the lighter members are more likely to deform and
vibrate due to the inertial and external forces, and they are also more
sensitive to hygrothermal environments. Hence the
hygrothermoelastodynamic analysis of articulating mechanisms becomes
necessary and it is an important embryonic field of investigation.

Considerable progress has been made in understanding the dynamic
response characteristics of polymeric composite materials under iso-
hygrothermal conditions [G5,B4,C4,S3,S4,SS,N1,A5,T2,T3]. Other work has
focused on the hygrothermal response characteristics of polymeric
composite materials [S6,W2,BZ,B3,G6,H1,R2,G3,G4], and also the optimal
design of flexible structures fabricated from polymeric composite
materials [T5,L3,Bl,T6,T7,T8,0l,K1,A4].

However research featuring a coherent combination of theoretical and
experimental work to provide a viable basis for subsequent optimal design
methodologies has not been undertaken for predicting the
hygrothermoelastodynamic or elastodynamic response of articulating
mechanism and robotic systems fabricated from fiberous polymeric
composite materials.

The first objective of this thesis, therefore, is to develop a
theoretical model, whose viability and limitations are investigated by
experimental work, in order to predict the elastodynamic response
characteristics of flexible linkage systems fabricated with a graphite-
epoxy prepreg material and subjected to hygrothermal environments.

Subsequently a single-objective optimal design methodology is developed

utilizing this computational model. The overall programming methods in
structural design is shown in Figure 1.1.

Flexible robotic arms fabricated with advanced composite materials
are increasingly being proposed, proved, and developed in order to
achieve high performances compared with heavy, rigid robotic arms.
Ramirez 1984 [R5] and Book, et a1. 1986 [BS] have proposed the design
strategies for high-speed advanced composite robots; Thompson, et a1.
1986 [T10] and Choi, et a1. 1988 [C3] have demonstrated both analytically
and experimentally significant results to prove the high performance of
advanced composite materials employed in robotic manipulators; Liao, et
a1. 1987 [L6] have developed a design methodology based on traditional
single-objective optimization problem such as the objective goal is to
maximize the damping.

In fact, to design the so—called high-performance robotic arms,
several, often conflicting objectives (design criteria) must be
considered simultaneously by the designer, such as lightweight robotic
arms, high stiffness, high damping capacity, and high end-point
positional accuracy, for example. Therefore, the problem must be cast as
a multi-objective optimization design task instead of considering it as a
single-objective optimization problem. This objective has been surveyed
in [R6]. However, there is a distinct absence of viable design
methodologies for synthesizing robotic arms in the context of multi-
objective optimal design.

Optimal solutions to multi-objective problems are typically infinite
in number and make up the so-called efficient set. In multi-objective
problems different designers may select different solutions from the

efficient set as the most satisfactory, even if the selection is based on

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Design Problem
v
Establish Design Criteria I Change Design Criteria
v:
Generate Alternate 7
Structural Configurations Change Structural I
’1
Develop Mathematical Models l Change Analysis Model I
y:
Analyze l Optimization I
it

 

 

 

 

it

Evaluate Design
Quantitatively

 

 

 

 

 

 

V

 

 

 

Carry Out Testing

 

 

Satisfactory ? NO > > e

 

 

 

 

YES

 

 

Stop

 

 

Figure 1.1 Programming Methods In Structural Design

identical performance constraints. Clearly, a satisfactory design must
agree with the every single objective criterion, but in true multi-
objective problems the manner in which every single objective criterion
contributes to the notion of satisfaction is not at all clear. In
addition, the designer's decision may be affected by an incomplete
knowledge of the efficient set. Therefore, the concept of a fuzzy set is
more suited to model the algorithm to solve multi-objective problems than
traditional algorithms such as the goal programming method, or the game
theory approach.

The second objective of this thesis, is to develop a design
methodology based upon employing fuzzy set theory to transform a multi-
objective optimization problem into an ordinary single-objective
optimization problem. A flexible single-link robotic system fabricated
from composite laminate is empolyed as an illustrative example of this
proposed methodology.

The increasing deployment of fiber-reinforced composite materials in
structures will further exacerbate the demand for superior design and
manufacture methodologies. The reason for this is that structures
fabricated with fiberous composite materials involve more design
variables than those manufactured in the commerical metals. These design
variables include lamina thickness, fiber orientation, and fiber volume
fraction. Moreover, these composite structures involve more complicated
manufacturing processes such as resin-transfer molding or braiding, and
each manufacturing process imparts distinct stiffness, damping, and mass
characteristics due to the distinct fiber volume fractions, spatial
distribution of the fibers, fiber orientations and lay-ups associated

with each process. These relationships are presented in Figure 1.2.

Figure 1.2

 

COMPOSITE -8ASED PARTISUB-ASSEMBLY SPECIFICATION

r———___.._..________.._.___.____—.__._-.______.

Loading specification : Mechanical, thermal, Impact
chemical, electrical. magnetic

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Performance specification : Stresses, deflections. settling-tine.
(requency response
Other specifications : Fatigue lite. costs. production rates.
production volume
V
MATERIAL SELECTION PART CEOMETRY
——————————————————— sq ——-———-———-—
Matrix materlals V A
For. synthesis
Fibres (contlnuous. dlscontlnuousl Synthesla of parts
I
I ll
HYBRIDS CHARACTERISTICS AND CONSTRAINTS
[_ _____________ IMPOSEO BY MANUFACTURING PROCESS

 

Contlnuous. dlscontlnuous

Glut "'m‘d- 9"Ph‘u Pultruslon. filament winding. lay-op.
moulding. laminating. mvlng.
mechanical fasteners. adhesive bonding.

antltrlctlon bearings

ll

MACROMECHANICAL CHARACTERISTICS
OF COMPOSITE-BASED PART

 

 

 

 

 

 

 

 

 

 

 

Fibre volume fraction, fibre
orientation. stacking sequence.
ply thickness

 

 

 

I

ELASTOOYNAMIC CHARACTERISTICS
OF COMPOSITE-BASED PART

_________________ .4

Mass, stiffness, damp-n9

 

 

 

 

 

 

V

FINITE ELEMENT ANALYSIS

b—-—-_————————————

 

Deflections. stresses.
settling-time. fatigue life

 

 

 

   

  
    
  

PART
PERFORMANCE SPECIFICATION
SATISFIED!

 

 

 

Materlal specification.
part geometry specification.
manufacturing process specification

1

 

 

 

Design-for-Manufacture Strategy for Components
Fabricated in Composite Materials

7
Thus, it is imperative that the designer be aware of these manufacturing
constraints when the manufacturing process is specified, otherwise the
part will be fabricated with mass, stiffness and damping properties
different from these specified by the design engineer.

Furthermore, with the hand-layup technique for example, if the
fabricator does not comply with the angle prescribed for the prepreg or
the prescribed stacking sequence is not developed then the properties of
the part will again be different from these prescribed by the design
engineer. There are numerious other examples where errors in fabricating
fiberous polymeric parts can adversely impact upon the global static and
dynamic properties of the part. Filament winding is another common
manufacturing technique where the control of fiber volume fraction and
helix angles are of crucial importance. Furthermore, since more design
variables and more complicated manufacture processes are associated with
the fabrication of fibrous polymeric composite materials, it involves
more tolerances as compared with the traditional monolithic materials
which have less number of design variables and restricted number of
manufacture processes. Hence, optimal design problems associated with
the design and fabrication of composite parts become much more complex
with more uncertainties than those associated with the fabrication of the
commercial metals. It is desirable, therefore, to pose this class of
problems as a robust stochastic, or reliability-based, optimization
design task instead of considering it as a deterministic optimization
task.

Therefore, the third objective of this thesis is to develop a design
methodology for the application of chance constrained programming to the

design and manufacture of composite structures by converting a stochastic

8
or reliability-based optimal design problem into an equivalent
deterministic nonlinear programming problem focused on the finite element
analysis of structures subjected to dynamic excitation. The finite
element method permits the applicability of probabilistic analysis and

optimal design to a wide variety of engineering problems.

1.2 Literature Review

Although this research program discusses on the analytical modeling
and optimization design strategies of single-objective, multi-objective,
and design-for-manufacturing for articulating mechanisms and robotic
manipulators, some of the major contributions on the modeling and optimal
design of flexible structures and articulating manipulators will be still
reviewed here in order to provide a flavor of applicability of the

proposed research to the relevant structural systems.

1.2.1 Dynamic Modeling

Several surveys of the literature on the dynamic behavior of
mechanism links with continuously distributed mass and elasticity are
presented by Lowen and Jandrasits [Ll], Erdman and Sandor [E3], and Lowen
and Chassapis [L2]. Smith and Maunder [Sl2] developed a method of
determining the inertia effects in four-bar linkages. General
expressions were derived for the forces acting at the joints of the
mechanism. The effects of balancing and of changes in the mass
parameters of the linkage were also discussed. Erdman et al. [E1]
developed a general method for kineto—elastodynamic analysis and

synthesis of mechanisms. Their paper described the initial phases in the

9
development of a general method of kineto—elastodynamics analysis and
synthesis based on the flexibility approach of structural analysis.

Low [L9] applied the Hamilton's principle in order to derive the
coupled, nonlinear integro-differential governing equations for a
manipulator containing both rigid and flexible links. The method
accounts for the distribution of the mass and flexibility and its
nonlinear dynamic effects. The formulation is based on expressing the
kinetic and potential energies of the robotic system in terms of a set of
independent generalized coordinates.

Thompson [T10] developed a dynamic model for industrial robotic
manipulators constructed from composite materials by utilizing a
variational theorem and a displacement finite element method. The
variational theorem was formulated by considering composite materials to
be linear viscoelastic homogeneous materials. Turcic and Midha [T11]
developed the generalized equations of motion for elastic mechanism
systems by utilizing modal superposition finite element and a
formulation. The derivation and the equations of motion provide the
capability to model a general two- or three-dimensional complex elastic
mechanism, to include the nonlinear rigid-body and elastic motion
coupling terms in a general representation. The authors also presented
the approach to the solution of the equations of motion and an
experimental investigation of an elastic four—bar mechanism in [T12,Tl3].

Reddy [R11] studied the large-amplitude, free, flexural vibration of
layered composite plates, incorporating the effects of transverse shear
and rotatory inertia using a finite element based on the displacement

field of a shear deformable theory.

10

The finite element method has been applied by many researchers to
produce an accurate and efficient model of flexible structures, flexible
manipulators, and articulating mechanisms. The justification for
employing this method is three fold. First, the finite element method is
known by many investigators. Second, the method is generally versatile,
hence the model can be easily adapted to include as many, or as few,
degrees of freedom as desired, and also the structural damping can be
accommodated easily. Third, this method can easily incorporate with
different material properties, different boundary conditions and changes

in cross section of geometries in a simple and straightforward manner.

1.2.2 Dynamic Response Characteristics

Clary [C4] studied the effect of fiber orientation on the flexural
vibrations of plates and beams, and experimentally determined their
damping characteristics under iso-hygrothermal conditions. Schultz and
Tsai [83,84] investigated the dependence of dynamic moduli and damping
ratios upon the fiber orientation and stacking sequence by undertaking
theoretical and experimental work. Suarez et a1. [SS] applied the
elastic-viscoelastic correspondence principle to investigate the dynamic
properties of laminates. Adams et a1. [A5] studied the anisotropic
properties of various laminates. Ni et a1. [N1] undertook similar
studies on symmetric laminated beams subjected to bending loads only.

Shen et a1. [S6] presented a summary of papers concerning the
effects of environmental conditions on the tensile and compressive moduli
of different composite materials. Whitney et a1. [W2] employed flexure
texts to determine the moduli of specimens with various moisture contents

on the temperature over a range of 72°F to 400°F. Barker and Vangerko

ll
[32,B3] investigated the dependence of the shear modulus and elastic
constants of CFRP in the temperature range of -100°C to 200°C. Gibson
et a1. [G6] studied the dynamic characteristics of chopped-fiber-
reinforced composite materials in various environments. Hancox [H1]
investigated the response of hybrid composite materials under different
environmental conditions, whereas Rehfield, Briley and Putter [R2]
presented the experimental results for graphite/epoxy composites in
several moisture and temperature environments.

Thompson et al. [T14] presented an experimental and computational
study of a flexible planar four-bar linkage fabricated from fiber-
reinforced composite materials. The fiber-reinforced composite materials
are modeled as an elastic solid by using a continuum theory which
accommodates laminae with orthotropic properties, and differing
densities, thickness, and elastic moduli. Gandhi [G3,G4] theoretically
investigated the dynamic response characteristics of laminated composites

structures in hygrothermal environments.

1.2.3 The Single—objective Optimal Design of Isotropic Elastic
Structures

Since some progresses of analytical model owing to the establishment
of the finite element method, significant work has been done on the
single-objective optimal design problems. The literature in this field
is enormous, then only some of the more representative and important work
is reviewed herein.

Several extensive surveys on the optimal structural design are
presented by Haftka [H4,H6], Vanderplaats [V3], and Ashley [A7]. Rao

[R6] has a extensive survey on the optimization design of robotic

12
manipulators. Haug and Arora [H2] have analyzed methods and techniques
for the optimum design of structures and mechanisms. Arora and Hung [Al]
have also proposed structural methods which are based on the design
sensitivity analysis methodology. According to their work, optimal
structural methods can be classified into two categories: namely,
numerical optimality criterion methods and mathematical programming
methods [813]. The numerical optimality criterion methods are based on
the numerical solution of the Kuhn-Tucker optimality criteria that are
necessary conditions for the existence of a local minimum solution. On
the other hand, the mathematical programming methods usually start from
an initial design and reach the optimum solution through appropriate
search strategies.

Arora and Belegundu [A2] have also proposed mathematical programming
methods for structural optimization. Numerical results for some methods
have been compared in terms of efficiency, accuracy, and reliability.

Kiusalaas [K3], Fleury [F4], and Lin [L8] have also proposed methods
which are based on the numerical optimality criterion methods. Fleury
[F4] used the minimum weight of structures as the objective function with
explicit constraints on the design variables, meanwhile, displacement and
stress as upper implicit constraints. Numerical illustrations of this
method are the minimum weight design of airplane wing structures.
Kiusalaas [K3] has presented a method based on finite element method for
the minimum weight of structures subjected to explicit constraints on the
design variables and implicit lower bound constraint on natural
frequency. The illustration of this method was the optimal design of
shafts. Same as Kiusalaas's method, Lin [L8] added upper bounds on

natural frequency and stress constraints in the problem. This optimal

13

design problem has applied to the minimum weight design of shafts and
trusses. Govil, et al. [G7] have also presented on the optimal weight
design of large scale structures. They added displacement and stress
implicit constraints and explicit constraints on the design variables.

On the problems of optimal sizing of flexible linkages, the authors
of reference papers [I1,T15,C6] have considered multiple configurations
which included the effect of dynamic loading in the analysis. They have
incorporated constraints on deflections and stresses induced by cyclic
dynamic loads. The consideration of dynamic loading is an excellent

decision for articulating mechanisms.

1.2.4 The Single-objective Optimal Design of Fiberous Composite
Materials

Schmit [814] has worked on the minimum weight of simply supported
laminated composite plates. The number of the laminaes and the fiber
orientations were predefined, only the laminae thickness have been
treated as design variables. Explicit constraints on the laminae
thickness and lower limits on the diagonal elements of the extensional
stiffness matrix have been considered. Soni [815] used fiber
orientations, laminae thicknesses, and the plate aspect ratio as design
variables in the minimum weight design of clamped plates. Implicit
constraints have been imposed on the displacements, principal stresses,
and first mode natural frequency.

Eschenauer [E2] optimized sandwich carbon fiber composite plates
used as the reflector surfaces of accurate radio telescopes. the
impressive point of this paper is the combined minimization of weight and

maximum deflection. The surface layer thicknesses, the fiber

l4
orientations, and the core height were design variables. Tauchert and
Adibhatla [T6,T7,T8] proposed optimal procedure for the design of
laminated fiber-reinforced plates having midplane symmetry. Strain
energy is taken as the objective function, while the fiber orientations
and/or laminae thicknesses represent the design variables. The
minimization of the strain energy is equal to the maximization of the
stiffness. Deformation analysis is carried out employing the finite
element method.

Adali [A9] presented the optimization of an antisymmetrically
laminated angle-ply strip subjected to transverse loads. The objective
function is to maximize the failure loads which is based on initial ply
failure and determined with respect to a given strength criterion. The
design variable is the fiber volume ratio of the composite materials.
Hygrothermal conditions and the presence of voids in the resin are taken
into account in computing the material and strength properties from
micromechanical relations. Liao, et a1. [L3] have presented the
optimization of symmetric composite beams for maximum damping capacity.
The design variables are the laminae thicknesses, fiber orientations, and
the stacking sequences. The paper indicated that the maximization of the
damping deteriorates the flexural rigidity and the stiffness of the
laminated beam. Thompson and Gandhi [T5] developed optimal composite
laminate design methodologies for investigating the high-performance

articulating robotic systems under iso-hygrothermal conditions.

1.2.5 Fuzzy Sets and.Multi-objective Optimal Design

The theory of fuzzy sets was developed for a domain in which

descriptions of activities and observations are fuzzy, in the sense that

15
there are not well defined boundaries of the set of activities or
observations to which the descriptions apply [R8]. The theory was
initiated by Zadeh [Z3], and later theoretical studies presented in many
mathematical papers. However, this theory applied to different practical
systems is still fewer. Herein the literature review is focused on the
papers which are proposed the fuzzy set theory and multi-objective
optimal design.

Adali [A8] optimized an antisymmetrically laminated angle-ply plate
with the multi-objectives of minimizing the maximum dynamic deflection,
maximizing the natural frequencies and/or maximizing the buckling load.
The design variables are the fiber orientation and the thickness of
individual laminaes. Numerical results are presented in the form of
optimal tradeoff curves which allow the designer to assess the various
possibilities open before deciding on a certain design.

Wang [W5] proposed fuzzy information in both objective and
constraint functions of optimum design of structures. Under this fuzzy
conditions, when all fuzzy constraints enclose a fuzzy feasible region in
decision-making space, the fuzzy optimum solution will be a sequence of
points in a small fuzzy optimum subregion. Optimum structural design
with fuzzy constraints is then transformed into a set of ordinary optimum
problems by a level cuts approach which results in a sequence of optimum
design schemes with different design levels. this method is applied to
solve the three bar truss and a symmetric two-story shear frame. Wang
[W4] also developed the procedure of fuzzy optimum design of aseismic
structures. The application of fuzzy sets to several civil engineering

problems was reviewed by Brown and Yao [B7].

16

Rao [R7,R8] also proposed fuzzy information in both the objective
and constraint functions of the optimum design of structures. A multi-
objective fuzzy programming method is outlined in order to solve
structural design problems which involve several, often conflicting,
objective functions. Then a method of solving a fuzzy multi-objective
structural optimization problem using ordinary single-objective
programming methods is presented. The computational approach is
illustrated with three-bar truss and and twenty-five-bar truss. Diaz
[D2] presented a rational and systematic procedure for aggregating
performance criteria based on fuzzy set theory. Optimal solutions
obtained from the procedure are formally related to solutions to
traditional formulations in multiobjective optimization. The procedure
is illustrated withimpact absorber problem. Dubois and Prade [D4]
provided an extensive survey on fuzzy set—theoretic operations, and
emphasizes the relevance of the theory of functional equations in the
axiomatical construction of classes of such operations and the derivation
of functional representations. Problems of modeling the importance of
criteria, and of choosing a proper aggregation connective in a given

situation are discussed in this paper.

1.2.6 Reliability-based Optimal Design

Ang [A10] brings out significant developments that have occured in
the reliability-based design area and emphasizes the importance of
probabilisticapproaches to design. Moses and Kinser [M3] have directed
their efforts to the automated optimum design of structures for a
specified reliability level by incorporating the probability of failure,

which is a function of the design variables, as a constraint in a general

17
nonlinear programming problem. Rao [R9] applied chance constrained
programming techniques for the optimum design of structures with random
parameters. Davidson et a1. [D1] considered the weight optimization of
indeterminate structures subjected to deterministic transient dynamic
loads and reliability constraints. Gross and Sobieski [G8] developed a
method to link deterministic and random loads for aircraft design by
treating probability criteria as a constraints on combined load
conditions at the structural element level. Rao [R10] presented a
methodology of formulating the optimum design problem for structural
systems with random parameters and subjected to random vibration. This
proposed method is applied to the optimum design of a cantilever beam
with a tip mass and a truss structure supporting a water tank.

Frangopol [F5] presented an overview of the major concepts and
methods used in reliability-based structural Optimization. In this
paper, new formulations related to multicriteria optimization were
developed. Comparative results are presented when different criteria are
used for the optimum design of a structure under service and ultimate
reliability constraints. Rao [R12] studied the application of
multiobjective optimization techniques to structural design problems
involving uncertain parameters and random processes. Several of the
problem parameters are assumed to be random variables and the structural
mass, fatigue damage, and negative of natural frequency of vibration are
considered for minimization. The design of a cantilever beam with a tip
mass subjected to a stochastic base excitation is considered for
illustration. The solution of this multi-objective design problem was
solved by using global criterion, utility function, game theory, goal

programming, goal attainment, bounded objective function, and

18
lexicographic methods. Jozwiak [J3] discussed the minimum weight design
of structures when design parameters have random variations. Belegundu
[B6] developed a practical approach to optimal design with random design
parameters for probability-based optimal design of structures. The
Hasofer-Lind second-moment design criterion is used to formulate the
general design problem. A method for calculating the sensitivity
coefficients was also presented in this paper.

The research fields om dynamic modeling, single-objecitve optimum
design with isotropic materials or fiberous composite materials, fuzzy
sets and multi-objective optimum design, and reliability-based optimum
design on flexible structures or articulating mechanisms have been
reviewed. Most research studies have been done on the dynamic modeling
and optimum design of flexible structures, articulating mechanisms, and
robotic manipulators fabrcated from commercial isotropic materials.
However research featuring a coherent combination of theoretical and
experimental work to provide a viable basis for subsequent optimal design
methodologies and design for manufacturing strategies for incorporating
manufacturing uncertainties has not been undertaken for predicting the
hygrothermoelastodynamic or elastodynamic response of articulating
mechanism and robotic systems fabricated from fiberous polymeric
composite materials. Therefore, this research program will provide a
overall investigation and study to fill this void in the literature. In
addition, this study will furnish viable guidelines and directions for

future work in this or relevant research fields.

19

1.3 Thesis Organization

In Chapter II, the theoretical model of flexible structures and
articulating mechanism fabricated from fiberous ploymeric composite
materials under hygrothermal environments is discussed for studying
optimal design strategies. The elastic moduli of fiberous polymeric
composite materials are first derived through the micromechanical model
of a laminae and the macromechanical model of a laminate. The governing
equations of motion are based on the modification of the shear-deformable
plate theory to account for midplane stretching due to large deflections,
large rotations, and dimensional and constitutive changes due to
moisture-induced swelling effects and temperature-induced expansions and
contractions. The in-plane hygrothermal dimensional changes are
accounted for by introducing fictitious forces and moments, and the
hygrothermal dimensional changes along the thickness of the plate are
explicitly incorporated in the assumed form of the deformation field.

The equations are then discretized using a finite element method. The
inherent damping of the structure is modeled in the finite element
formulation based on the hypothesis that the damping matrix is
proportional to the stiffness matrix of the structure.

Chapter III presents a theoretical and experimental study to
investigate the performance of articulating linkage mechanisms
incorporating symmetric composite laminated links operating in various
hygrothermal environments. The theoretical investigation is based upon
the developed model in Chapter II to predict the hygrothermoelastodynamic
response of four-bar linkage mechanisms. In a complementary
investigation, an experimental program focuses on the elastodynamic

response of a four-bar mechanism with composite laminated coupler links

20
fabricated from various configurations of Hercules AS4/3501-6 prepreg and
subjected to a variety of hygrothermal conditions. A detailed
description of the experimental apparatus and procedures of the proposed
four-bar mechanism featuring hygrothermoelastodynamic response is also
presented.

Chapter IV presents the design strategy for single-objective
optimization design of flexible structures and robotic manipulators
fabricated from fiberous composite materials. The mathematical
background for single-objective optimization design problems is discussed
with the aid of the method of Lagrange Multipliers. The generalized
reduced gradient algorithm is then introduced for the optimization
design endeavor. The numerical example of a flexible single-link robot
arm is illustrated through this design strategy.

A strategy for the application of fuzzy set methodologies to multi-
objective optimal design requirements of a composite-based robot arm is
presented in Chapter V. The formulation of the fuzzy multi—objective
optimization design problem is accomplished by integrating design
variables such as, laminae thicknesses, fiber orientations, and fiber
volume fractions, which characterize the mass, damping and stiffness
properties of the structure, and by defining the fuzzy domains
corresponding to the objective functions and the constraints. A method
of solving the fuzzy multi-objective optimization problem empolying
conventional single-objective function programming technique such as
generalized reduced gradient method is presented. This fuzzy-set-based
strategy is illustrated by considering the design of a single-link

flexible robotic arm fabricated by laminated composites.

21

Chapter V1 is focused on developing viable design for manufacturing
strategies for incorporating manufacturing uncertainties in the robust
design of composite structures. A methodology for the application of
chance constrained programming to stochastic design optimization of
composite-based structural and mechanical systems is presented herein.
The formulation of the stochastic optimization problem which addresses
the design and manufacture of fiberous polymeric parts is accomplished by
integrating the design and manufacture random variables such as, laminae
thicknesses, fiber orientations, and fiber volume fractions for example.
These random design and manufacture variables characterize the mass,
damping and stiffness properties of the structure. By adopting chance
constrained programming these random variables map the stochastic
programming problem into an equivalent deterministic nonlinear
programming problem. A sensitivity analysis of the dynamic response of
composite structures is discussed herein as a pre-requisite to obtaining
the optimal solution. This formulation is then employed to study the
transient response of cantilevered beams in order to demonstrate the
significance of incorporating random parameters into traditional
deterministic optimization design methodologies.

Finally, conclusions of this research program and recommendations

for future work are presented in Chapter VII.

1.4 Principal Contributions

1. A generalized analytical model is developed for predicting the
hygrothermoelastodynamic response of articulating linkage mechanisms

fabricated from fiber reinforced composite materials. This model is also

22
employed to flexible structures and robotic manipulators for the
optimization design endeavor.
2. Experimental works have been done to investigate the performance of
articulating linkage mechanisms incorporating symmetric composite
laminated links operating in various hygrothermal environments. The
experimental results from this mechanism system clearly demonstrates the
necessity to incorporate hygrothermal effects into mathematical models
for predicting the behavior of polymeric composite materials exposed to
variable hygrothermal conditions. It is anticipated that the theoretical
predictions based on the finite element model developed herein and also
the experimental response-data developed from this laboratory
investigation will provide crucial guidelines for realistically designing
articulating mechanism and robotic systems for defense, aerospace and
manufacturing environments where severe variations in the ambient
moisture and temperature are frequently encountered.
3. The single-objective optimization design methodology is developed
for robotic manipulators fabricated from fiberous composite materials. A
numerical simulation demonstrates that the optimal design methodology is
an efficient design process compared with the performance of a
arbitrarily intuitive design.
4. A strategy for the application of fuzzy set theory to multi-
objective optimal design requirements of a composite-based robot arm is
developed to choose the compromise solution. This work is the first of
its kind to apply multi-objective optimization techniques to robot
design. This work will also stimulate the evolution of an innovative
class of optimization strategies which will be crucial for the design and

manufacture of 'real-world' articulating systems.

23
5. Design for manufacturing strategies for incorporating manufacturing
uncertainties in the robust design of composite structures are proposed.
A methodology for the application of chance constrained programming to
stochastic design optimization of composite-based structural is also
presented. A sensitivity analysis of the dynamic response of composite
structures is developed as a pre-requisite to obtaining the optimal
solution. This formulation is then employed to study the transient
response of cantilevered beams in order to demonstrate the significance
of incorporating random parameters into traditional deterministic
optimization design methodologies. The sensitivity analysis can not only
offer the first-order derivative for stochastic optimal design but it
also provides several other applications in structural mechanics such as
approximate analysis and reanalysis techniques, analytical model

improvement, and assessment of design trends.

CHAPTER II

MODELING OF FLEXIBLE STRUCTURES AND ARTICULATING
MECHANISMS FABRICATED FROM FIBER-REINFORCED!COMPOSITE MATERIALS

UNDER HYGROTHERMAL ENVIRONMENTS

2.1 Introduction

The most important issue for an optimization design methodology is
to establish a reliable model upon which the optimal computational
algorithm is based. Since the core of this thesis discusses flexible
structures and articulating mechanisms fabricated from fiberous polymeric
composite materials, there are two distinct parts in this chapter. The
first portion of the discussion focuses on the calculations of elastic
moduli of fiber-reinforced composite materals, which the micromechanical
model for a laminae and the macromechanical model for a laminate are
necessary. Those calculations are based on classical lamination theory

which incorporates the following assumption.

Assumption 2.1 : (l) The laminate consists of perfectly bonded
laminae. (2) The bonds are infinitesimally thin and nonshear-deformable,

i.e., displacements are continuous across laminae boundaries so that no

laminae can slip relative to another.

The second portion involves the governing equations of motion as

well as associated boundary conditions which derived by using continuum

24

25
mechanics, and also the reduced-order dynamic model which is accomplished

by using finite element method.

 

 

[OOOOOOOOOOO

 

 

/

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[OOOOOOOOOOO

Figure 2.1 Configuration of Composite Laminae

2.2 Micromechanical Model of a Laminae

In order to establish a viable theoretical model for studying
optimal design strategies of flexible structures and articulating
mechanisms fabricated from fiber reinforced composite materials, the
micromechanical model of a laminae and the macromechanical model of a
laminate must be established. The elastic moduli of fiber reinforced

composite materials are first expressed as a function of the properties

26
of the constituents, which are the fiber and matrix properties, and also
fiber volume fraction. Subsequently, these variables are incorporated
into macromechanical models for the global mechanical properties of the
laminate which include such optimal design variables as fiber
orientation, laminae thickness, and the stacking sequence. These
variables serve as manufacturing specifications for the fabrication of
the laminate and also govern the dynamical response of the fabricated
part.

Considering the fiber-reinforced composite laminae as shown in
Figure 2.1, a viable formulation for the elastic moduli of each
moderately-thick laminae is accomplished by utilizing the modified rule-
of-mixtures approach for fibers which are transversely isotropic [T4].
The details are shown in Appendix A. Therefore, the reduced stiffness

quantities Qij of a moderately-thick laminae are

Q11 - m ( 1 ' ”23 ”32 ) E1

Q12 ' m ( "21 + ”31 ”23 ) E1

Q22 ' m ( 1 ' ”13V31 ) E2

Q13 - m ( v31 + u21u32 ) E1 (2.1.1)
Q23 ‘ m ( ”32 + ”12V31 ) E2

Q33 ' m ( 1 ’ ”12V21 ) E3

Q44 ' G23 ' Q55“ G13 ' Q66=612

-1
Where m‘[ 1"’12"21"’23"32“’31"13'(2 ”21V32V13) ]

 

 

 

E3 ' E2 , ”13 ' ”12 ' G12 ' G13
u = ”12E2 , y = ”13E3 , v = ”23E3
21 E 31 E 32 E
1 1 2

In equation (2.1.1), El and E2 denote the longitudinal Young's

modulus and the transverse Young's modulus respectively, G12 defines the

shear modulus in the 1-2 plane, and G presents the shear modulus in the

13

1-3 plane. The Poisson's ratio, Vij, is defined by

- - __i (2.1.2)

and under hygrothermal conditions the Young's moduli E1 and the shear

moduli Gij will be functions of both moisture, M, and temperature, T;

namely, Ei(M,T) and Gij(M’T)' The details of these terms will be

described in Chapter 111.

Note that the longitudinal direction is defined to be the 1-
direction. Similarly, the 2-direction is in the laminae transverse to the
logitudinal direction. The 3-direction is defined to be perpendicular to

the laminae.

2.3 Macro-eehanical Model of a Laminate

After determined the so-called reduced stiffness quantities Qij of

fiber-reinforced composite materials by employing micromechanical model

28
for a moderately-thick laminae, consider the composite laminate as shown

in Figure 2.2. It is evident from Figure 2.2 that hk is the vertical

distance from the lamiminate middle surface (z=0) to the upper surface of
the kth laminae and h is the total thickness of the laminate.

. . . th . . .
Const1tut1ve equations of the k lam1nae w111 be stated by assum1ng
that the strains due to the moisture and temperature environments are
decoupled from the strains due to mechanical loading; and therefore,

these strains may be superposed to yield the total strains as follows:

6.. = e. + 6.. (2.2)

t .
where Cij’ eij and 6?. represent the components of the k h lam1nae
of the total strain, the mechanical strain, and the non-mechanical strain
due to the superposition of the hygroscopic and the thermal strains,
respectively. The hygroscopic and the thermal strains induced in a

homogeneous orthotropic body due to a uniform change in temperature AT

and moisture content AM are assumed to be given by

a? = a. AT, and (2.3)
1 1

Furthermore

H-E:

(2.4)

T
e - e. + e
1.

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

b
__‘L_
z/<-—7 A +__
x L
h
z
A
A
n
\
hn
A h“-
k 11!: 1
A
hk-l
Middle Plane
hl ho
2 v
1 v
Section A-A

Figure 2.2 Configuration of Composite Laminate

3O

. . . th .
The three-dimens1onal stress-stra1n relat1on of the k lam1nae

including hygrothermal effects in matrix form is therefore given by

 

 

 

 

 

 

r ‘k r «k , ,k
”1 Q11 Q12 Q13 0 O O ‘1' “1 AT’ 51AM
”2 Q21 Q22 Q23 ‘2’ “2 AT’ 32AM
“3 Q31 Q32 Q33 ‘3' “3 AT' 53A”
4 r — 4 r
023 o o 2Q44 0 o 623
031 O O 2Q55 0 e31
012 0 O 0 O 2Q66 £12
. 1 _ 1 . 1
9 [Qij]k{eij}:-2_3 (2.5)

where ai is the coefficient of thermal expansion in the i direction and

AT is the change in temperature from some prescribed reference
temperature at which there are no thermal stresses or thermal strains.

The coefficient of hygroscopic expansion in 1 direction is denoted by fli

and AM is the increase in moisture content by weight from the condition
of zero moisture. The normal stress in the longitudinal fiber direction

is represented by 01 while 02 and 03 represent the normal stresses in

the direction perpendicular to the fiber.

. . th . .
The stress-stra1n relat1on for the k lam1nae of arbitrary
orientation can be obtained by employing the transformation matrix
relating the 1-2-3 coordinate system and the x-y-z coordinate system,

which has the following form

31

 

 

F -
m2 n2 0 2mm
n2 m2 0 O -2mn
O O l O O 0
[T] = (2.6)
O O 0 m -n O
O 0 O n m 0
2 2
-mn mn O O (m -n )

where m— c056, n=sin0. The fiber orientation relative to the oxy plane
is denoted by 0, which is defined as positive in the counter-clockwise
direction. This transformation matrix provides the stress-strain

relation in the x-y-z coordinate on which the laminate is based. Thus:

, ‘k r .k
a e - a AT - fl AM
x x x x
a e - a AT - fl AM
Y Y Y Y
a e — a AT - fl AM
2 z z z
-1 k -T
< > — [T1 [Qij] [T] l >
a 5
Y2 yz
a e
zx zx
oxy oxy -l/2axyAT-l/2fixyAM
t J t J

 

 

 

 

A -k “k
=[Q] {e}x_

y-z (2.7)
2 2 2 2
where ax= alm +a2n , fix: film +fi2n
a = a2m +aln , fly flzm +flln
02: a3 ’ flz‘ fl3
axya (01° a2)mn : fixy= (Bl- fl2)mn

32

[(2]k is the transformed reduced stiffness matrix of the kth

laminae. Therefore, the constitutive relationship for a laminate is

given by
n n
k - k N k
E aij - E ( [Q] {eij' Eij} ) (2 8)
k=1 k-l

The above constitutive relationship may be re-written in terms of

in—plane and out-of-plane stresses as follows:

n n

Ea: -§ (Qagufi - a: )k + QafiuA - .59“) (2.9)
k=l k=1

n n

Ea: -§ (QAficea - a: )k + Gain}, - e§>k) (2.10)
k=1 k-l

where, a,fl = 1,2,6, and

A,B - 3,4,5

where 0a represent the in-plane stresses, and the first and second term

on the right-hand-side represent the in-plane strains and out-of-plane

strains, respectively. Similarly, 0 represent out-of—plane stresses,

A
and the first and second term on the right-hand-side represent the in-

plane and out-of—plane strains, respectively.

33

2.4 Governing Equations of Motion

The governing equations of motion for the moderately-thick laminated
structures in hygrothermal environments will be derived from the three-
dimensional equations of motion of a continuum. The equations are based
on the modification of the shear-deformable plate theory to account for
midplane stretching due to large deflections, large rotations, and
dimensional and constitutive changes due to moisture-induced swelling
effects and temperature-induced expansions and contractions. The in-
plane hygrothermal dimensional changes are accounted for by introducing
fictitious forces and moments, and the hygrothermal dimensional changes
along the thickness of the plate are explicitly incorporated in the
assumed form of the deformation field.

Let xi denote the components of the position vector of a typical
particle in the undeformed configuration of the continuum denoted by 00.
The components of the position vector of the same particle in the
deformed configuration 0 are denoted by ii. The equations of motion for

the continuum are given by

:‘LiJ_+bi-p fi (2.11)
axi
where, aij are the components of the mechanical stress tensor, bi are the

components of the body force, p is the mass density of the continuum, and

fi are the components of the acceleration vector. The above equation may

34
be re-written in terms of the coordinates in the undeformed configuration

denoted by xi to yield

 

332.
g: (”kg—1'] +bi-p fi (2.12)
J J "k

By employing the following deformations,

xi= xi + Ui (2.13)
1-1_[.a£_i.+§fli] (2.14)
3 2 ax. ax
3 i
wi.-1_ [flflii] (2.15)
J 2 axj 6x
6 .- e.. + 1 e . + w . e. - w. 2.16
13 13 ET' [ k3 k3] [ 1k 1k] ( )
where, U., e... w.. and 6.. are the components of the displacement
1 13 13 13

vector, the linear stretch tensor, the linear rotation tensor and the
mechanical strain tensor, respectively, the equations of motion may be

re-written as

3;; [03k [ 61k + eik + wik] ] + bi - p fi (2.17)

35
For a moderately-thick lamilated structures, by assuming the strains
to be much smaller than the rotations, the strain displacement equation

may be re-written as

 

 

 

eij - eij + %_ U3’iU3’j (2.18)
and the equations of motion may be written as
+h/2 +h/2 +h/2
N , + N , + a + I b dz = I p f dz (2.19)
11 l 21 2 13 —h/2 -h/2 l -h/2 l
+h/2 +h/2 +h/2
N , + N , + a + J b dz - I p f dz (2.20)
21 1 22 2 23 -h/2 -h/2 2 -h/2 2
+h/2
Q, +QI +0
1 1 2 2 33 -h/2
+h/2
+ fi__ I (a U , + a U , ) dz
ax -h/2 11 3 1 l2 3 2
+h/2
+ Q__ I (a U , + a U ) dz
6y -h/2 21 3 1 22 3,2
+h/2
+ (”31U3'1 + ”32U3'2)

-h/2

36

 

+h/2 +h/2
M , + M , - Q + m + I z b dz = I p 2 f dz
11 1 12 2 1 l -h/2 1 -h/2 l
+h/2 +h/2
M , + M , Q + m + I z b dz = I p 2 f dz
21 1 22 2 2 2 -h/2 2 -h/2 2
and
Q1'1 Q2'2 N<U3’N1) + m3
[+h/2 +h/2
- 0 dz - I (a U , + a U , ) dz
-h/2 33 -h/2 31 3 1 32 3 2
+h/2 +h/2 +h/2
+ z a + I z b dz - I p 2 f dz
33 -h/2 -h/2 3 -h/2 3
where,
+h/2
N - I 0 dz, a,fl = 1,2.
afl -h/2 afl

(2.21)

(2.22)

(2.23)

(2.24)

37

 

 

+h/2
m - z a ,
l 13 3 -h/2
+h/2
m - z a ,
2 23 3 -h/2
+h/2
m - z(a U , + a U , )
3 31 3 l 32 3 2 -h/2
In equations (2.19) - (2.24), which govern the nonlinear vibrational

response of moderately-thick plates, are specialized to study the problem
of vibration of moderately-thick laminated composite plates

in hygrothermal environments. This is accomplished by modifying the
shear-deformable plate theory [Yl,F2] in order to account for midplane
stretching due to large deflections, large rotations, moisture-induced
swelling effects and temperature-induced expansions and contractions.

The in-plane hygrothermal effects are accounted for by introducing
fictitious forces and moments; and the hygrothermal dimensional changes
along the thickness of the laminated structures are explicitly
incorporated in the assumed form of the deformation field. A deformation
field of the following form is considered in order to accommodate these

effects:

XI

(x,y,z,t) - X(X.y,t) + A z 6(X.y.t)

§ (x,y,z,t) - Y(x,y,t) + A z n(x,y,t) (2.25)

38

E (x,y,z,t) - Z(x,y,t) + A z

In equations (2.25), the position of a typical particle in the
reference undeformed configuration is represented by the coordinates

(x,y,z), and the coordinates of the same particle in the deformed and

swollen state are denoted by (§,§,E). The midplane of the plate in the
undeformed state is assumed to be at z = O, and (X,Y,Z) represent the
deformed position of a typical particle on the midplane of the plate in
the undeformed position. A herein denotes the stretch ratio in the z
direction which accounts for hygrothermal dimensional changes along the
thickness. f and 0 represent the slopes in the xz and yz planes,
respectively, due to bending only.

The displacement field corresponding to equations (2.25) is given by

U1 - U(X.y.2.t) - U°(X.y.t) + A z €(X.y.t)

U2 - V(x,y,z,t) = Vo(x,y,t) + A z n(x,y,t) (2.26)

U3 - W(x,y,z,t) - W°(x,y,t) + (A - 1)z

where, U0, V0, W0 are midplane displacements given as

U - X - x

39

v°-Y - y (2.27)

Using equations (2.26) and (2.18) into equation (2.2) yield the

following set of total-strain-displacement equations:

6 = 5° + Azk + 5N, a = 1,2,6
0 a a

6 - 1_ ( A - 1) + c (2.28)

N

m
b
I
y
A
m
+
C!
u:
H
v
+
m
b

m
I
y
A

é
+
C‘.

40

k6 - £32 + ”.1

The equations of motion (2.19) - (2.24) and the strain-displacement
relations given by equations (2.28) represent a general three-dimensional
problem and can not be solved in a two-dimensional plate-like sense. The
problem may be considered in two steps. In the first step, the
anisotropic laminated plate may be assumed to be deformed to an
intermediate state due to out-of—plane moisture and temperature effects.
In this intermediate state the thickness is assumed to be changed by a
stretch ratio A. In the second step, the deformation from the
intermediate state to the final state is assumed to be due to mechanical
loads, on which are superposed fictitious loads due to in-plane
hygrothermal effects. Once the hygrothermal environment is specified, A,

and, therefore, 6 can be determined; and for a constant value of A the

3
equation of motion (2.24) is trivially satisfied,which reduces the
problem to a plane stress problem. Now the strain-displacement relations

are given by

61 — a: + 2 k1 + 6? (2.29.1)
62 - e? + 2 k2 + 65 (2.29.2)
e6 - £2 + 2 k6 + 6: (2.29.3)
64 - (5 + U;.1) (2.29.4)

41

-(17+U

0
‘5 3'2)

and the equations of motion reduce to

N1’1 + N6’2 ‘ P Ul’tt + Ré’tt
N6’1 + N2'2 ‘ P U2'cc + R"'cc
Q1’1 + Q2’2 + N(U3’N1) ' P U3’tt
M1’1 + M6’2 ’ Q1 - Ig’tt + R Ul’tt
M6’1 + M2’2 ' Q2 ’ In’tt + R Uz'cc
In equations (2.30),
+h/2 . .2 .
(P.R,I) ‘I (1. Z, Z )P dz
-h/2
n hk+1 A A2 m A
-§J (1.2.Z)pdz
k-l h

(2.29.5)

(2.30)

(2.31)

42

a a A
-h/2
“ hk+l . A
—E J (1, z ) 0: dz (2.32)
k-l hk
+h/2 A
QC - I A 0C dz
-h/2
n hk+1 A
- E I 0: dz (2.33)
kal hk
and
6x 6y
+ L016. “312) + a_ (N2, 113.2) (2.34)
8x By

The definitions given by equations (2.31) - (2.33) are based on the

assumption that the laminated composite plate is composed on n laminaes,

and hk and hk+1 are the coordinates of the lower and upper boundary of

each individual laminae. In addition,

43

A

h = A h, and z = A 2

For one plane of material symmetry parallel to the plane of the layer,
the constitutive equations in terms of the stress and moment resultants

which appear in the equations of motion (2.30) may be written as

N A B so - 6N
[ a ] = [ “fl “5 J [ fl 8 J , 6,6 = 1,2,6 (2.35)
Ma 3&6 D06 Kfl
N
QC = ACD (5D - 5D ), C,D = 4,5 (2.36)

The fictitious loads due to hygrothermal loading may now be written as

N N
N A e
a 093 13

N ‘ * N
Q A A 6
[ 1 ] - [ 44 45 ] I 4 J (2.38)
N ‘ “ N
Q2 A54 A55 65
where,
n h
k _ § 2
Aafi’ Bafl’ Dafl - E I (Qafl (1. z, z ) dz (2.39)

44

n h
A k _ k
ACD - E I KC KD (QCD) dz (2.40)

k-1 hm

In equation (2.40) KC and KD are the shear correction coefficients.

2.5 Finite Element Formulation

The general equations of motion for the laminated composite
structures under hygrothermal environments in Section 2.4 is specialized
here-in to develop a finite element formulation for beam-like flexible
structures and articulating mechanisms. Since the general equations of
motion in equations (2.30) included transverse shearing deformations and
rotary inertias, the details of finite element model with these effects

will present in this section.

92

 

 

NV

0

Figure 2.3 The Nodal Displacements of One Element

45
Therefore, the solution to this class of boundary-value problem can

be obtained by discretizing the displacement field as follows:

w(x.t) - [N] {U} (2.41)

where [N] is the row vector containing the shape function which is
dependent upon the position only and {U} is a column vector containing

the nodal d1splacements U1, W1, 01, U2, W2, and 02. The degrees of

freedom of this two-node element is shown in Figure 2.3 along with the
local and global reference frame.

The deformation of the flexible beam-like structures in the e and

1
e2 directions can now be expressed as
wx = [Nx] {U}, and
w = N {U} (2.42)
Y [ Y]

where [Nx]= [ N1 0 0 N4 0 0 ]

[Ny]- [ 0 N2 N3 0 N5 N6]

T
{U} - { Ul W1 01 U2 W2 62}

The procedures in order to derive the row vector [Ny] will be

presented herein. A cubic interpolating polynomial is employed to

represent the transverse deflection wy(x,t) of a typical point x on a

beam element of length 2.

46

Thus,

2 3
wy(x,t) - a0 + alx + azx + a3x (2-43)

In equation (2.43) the deflection wy corresponds to W0 defined in

equation (2.27). Furthermore, on assuming that the transverse shear

strain 1 is independent of x, it can be shown that equation (2.29.4)

reduces to

g3 (x,t) = 0 + 7 (2.44)
dx

In equation (2.44) the transverse shear strain 1 and the rotation 0

correspond to e and -€ respectively, defined in Section 2.4. Based on

A
the moment equilibrium of an infinitesimal beam element, it can be shown

that

7 .. _ 6 a3 8 (2.45)

where

47

where K5 is the shear correction coefficient which depends upon the shape

of cross section of beam. Furthermore, on substituting equation (2.45)

into equation (2.44) it can be shown that

0 - a + 2 a x + (3x2 + 6g) a

1 2 (2.46)

3

a and a

Based on equations (2.43) and (2.46), the coefficients a 2, 3

0’ a

1’
can be determined. Then the shape functions for the transverse

deflection wy(x,t) are given by

N - 1 [( 22+l2g) - 123 x - 3x2 + _E_ x3]
2 2 2 2
2 + 12g
2 (222 + 6 ) 2 3
N3 - 1 [(2 + 6g)x - g x + x ]
22 + 12g 1
(2.47)
NS - [ 12g x + 3x - _3_ x3]
22 + 12g 2 I
N - [ -6gx - (2 - 6g) x2 + x3 ]
6 2 2
2 + 12g

Similarly, by employing an interpolating polynomial again the shape

functions for axial deflection wx(x,t) are given as

48

N1 = l - x/2

(2.48)

NA - x/2

The effective mass density of the kth laminae is expressed by

k k k
p = pm(1-Vf) + prf (2.49)

where pm and pf are the density of the matrix and the fiber respectively

, and Vf is the fiber volume fraction of the kth laminae. The mass per

unit length of the beam-like structure for translational inertia can be

expressed by

B!

n
k
t - b E p (hk- hk-l) (2.50)
k-l

Furthermore, the product of the mass density and the moment of inertia

per unit length of the beam for rotational inertia is defined by
n
{xi-=15 k1(h-h) (251)
P— k k-l '
k

Therefore, the local mass matrix of the laminate for translational

inertia is given by

49

[N 1 a ” I£( N T N + N T N )d d (2 52)
t mto[y][y] [x][x]x’an -

the local mass matrix of the laminate for rotational inertia is given by

A ~ 2 , T ,
- 2.53
[Mr] mr JO [NY] [NY] dx < >

, and the local stiffness matrix of the laminate is

A

2 n n I r "
[K] = bIO([Ny]TD11[Ny] + [NX]TA11[NX] + [NS]TA55[NS])dx (2.54)

the flexural stiffness,D and

where the extensional stiffness,A 11.

11’

transverse shearing stiffness A are defined in equations (2.39) and

55

(2.40),while

[N"] = 2 [o -3+§_ x -(222+6g)+3x o 3-6_x -(22-6g)+3x 1
y 2 2 2 2 2
2 +l2g
' 1
[N 1 = ___ [-1 o o 1 o o ]
X
2
[NS] - ~6g [ o 2_ 1 o -2 1]

,2 +l2g 2 2

50
The formulation for the damping matrix is based on the hypothesis
that structural damping is a function of the strain energy dissipated
during each stress cycle [L4,N2]. The structural damping is therefore
transformed to equivalent viscous damping in order to create a model for
a Kelvin solid [F2,M1]. Following the procedure presented in [L4], the
effective local viscous damping matrix of the laminate, which is

proportional to the stiffness matrix, is represented as

 

r w
m
A n (R )k 2 A: A
[ c 1 - E 11 1 < i=1 > [K] (2.55)
k 2« m
H (Q11) 2 w. A?
. 1 1
\ 1:1 J

 

 

where

(R11) - 611 $1 c0540 + 612 (¢1+¢2) c0520 sin20
+ ' 2 ' 40 (2 56)
022 2 Sln .

+ 4 Q66 $12 c0520 sin26

In equation (2.55), n is the total number of laminae, i is the it mode,

m represents the number of active modes in the flexural dynamic response
of the beam-like structure and A1 denotes the amplitude of the ith mode
which corresponds to the natural frequency wi. In equation (2.56), $1,

¢ , and p represent the specific dampin capacities in the
2 12 g

51

longitudinal, transverse and longitudinal shear directions respectively,

and all of these parameters are functions of both moisture and

temperature [L5].

A

The nodal fictitious hygrothermal forces {f} are given by

A 2
" T 0N N
{f}- bIo [Ny] (81161 + Dllnl ) dx
2 ' T 0N N
+ b} [Nx] (A1161 + Bllnl ) dx

0

where the hygrothermal strain eiN

written as

oN -1 N

-1 N
61 = F11 (N1 ’ B11 D11 ”1)
N -1 N oN
”1 ' D11 (M1 ‘ B11 61 )
F A - B D"1 B
11 11 11 11 11
NN - NT + NM
1 x x
MN - MT + MM
1 x x

and hygrothermal curvature 5N

(2.57)

1 can be

(2.58)

52
In some mechanism systems such as the four-bar linkage and slider-
crank mechanism, which flexible links are subjected an external constant
axial force P, an initial stress stiffness matrix [C5] for an element has

to be invited into the finite element model for no accuracy being lost.
A
This initial stress stiffness matrix, [K0], is independent of elastic

properties, however, it depends only upon the geometry of the element,

displacement field, and state of stress. There, the initial stress

A

stiffness matrix [K0] for an element can be written as

A 2,T,
[K0] = bPJ [Ny] [Ny] dx (2.59)
O

The global finite element equations for predicting the elastodynamic
response of flexible linkage mechanisms and robotic systems with fiberous
polymeric links and operating under various hygrothermal environments are
obtained by incorporating the relevant boundary conditions and using the
transformation matrices relating the local and global reference frames.

These equations are:

[M] {U} + [C] {1'11 + [K] {U} = {Q} + {F} (2.60)

where the global mass, damping and stiffness matrices are represented by
[M], [C], and [K], respectively, {Q} is the generalized force vector
excluding hygrothermal effects and {F} is the global fictitious

hygrothermal force vector.

53

2.6 Su-lery of the Chapter

The elastic moduli of fiber-reinforced composite materials are
obtained by employing the micromechanical model of a laminae and the
macromechanical model of a laminate. The governing equations for the
moderately-thick laminated structures under hygrothermal environments
were derived from the three-dimensional equations of motion of a
continuum, and these equations were discretized by using the finite
element method incorporating the rotary inertia and transverse shearing
deformation for beam-like structures. The external axial force effect
which will raise the strength of the stiffness matrix of finite element
modeling for some structures such as four-bar linkage and slider-crank
mechanism was also considered herein for the development of a precisely
theoretical model. In the following chapters, the main frame of analysis
models for investigating the hygrothermoelastodynamic response and

optimization design strategies will adopt the study in this chapter.

.F1

»1

.h»

*4
«I:

CHAPTER III

AN EXPERIMENTAL INVESTIGATION OF THE
HYGROTHERMOELASTODYNAMIC RESPONSE OF LINKAGE MECHANISMS

EABRICATED FROM POLYMERIC COMPOSITE MATERIALS

3.1 Introduction

The extensive utilization of fiber reinforced composite materials
in the defense, aerospace and manufacturing industries mandates that the
dynamic response characteristics of these advanced materials be carefully
studied under severe hygrothermal environments in order that reliable
design guidelines be distilled for the advanced optimal design of
composite parts. Since polymeric composite materials are relatively
sensitive to hygrothermal environments compared with the commercial
metals, it is imperative that the hygrothermoelastodynamic
characteristics of these materials be carefully investigated and
incorporated into design methodologies.

The theoretical model which was already presented in Chapter II
will be employed herein in order to predict the elastodynamic response
characteristics of flexible linkage systems fabricated with a graphite-
epoxy prepreg material and subjected to various diverse hygrothermal
environments. Section 3.2 is focused on developing three-dimensional
graphical design tools which present the variation of effective Young's

moduli with both temperature and moisture content; and specific damping

54

55
capacity with both temperature and moisture content. These three-
dimensional plots are obtained by developing surface-fit functions which
interpolate the experimental scattered data. In Section 3.3, the details
for fabricating the flexible composite coupler links are described in
addition to describing the design of the four-bar mechanism and the
experimental apparatus. The experimental procedures are also described
in Section 3.3. In Section 3.4 the computational simulations and
experimental results of the four-bar mechanism subjected to various

hygrothermal environments are presented in a comparative format.

3.2 Surface-fit Functions of E1 and¢i

Having established the model for predicting the elastodynamic
response of articulating mechanisms and robotic systems operating under
variable hygrothermal environments in Chapter 2, attention now focuses on
experimentally determining the specific damping capacities ¢1(M,T),
¢2(M,T), and $12(M,T), the Young's moduli E1(M,T), E2(M,T), and the shear
modulus 612(M,T). Laboratory tests were performed on coupons of Hercules
AS4/3501-6 with 0 degree, 90 degree and 45 degree fiber orientations.

The shear modulus 612 (M,T) was determined by using the formula

1 - 2V12(M,T)

 

 

1 = l‘ - - —l—- (3.1)
612(M,T) E45(M,T) E1 (M,T) E2(M,T)

where E45 (M,T) is the Young's modulus of the [4512] specimens and
u12(M,T) herein is assumed to be 0.3 using reference [B2].
The specific damping capacities ¢1(M,T) and ¢2(M,T) were

determined from the tests of [012] and [9012] specimens. The specific

56
damping capacity $12(M,T) can be inferred from the [4512] test data. The
details of the experimental investigations for determing the specific
damping capacities are presented in reference [L5]. The experimental

scattered data for E 612, $1, $2, and $12 in [L5], provided the

1' E2'
basis for determining the numerical values of the coefficients for the
surface-fit functions for the application of this class of graphite epoxy
polymeric material under hygrothermal environments. The form of the

interpolation formula for the individual surface-fit function Ei(M,T),

Gi3(M’T) or wi(M,T) are chosen as
Ei(M.T). Gij(M.T) 01' ¢i(M.T)

- (al)i + (a2)iT + (a3)iM + (a4)i T2 + (a5)i TM

2

+

2 3 2
(a6)i M + (a7)i T + (a8)i T M + (a9)i TM

3 4 3 2 2
(a10)iM + (a11)iT + (alz)i T M + (a13)i T M

+

T5 + (a

+

3 4 4
(314)1TM + (a15)1M + (a16)i 17)1T M

3 3 4 5
(a18)iT M + (alg)i TM + (azo)iM (3.2)

+

where T is temperature and M represents the moisture content. The

surface-fit functions of E1 and ¢i are shown in Figure 3.1, 3.2, 3.3,

3.4, 3.5, and 3.6 and these design tools are discussed in Section 3.4.

57

 

 

 

\Nmmmm

nun—Lu 321.09 Q.§>

 

s Modulus E1

I

The Surface-fit Function of Young

Figure 3.1

 

 

 

 

..
0

's Modulus £2

The Surface-fit Function of Young

Figure 3.2

58

/

 

 

 

Anna): 3:34 0.8d>

The Surface-fit Function of Shear Modulus 612

Figure 3.3

 

 

 

 

The Surface-fit Function of Specific Damping

Capacity $1

Figure 3.4

“’ 5‘
w.
9’2

(2‘6

69,

’ ’ 9’ \
1’”
’7”

222.2
’2
'o
10’
Q,
’9
A

 

aneclf‘lc: Davplhg Oepamty

 

 

Figure 3.5 The Surface—fit Function of Specific Damping
Capacity $2

9"
i ,
6,4

:;’ 25
.«i‘m
{32 \
nu

'39?”
1 //
’22”

M
l//

2"
2
'9
'9:
O
I N
i
6
[I
1
I
1

 

Saeclf‘lc Darplr‘g aneclty
0
, £7
‘3.

 

Figure 3.6 The Surface-fit Function of Specific Damping
Capacity $12

 

60

3.3 Experimental Apparatus and Procedures

The previous section focused upon establishing the experimental
protocols for investigating the elastodynamic response of articulating
mechanisms and robotic systems subjected to various dynamic hygrothermal
environments. Herein, details are presented for fabricating the fiberous
composite flexible links for linkage mechanisms, in addition to
describing the experimental apparatus, and outlining the experimental

procedures.

3.3.1 Fiberous Polymeric Composite Materials

The material utilized to fabricate the beam-like links employed in
the experimental investigation was Magnamite graphite prepreg tape, type
AS4/3501-6, which is manufactured by Hercules Inc.. The prepreg tape was
cut in order to fabricate 11.5 inches square laminas with fiber
orientations dependent on the design requirements. Then, these square
laminas were layed up by hand in order to fabricate square laminates
which featured several different stacking sequences.

The autoclave was programmed to impose a pressure and thermal
cycle upon the vacuum-baged laminates as specified by the Hercules data
sheet as shown in Figure 3.7. Subsequent to this critical manufacturing
procedure, a postcure period was initiated to allow for final cross
linking of the polymer chains. The postcure process was performed in a
laboratory oven at 350°F for 3 hours. The fully cured laminate was then
cut using a diamond grinding wheel in order to manufacture a set of beam-

like links for the hygrothermoelastodynamic response investigation.

61

 

 

 

 

 

 

 

 

 

300-

—PRESSURE=85 PSI, VACUUM=28 IN HG

----PRESSURE=1OO PSI, VACUUM=VENT
g: 200—
E1“ , .................................
Di
3
F- -
<(
0:
LL] 1
g .
uJ 100-
[_—

O r I I I T 1 I I I j I T I I T I I I
o 60 120 180 240 300

TIME(sec)

Figure 3.7 Curing Cycle for AS4/3501-6 Graphite-Epoxy
Prepreg

3.3.2 Four-bar Linkage and Hygrothernal Chamber

A four-bar mechanism was employed as a test bed in order to
investigated the ability of the theoretical model to predict the
hygrothermoelastodynamic response of this articulating mechanical
system. A photograph of the experimental four-bar linkage used in this
investigation is presented in Figure 3.8. A detailed schematic diagram
of the experimental four-bar linkage is presented in Figure 3.9, from
which it is evident that it comprised one flexible link, the coupler.
While the mechanism was designed to accommodate a range of link lengths,
this investigation was undertaken with a fixed ground link length of
406.4 mm, while the length of the coupler was 313.4 mm, the rigid

follower was 203.2 mm and the rigid crank was 63.5 mm.

62

 

Figure 3.8 Photograph of the Experimental Four—bar Linkage

 

 

Figure 3.9 Schematic Diagram of the Experimental
Four—bar Linkage

63

The follower and coupler links articulated in the same plane.
This was achieved using a cleavage bearing design that is clearly evident
in the center of the photograph, Figure 3.8. The flexible strip of link
material was clamped to the stainless steel bearing housings by two
socket screws at each end of the specimen and the clamping loads were
distributed over the ends by flat plates which are again clearly visible
on the coupler link in Figure 3.8. These small plates were found to be
essential components of the mechanisms since they ensured a smooth load
transfer between the three principal components of each link. They
represent only one of the many detail design features incorporated into
each mechanism.

The coupler and follower links were supported on matched pairs of
instrument ball bearings, type SR4C HH5P58LY283, supplied by Miniature
Precision Bearing Inc. Each bearing housing was preloaded using a
Dresser Industries torque limiting screw driver calibrated to i0.113 N-m
(i 1 in-lbf). This procedure permitted bearing clearance to be
eliminated since the impactive loading associated with bearing clearance
would have resulted in larger link deflections. Conversely, if the
bearing were subjected to large axial preloads, then the deflections
would be attenuated, hence the need for an accurate torque limiting
device.

The mechanism, which featured a flexible graphite-epoxy link
coupler, was bolted to a 35 in x 33.625 in x 1.5 in aluminum plate which
was housed inside a BMA, model TH-27S computer-controlled hygrothermal
chamber. The plate was pressed against the floor of the chamber by 4
long expander bolt arrangements in the four corners of the plate. These

members are evidenced in Figure 3.8 and 3.10. It is a low temperature

64
and high temperature humidity test chamber which has 27 cubic feet of
working volume. The TH-27S is designed to generate and control low and
high temperature environments in the range of —35 0F to +350 0F. For
temperature control, this chamber incorporates an all solid state
proportional controller which features i 0.5 0F temperature control
stability. Fast response Nicrom wire heaters are used to raise the
temperature and a single stage mechanical refrigeration system lowers the
temperature. A continuous water line feed humidity system provides a

range of 20% to 95% t 5% relative humidity.

 

Figure 3.10 Photograph of the Hygrothermal Chamber and Associated
Instrumentation

 

 

 

 

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l':.yc|1l 01111:! (‘1 11,—. 7
12 Volts
D. C.
Micro-
Processor
. Controller
‘rx . 7 for
Temperature
Strain 6 flunidity
Gauges
1
_r
Electra Corp.
MAX-MAC Specimen
Sensor f
EMA Environmental
Chamber Model TH-27S
Micromeasurementa Vavetek
Conditioner-Amplifier Dual Hi/Lo Filter ___
Alplifiet Model 2100 Model 432
! IX Trigger
A/D data
50 T°°th Acquisition System
Spur Gear Experimental Results ‘—_'
Micro VMS II/GPX 21

 

 

 

 

 

 

Electra Corp.
MAX-MAG
Pickup

 

 

 

 

 

Electra Corp.
MZA 101 MAXI-MAG
Amplifier

 

 

 

 

12 Volts
D. C.

 

 

 

 

 

 

HP 5314A
Universal
Counter

 

 

 

" Visual Monitoring of Crank Frequency (rpm)

 

 

Figure 3.11 Schematic Diagram of the Experimental Apparatus and
Associated Instrumentation

 

66

A schematic diagram of the this chamber and associated
instrumentations is shown as in Figure 3.11 and Figure 3.10 presents a
photograph of the environmental chamber and associated instrumentation.

A 0.75 HP adjustable speed permanent magnet d-c motor, model
2Z846B, manufactured by Dayton Electric Mfg. Co., was bolted in a
structure located in the outside of the environmental chamber and was
clearly evident in the left hand side of the photograph, Figure 3.11.

The motor powered the linkage through a 15.9 mm diameter driving shaft to
both 19.2 mm and 25.4 mm diameter shafts connected by both 5/8 inch bore
L070 and 1 inch bore L095 flexible couplings, which every flexible
coupling is the combination of two coupling bodies and one Buna-N spider,
manufactured by Lovejoy. Every Buna-N spider installs in center of
coupling bodies. The long-life material spiders is oil resistant and
highly resilient. Its serviceable temperature range is from -40 o to 212
0F and accommodates 1o angular and 0.015 in of parallel misalignment.

The reason of choosing flexible couplings instead of other rigid joints
is to eliminate large coupler link deflections due to possibly inaccurate
alignment for different diameter shafts. The 25.4 mm diameter shaft
supported by Fafnir cast-iron pillow block bearing, type LAK-l. A 125 mm
(5 in) stainless steel diameter flywheel was keyed to the shaft thereby
providing a large inertia to ensure a constant crank frequency, when
operating in unison with the motor’s speed controller.

The rated speed of the electric motor was 2500 rpm and this was
directly measured in revolutions per minute by a Hewlett Packard 5314A
universal counter which was activated by a digital-magnetic MAX-MAG
pickup, model 3040HTB, operating temperature range from -100 0F to +450

oF, manufactured by Electra Corporation. This pickup was sensing a sixty

67
tooth spur gear mounted on the drive shaft of the rig and converted
mechanical motion of spur gear into electrical control signals. The
large spur gear and the MAX-MAG pickup are clearly visible in Figure 3.8.
This arrangement provided visual feedback to the operator when the speed
controller of the motor was being adjusted while establishing a desired

speed.

3.3.3 Strain Gauges

Strain gauges, type WD-DY-250BG-350, manufactured by
Micromeasurements Group Inc., were mounted in a half-bridge configuration
at the mid-span location of the flexible coupler in order to measure the
transverse deflection. Multistranded, flat cables, type 330-FFE,
supplied by Micromeasurements were employed to avoid the high temperature
effects to the cable since these type cables can use from -452 to +500
oF. Protective coatings, type 3140 RTV, supplied by Micromeasurements,
were covered on strain gauges after installation since strain gauge
performance is easily degraded by the effects of high humidity in which
strain gauges have to operate for hygrothermoelastodynamic responses in
laboratory tests. The detail-design features of the coupler are
presented in Figure 3.12 and its geometrical and material properties are
presented in Table 3.1.

The measured strain was conditioned and amplified by a wheatstone
bridge conditioner/amplifier, model 2100, manufactured by
Micromeasurements Group Inc. The amplified signals then were fed to a
dual analog filter, model 432, manufactured by Wavetek Inc., in order to

eliminate spurious high frequency noise. An 80 Hz cutoff frequency was

employed during the laboratory tests.

68

Table 3.1 Geometrical and Material Properties of Flexible Coupler

 

Parameters and Properties of Flexible Coupler Fabricated

By Graphite Epoxy Composites

 

 

 

 

 

 

 

 

 

 

 

 

Lay-UPS and [(0/0/0) ] [(0/30/-30) ] [(0/45/-45) ]
. 2 s 2 s 2 s
Stack1ng Seq.
Length 23 cm 23 cm 23 cm
Width 1.875 cm 1.875 cm 1.875 cm
Thickness 1.52 mm 1.56 mm 1.56 mm
3 3 3
pfiber 1750 kg/m 1750 kg/m 1750 kg/m
Density 3 3 3
matrix 1200 kg/m 1200 kg/m 1200 kg/m
Number of
Prepreg tape 12 12 12
Fiber Volume
Fraction (%) 65 62 62
4 6 10 12 14

 

 

 

31.40 cm

 

 

 

I
VI}

Figure 3.12 The Detail-design Features of Coupler Link

 

69

In order to relate the strain gauge signal to the configuration of
the four-bar mechanism being studied, a transducer arrangement was
established. An Airpax type 70085-1010-048 zero velocity digital MAX-MAG
sensor was employed to sense the bolt head at the end of the crank when
the mechanism was in the conventional zero-degree crank angle position.
This hexagonal transducer is clearly visible in Figure 3.8.

This mechanism configuration signal and the output from the gauges
were either fed to a Tektronix model T912 dual trace storage oscilloscope
with a C—5C camera attachment for photographically recording on Type 667
Polaroid coaterless land film, or alternatively, the signals were fed to
an A/D connecter on a digital data acquisition system which has 15
channels of A/D converter, Model Micro VAX VMS II/GPX 21, manufactured by
Digital Equipment Corporation. Using software developed specifically for
digital data acquisition purposes, the flexural response signal was
recorded from the zero crank angle position through 360 degrees by firing
the trigger. In order to activate the trigger a two channel operational
amplifier producing the amplification factor of 20 was used to modify the
square-wave output from the Airpax pickup.

A calibration factor relating the amplified voltage from the
strain gauges output and the transverse deflection of the flexible
coupler link was obtained by supporting the link in the calibration-
fixture presented in Figure 3.13 and 3.14. In Figure 3.14, the coupler
link was supported on the top of two pins of the calibration-fixture
since the boundary conditions of coupler link are rollers in the four-bar

linkage configuration. Figure 3.15 presented the calibration system.

 

Figure 3.14 Photograph of the Calibration-fixture with Caupler
Link

 

Figure 3.15 Photograph of the Calibration System

 

Mid—span Deflection(mm)
O
00
l

 

 

 

' OO 7 r I I I r I I I
O 1 2 3 4 5

Amplifier Output(volt)

Figure 3.16 Calibration Relationship of the Amplified Voltage
and the Mid—span Deflection of Coupler

72
The midspan of the link specimen was subjected to a series of
monotonically increasing transverse deflections which were accurately
measured by the micrometer attachment on the central support structure.
The corresponding voltages from the strain gauges bonded to the link
specimen were also recorded in Figure 3.16 to complete the calibration

procedure.

3.3.4 Moisture Raising of Coupler Links

Since the experimental investigations were undertaken over a range
of moisture and temperature conditions, the flexible graphite-epoxy
coupler links were initially subjected to an environment of 90°C and -80
kPa in a vacuum oven, model 1410, manufactured by VWR Scientific Inc..
The weight of each coupler was determined by using a Satorious GmbH,
model H120 laboratory balance with a sensitivity of 0.001 gm until no
further change in the specimen weight was observed. This equilibrium
weight of each coupler under the conditions specified is hereafter
referred to as the dry weight of the specimen. Under this dry weight
coupler links with different stacking sequences were subjected to the
forced vibration conditions imposed by the four-bar mechanism operating
in the hygrothermal chamber at a temperature of 5°F and the dynamic
responses were recorded by the digital data acquisition system.

Upon completion of the 5°F tests the coupler links were then
placed in a Cole-Farmer laboratory oven, model 5015-15, and the moisture
content of the couplers increased. This oven was capable of providing an
hygroscopic environment of up to 80% RH at 70°C in order to accelerate

the moisture uptake of each coupler link.

73

Thermometer Psychrometer
Specimen
Rack .

 

 

' 'd'llllal i'lll’l. ! I'll,

 

 

 

 

 

 

 

 

 

 

 

2
11 11111111111111.
Water W\’ I I g
1’3“ [11111111111111 /
/
KL;\x\\\\\\\\\\\xux\\m\\\\\\u :;
' ' ¥'7<. § Q 'C
y § § § 3 S s /
Distilled A
Water j

 

 

 

   
  

Water Tank Cloths Heater

Cole-Farmer
Laboratory Oven
Model 5015-15

Figure 3.17 Specimen Preparation: Environmental Chamber

74

This apparatus is shown schematically in Figure 3.17, and its
photograph presents in Figure 3.18. In Figure 3.18, the right hand side
on the table is the water tank which provides water into the pan located
inside the chamber. The weight of the coupler specimens was monitored
periodically by employing the Satorious balance until a desired set of
moisture contents were obtained sequentially. The couplers were returned
to the four-bar mechanism located inside the BMA environmental chamber
and the dynamic responses were again measured under the hygrothermal

conditions of 75°F and 100 RH%, and 200°F with 100 RH%.

 

Figure 3.18 Photograph of Specimen Preparation Environmental
Chamber

75

3.4 Results and Discussions

Having described, in Section 3.3, the laboratory apparatus and
experimental procedures for investigating the hygrothermoelastodynamic
response of flexible linkages fabricated with polymeric composite
materials, attention now focuses on the comparison of the experimental
and theoretical results obtained from this investigation.

Figure 3.19 presents the transverse acceleration characteristics
of the system operating at 130 rpm and employing a rigid coupler for
accelerometer locations 2, 4, 8, 12, 14 indicated on Figure 3.12. Figure
3.20 presents the transverse acceleration characteristics of the system
by using kinematics fomulas under the same conditions in Figure 3.19.
Herein, Brfiel & Kjer piezoelectric accelerometer, type 4371, diameter 14
mm, was employed which gives domain data not point data and makes the
transverse acceleration characteristics of DF 8 a quite different in
Figures 3.19 and 3.20. From Figure 3.19 and 3.20, it is credit that
matches very well in most crank angle locations. Figure 3.21 shows the
correlation between the finite element results and the experimental
results of the transverse mid-span deflections for a [(0/45/-45)2]S
flexible coupler subjected to a temperature of 75 0F and a moisture
content of 0.5%, for example and operating at a crank frequency of 130
rpm and c.c.w. direction. Herein the Newmark method was employed to
solve the finite element based equations of motion. The time step size
of Newmark method was chosen as 2n/360/wf. The wf is the crank frequency

which is 130 rpm. Parameters a and 6 for numerical integration are 0.5

and 0.25 respectively.

76

 

 

 

 

 

 

4,444 .1

 

 

. —— DF 14l
A 10“ CA /"\ DF12
_, /J x \ ”A \
8 _ N \H ,0 \‘n ora
/ r\v\ / I"\ l _ __
U) ‘ // I, ‘ \\\ // ”I \\ \]\ DF 4
E ET ”I, \ \\ \\ /,1". "A ‘1 L‘— - DF 2
Q) «4 ”f -\ \\ \\ //,r I K \ \\
/
m 4 fl, , ,\\ If; ‘.\\\
\ .3 II, \ \\ \\ III'! ‘.‘ \ \\\
E 2— 1,1,: x 1 “1 m,’ '1. ‘, \1
V - 1].] x 1‘ ,1 /,’;’ 1 ,1 A- .
I ‘ l “ I . \ ’
C d I, || | \\ ‘x I, “ ‘ \\ I, \
O " i \ \\ I, \ :1 \ \\ I, "I
o— I. I II . ‘ \ ',
6 ‘2“: '. ‘ ‘3 ‘7 '- \ l x
E " ‘1 \\ lf~J/~v /I “ \ \ Ir.“ 1
“ v l‘ \- ~ I
Q .4 ‘1 \ ’I ‘. \ / I
O —6_ “1 I, \\ /I'/ “ ll, \\ //
<1 . ‘4 \\av/ / ‘« \\\ ,//
.. \ / \ /
A \ / \ /
K J v I
”—1 O T I I I I l I I I I l I r I
0 100 200 300 400 500 600 700

Crank Angle (degree)

Figure 3.19 The Experimental Transverse Acceleration

Characteristics of the Coupler for Four-bar
Mechanism Operating Speed 130 rpm

 

 

 

 

 

Acceleration (m/sec/sec)

 

 

l I I
0 100 200

j I I i T l I I I
300 400 1500 600

Cronk Angle (degree)

Figure 3.20 The Numerical Transverse Acceleration Characteristics

of the Coupler for Four-bar Mechanism Operating
Speed 130 rpm

 

 

I
700

77

Figure 3.22 to 3.24 show the correlation between the finite
element results and the experimental results of the transverse mid-span
deflections for a [(0/30/-3O)2]s flexible coupler subjected to a moisture
contents of 0.5%, and diverse temperatures of 8 oF, 75 0F, and 200 0F,
which operating at a crank frequency of 130 rpm and c.c.w. direction.

The good correlation between theory and practice provides credibility for
the proposed approach especially when the analysis focuses on predicting
the phase and frequency of the elastodynamic response.

Figure 3.25 shows the experimental results of the mid-span
transverse deflections of a [(0/3O/-3O)2]s flexible coupler subjected a
moisture content of 0.5%, and diverse temperatures of 8 0F, 75 0F, and
200 oF, which operating at a crank frequency of 130 rpm and c c.w.
direction. The experimental results demonstrate that at 200 0F, the
specimen has a maxium deflection 15% greater than at room temperature.
Furthermore, the specimen has a maximum deflection at 8 0F which is 5%
greater than when operating at room temperature. The results can be
anticipated from the design charts presented in Figures 3.1 - 3.6 which

show that the Young's moduli E and specific damping capacities ¢i are

i
not monotonically increased or decreased with respect to temperature and
moisture content. In addition, the hygrothermoelastodynamic response of
a flexible coupler link is dependent upon the global properties of the
four-bar mechanism, such as the effect of temperature upon the
characteristics of the bearings and the dependence of the kinematic
properties upon the thermal properties of the link materials. Figure
3.26 presents the experimental results of mid-span deflections of the

hygrothermoelastodynamic response of a [(O/O/O)2]s flexible coupler

subjected to various temperatures and 0.5% moisture content.

 

Deflection (mm)

 

 

 

 

 

 

4I - - Experiment
L -— Simulation
._3 , ' I fl, ”r
0 100 200 300

Crank Angle (degree)

Figure 3.21 The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/45/-45)2]s

Coupler Link: Mechanism Operating Speed 130 rpm

 

Deflection (mm)

 

 

-- Simulation
-- Experiment

"3 I I T T
O 1 00 200 300

Crank Angle (degree)

Figure 3.22 The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/3O/-30)2]S

 

 

 

 

 

Coupler Link: Temperature of 80 F

79

 

 

 

 

 

Deflection (mm)

 

 

 

 

 

——- Sanafion
_3 -- Experiment I

o 100 260

 

Crank Angle (degree)

Figure 3.23 The Comparison of the Theoretical and EXperimental
Mid-span Transverse Deflection of the [(0/3O/-3O)2]s

Coupler Link: Temperature of 750 F

 

Deflection (mm)

 

 

-- Simulation
-- Experknent

 

 

 

 

 

o 100 250 360
Crank Angle (degree)

Figure 3.24 The Comparison of the Theoretical and Experimental
Mid-span Transverse Deflection of the [(0/30/-3O)2]S

Coupler Link: Temperature of 2000 F

80

 

 

 

 

 

 

 

 

E .
i5, 0—
C
.9 '
8 ~1—
“a?
Q .
._2fl
—- T= 8°F
‘ --- T= 75°F
—— T=ZOO°F
"3 I I I I I I I
O 100 200 300

Crank Angle (degree)

Figure 3.25 Experimental Results for the Mid-span Transverse
Deflection of the [(0/3O/-3O)2]S Coupler Link:

Mechanism Operating Speed 130 rpm

 

Deflection (rnrn)

   

 

 

 

 

 

 

‘ —- T: 9°F
‘ --- T: 75 °F
‘ — T=198°F

_1.0 j I I I I I Tw
0 100 200 300

Crank Angle (degree)

Figure 3.26 Experimental Results for the Mid-span Transverse
Deflection of the [(O/O/O)2]S Coupler Link: Mechanism

Operating Speed 130 rpm

81

The stiffness of [(0/0/0)2]S specimen is greater than the [(0/30/-
30)2]s specimen and the dynamic response of the [(0/0/0)2]S specimen is
less sensitive to hygrothermal environments than the [(0/30/-30)2]S
specimen.

Figure 3.27 presents the experimental results which demonstrate
the effect of the layup of the laminae upon the elastodynamic response
profile at room temperature of 75 0F and relative humidity of 100%. The
results show that by experimentally changing some of the parameters, such
as fiber orientation and ply thickness for example, the elastodynamic
response can be gradually changed in order to reduce the mid-span
transverse deflections of the flexible coupler. This optimal design
philosophy can be systematically accomplished under hygrothermal
environmentals by implementing numerical modeling in order to be
acceptable in the efficiency and the cost compared with experimental

tests .

3.5 Summary of the Chapter

An experimental and theoretical investigation of the performance
of articulating mechanisms incorporating flexible composite links and
operating in hygrothermal environments has been presented. A generic
theoretical model has been presented in Chapter II for predicting the
elastodynamic response characteristics of flexible linkages operating
under hygrothermal environments and this theoretical model has applied to
study the behavior of four-bar linkages. The experimental results for
these mechanism systems clearly demonstrate the necessity for
incorporating hygrothermal effects into the design algorithms for

analysing machine members fabricated from polymeric composite materials.

Deflection (mm)

82

 

 

 

 

 

 

2
in r
a :9 1L~ -_a \
0— ‘\\ \\ /,- ’I‘ --
\ /" " I
\ / __'
~ \\ / f l ’
_1 —-1 \“‘ ’I’,’
__24 \ --”
. - — [<o/o/o>.Is
"“ [(0/3O/-3O)2]S
_ 3 -- [(9145/-4§).is I I I I I
O 100 200 300

Crank Angle (degree)

Figure 3.27 The Experimental Results of the Effects of

Layup of the Laminae upon the Elastodynamic
Response Profile

 

CHAPTER IV

SINGLE-OBJECTIVE OPTIMIZATION DESIGN METHODOLOGY

4.1 Introduction

This chapter presents the design strategy for single-objective
optimization design of articulating structures and robotic manipulators
fabricated from fiberous polymeric composite materials. For designing
structures fabricated from composite materials, there are two features
have to draw more attention on this issue. First of all, it is obvious
that the number of involving design parameters for structures fabricated
from composite materials is usually larger than those associated with the
fabrication of the commercial metals. Secondly, if optimality criteria
based on elastodynamic finite element analysis simulation, the high
computational cost have to be encountered during optimization process.
Therefore, efficiency in terms of the number of design variables,
objective function, and constraints evaluations is a significant
requirement for the selection of the optimization routine that will be
involved in the optimum design of articulating structures and robotic
arms.

Based on the mentioned features for structures fabricated from
composite materials, the generalized reduced gradient method is adopted
for optimal algorithm since this method has superior speed and robustness
under a wide variety of problem conditions. In the subsequent section,

the mathematical formulation for single-objective optimization design

83

84
will be discussed with the aid of the method of Lagrange Multipliers. In
Section 4.3, the generalized reduced gradient method is introduced for
optimal design. In Section 4.4, the numerical example of a flexible
single-link of the robotic system is illustrated through the methodology.
It should be mentioned that the finite element analysis model employs the
one derived in Chapter II but excludes the consideration of rotary

inertia and transverse shear effect of beam-like structures.

4.2 Formulation of Single-objective Optimization Design

In order to develop a methodology for the single-objective optimal
design of articulating mechanisms and robotic manipulators fabricated
from fiberous polymeric composite materials, the general optimization
design problems can be mathematically formulated in the following general
form including objective function, explicit constraints and implicit

comstraints.
Find X C R? such that
minimizes objective function
F(X) (4.1)

subject to implicit constraints which involve

J nonlinear inequality constraints,

gj(X)$0 ,j-1,...,J (4.2)

85

or K nonlinear equality constraints,
hk(X) =0 ,k=1,...,1< (4.3)

and explicit constraints which describe the upper and lower

acceptable I design variables,
U .

x. S x s x. , 1= 1,...,I (4.4)

To solve the above optimal design problem, the method of Lagrange
Multipliers which introduce one additional variable to the problem for
each constraint is typically employed. First of all, the inequality
constraints can be transformed to equality constraints by adding
nonnegative slack variables, wg, as

gJ. (X)+w§=0 ,j=1,...,J (4.5)

where the values of slack variables are yet unknown. Then, equations

(4.1) to (4.4) can be written by:
Find X such that
Minimize F(X) (4.6.1)
subject to

2
x - .x .-o ,'- ,...,J 4.6.2
Gj( .Y) gJ( ) + wJ J 1 ( )

86

hk(X) - 0 , k= 1,...,K (4.6.3)
x? S x. S x? , i= 1,...,I (4.6.4)
1 1 1
w. z 0 , '. l, ,J
J J
where YT - { w1, w2,..., wj} is the vector of slack variables.

Therefore, the Lagrange function L can be written as

J K
L(X,Y,A) = F(X) + E Aj Gj(X,Y) + E ”k hk(X) (4.7)
j=l k=l
where A - { Al, A2,..., AJ } and { p1, p2,..., #K } are vectors of

Lagrange Multipliers.

The stationary points of the Lagrange function can be calculated by

solving the following equations:

J K
11.4mm) - 35.00 +) Aj £100 +) ukflm = 0. i=1....,I
6x1 6x1 j=l 8xi k=1 6x1 (4.8)
gum“) - G.(X,Y) - g.(X) + w2 - o, j=1,...,J (4.9)
ax. J J j

J
gum“) hk(X) = o, k=1,...,1( (4.10)

8pk

87

QL (X,Y,A) - 2A.w. - 0, j=l,...,J (4.11)
aw. J J
J
In equation (4.9), the constraints gj(X) S 0 ,j=l,...,J, are still

satisfied, while equation (4.11) imply that either Aj = 0 or wj = 0. If

Aj- 0, it means that the constraint is inactive and hence it can be
ignored. The active and inactive constraints defined that those
constraints which are satisfied with equality sign, gj = 0, at the
optimum point are called the active constraints while those that are
satisfied with strict inequality sign, gj < 0, are defined as inactive
constraints. On the other hand, if wj - 0, it means that the constraint
is active, i.e. gj - 0,at the optimum point. Now consider the division

of the constraints into two subsets S1 and 82 while SlU 82 represents the

total set of constraints. Let the set T1 denote the indices of those

constraints which are active at the optimum point and let T represent

2

the indices of all the inactive constraints.

Therefore, for jETl, yj= 0, and for jET2, Aj- 0. Equations (4.8)

may be simplified by

J+K
6300+ A E<x>+)p ihim>=0,1=1,....1 (4.12)
ax j 3X k 6x
1 jeT i k=J+1 i

1

Similarly, equations (4.9) may be presented by

gj(X) = o, je T (4.13)

l

gJ(X) + w; - o, je T (4.14)

2

88

Equations (4.12) to (4.14) indicate I+R+(J+K-R) = I+J+K equations

including the I+J+K unknowns x1 (i-l,...,I), Aj(jeT1), wj(jET2), and
pk(k-l,...,k) while R represents the number of active constraints.
Assuming that the first R constraints of subset S1 are active,
equations (4.12) may be presented by
- if = 11§§1 +...+ 1R§§g + ”13E; +...+ pK 3E5 (4.15)
6x. 8x. 8x. 8x. 8x.
1 1 1 1 1
where i- 1, 2,..., I
Equations (4.15) may be expressed as
- VF = AlVgl + ... + ARVgR + pIVh1 + AKVhK (4.16)
where
6F/6x1 6g./axl ah/ax1
aF/ax2 egg/3x2 ah/ax2
VF - i ng= th- E (4.17)
aF/axI agj/axI ah/axI

The negative of the gradient of the objective function can then be
presented as a linear combination of the gradients of the active
constraints at the optimum point.

In the case of a minimization problem, the A in equation (4.16)

J
have to be positive by introducing the definition of a feasible
direction 8. This definition denotes that a vector 8 is called a

feasible direction from a point X,if at least a small step can be taken

89
along it without immediately leaving the feasible region. The details
are shown in reference [R4]
The conditions to be satisfied at a constrained optimum point, i ,

of the problem stated in equations (4.2) and (4.3) may be presented by

 

 

K
aF(x)+ E Ajagj(x) + E pkahk(x) = o , is 1.2....,1 <4-18>
ax1 jeTl 6X1 k=l 8x1

(4.19)

These are the necessary conditions to be satisfied for optimality of
a function F(X) with equality and inequality constraints. However, if
F(X), gj(X), and hk(X) are convex functions, equations (4.18) and (4.19)

are also sufficient conditions for optimality of the function F(X)

4.3 Generalized Reduced Gradient Method

In the previous Section the formulation of general single-objective
optimization design problems with both equality and inequality
constraints was presented, and also the necessary and sufficient
conditions for optimality of the single-objective function with
constraints were briefly discussed. There several methods to solve these
constrained optimal design problems such as linear programming method,
penalty method, generalized reduced gradient method, and sequential
quadratic programming method. It has been shown [S8] that the
generalized reduced gradient method has superior speed and robustness

under a wide variety of problem conditions. Furthermore, the optimal

90

structural design of a flexible robotic arm fabricated from fiberous
polymeric composite materials has the feature that the number of design
variables is bigger than one of constraints, it is desirable to adopt the
generalized reduced gradient method instead of choosing other methods.
Thus, the algorithm of the generalized reduced gradient method will be
introduced herein.

Most optimization algorithms require that an initial set of design
variables be specified. Beginning from this starting point, the design
is updated iteratively. The most common form of this iterative

procedures is given by

X = X + a S (4.20)

where k is the iteration number and S is a vector search direction in the

design space. The scalar quantity a* defines the step size that we wish
to move in direction S. The following task is first to decide the search
direction 8 by employing the generalized reduced gradient method.

The generalized reduced gradient method [A3,Gl] is an extension of
an earlier reduced gradient method which can solved equality-constrained
optimal design problems only [W3]. Then, by adding one slack variable to
each inequality constraint, the general form of optimal design problems
can be presented as equations (4.6.1) to (4.6.5) which become to be a
equality-constrained problem. The unknown X.vector therefore contains
the original I design variables and the J slack variables.

The basic concept of the generalized reduced grident method is to
recognize that for each equality constraint, one can define one dependent

design variable, thereby reducing the total number of independent design

91
variables [V1]. Based on the concept, the unknown X can be partitioned

into two vectors as follows:

where Z is a vector which has the I-K independent variables, and Y is a
vector which has the J+K dependent variables. Now, for convenience,
combined equations (4.6.2) and (4.6.3) under a single constraint

definition h(X), therefore

hJ.(X) = o ,j= 1,2,...,J+K (4.21.1)

and combine equations (4.6.4) and (4.6.5) to get

5 x. S x , i= l,2,...,I+J (4.21.2)

where the upper bounds associated with slack variables canbe set as very
large number.
The optimization problem of equations (4.6.1) to (4.6.5) can be

written by

Find Z,Y z c RI’K , Y c RJ+K

minimizes F(X) a F(Z,Y) (4.22.1)

92

subject to

h (X) = 0 , j= 1,2,...,J+K (4.22.2)

Ht"
(2‘.

. . , i= 1,2,...,I+J (4.22.3)
1 1

Differentiating equations (4.22.1) and (4.22 2) with respect to Z

and Y may be given the following form:

dF(X) = VZF(X)-dZ + VYF(X)-dY (4.23.1)
dhj(X) - Vzhj(X)-dZ + Vth(X)-dY , j=l,...,J+K (4.23.2)

where the subscripts Z and Y mean the gradient with respect to the
independent and dependent variables, respectively.

Assuming the equality constraints h(X) has feasible initial point
0

X , a linear approximation of h(X) can expressed by Taylor expansion

about point X9. Thus,

7101) = h(x°) + Vxh(X-X°) (4.24)

where h(Xp) = 0 since X9 is a feasible point. However, for the goal of
finding a optimum point such as h(X) = 0 or h(X) - 0, the condition of

Vxh(XEXP) = 0 has to be satisfied. That is

93

Vxh(XrX9) = [th](Y—Y°) + [Vzh](z-Z°) - o (4.25)

Equation (4.25) can be solved as the dependent variables Y'to be a

function of independent variables 2.
‘Y - y° - [thI’l [Vzh] (z-z°) (4.26)
Based on equation (4.26), equation (4.23.1) can be expressed by

T -1
dF(X)=VZF(X)odZ - VYF(X) [th] [thI-dz

={VZF(X)T - vYF(X)T[th]'1[th])-dz (4.27)
there
a - v F(X) - ([v hI‘1[v h])TV F(X) (4.28)
R Z Y z y
Equation (4.28) defines the generalized reduced gradient GR. The

generalized reduced gradient can now be used to determine a search

direction S = -GR to find out the optimum point step by step until GR -

0. However, it must be noticed that equation (4.20) is really a linear

approximation to the original nonlinear constrained problem, and so when

*
the constraints are evaluated for this a , they may not be precisely

zero. That is, the dhj(X) in equation (4.23.2) might not be zero. This

94

issue can be solved by using the Newton-Raphson approach to determine the

*

step size a

The essential features of this non-linear programming algorithm

employed are as following.

1)

2)

3)

4)

5)

6)

7)

8)

9)

Input the initial design variables.

Partition the design variables as dependent and independent
variables.

Search for a feasible starting point by using the constraint
function as a temporary objective function and minimize the
error.

Calculate the reduced gradient and search direction.

Check the convergence criterion; if this is satisfied, the
results are stored and the iterations stoped, otherwise the
iterations continue.

Search for the restricted line using the Newton-Raphson approach
to determine the step size.

Develop a better partition if nessary; change independent and
dependent variables.

Update the Hessian matrix using quasi-Newton procedures.

Check for the limit of the maximum number of iterations; if the
iterations are less than a user-prescribed limit, then the
control returns to step (4) above and the cycle is repeated;

otherwise iterations stop.

The algorithm for optimal design problems is schematically shown in

Figure 4.1.

 

 

95

START

l

Input Feasible

 

 

Starting Point X°

l

Iteration Counter
k = 0

 

 

 

 

 

 

Find Feasible Descent Direction

1: = [(+1
Sk

 

 

 

Find Optimum Step Size
in Line Search Along

xk+ask+x

. 1

New Design
xk + a Sk _, Xk+l

 

 

 

 

 

 

 

      
 

 

  

NO Does X

Converge ?

 

Figure 4.1 Algorithm for Generalized Reduced Gradient Method

 

96
If problem in the optimal structure design is a non-convex
programming one, then this optimization method will only converge to a
local optimum. It is therefore evident that several different update
design variables must be chosen in order that an approximate global

solution be determined.

4.4 Illustrative Example

Having discussed the algorithm of the generalized reduced grident
method, attention now focuses on the numerical example of a single-
objective optimal design. The task of designing a planar-motion single-
link flexible robotic arm with optimal requirement which is to be
fabricated from a symmetric composite laminate, is demonstrated herein as
the application of the proposed methodology. The geometrical and
material properties employed in this simulation is presented as Table
4.1.

The configuration of a single—link flexible robotic arm on the
planar motion is shown in Figure 4.2. The arm is modeled as a continuous
and uniform beam of length L whose moment of inertia about the root is

I with an additional lumped inertia I

B’ at the actuator assembly, i.e.

H
hub. The ox, is the fixed reference line and ox is the tangential line
to the beam's neutral axis at the hub. The total displacement of any
point along the beam's neutral axis at a distance x from the root of the
beam is the sum of the small elastic deflection w(x,t) and hub angle

9(t). The arm was manufactured with composite laminate which already

introduced in Figure 2.2.

97

Table 4.1 Geometrical and Material Properties of Robotic Arm

 

Parameters and Properties
of Graphite Epoxy Composites

 

 

 

Single-link length (L) 1.5m
Single-link width (b) 3.5cm
Payload (mt) 0.13 kg

 

Hub Moment of Inertia (IH)

0.004962 kg.m2

 

 

 

 

 

No. Lamina 6
Efiber - 220 GPa
Young's Modulus
E . = 4.27 GPa
matr1x
3
Density p . = 1750 kg/m
fibei = 1200 kg/m
matr1x
u . = 0.20
Poisson's Ratio f1ber
V . = 0.34
matr1x

 

 

Cl

 

Figure 4.2
Robotic Arm

 

Schematic Diagram of a Single-link Flexible

 

98

START

 

Control Parameters
(tip deflection, tracking error)

 

 

 

 

V

 

 

. Determination of Objective <
Function and Test Input Torque

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l 4, Initial Design
Design Variables
and Constraints
; Optimal Algorithm
V ,
1 Evaluation of
Objective Function
Select New Design Evaluation of Objective
Variables Function
1
Unsatisfactory Comparison of

 

   

 

OIIOI'O’OIIJOD"lllloll’Clotf'l'lo'tl"l"l"""’-”

\\\\\\\\\\\\\\\\\\\\\\\\\\\\

“““ Update Local Solutions

Optimization
Criterion

     
 
  

Satisfactory

 

 

 

 

 

i7

Global Solution

 

 

 

 

 

STOP

Figure 4.3 Methodology for Optimal Structure Design of a
Flexible Single-link robotic Arm Fabricated

from Composite Laminate

 

99

An optimization design algorithm is proposed as schematically shown
in Figure 4.3 for the flexible robotic arm fabricated from symmetric
fiberous composite laminates. In this algorithm, the objective function
can include the control parameters of robotic systems which feature the
characteristics of flexible links such as tip deflection, rise-time,
overshoot, settling time etc. The test input torques or forces also can
be determined from the control parameters which characterize the control
performances. Since a flexible single-link of robot is considered in
this study, the tip deflection of the arm was chosen as an objective
function without loss of generalities, and a square-wave torque which
characterizes the step input response such as rise-time, overshoot, and

settling-time was chosen as a test input shown in Figure 4.4.

2i)

 

i.0~

 

0i)——- 4

 

INPUT TORQUE (N — m)

 

 

-1.0— L.

—2L0

 

 

 

fl
a:
..l

.—

CLO C15 1.0 1.5 2J3 2J5 31) 3.5

TIME (see)

Figure 4.4 Test Input Torque for Evaluation of the
Objective Function

100
Thus, the optimization criterion is to minimize tip deflection of
the link subjected to constraints of the lower bound on the first-mode
natural frequency, static stiffness of the cross section, and upper and
lower bounds on the design variables for the composite laminate as
represented by the thickness of h , fiber orientation of 6 and fiber
k k
. th .
volume fract1on of ka for the k lam1nae.
Since the robotic arm is a symmetric laminate robotic arm and the
total laminae is 6 shown in Table 4.1, 3 laminaes are considered in

design processes. Every laminae has three design variables hk’ 0k, ka,

k- 1,2,3. This optimal design task may be formulated as follows:

Find the design variables {h1, h2, h3, 01, 02, 93, Vfl’ Vf2, Vf3}

such that

minimize the tip deflection of a sigle-link flexible robotic arm,

T
U . d
l. I ....ml .

subject to

(wl)op Z (wl)in
( xx)op 2 xx in
h 2 h 2 h . (4.29)

101

max k min

> >
meax — fk _ mein

A

U‘

V
Ii

(b).

op in

A
5
il

(L).

op in

where subscript (op) denotes optimal design, (in) represents initial

design and w b and L are the first-mode natural frequency, width and

1 ’

length of the robotic arm, respectively.

The upper bound and lower bound of design variables hk’ 6k, ka in

equation (4.29) for single-objective optimal design problem are

h . = 0.35 mm, h = 0.7 mm
min max

0 . = -90°, 9 = 90°
min max
fmin 3 0'4’ fmax = 7

The constraints (601)in and (Xxx)in in equation (4.29) are 22.0
rad/sec and 5.5125 Nom, respectively. Figure 4.5 shows the tip
deflection of the arm without rigid-body motion induced by torque in
Figure 4.4 for the initial design and optimal design whose manufacturing
specifications are presented in Table 4.2. From Figure 4.5, it is
evident that the tip deflection of robot arm for optimal design really

has a spectacular decrease associated with those of arbitrary initial

102

Figure 4.6, 4.7 show the tip positions of the flexible motion

selection.

initial design of robotic arm

and rigid body motion for optimal design,

From the Figure 4.6, it can be observed that by using

respectively.

optimal design method the tip position can reduce its deflection but it

can not completely be eliminated unless the control approach involved in

the design strategy.

 

 

Optimal Design

 

 

 

 

3.0

{—1
2i) 215

1.5

 

 

AEov cozomzmo n5

 

Time (sec)

Tip Deflection Between Initial and Optimal

Design Due to Test Input Torque

Figure 4.5

103

Table 4.2 Single-objective Optimization Results of Robotic Arm

 

 

 

 

 

 

 

 

 

 

 

Single-objective
Optimization Design
Quantity
Optimal Initial
l . h1 0.687 mm 0.500 mm
am1na
thickness h2 0.687 mm 0.500 mm
h3 0.667 mm 0.500 mm
61 0.94° 30.0°
fiber 02 3.78° 30.0°
orientationfl3 0.08° 30.0°
fiber Vf 0.650 0.500
1
volume Vf 0.603 0.500
2
fraction Vf 0.694 0.500
3
first mode
nature 55.66 rad/sec 26.85 rad/sec
frequency
static
stiffness 29.8453 N-m 5.4968 N-m
tip
deflection 2.975x10-3m-sec 7.735x10-2m-sec
(3 seconds integration) (3 seconds integration)
damping
-3 _3
capacity 1.8595x10 2.695x10
total
weight 0.3336 kg 0.2844 kg

 

 

 

 

 

 

104

 

 

3T)

 

 

 

 

 

Ac; conned ac.

 

15.
.U0 oz
|1l\ 10.
it... 2
11. ..
..u 5.
\ts: ”1.1
in .. .wi
-.,. m to...
I.
I“ w M]
1.. I. Y:
M w 0.
H II\ R 8'1.
..m .m.
n .1 e ...nNT.
.m- _r_K1
.m. m 5
nu l .
.-- Mann nu
m- 9.9-
... O O.
111111111 . _ fi-
1 . I
O
a a a _ a _ a _ 4 “U.
3 2 1| 0 1n
0 0. 0. 0. 0.
O O O O n_U

Time (sec)

Tip Position for Optimal Design

Figure 4.6

 

 

IIIIIIIIII

l.
'10..--
II I I-

VIII
Ill
Ill
0

I. |
II 0
I I D |
‘ .l
.l

I I
I. C 8 |
|||||

II-
I.--
.IDIIIII
|

all-
' 'lv' 0
9
8
'

Ir
"""""
' A

.....
'''''
II|-|
IIIIIIIIIIII
Ill
IDDS

""
---,

'
"'
D

-‘
I'||
.........
I---
'0'

llllll
"
----
D
I'D'
............
|nl|lu
--..

t
I.
'''''''
'

VIII
"""
Ill
‘
-
---
|||||||||||
II
----
||

"I
"'
"
In-
I.
I

I
""
'II"’II.I

''''''
''''
'
l

I
|
II
8"!-
888'

IIIl
IIIIII
\‘nl

 

 

 

1 5
Time (sec)

Tip Position for Initial Design

 

 

 

are common a:

 

1
”I
.m'
t
0...
Mr
no
«6.11.
O:
n.
:07
”w.
R;
limo
.AU
1 7
. 4
1 e
0 m
nznu g
.1
at w.
nu
_

105

4.5 Sulnary of the Chapter

A methodology for design flexible structures and robotic arms
fabricated with fiberous polymeric composite materials under single-
objective optimization criterion has been proposed. The mathematical
formulation of single-objective optimization design and the generalized
reduced gradient method have also been discussed. A flexible single-link
robot arm design by employing this methodology shows that the optimal
solutions really have a spectacular performance associated with the
performance of arbitrary initial selection. This proposed single-
objective optimal design methodology will be continued to employ in
Chapter V and Chapter VI for fuzzy multi-objective optimization
requirements problems and stochastic optimization design problems,

respectively.

CHAPTER V

A.FUZZY SET AND HULTI-OBJECTIVE OPTIMIZATION DESIGN METHODOLOGY

5.1 Introduction

A strategy for the application of fuzzy set methodologies to
multi-objective optimal design requirements of a composite-based robot
arm is presented in this chapter. The formulation of the fuzzy multi-
objective optimization design problem is accomplished by integrating
design variables such as, laminae thicknesses, fiber orientations, and
fiber volume fractions, which characterize the mass, damping and
stiffness properties of the structure, and by defining the fuzzy domains
corresponding to the objective functions and the constraints. The fuzzy
multi-objective optimization problem is solved by using conventional
single-objective function programming algorithm such as generalized
reduced gradient method. This fuzzy-set-based strategy is illustrated by
considering the design of a single-link flexible robotic arm fabricated
by laminated composites.

The governing equation of motion for the flexible-link which
features homogeneous viscoelastic material are developed in Chapter II.
In Section 5.2, the generalized formulation for multi-objective
optimization for engineering design based on fuzzy set theory is
presented. In Section 5.3, the computational procedure of fuzzy multi-

objective optimization design of a flexible single-link robotic arm is

106

107
represented. In Section 5,4, the numerical example of a flexible single-
link of the robotic system is illustrated through the methodology.

Conclusions are drawn in section 5.5.

5.2 Fuzzy Set and Hulti-objective Optimization Problem

In this section, the concepts of fuzzy set and the utilizations
of it in the fuzzy multi-objective optimization design will be discussed
herein. A Fuzzy set A contained in a universe U is a set in which
transition between membership and non-membership is gradual rather than
abrupt and it is measure by ”A' In traditional (crisp) set theory, a
membership function pA(a) can be defined for all elements a in U such

that

1 if aeA
”A(a) ' { 0 if a§A (5’1)

In Fuzzy set theory [21], #A is a continuous function defined on A, with
range [0,1]. Furthermore, pA(a) may be interpreted as the degree of
membership of a to A. Formally, A can be identified with a set of

ordered pairs
A — (a, p (a)} for aeA (5.2)

A typical example of a membership function defined on the

universe A - R is

108

r1 ifaSaL
U
pA(a) - 4 ;§Ei—é—E if aL S a S aU (5.3)
- a
k if aU S a

 

The above function, shown in Figure 5.1, represents that a is fully
inside A if it is below the threshold value aL and fully outside A if it
is above aU. If a is between aL and aU, it belongs to A with varying
degrees of membership. The concept of fuzzy set can be associated easily
with a performance criterion in design evaluation. For instance, one may
informally make reference to the ‘set of stiffer structures' or the 'set
of perfect designs' when stiffness and perfectness are desirable
attribute of a design. These loosely described sets are fuzzy sets in

the universe of all real-world designs.

1.0

 

Lfla)

 

Figure 5.1 Diagram of Membership Function For Fuzzy Set

109
A typical multi-objective optimization design problem can be

stated mathematically as follows:
n
Find x c R that

minimizes F -{f1(X), f2(X),---, fp(X)]
(5.4)

subject to gj (X) S bj' j - l, 2,---,m
where X = {X1, X2,---, Xn)T is the set of design variables, {f1, f2,---,
fp} are real valued, continuously differentiable function on Rn, fi is
individual performance criterion (objective function), and bj denotes the
upper bound value on the explicit or implicit constraint gj(X) with bjz
0. The functions gj(X) form a set of constraints which delimit the
feasible design space.

A fuzzy set may be associated with each measure of design
performance if a design problem needs several performance criteria [D2].

This leads to the definition of individual goals.

Definition: A fuzzy set A1 is an individual goal if

i. The membership function of A1 is a known function of the ith

performance measure and no other measure.

ii. pi does not decrease with improving performance.

110
In terms of the problem form of Eq. (5.4), ”i is a function of fj if and
only if i - j.

The advantage of introducing individual goal sets is that a
single performance criterion is much easier to measure than to evaluate
the overall performance. However, the fuzzy set theory in the evaluation
of overall performance is most useful. Fuzzy sets have been extensively
used to represent the imprecise information that characterizes the
selection of satisfactory designs. This leads to the definition of the

overall goal.

Definition: A fuzzy set A is an overall goal if

i. A depends only on and is fully characterized by the sets A1, A2,
00-, Ap, i.e., A - H (Al, A2, ---, AP) where H indicates aggregation
operator and can be a combination of intersection (non-cooperative),

union (cooperative), or averaging (trade-off) operations on its

arguments [D2].

ii. The membership function of A, ”A’ is a function of the individual

memberships #1, p2, ---, up, i.e.,

#A - h (#1. p2. °°'. #p)

h is aggregation operator and is a function from [0,1]P to [0,1]

that satisfies the axioms [D3]:

A1. h (l, 1, °--, 1) - l and h (0, 0, 00-, 0) - 0

111

A2. If pj(x1) > pj(x2) for all j - l, 2, 00°, g,

h (#1 (X1), °": Fg (X1) ) > h (#1 (X2):

°°'. (X ) )
#g 2
A3. h is continuous.

Axiom Al states that a design is fully acceptable if it is fully
acceptable by all individual criteria, otherwise it is fully unacceptable
if it is fully unacceptable by all individual criteria. A2 requires that
a design be considered inferior overall if it is not preferable according
to at least one criterion. A3 demands that small changes in one
criterion do not produce large changes in the global rating. Under these
axioms one can define the following binary operations, for example

[92,22,94].

Minimun: A symmetric mapping I from [0,1]2 to [0,1] satisfying A1-A3 is a

minimum if

I(#1.#2) S min<u1.#2) . for all #1.#2 [0.1]

Maximum: A symmetric mapping U from [0,1]2 to [0,1] satisfying Al-A3 is a

maximum if

U(p1,p2) Z maX(pl,p2) 3 for all pl’pz E [091]

112
Average: Asymmetric mapping M from [0,1]2 to [0,1] satisfying A1-A3 is an

average if

min(#1,#2) S M(#1.p2) S maX(#1.#2) . for all #1.u2 6 [0.1]

From the previous definitions it follows that membership in A is
associated with the individual measures of performance. The precise,
unfuzzy relation between ”A and overall performance (goal) depends on the
operators H and h. Therefore, the mechanism to produce the most

satisfactory solution for fuzzy multiobjectives optimization problem is:

Find x e X that
maximizes pA(X) - h(pl, p2, ..., pp) (5.5)
subject to x e X C Rn

Equation (5.5) is a form of fuzzy multiobjective optimization. Its
advantage over the traditional multiobjective optimization methods is
that H, h, and the fuzzy sets A1 are powerful to represent the subjective
intervention of the deSigner in the selection process. A diagram
illustrating the fuzzy multiobjective optimization problem is shown in

Figure 5.2.

5.3 Co-putational Strategy

In the previous section, the concept of a fuzzy set and the
generalized formulation of fuzzy multi-objective optimization problems
were discussed. Herein, the details for defining the overall

performance, establishing the fuzzy multiobjective optimization

113
formulation, and sketching the computational procedure of a flexible

single-link robotic arm will be represented.

 

 

 

 

 

 

 

Wle"‘—x‘\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

XI
5 x e )(
\_. fp _, ,.p _/
I EVALUATION I DECISION | OPTIMIZATION J
r l | I

Figure 5.2 Methodology for the Fuzzy Multi-Objective
Optimization Problem

114
The overall performance criteria desire to "minimize" the tip
deflection of the robotic arm, "minimize" the total weight of the robotic
arm, and "maximize" the damping capacity of the robot subjected to
implicit constraints of lower bound on the first mode natural frequency,
static stiffness of the cross section, and explicit constraints on the
design variables represented by thickness of hk’ fiber orientation of 0

k

and fiber volume fraction of ka for the kth lamina.
The formulation of above "multi-objective" optimization problem

may be written as follows:

Find design variables

T
X ' Ihl: h27...: hn: 61: 02,°°°: on» Vfl’ Vf2,°", vfn}
which minimizes
the tip deflection of the link of robotic arm,
t
f1(X) - I.| Utip (c)| dt (5.6)
the total weight of robotic arm,
n
k
f2(X) - b E: p (hk - hk-l) L (5.7)
k-l

and the damping capacity,

 

m
n k 2
f (X) - - EEill- l— 1&1 A1 (5 8)
3 (— )k 2n m A2 '
k=1 Q11 1E1 ”1 1

subject to

(0)1)01) Z (‘01)in

max k — min

(5.9)
omax Z 0k 2 0min
f max fk _ f min

where subscript (op) represents optimal design requirement, (in) denotes
the lower bound of implicit constraint. ml is the first mode natural
frequency.

Since the overall satisfaction (optimization) of the flexible
single-link robot requires consideration of all three individual
objective function, the decision in a fuzzy environment can, therefore,
be viewed as the intersection (non-cooperative) of individual objective
functions. That is, the design whose overall performance depends on
three measures (fl, f2, f3} and corresponding individual goals A1, A2,

A3, the overall goal of a fuzzy feasible set is

116

A - A1 n A2 n A3

and the membership function ”A of any set of design variables X to fuzzy

feasible set is given by

#A(X)-min { #fl(X). #f2(X). #f3(X) } (5 10)

where ”fi denote the membership functions of the ith objective function,
i - 1,2,3.

*
then, the optimum solution X is selected as

*
#A (X ) - max “A (X) (5.11)

T

*
whereX -{A,h1,h2,---,hn,01,02,---,0 V }

n’ fl'Vf2""'an
The essential features of the computational argorithm for solving
the fuzzy multiobjective optimization problem defined in Eq. (5.11) are

as follows [R8].

1). Searching for the optimal solutions of the individual objective
function fi(X) subject to the same constraints shown in Eq. (5.9)
using traditional optimization procedures (Herein, the generalized
reduced gradient algorithm is employed). An optimization algorithm
is proposed as schematically shown in Figure 5.3. Let the solutions

be Xi, i-1,2,3.

117

2). Depended on the solutions X of every individual objective function

i
L U
fi(X) to determine the threshold value fi (pfi = l) and fi (pfi -

0) for each of the objective functions.

L .
f1 - min fi (Xj) = fi (Xi)

U
fi - max fi (Xj) (5.12)

where i, j = 1,2,3

3). From the threshold value of f1 determined in equation (5.12), the
membership functions of the individual fuzzy objective function are

constructed as

 

r o, if 131 (X) > f?
"f1 (x) + f? L U
pfi(X)- « f0 L , 1f £1 < fi (X) 5 fi (5.13)
.. f.
i 1 L
L 1’ 1f fi (X) 5 fi

 

4). Finally, the solution of the fuzzy multiobjective optimization for

flexible single-link of robot can be solved as follows:

*
Find X that
*
maximizes g(X ) - A (5.14)
subject to

A 5 “£1 (X) , 1 = 1,2,3

 

 

118

START

],

Determination of Individual
Objective Function

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> Initial Design
Design Variables
and Constraints
:I Optimal Algorithm
=I (Generalized Reduced
l Gradient Method)
7
Evaluation of
Objective Function
Select New Design Evaluation of Objective
Variables Function
‘i
Unsatisfactory Comparison of A

 

 

 

 

Optimization
Criterion

     
 
  

Satisfactory

 

Update Local Solutions

 

 

 

 

17

Global Solution

Methodology for Optimal Structural Design

 

 

 

 

Figure 5.3

INPUT TORQUE (N - m)

119

 

Figure 5.4

2.0

 

Configuration of the Planar Motion of a
Single-Link Robotic Arm

 

1.0~

 

0.0-*—

_1'0...

 

 

—2.0 I

 

 

 

0.0

Figure 5.5

I ' I ' I ' I ' I ' I
0.5 1.0 1.5 2.0 2.5 3.0

TIME (sec)

Input Torque for Operating the Robotic Arm

 

3.5

120
Equation (5.14) can be solved by employing traditional single-
objective non-linear programming method (Herein, the generalized reduced

gradient algorithm is used again).

5.4 Numerical Example

The task of designing a flexible single-link robotic arm with
optimal multi-objective requirement which is to be fabricated from
symmetric composite laminate, is demonstrated herein as the application
of the proposed methodology. The geometrical and material properties
employed in the computaional simulation are presented in Table 5.1. The
configuration of the planar motion of a single-link robotic arm is shown
in Figure 5.4. The input torque to operate the robotic arm is presented
in Figure 5.5. Since the robotic arm is a symmetric laminate robotic arm
and the total lamina is 6 shown in Table 5.1, 3 laminas are considered in

, V i =

design process. Every lamina has three design variables hi’ 6. fi’

1
1,2,3.
The upper bound and lower bound of design variables hi’ 01, vfi

in equation (5.9) for the above optimal multi-objective design problem

are

h = 0.2 mm, h = 0.7 mm
min max
9 - -90°, 0 - 90°
min max
mein - 0.4, meax - 0.7

121

Table 5.1 Geometrical and Material Properties of Robotic Arm

 

Parameters and Properties
of Graphite Epoxy Composites

 

 

 

Single-link length (L) 1.5m
Single-link width (b) 3.5cm
Payload (mt) 0.13 kg

 

Hub Moment of Inertia (1H)

0.004962 kg.m2

 

 

 

 

 

No. Lamina 6
Efiber = 220 GPa
Young's Modulus
. = 4.27 GPa
matr1x
3
. pfiber ' 1750 kg/m
Den51ty 3
pmatrix - 1200 kg/m
u - 0.20
Poisson's Ratio fiber
- 0.34

 

V .
matr1x

 

 

122

T
The constraints (w1)in and (Kxx)in in equation (5.9) are 34.0

rad/sec and 5.5125 N.m, respectively. The individual objective functions
shown in equations (5.6), (5.7), and (5.8) are optimized in the presence
of the above constraints using generalized reduced gradient algorithm
[88] and the results are given in Table 5.2. The best and worst possible
value of each of the objective functions shown in equation (5.12), which
aid in the construction of membership function of individual objective
function in the fuzzy formulation, can be identified from these results.
Thus the fuzzy multi-objective optimization problem for single-

link robotic arm is:

*
find X -{A, h , h , h , 6 , 6 , 6 , V , ,
1 2 3 1 2 3 f1 f2 f3

*
that maximizes g(X ) - A (5.15)
subject to A S pfi(X), i = 1,2,3.

where the membership functions #fi are given by equation (5.13) and X -

{h1, h2, h3, 61, 62, 63, V } . The solutionn of equation

Vf,’ Vf2’ f,

(5.15) is optimized with generalized reduced gradient algorithm (GRG)
[88] again and is given in Table 5.2. It should be noted that the class
of equation (5.15) problem in the optimal multi-objective design of
composite laminates is a non-convex programming problem, hence this
optimization solution will only converge to a local minimum. It is
therefore necessary that several operations of GRG must be processed in

order that an approximation global solution be determined.

123

Table 5.2 Multi-objective Optimization Results of Robotic Arm

 

Traditional Single-objective

Fuzzy Multi-Objective
Optimization

 

 

 

 

 

 

 

 

deflection

,3
3.190x10 m-sec

_2
5.033x10 m-sec

,2
4.834x10 m-sec

Optimization Design Design
Quantity
Minimization Minimization Maximization Maximization
of of of of
Tip Deflection Total Weight Damping Capacity A
. 0.687 mm 0.449 mm 0.451 mm 0.694 mm
lamina 1
thickness h2 0.687 mm 0.423 mm 0.438 mm 0.519 mm
3 0.667 mm 0.456 mm 0.440 mm 0.423 mm
61 0.94° 4.34' 0.0° 2.8°
fiber 62 3.78° 12.93° 0.30° l.0°
orientation 63 0.08° -l.84' 0.56° 0.0
Vf 0.650 0.484 0.473 0.500
1
fiber
volume Vf O 603 0.652 0.670 0.688
2
fraction
Vf 0 694 0.700 0.693 O 700
3
A -- -- -- 0.6221
first mode
nature 55.66 rad/sec 34.00 rad/sec 35.68 rad/sec 44 92 rad/sec
frequency
static
stiffness 29.8453 N‘m 8.3836 Nom 8.1869 Nom 16.5084 N-m
tip

_2
1.479x10 m-sec

 

damping

capacity

 

total
weight

 

 

_3
1.8595x10

,3
3.195x10

_3
3.394x10

 

0.3336 kg

0.2144 kg

 

 

0.2245 kg

 

_3
2.365x10

 

 

0 2595 kg

 

 

124

The solution corresponds to the maximum level of degree of
membership A - 0.6221 with an associated weight of 0.2595, a tip
deflection of 1.4789 x 10-2 m sec, and a damping capacity 2.3648 x 10.3
Above solution is the best compromise solution that can be achieved in
the existing of the three conflicting objective functions. The part of
the iteration history of individual membership function is shown in
Figure 5.6. It represents that the contribution of individual objective

function to overall performance is different.

MEMBERSHIP FUNCTION

 

HISTORY OF ITERATION

Figure 5.6 Iteration History of Individual Membership
Function

125

5.5 Summary of the Chapter

A methodology for designing the flexible robot arms fabricated
with laminated composite under fuzzy multi-objective optimization
requirements has been proposed. The design parameters employed in this
methodology are the fiber and matrix properties, fiber volume fractions,
lamina thickness, and fiber orientations. An single-link robot arm
design demonstrates that fuzzy set can be effectively used to model
multi-objective optimization problems flexibly and transform the solving
process from complex multi-objective optimization problem into an simple
ordinary single-objective optimization problem to get best compromise
solution. The methodology presented here is generic in static and
dynamic optimal design and can be extended to multi-level optimization

problem.

CHAPTER‘VI

DESIGN-FOR-HANUFACTURING STRATEGIES
FOR INCORPORATING MANUFACTURING UNCERTAINTIES

IN THE ROBUST DESIGN OF COMPOSITE STRUCTURES

6.1 Introduction

Over the last three decades significant progress has been made in
the development of deterministic engineering optimization strategies
[V2,K2,V3,S4] based on the assumption of precise information pertaining
to design and manufacture variables. However, in real-world problems
random or uncertainty effects exist more or less in every circumstance.
For instance, in the design of mechanical systems, the actual dimension
of any machined part has to be taken as a random variable since the
dimension may lie anywhere within a permissible and specified tolerance
band. Considering only deterministic optimal design philosophy, there is
strong probability of the structures of machine and mechanism systems
being pregnant with danger or over-design. Therefore, it would be much
more rational to treat loading, strengths, and other design parameters
characterizing the system as random variables through the use of a
stochastic optimization philosophy.

A stochastic optimal design problem is one in which some or all of
the parameters are described by random variables rather than by
deterministic variables. The chance constrained programming technique,

which was originally developed by Charnes, et. a1. [C2], can be employed

126

127
to convert a stochastic optimal problem into an equivalent deterministic
problem. Rao [R9,R10] adopted the above technique for the optimum design
of structures with random parameters subjected to both static and dynamic
loads. Davidson et. a1. [D1] considered the weight optimization of non-
determinate structures subjected to deterministic dynamic load and
reliability constraints. Jozwiak [J3] discussed the minimum weight
design of structures when design parameters have random variation.

This chapter is focused on developing viable design for
manufacturing strategies for incorporating manufacturing uncertainties in
the robust design of composite structures. A methodology for the
application of chance constrained programming to stochastic design
optimization of composite-based structural and mechanical systems is
presented herein. The motivation for developing this methodology is the
uncertainties associated with the manufacture of fiberous polymeric
composite parts where, for example, errors in laying up prepreg tape can
adversely affect the global mechanical properties of a part fabricated by
the hard layup technique, and the task of accurately controlling the
fiber volume fraction of a filament-wound part is a challenging task.

The formulation of the stochastic optimization problem which addresses
the design and manufacture of fiberous polymeric parts is accomplished by
integrating the design and manufacture random variables such as, laminae
thicknesses, fiber orientations, and fiber volume fractions for example.
These random design and manufacture variables characterize the mass,
damping and stiffness properties of the structure. By adopting chance
constrained programming these random variables map the stochastic
programming problem into an equivalent deterministic nonlinear

programming problem. A sensitivity analysis of the dynamic response of

128
composite structures is discussed herein as a pre-requisite to obtaining
the optimal solution.

The modeling of structures fabricated with advanced composite
materials is followed the proposed model in Chapter II, and in Section
6.2 the general formulations of stochastic optimization problems are
presented. In Section 6.3 the sensitivity analysis of dynamic structures
is presented in order to develop the first-order derivative for the
stochastic optimal design. Then ideas are integrated in Section 6.4 in
order to demonstrate the significance of the proposed stochastic optimal
design philosophy by considering a composite cantilevered beam as an

illustrative example.

6.2 Formulation of the StoChastie Optimization Problem

Having established analysis model for the dynamic response of
structures fabricated from advanced composites in Chapter II, attention
now focuses on the formulation of the stochastic optimization problems.
For a structure with random variables, the optimum design problem can be

stated as follows:

Fine X, X C Rn such that

minimize F(X,X) (6.1)

subject to the constraints

i = 1,2,..., 2 (6.2)

129

- P [gj <§.3> z 01 2 p . (6.3)

where X represents vector of n random variables, X represents vector of m
deterministic decision variables, X denotes vector of mean values of n
random variables, 2 is the number of deterministic constraints, and k is
the number of probabilistic constraints.

The objective function F(X,X) is taken as
F<§.z) - CI? (3.3) + czaf (x.Y) (6.4)

where

E(Z.z) - Ef<§.z) - [m f <§.z) h(x)dx

-d3

and
2 - — 2

f(X,X) is a function of the random variables X and deterministic
variables X. E is expected value and h represents probability density
function. Equation (6.2) defines the deterministic constraints of the

problem, while Equation (6 3) defines the probability of obtaining

130
gj(X,I) greater or equal to zero must be greater than or equal to the
specified probability pj.
The stochastic optimization problem stated in equations (6.1),
(6.2) and (6.3) can be converted into an equivalent deterministic optimal
design problem by employing the chance constrainted programming method

[C2]. The method requires the constraint functions gj(X,X) to be first

expanded in a Taylor series about the mean value of g .

 

gj (51X) - gj (EIY) + E (XI-£1)
' i=3
l 82 g _
+ 2 § § _____1_ (xi - x )(xk - Xk) (6.5)

i- _1 k= 1 8x1 axk

+ higher order terms

If standard derivations ox of x1 are small, gj (X,Z) can be approximated
j
by a first-order approximation of equation (18):

6g
3 — ——J- -
gj(§.3) gj(§.Z) + E 8x1 (x x.) (6.6)

From equation (6.6), the mean value, gj, and the standard

deviation, 0 , of gj can be obtained as

61

Ej ' SJ (§:Y) (6.7)

131

 

and
n 2 1/2
a g. 2
o - E ———l' a (6.8)
g. 3 xi _ xi
J 1-1 2‘2
with the transformation of variables
8 ’ 5-
e - 40—4 (6.9)
81
and from standard normal distirbution
_t_2.
_l_ 2
P(-w<t<w) - e dt - 1 (6.10)
-00 ./ 2n

Equation (6.3) defining the constraint inequality can be written

as

f . d . _ P. 6.11
[m g (g3) g > ( )

Since fg (gj) is the probability density function of the random variable
J
gj whose range is assumed to be -m to m, the following relation can be

obtained.

i
2

[m -l- 6 d6 2 [m -l- 6 dt (6.12)
-(gj/agj)J 21 _¢31(PJ)J 2n

hfln

132
where ¢31 (Pj) is the value of the standard normal variate corresponding
to the probability Pj. Their relationship is presented in Figure 6.1 and

Table 6.1. Therefore
- “‘ S - ¢ (Pj) (6.13)

g], - 0g ¢J (Pj) 2 0 (6.14)

 

 

 

Figure 6.1 Standard Normal Distribution

133

Table 6.1 Standard Normal Distribution Table

 

 

 

fIPI, ¢(P)
0.0 0.398942 0.500000
0.1 0.396952 0.539828
0.2 0.391043 0.579260
0.3 0.381388 0.617912
0.4 0.368270 0.655422
0.5 0.352065 0.691463
0.6 0.333225 0.725747
0.7 0.312254 0.758036
0.8 0.289692 0.788145
0.9 0.266085 0.815940
1.0 0.241971 0.841345
1.2 0.194186 0.884930
1.4 0.149727 0.919243
1.6 0.110921 0.945201
1.8 0.078950 0.964070
2.0 0.053981 0.977250
2.2 0.035475 0.986097
2.4 0.022395 0.991802
2.6 0.013583 0.995339
2.8 0.007915 0.987445
3.0 0.004432 0.998650
3.5 0.000873 0.988767
4.0 0.000134 0.988968
4.5 0.000016 0.999996
5.0 0.000015 0.999997

 

 

 

 

134

Similarly, the f and a of equation (6.4) can be obtained as

f

(6.15)

 

Finally, the original stochastic optimal design problem can be expressed

as the equivalent deterministic optimal design problem.
Find X, X C Rn such that

minimize F = C f + C o

l 2 f

subject to the constraints

qi (g,g) 2 h 1=1,2,..., 2 (6.16)

i,

)‘ a: Z 0; j-l,2,..., k

-1
g.-¢ (P
J j

J
Equation (6.16) has been reformulated as a deterministic nonlinear
programming problem and can be solved by employing single-objective

optimal design methods such as the generalized reduced gradient

algorithm.

135

6.3 Design Sensitivity Analysis of Dynamic Prdblems

In the previous section, the general formulation of stochastic
optimization problems was discused. Since the first-order derivative of
either the objective function or the probabilistic constraints are
necessary for a stochastic optimization problem, the sensitivity for
every random variable must be calculated. The sensitivity analysis can
not only offer the first-order derivative for stochastic optimal design
but it also provides several other applications in structural mechanics
such as approximate analysis and reanalysis techniques, analytical model
improvement, and assessment of design trends. A review of the state of
the art in sensitivity is contained in survey papers [H3,A6,Cl].

There are various approaches to obtain the desired sensitivity
information for structure design problems. The variational approach
[H6,Y2], which is particularly well-suited to continuum structures,
borrows the concept of material derivative from continuum mechanics. The
finite difference method [H7,H8], direct or design space method [811],
and adjoint variable or state space method [H2] are quite popular for
structural models derived through finite elements. In finite difference
method, the original response and the modified response are computed by
repeating the analysis for a perturbation in design variable. The
approximate derivative can be calculated using an appropriate difference
formula. Analysts tend to avoid finite difference methods due to
potential accuracy problems and high computational cost.

Therefore, both direct and adjoint methods produce the desired
sensitivity information. The essential difference is in the
computational effort needed as a result of the order of the matrix

operations. If the number of response quantities is small compared to

136
the number of design variables, the adjoint method is usually best.
Otherwise, the direct method is applied [V4]. In this section, the
theory underlying the dynamic response sensitivity analysis using the
adjoint method and the natural frequency sensitivity analysis using the

direct method is presented herein.

6.3.1 Dynamic Response Sensitivities

For a structure with design variables including random and

deterministic variables, the optimization design problem can be stated as

follows.
Find 9 c R“ such that
_ T
minimize W (b) - I C (2,9) dt (6.17)

0

subject to the constraints

3 (g,§) s 0, i=1,2,..., 2 (6.18)

b. s b s S , j-1,2,..., n (6.19)

10>z+90>2+50>z-§e> <9”)

and initial conditions

137

g(0> = §°. 2(0) - 2° . g c R”

(6.21)
where 2 represents vector of n design variables, 2 represents vector of m
state variables, Ej and SJ are the lower and upper bound respectively of
the jth side constraint, and the function G is continuously
differentiable with respect to its parameters. M, 9 and 5 are square

matrices and present mass, viscous damping, and stiffness matrix,

respectively.

T
I (b+r6b) - I C(g (9 + 16b), 6 + 169) dt
0

Taking the variation of equation (6.17) with respect to the design

variables yields

d0 a- T as 82 ac db

55 69 - 5: - JO[ 3; 5: + 5E d:_] dt (6.22)
where

6% dz db d; db

5: - db -: - 58 6b , Slnce a; a 62

Then, equation (6.22) can written as

‘LEEI

- T 3G dZ
”-H "

6b -

~ 87

___.as
6% db + 82 J 62 dt (6.23)

I

O

138

In equation (6.23), the calculations of 3% and 3% are not difficult, but
62
the calculations of db are not straightforward and explicit. To solve

this problem, it is appropriate to implement the state equation shown in

equation (6.20). Now, equation (6.20) may be written as

It!
1N

+
In
1N

+
I7:

2 - g = 0 (6.24)

If a row vector 1T which has m arbitrary real numbers is introduced and
both sides of equation (6.24) are multiplied by IT, then upon integration

this yields

T
I 1T(§Z + 9% + E; - E) dt - 0 for all t>0 (6.25)
o

where 1 C Rm

let 2° . b°

+ 162

where 20 is a vector of the reference design. Equation (6.25) then

becomes

T a
I 2T [1 <2°+r62>z<2°+r62>
0

+ g(bo+r6b)2(b°+16b)+R(bo+16b)E(b°+r6b)

- g (§°+75§)] dt - 0

139

when r=0, the above equation becomes

Q

d2 d2
0 ~ 0 ~
3%1[{1T[M(b°db)d—Z + 903 >712 + 1503 >—db

~

T a» o d9.o s o 65
+1[d—E§(b)+-(i—E§(b)+EE-Z_(b)-3E]}613dt (6.26)

" o - o 0
Let -8 = vz<2 > + 9§<2 > + 82(2 > - E <6-27>

since equation (6.27) has the same form with equation (6.20) and this

definition can also easily establish the computational programming.

Equation (6.26), then becomes

T T 0 dz 0 dz 0 a;
[g 80>g+90>g+80>5-

~ ~ ~

] 59 dt = 0 (6.28)

Gala)
10" 171

Integrate equation (6.28) by parts to remove the following terms

d2 82

 

db and 5E terms, then
T T dZ
I Z M (b0 )d -“ dt
0 ~
. T T dZ
T g; T __
- 1 M db - I 1 M (b0 )d bdt
... o o ...

140

T 0 d2 d2
- 1 (T) M (b ) a; (T) - g(T) M (b0 )— b(T)
T "T o d;
+ I I M (b ) Hg dt
0 ~ ~
dz

Note that 1T(0) M (b0 )— b(0)-

.T 0 dz
and: (0)5(E)$ (0)30

~

Since initial conditions Z and Z are independent of b,

_‘L-Z-_<0>=o and _‘LE_<0>=0.
d2 db

T T d2
Similarly, I 7 C (b0 )d —— dt
0 ~

dZ T T dZ
-1T(T>c<b°5-> bm-chafl—dt
~ 0 0.

Therefore, equation (6.28) can be simplified as

d2 “T dZ T d2
1 T(T) M(bo )d g(T) - 1 (T) M(bo )— b(T) +1 (T) C(bo )— g(T)

~

T “ dZ
T -T T ~ T
+J{IIM-1£+1§]g-z
o ..

Q1 o:
HT WU

} 6b dt - 0 (6.29)

If 7 for all t > 0

I:
u
34
IO
+
'4
I 7‘:
I

141
and IT (T) = o, iT (T) = 0

Then equation (6.29) becomes

T ac d; T aR
I [ 52 53 ’ ~ ab
0 .... ... ...

since 6b is not equal to zero, therefore,

ac d; T 33
- (6.30)

@@‘1®
Put the relation of equation (6.30) back to equation (6.23), the

derivation of objective function i with respect to design variables can

be obtained as
- T 6R
dW T ~ BC
:1. I [1 $9.73,] .1. (6.31)

6.3.2 Natural Frequency Sensitivities

Since first mode natural frequency was employed as a constraint of
the stochastic optimal design problem, herein the direct method is
employed to obtain the natural frequency sensitivities.

The free vibration response of the structure is governed by an

eigenvalue problem:

K b A. = A.M b . 6.32
()3 J_<__>AJ ( >

142
where M is the symmetric nonsingular structural mass matrix of order j, M
is the symmetric nonsingular structural stiffness matrix of order j, M is

the design variable vector of order i, A is the eigenvalue related to

J

the jth natural frequency, and A is the associated eigenvector. The

J
order j corresponds to the degrees of freedom of the model and the order
i refers to the number of design variables.

Assuming that the mode shapes are normalized with respect to the

mass matrix M, then

A.MA. = 1 (6.33)
J' J
Differentiate equation (6.32) with respect to design variable bi’

thus

65 A.+ x aAj - 321 MA. - A 6” A. - A.M aAj - 0 (6.34)

ab. 3 ab. 66. ' 3 J66. J 1' ab
1 1 1 1

After premultiplying by A§ and substituting equation (6.33), the

above equation can be simplified to:

 

A§[?g—'Aja—b}—]AJ
321 - ab: abi (6.35)
abi AT M A.
J“ J
Therefore,
6w 1 6A

__1 - ______.__i (6.36)

143

6.4 Nu-erical Example

In the previous two sections, the formulation of stochastic
optimization problems and design sensitivity analysis were discussed.
The task of designing a cantilevered beam with stochastic optimal
requirement which is to be fabricated from a symmetric composite
laminate, is demonstrated herein as the application of the proposed
methodology. The geometrical and material properties employed in the

computational simulation are presented in Table 6.2.

Table 6.2 Geometrical and Material Properties of
Beam-like Structure

 

Parameters and Properties of Bea-rlike Structures
Fabricated by Graphite Epoxy Couposites

 

 

 

Length (L) 0.4 m
Width (b) 2.54 cm
Initial Diplacement of 6.0 mm

Tip Position

Number of Laminae 6

 

E er - 220 GPa
Young's Modulus
E . - 4.27 GPa

 

3
. pfiber - 1750 kg/m
Den31ty

- 1200 kg/m3
- 0.20

pmatrix

u .
Poisson's Ratio fiber

0.34

 

 

 

V -
matrix

144
The configuration of the transient vibration of a cantilevered
beam is shown in Figure 6.2. Since the cantilevered beam is a symmetric
composite laminate and the total laminae is 6 shown in Table 6.2, 3 plies
. . . th .
are con51dered as in de51gn process. Therefore, every the k laminae

has three design variables h 0

k’ k’ ka, where k 18 1,2,3.

//

/ .7?

 

Figure 6.2 Transient Response of a Cantilevered Beam

The formulation of the stochastic optimization problem for a
cantilevered beam may be written as follows:

Determine the random variables in the vector X

T

x-(hhhooo f3}

1’ 2’ 3’ l’ 2’ 3’ Vf1 vf2’

which minimizes the expected value of the lateral tip deflection of

a cantilevered beam

t
Fog) = I

0

 

Utip (t) l dt (6.37)

145
subject to the constraints
P [(w

l)op ‘ (”1) in Z 0 1 2 p1

P [(Dll) op - (D11) in 2 0 ] 2 p2 (6.38)

P [(M) o

p (M) in s O ] 2 p3

min k max

w

min max

mein S ka 5 meax
where subscript (op) represents the optimal design requirement, (in)
denotes the lower or upper bound of probability constraints, ml, is the

first mode natural frequency, the effective stiffness of the

”11’
laminate in the longitudinal direction, is defined in Chapter II, M, the
mass of cantilevered beam, is the product of L and E where defined in

Chapter II. It should be noted that w and M are also random

1’ 611’
variables, since a function of several random variables is also a random
variable.

Equations (6.37) and (6.38) then can be converted into the
equivalent deterministic optimal problem by employing the chance

constrainted programming method and can be expressed as follows:

Determine the mean values of the random variables in the vector i

146

— T
1' 2' 3' 1' 2' 3’ fl’ sz' Vf3}

:xl
J.
:1
:1
:1
an
a"
an
<1

which minimizes the mean value of the tip deflection of a cantilevered

 

beam
_ t
f (g) - I Utip (t) I dt (6.39)
0
subject to the constraints
— -l ,—-——
w
l
= -1 _
(Dll)op ' ¢ (pz) 4 03 ' (D11)in Z 0 (6'40)
D1
- -1 _
(mop + «6 (p3) ./ 03 - (M)in s o
M
h . s E s h
min k max
0min 5 0k S omax
mein S ka S meax
where wl, D11, M, hk’ 0k, and ka are the mean values of wl, D11 and M11,

hk’ 0k, and ka respectively. Note that it was assumed that random

variables are statistically independent and follow a normal distribution.

147

The upper bound and lower bound of random variables hk’ 0k, ka in

equation (6.40) are

h . = 0.3 mm, h = 0.7 mm
m1n max

0 - -90° , 0 = 90°
min max

mein - 0'4 ’ meax- 0'7

The standard deviations of hk’ 0k, v are chosen as 5 percent of

fk
their own values for the numerical demonstration. The constraints

(w and (M)in in equation (6.38) are 120.0 rad/sec, 6.5 N.m,

l)in’ (511)1n’
and 0.0625 kg, respectively. The objective function and corresponding
constraints shown in equation (6.38) and (6.40) are optimized by using
generalized reduced gradient algorithm [88] and the results are given in
Table 6.3 under different probabilities. It should be noted equations
(6.39) and (6.40) define a class of non-convex programming problem in the
optimal design of composite laminates. The optimal solution will,
therefore, only converge to a local minimum. It is necessary, therefore,
to try several GRG operations in order that an appropriate global
solution be obtained. Based on Table 6.3, the utilization of the
stochastic optimal design methodology in dynamic structures are already
automatically involved safety factor on it, since that the performance of
requirement will decrease if the specified constraints have to be

satisfied with a higher probability. Note that the condition P = 0.5 is

equal to the case of deterministic optimal design problem.

 

148

Table 6.3 Stochastic Optimization Results of a Cantilevered Beam

 

 

 

 

 

 

 

 

 

Quantity Probabilityi
P=O.5 P=0.95 P=O.9999997
Laminae Bl 0.651 mm 0.586 mm 0.556 mm
E2 0.651 mm 0.568 mm 0.574 mm
Thickness h3 0.651 mm 0.552 mm 0.541 mm
Fiber 51 -18.5° 10.5° 14.2°
52 - 1.2° 3.7° -6 2°
Orientation 83 - 5.2° ~0.8° 8.9°
Fiber GfI 0.610 0.649 0.453
Volume sz 0.650 0.644 0.440
Fraction va 0.669 0.631 0.438
ObJeCtive f 4.16 E-3 4.344 E-3 4.437 E-3
Function
(m-sec)
First Mode 51
Nature 238.68 205.62 172.47
Frequency
(rad/sec)
Effective 511
18.57 12.03 7.665
Stiffness
(Nam)
Total M
0.06295 0.0550 0.0498

Weight (kg)

 

 

 

 

 

149
Figure 6.3 shows the dynamic responses of the cantilevered beam
under P - 0.5 and P - 0.9999997. Table 6.4 and Table 6.5 present the
first-order derivatives of the objective function and corresponding
constraints by using the sensitivity analysis method with P - 0.5 and P =
0.9999997, respectively. From Table 6.4 and Table 6.5, it can be
observed that the design parameters of laminae thickness h have the

k

largest effect on the objective function and corresponding constraints.

6.5 Summary of the Chapter

A design-for-manufacturing methodology for incorporating
manufacturing uncertainties in the design of structures fabricated with
fiberous laminated composite has been proposed. The random parameters
employed in this methodology for a laminate are laminae thickness, fiber
orientation, and fiber volume fraction. The objective function and
corresponding probabilistic constraints were transformed into equivalent
deterministic quantities using Taylor's series expansion and probability
principles. The method for undertaking a sensitivity analysis of the
dynamic response of composite structures can not only provide information
on the first-order derivative for a stochastic optimization design
process but it also provides the trend information of design variables
with respect to the objective function during the re-optimization design
process. The illustrative transient response studies of cantilevered
beam demonstrate the significance of optimizing composites structures
with random parameters. The extensive utilization of composite-based
structures will mandate that this robust stochastic optimal design
methodology will assume a critical role in the integrated design and

manufacture of these parts.

150

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.006
0.004-
5 0.002_
\../
g .
1:, 0.000-
O
2 -4
E
O —0.002-
-0.004~
"0.005 I I I I I I I f fi’ 1' x I I r
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (sec)
0.006
0.004—
\E/ 0.002-
C -
.3 0.000~
o
2 —(
“<13 _
O —0.002
-—0.004~
‘0005 ' 144* I ' I ' I I I r* 1 Vi
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Time (sec)
Figure 6.3 The Comparison of Dynamic Responses for P-0.5 and

P-O. 9999997

 

151

Table 6.4 First Order Derivatives of Objective Function and
Constraints with respect to Random Variables

 

 

 

 

when P=0.5
P - 0.5
Quantity Q: 331 6511 3%
6x 6x 6x
El ~0.70675 1.196 85 2.798 84 3.181 81
Hz -0.72742 1.231 E5 2.868 E4 3.231 81
H3 -o.7317o 1.238 85 2.886 E4 3.255 81
when x is: 71 -2.3374 8-5 3.967 0.6173 0.0
72 -1.311 E-5 2.226 0.3463 0.0
73 -1.592 8-4 2.702 81 4.204 0.0
;fl 5.523 8-5 -9.482 0.8664 7.94 E-3
sz -1.832 E-4 3.100 81 7.1653 7.165
Gf3 -6.399 E-4 1.086 81 1.923 81 7.94 8-3

 

 

 

 

 

 

 

152

Table 6.5 First Order Derivatives of Objective Function and

Constraints with respect to Random Variables
when P=0.9999997

 

 

 

P=0.9999997

Quantity Q: 331 i§11 3%
6x 8x 6x

E1 0.1252 1.030 85 1.376 84 2.891 81

£2 0.1267 1.043 85 1.385 84 2.975 81

E3 0.1245 1.024 85 1.367 84 2.972 81
71 -3.164 E-6 -2.697 -O.2387 0.0
when x is: 72 1.089 E-5 9.278 0.8248 0.0
73 -3.853 E-S -3.285 81 -0.2919 0.0

Gfl -5.185 E-6 -5.271 0.5754 6.778 E-3

sz 4.806 8-5 4.009 81 4.641 6.898 E-3

Gf3 1.394 E-4 1.180 82 1.15 81 6.592 E-3

 

 

 

 

 

 

CHAPTER.VII

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions of the Thesis

Five schemes for articulating mechanisms and robotic manipulators
fabricated from fiberous polymeric composite materials were developed in
this thesis; the analytical model incorporating micromechanical model of
a laminae and macromechanical model of a laminate, the theoretical and
experimental investigation of the hygrothermoelastodynamic resopnse of
linkage mechanisms, the single-objective optimal design, the multi-
objective optimal design employed fuzzy sets, and the stochastic
optimization design.

A theoretical and experimental study was undertaken to investigate
the performance of articulating linkage mechanisms incorporating
symmetric composite laminated links operating in various hygrothermal
environments. The theoretical investigation is based upon the finite
element method developed in Chapter II to predict the
hygrothermoelastodynamic response of four-bar linkage mechanisms. In a
complementary investigation, an experimental program focuses on the
elastodynamic response of a four-bar mechanism with composite laminated
coupler links fabricated from various configurations of Hercules
AS4/3501-6 prepreg and subjected to a variety of hygrothermal conditions.
The experimental results for these four-bar mechanism clearly demonstrate

the necessity for incorporating hygrothermal effects into the design

153

154
algorithms for analysing machine members fabricated from polymeric
composite materials.

The design strategy for single-objective optimization design of
flexible structures and robotic manipulators fabricated from fiberous
composite materials was developed. The generalized reduced gradient
algorithm was introduced for the optimization design endeavor. For this
problem, the transverse deflection of dynamic performance was the
objective function with explicit design variables such as laminae
thicknesses, fiber orientations, and fiber volume fractions and implicit
constraints such as static stiffness of the structure and first mode
natural frequency. A flexible single-link robot arm design by employing
this methodology shows that the optimal solutions really have a
spectacular performance associated with the performance of arbitrary
initial selection.

A strategy for the application of fuzzy set methodologies to multi-
objective optimal design requirements of a composite-based robot arm is
developed for choosing a compromise solution. The formulation of the
fuzzy multi-objective optimization design problem is accomplished by
integrating design variables such as, laminae thicknesses, fiber
orientations, and fiber volume fractions, which characterize the mass,
damping and stiffness properties of the structure, and by defining the
fuzzy domains corresponding to the objective functions and the
constraints. The systematic procedure for aggregating operations based
on fuzzy set theory was also discussed. The method of solving a fuzzy
multi-objective optimization problem employing conventional single-
objective function programming technique such as generalized reduced

gradient method was presented. This fuzzy-set-based strategy is

155
illustrated by considering the design of a single-link flexible robotic
arm fabricated by laminated composites.

Design for manufacturing strategies for incorporating manufacturing
uncertainties in the robust design of composite structures were
developed. A methodology for the application of chance constrained
programming to stochastic design optimization of structures with dynamic
performance as the objrctive function and fabricated from composite
materials was presented. The formulation of the stochastic optimization
problem which addresses the design and manufacture of fiberous polymeric
parts is accomplished by integrating the design and manufacture random
variables such as, laminae thicknesses, fiber orientations, and fiber
volume fractions for example. These random design and manufacture
variables characterize the mass, damping and stiffness properties of the
structure. By adopting chance constrained programming these random
variables map the stochastic programming problem into an equivalent
deterministic nonlinear programming problem. A sensitivity analysis of
the dynamic response of composite structures was presented as a pre-
requisite to obtaining the optimal solution. This formulation is then
employed to study the transient response of cantilevered beams in order
to demonstrate the significance of incorporating random parameters into
traditional deterministic optimization design methodologies.

The author believes that the research prosecuted in this thesis will
stimulate the evolution for the integrate design and manufacturing of
composite-based structures in aerospace and manufacturing industries.
Moreover, this complementary theoretical, computational investigation on
the optimum design methodologies for the flexible structures and robot

arms will furnish fundamental and very crucial guidelines as well as

156
valuable information for the development of the much more complex
structures and the new generation robotic system fabricated from advanced

composite materials.

7.2 Recommendations for Future Wbrk

7.2.1 The experimental scattered data for E1, E2, 612, $1, $2, and $12
provided the bais for determining the numerical values of the
coefficients for the surface-fit functions may need to take more in order

to establish more accuracy surface-fit functions.

7.2.2 In this thesis the dynamic performance was focused on linear
vibration for optimal design, the nonlinear effects may be involved in

the design precess.

7.2.3 The approach of the structures fabricated from advanced composite
materials subjected to multi-objective requirements with random design

variables is an area to be studied.

7.2.4 Experimental verification of the accuracy of the optimum design
methodologies with composites structures is an area requiring
investigation. This issue has to be concerned especially in complicated
structures such as three-dimensional composites structrue since the
accuracy of the optimal design approach is based on the analytical

modeling.

7.2.5 With the increasing utilization of smart materials such as

electro-rheological fluids, piezoelectric materials, and shape memory

157
alloys, how to incorporate the self-inspection and inherent adaptive
capabilities of smart materials to the structrue fabricated with the higt
stiffness-to-weight ratio fiberous composite materials for optimal design

is a challenging study.

7.2.6 The philisophy of optimal design can applied to find out the best
laminar flow field in a wind tunnel without turbulence by using shape
optimal design techniques. Obviously, this class of problems has
features of large number of design variables and higher computational

time.

APPENDIX

APPENDIX.A

The Modified Rule-ofeflixtures Approach

Based on work reported in reference [T4], the following expressions
are sufficient for the calculation of the Young's moduli E1, E2, and the
shear modulus G12 for the micromechanical model of a fiberous polymeric
composite laminae. Assuming that the fiberous polymeric composite
materials are transversely isotropic and the materials are free of voids,
the longitudinal Young's mudulus, E1’ of a unidirectional laminae is
defined by

E1 = Vf Ef + Vm Em (A 1)

where Ef and Em are the elastic moduli of the fiber and matrix

respectively, V and VIII are the fiber and matrix volume fractions and are

f

relative by

V - 1. - V (A.2)

The density of the laminae is defined by

p - pf Vf + pm Vm (A.3)

158

159
where pf and pm are the density of the fiber and matrix, respectively.

The longitudinal Poisson's ratio, u is presented by

1’

V - V v + V v (A.4)
m m

where uf and um are the Poisson's ratios of fiber and matrix,

respectively. The shear modulus of the matrix, Gm’ is defined by

0 _ (A.5)

 

Therefore, the bulk modulus of the matrix is

 

Gm
k - (A.6)
m l - 2 V
The transverse plane strain bulk modulus of the fiber, kf2’ is
defined by
E
kfz - f2 (A.7)
2 ( l - uf)

where Ef2 is the transverse Young's modulus of the fiber. The stress
partitioning parameter "k’ which measures the relative magnitude of the
average stress in the fiber for conditions of plane strain hydrostotic

uniaxial loading, is defined by

160

] (A.8)

The transverse plane strain bulk modulus of the laminae, k2, is

defined by

'7
1 - 1 [ f + k v 1 (4.9)
k (Vf+”kvm) kf2 km

 

The stress partitioning parameter in the transverse shear "C’ is

defined by

l Gm

"0 - [ 1 [ 3- 4um+

___—___ 1 (4.10)
4(l-vm) 0f2

The transverse shear modulus of the laminae, G is defined by

2,

1 1 Vf Vm
- [ _______ ] [ ____ + 06
G Vf+”cvm Gf2 Gm

] (A.ll)

 

 

Therefore, the transverse Young's modulus, E2, of a unidirectional
laminae is defined by
4k G

E - 2 2 (A.l2)

k + nG

Where

161

 

4 k2 vxz
E1
The shear modulus of fiber, Gf, is denoted by
E
0f - f (A.14)
2(1+ Vf)

 

 

THe stress partitioning parameter in shear. 012, is defined by
Gm
n12 - 1/2 [ l + ] (A.15)
Cf
Therefore, the shear modulus of the laminae, 612, may be defined as
a function of the previous expressions. thus
V V
1 l f
'[___][__+’712m] (A.l6)
G V G G

12 f+"12Vm f m

BIBLIOGRAPHY

[A1]

[A2]

[A3]

[A4]

[A5]

[A6]

[A7]

[A8]

[A9]

BIBLIOGRAPHY

Arora, J. S., and Haug, E. J., "Methods of Design Sensitivity
Analysis in Structural Optimization," AIAA Journal, Vol. 17, No. 9,
1979, pp. 970-974.

Arora, J. S. and Belegundu, A. D., "Structural Optimization by
Mathematical Programming Methods," AIAA Journal, Vol. 22, No. 6,
1984, pp. 854-856.

Abadie, J. and Carpentier, J., "Generalization of the Wolfe Reduced
Gradient Method to the Case of Nonlinear Constraints,"
Optimization, R. Fletcher (ed.), Academic Press, London, 1969, pp.
37-48.

Adali, S., "Optimization of Fiber Reinforced Composite Laminates
Subject to Fatigue Loading," Composite Structures 3. Proceedings of
Third International Conference on Composite Structures. 9-11.
September 1985, Paisley College of Technology, Scotland, pp. 43-55.

Adams, R. D. and Bacon, D. G. C., "Effect of Fibre Orientation and
Laminate Geometry on the Dynamic Properties of CFRP," Journal of
Composite Materials, Vol. 7, pp. 402-428, 1973.

Adelman, H.M., and Haftka, R.T., "Sensitivity Analysis of Discrete
Structural Systmes," AIAA Journal, Vol. 24, No. 5, 1986, pp. 823-
832.

Ashley, H., "On Making Things the Best - Aeronautical Uses of
Optimization," Journal of Aircraft, Vol. 19, No.1, 1982, pp. 5-28.

Adali, S., "Multiobjective Design of an Antisymmetric Angle-ply
Laminate by Nonlinear Programming," Transactions of the ASME, Vol.
105, 1983, pp. 214-219.

Adali, S. and Nissen, H., "Optimization of Antisymmetric Angle-ply
Strip for Maximum Strength Including Hygrothermal Effects,"
Transactions of the ASME, Vol. 109, 1987, pp. 392-397.

[A10]Ang, A. H. S.(Chairman), Task Committee on Structural Safety,

[131}

"Structural Safety - A Literature Review," JListructural Division,
Proc. ASCE, Vol. 98, 1972, pp. 845-854.

Banichuk, N. V., and Kobelev, V. V., "Optimizing the Effective
Characteristics of Granular Composites in Component Design
Problems," Mechanics of Composite Materials, Vol. 17, pp 175-180,
1981,

162

[82]

[B3]

[B4]

[B5]

[136]

[B7]

[Cl]

[02}

[C3]

[C4]

[C5]

[C6]

[DI]

[D2]

[D3]

163

Barker, A. J. and Vangerko, R., "Temperature Dependence of Elastic
Constant of CFRP," Composites, Vol. 14, No. 2, pp. 52-56, 1983.

Barker, A. J. and Vangerko, H., "Temperature Dependent Dynamic Shear
Properties of CFRP," Composites, Vol. 14, No. 2, pp. 141-144, 1983.

Bert, C. W., "Research on Dynamic Behavior of Composite Sandwich
Plates," e S ock and Vibra on D est, Vol. 17, No. 11, pp. 3-15,
1985.

Book, W.J., Alberts, T.E., and Hastings, G.G., "Design Strategies
for High-Speed Lightweight Robots," Computers in Mechanical

Engineering, Sept. 1986, pp. 26-33.

Belegundu, A. D., "Probabilistic Optimal Design Using Second Moment
Criteria,” Transactions of the ASME, Vol. 110, 1988, pp. 324-329.

Brown, C. B., and Yao, J. T. P., "Fuzzy Sets and Structural
Engineering," ASCE Journal of Structural Engineering, Vol. 109, No.
5, 1983, pp. 1211-1225.

Cardoso, J.B., and Arora, J.S., "Variational Method for Design
Sensitivity Analysis in Nonlinear Structural Mechanics," AIAA
Journal, Vol. 26, No. 5, 1988, pp. 595-603.

Charnes, A. and Cooper, W.W., "Chance Constrained Programming,"
Mgmt, Sci, Vol. 6, 1959, pp. 73-79.

Choi, S.B., Gandhi, M.V., Thompson, B.S., and Lee, C.Y., "An
Experimental Investigation of an Articulating Robotic Manipulator

with a Graphite-Epoxy Arm," ournal oi Robotic Systems, Vol. 5, No.
1, 1988, pp. 73-79.

Clary, R. R., "Vibration Characteristics of Unidirectional

Filamentary Composite Material Panels," Composite Materials; Testing

and Design (Second Conference), ASTM STP487, American Society for
Testing and Materials, pp. 415-438, 1972.

Cook, R. D., Concepts and Applications of Finite Element Analysis,
John Wile & Sons Inc., 1981.

Cleghorn, W. L., Fenton, R. C., and Tabarrok, B., "Optimum Design of
High-Speed Flexible Mechanisms," Meghapism and Machine Theory, Vol.
16, No. 4, 1981, pp. 399-406.

Davidson, J.W., Felton, L.P. and Hart, G.G., "Reliability Based

Optimization for Dynamic Loads," , oi Structural Division, proc.
ASCE, Vol. 103, 1977, pp. 2021-2035.

Diaz, A., "Goal Aggregation in Design Optimization," Engineering
Qppipizggipp, Vol. 13, 1988, pp. 257-273.

Dubois, D., and Prade, H., "Criteria Aggregation and Ranking of
Alternatives in the Framework of Fuzzy Set Theory," in H.J.

164

Zimmermann, L.A. Zadeh and B.R. Gaines, (eds.),Fuzzy Sets and
Decision Analysis, Elsevier Science, Amsterdam, 1982.

[D4] Dubois, D., and Prade, H., "A Review of Fuzzy Set Aggregation
Connectivities," Information Sciences, 36, (1), pp. 85-121, 1985.

[E1] Erdman, A. G., Sandor, G. N., and Oakberg, R. C., "A General Method
for Kineto-Elastodynamic Analysis and Synthesis of Mechanism," ASME

Journal of Engineering for Industry, Vol. 948, 1972, pp. 1193-1205.

[E2] Eschenauer, H. A. and Fuchs, W., "Fiber-Reinforced Sandwich Plates
under Static Loads - Proposals for their Optimization," ASME, paper
85-DET-82.

[E3] Erdman, A. G. and Sandor, G. N., "Kineto-Elastodynamics - A Review
of the State of the Art Trends," Mechanism and Machine Theory, Vol.
7, 1972, pp. 19-33.

[F1] Fiacco, A. V., Introduction to Sensitivity and Stability Analysis in
Nonlinear Programming, Academic Press Inc., 1983.

[F2] Flugge, W., Viscoelasticity, Springer-Verlag, Berlin, 1975.

[F3] Fox, R. L. and Kapoor, M. P., "Rates of Change of Eigenvalues and
Eigenvectors," AIAA Journal, 6, No. 12, 1968, pp. 2426-2429.

[F4] Fleury, C., "An Efficient Optimality Criteria Approach to the
Minimum Weight Design of Elastic Structures,” Computers app
Structures, Vol. 11, 1980, pp. 163-173.

[F5] Frangopol, D. M., "Structural Optimization Using Reliability
Concepts," 4, Str, Engr,, ASCE, Vol. 111, 1985, pp. 2288-2301.

[G1] Gabriele, G. A. and Ragsdell, K. M., "The Generalized Reduced
Gradient Method: A Reliable Tool for Optimal Design," ASME J, Engin.
in ,, series B, Vol. 99, No. 2, 1977, pp. 394-400.

[G2] Gandhi, M. V., Thompson, B. S., and Fischer, F., "Manufacturing-
process-driven Design Methodologies for Components Fabricated in

Composite Materials," Composites Manufacturing, Vol. 1, No. 1, 1990,
pp. 32-40.

[G3] Gandhi, M. V., Usman, M. and Chao, L. P., "Nonlinear Vibrations of
Laminated Composite Plates in Hygrothermal Environments," ASME

Journal pi Engineering Materiala and Technology, Vol. 10, No. 2, pp.
140-145, 1988.

[G4] Gandhi, M. V., Chao, L. P., Usman, M. and Thompson, B. S., "The
performance of Articulating Machine and Robotic Systems in Severe
Environments,” Proceedings of the Tenth QSU Applied Mechanisms
Conference, New Orleans, December 1987, Vol. 3, pp. 78 3-1 to 7B 3-
9.

[G5]

[G6]

[07}

[GS]

[H1]

[H2]

[H3]

[H4]

[H5]

[H6]

[H7]

[H3]

[H9]

165

Gibson, R. F., "Dynamic Mechanical Properties of Advanced Composite
Materials and Structures: A Review," The Shock and Vibration
Digest, Vol. 19, No. 7, pp. 13-22, 1987.

Gibson, R. F., Yau, A., Mendi, E. N. and Osborn, W. E., "The
Influence of Environmental Conditions of the Vibration
Characteristics of Chopped-Fiber-Reinforced Composite Materials,"

Journal of Reinforced Plastics and Composites, Vol. 1, pp. 225-241,
1982.

Govil, A. R., Arora, J. S., and Haug, E. J., "Optimum Design of Wing
Structures with Substructuring," Computers and Structures, Vol. 10,
1979, pp. 889-910.

Gross, D. W. and Sobieszczanski-Sobieski, J. E., "Application to
Aircraft Design of Non-linear Optimization methods which include
Probabilistic Constraints," AIAA Paper No. 80-0153. 1980.

Hancox, N. L., "The Environmental Response of Hybrid Composites,"
Journal of Materials Science, Vol. 19, 1984, pp. 1969-1979.

 

Haug, E. J., and Arora, J. S., Applied Optimal Design, John Wiley &
Sons Inc., 1979.

Haug, E.J., Choi, K.K., and Komkov, V., Design Sensitivity Anaiysis
of Structural Systems, Academic Press, Orlando, FL, 1986.

Haftka, R. T. and Prasad, B., "Optimum Structural Design with
Bending Elements- A Survey," AIAA Journal, Vol. 19, No. 4, 1981, pp.
517-522.

Haftka R. T. and Adelman, H. M., "Resent Developments in Structural
Sensitivity Analysis," Presented at the Third International
Conference on CAD/CAM Robotica_and Factorieaiof the Future,
Southfield, Michigan, August 14-17, 1988.

Haftka, R. T. and Grandhi, R. V., "Structural Shape Optimizations -
A Survey," Computer Methods Appl, Mech, Eng,, 57, 1986, pp. 91-106.

Haftka, R. T. and Malkus, D. S., "Calculation of Sensitivity
Derivatives in Thermal Problems by Finite Differences,"

International journal for Numerical Methods in Engineering, 17,
1981, pp. 1811-1821.

Haftka, R. T., "Sensitivity Calculations for Iteratively Solved

Problems," International Journal f0; Numerical Methoda in
Engineering, 21, 1985, pp. 1535-1546.

 

Huang, T. C., "The Effect of Rotatory Inertia and of Shear
Deformation on the Frequency and Normal Mode Equations of Uniform
Beams with Simple End Conditions," , of Applied Mechanics, 1961,
pp. 579-584.

166

[II] Immam, I. and Sandor, G. N., "High-speed Mechanism Design - A

General Analytical Approach," AMSE Journal of Engineering in
industry, Vol. 97, 1975, pp. 609-628.

[J1] Jones, R. M., Mechanica of Composite Materials, McGraw-Hill, Inc.,
New York, 1975.

[J2] Jones, R.M., Mechanica of Composite Materials, McGraw-Hill, New
York, 1975.

[J3] Jozwiak, S.F., "Minimum Weight Design of Structures with Random

Parameters," omputers Q Structures, Vol. 23, No. 4, 1986, pp. 481-
485.

[Kl] Kartvelishvili, V. M., and Kobelev, V. V., "Analytical Solutions to
Problems of Optimum Reinforcement for Multilayer Plates Made of
Composite Materials," Mechanics of Composite Materials, Vol. 19, pp.
571-576, 1983.

[K2] Krishnamoorth, C.S. and Mosi, D.R., "A Survey on Optimal Design of
Civil Engineering Structural Systems," Engineering Optimization,
Vol. 4, 1979, pp. 73-88.

[K3] Kiusalaas, J. and Shaw, R. C., "An Algorithm for Optimal Structural
Design with Frequency Constraints," International Journal for

Numerical Methods in Engineering, Vol. 13, 1978, pp. 283-295.

[L1] Lowen, G. G. and Jandrasits, W. G., "Survey of Investigations into
the Dynamic Behavior of Mechanisms Containing Links with Distributed
Mass and Elasticity," Mechanism and Machine Theory, Vol. 7, No. 1,
1972, pp. 3-17.

[L2] Lowen, G. G., and Chassapis, C., "The Elastic Behavior of Linkages:
An Update," Mechanisp_and Machine Theory, Vol. 21, No. 1, 1986, pp.
33-42.

[L3] Liao, D. X., Sung, C. K. and Thompson, B. S., "The Optimal Design of
Symmetric Laminated Composite Beams Considering Damping,” Journal of
Composite Materiaia, Vol. 21, No. 5, pp. 485-501, 1986.

[L4] Lin, P. X., Ni, R. G., and Adams, R. D., "Prediction and Measurement
of the Vibrational Damping Parameters of Carbon and Glass Fiber-
Reinforced Plastics Plates,” Journal of Composite Materials, Vol.
18, March 1984, pp. 132-152.

[L5] Lee, C. Y., Pfeifer, M., Thompson, B. S., Gandhi, M. V., "The
Characterization of Elastic Moduli and Damping Capacities of
Graphite/Epoxy Composite Laminated Beams in Hygrothermal

Environments, Journal of Composite Matepials, Vol. 23, pp. 819-845,
1989.

[L6] Liao, D.X., Sung, C.K., and Thompson, B.S., "The Design of Flexible
Robotic Manipulators with Optimal Arm Geometrics and Fabricated from
Composite Lamintaes with Optimal Material Properties," Internationa1_

Journal oi Robotics Research, Vol. 6, No. 3, 1987, pp. 116-130.

[L7]

[L8]

[L9]

[M1]

[M2]

[M3]

[N1]

[N2]

[N3]

[01]

[R1]

[R2]

[R3]

167

Lynch, F. De 8., "A Finite Element Method of Viscoelastic Stress
Analysis with Application to Rolling Contact Problems,"
International Journal for Numerical Methods in Engineering, Vol. 1,
1969, pp. 379-394.

Lin, J. H., Che, W. Y., and Yu, Y. S., "Structural Optimization on
Geometrical Configuration and Element Sizing with Static and

Dynamical Constraints," omputera and Sppucpuges, Vol. 15, No. 5,
1982, pp. 507-515.

Low, K. H., "A Systematic Formulation of Dynamic Equations for Robot
Manipulators with Elastic Links," Journal of Robotic Systepa, Vol.
4, No. 3, 1987, pp. 435-456.

Meirovitch, L., Analytical Methods in Vibgations, MacMillan, New
York, 1967.

Malone, D. W., and Connor, J. J., "Finite Elements and Dynamic

Viscoelasticity," Journal of the Engineering Mechanics Division.
ASCE Proceedings, Vol. 97, No. 2M4, August 1971, pp. 1145-1158.

 

Moses, F. and Kinser, D. E., "Optimum Structural Design with Failure
Probability Constraints," AIAA J,, Vol. 5, 1967, pp. 1152-1158.

Ni, R.G. and Adams, R. D., "The Damping and Dynamic Moduli of
Symmetric Laminated Composite Beams-Theoretical and Experimental

Results," Journal of Composite Materials, Vol. 18, pp. 104-121,
1984.

Ni, R. G., and Adams, R. D., "A Rational Method for Obtaining the
Dynamic Mechanical Properties of Lamina for Predicting the Stiffness
and Damping of Laminated Plates and Beams," Composites, Vol 15, No.
3, 1984, pp. 193-199.

Narayanaswami, R. and Adelman, H. M., "Inclusion of Transverse Shear
Deformation in Finite Element Displacement Formulations," AIAA J,,
Vol. 12, No.11, 1974, pp. 1613-1614.

Oda, J., "Minimum Weight Design Problems of Fiber-Reinforced beam
Subjected to Uniform Bending," ASME Journal of Mechanismay
Transmissions. and Automation in Design, Vol 107, pp. 88-93, 1985.

Reuleaux, F, "Theoretische Kinematik: Grundzuge einer Theorie des
maschinenwesens," Vieweg, Brunswick, Deutschland (1875). English
translation by A. B. W. Kennedy, Kinematics of Machinery, Macmillan,
London(1876). Reprinted by Dover Publications, New York (1963).

Rehfield, L. W., Briley, R. P. and Putters, S., "Dynamic Test of
Graphite/Epoxy Composites in Hygrothermal Environments," Composites
i0; Extreme Environments, ASTM STP 768, N.R. Adsin, Ed., American
Society for Testing and Materials, pp. 148-160, 1982.

 

Reddy, J. N., An Introduction t9 the Finite Element Method, New
York, 1986.

1",

[R4]

[R5]

[R6]

[R7]

[R3]

[R9]

168

Rao, 8. 8., Optimization; Theory and Applications, John Wiley & Sons
Inc., 1984.

Ramirez, R. B., "Design of a High Speed Graphite Composite Robot
Arm," SME Paper No. 84-DET-SO, 1984.

Rao, S. S. and Bhatti, P.K., "Optimization in the Design and Control
of Robotic Manipulators: A Survey," Applied Mechanics Review, Vol.
42, No. 4, 1989, pp. 117-128.

Rao, S. S., "Optimum Design of Structures in a Fuzzy Environment,"
AIAA Journal, 1987, pp. 1633-1638.

Rao, S. S., "Multi-Objective Optimization of Fuzzy Structural
Systems," International Journal for Numerical Methods in
Engineering, Vol. 24, pp. 1157-1171, 1987.

Rao, S. S., "Structural Optimization by Chance Constrainted

Programming Techniques," omputers Q Structures, Vol. 12, 1980, pp.
777-781.

[R10]Rao, S.S., "Reliability - Based Optimization under Random Vibration

Environment," Computers é Structures, Vol. 14, No. 5-6, 1981, pp.
345-355.

[R11]Reddy, J. N., "Nonlinear Vibration of Layered Composite Plates

Including Transverse Shear and Rotatory Inertia,” ASME Publication,
No. 81-DET-144, 1981, pp. 1-10.

[R12]Rao, S. S., "Multiobjective Optimization in Structural Design with

[51]

[32]

[S3]

[S4]

[35]

Uncertain Parameters and Stochastic Processes," AIAA J , Vol. 22,
No. 11, 1984, pp. 1670-1678.

 

Shigley, J. E., and Uicker, J. J., Jr., Theory of Machines and
Mechanisms, McGraw-Hill Book Company, New York, 1980.

Sung, C. K., Thompson, B. S., Crowley, P. and Cuccio, J., "An
Experimental Study to Demonstrate the Superior Response
Characteristics of Mechanisms Constructed with Composite Laminates,"

Mechanism and Machine Theory, Vol. 21, No. 2, 1986, pp. 103-119.

Schultz, A. B. and Tsai, S. W., "Dynamic Moduli and Damping Ratios
in Fiber-Reinforced Composites, "Journal pi Composite Materiaia,
Vol. 2, No. 3, pp. 368-379, 1968.

Schultz, A. B. and Tsai, S. W., "Measurements of Complex Dynamic
Moduli for Laminate Fiber-Reinforced Composite," Journal of
Composite Materials, Vol. 3, pp. 434-443, 1969.

Suarez, S. A., Gibson, R. F., Sun, C. T. and Chaturvedi, S. K., "The
Influence of Fiber Length and Fiber Orientation on Damping with
Stiffness of Polymer Composite Materials," Experimental Mechanics,
Vol. 26, No. 2, pp. 175-186, 1986.

w

[S6]

[S7]

[S8]

[S9]

169

Shen, C. H. and Springer, G. 8., "Environmental Effects on the
Elastic Moduli of Composite Materials,” Journal of Composite
Materials, Vol. 11, pp. 250-264, 1977.

Smith, M. R., and Maunder, L., "Inertia Forces in A Four-bar

Linkage," Journal Mechanical Engineering Science, Vol. 9, No.3, pp.
218-225, 1967.

 

Sandgren, E. and Ragsdell, K.M., "The Utility of Nonlinear
Programming Algorithms: A Comparative Study-Parts l and 2," ASME

Journal of Mechanical Design, Vol. 102, 1980, pp. 540-551.

Schmit, L.A., "Structural Synthesis - Its Genesis and Development,"
AIAA Journal, Vol. 19, No. 10, Oct. 1981, pp. 1249-1262.

[SlO]Sandgren, E., and Ragsdell, K.M., "The Utility of Nonlinear

Programming Algorithms: A Comparative Study - Parts 1 and 2," ASME
Journal of Mechanical Design, Vol. 102, 1980, pp. 540-551.

 

[Sll]Schmit, L. A. and Miura, H., "A New Structural Analysis and

Synthesis Capability-ACCESS l," AIAA Journal, 14, 1976, pp. 661-671.

[Sl2]Smith, M. R. and Maunder, L., "Inertia Forces in a Four-bar

Linkage," , of Mechanical Engineeping Science, Vol. 9, No. 3, 1967,
pp. 218-225.

[Sl3]Saravanos, D. A., "Structural Optimization of Robotic Manipulators

with Fiber Reinforced Composite Materials," Ph.D. Thesia, The
Pennsylvania State University, 1988.

[Sl4]Schmit, L. A. and Farhsi, B., "Optimum Design of Laminated Fibre

Composites Plates," International Journal for Numerical Methods in
Engineering, Vol. 11, 1977, pp. 623-640.

[S15]Soni, P. J. and Iyengar, N. G. R., ”Optimal Design of Clamped

[T1]

[T2]

[T3]

[T4]

Laminated Composite Plates," Fibre Scienpe and Technology, Vol. 19,
1983, pp. 281-296.

Thompson, B. S., and Sung, C. K., "A Survey of Finite Element

Techniques in Mechanism Design," Mechanism and Machine Theory, Vol.
21, No. 4, 1986, pp. 351-359.

Thompson, B. S., and Sung, C. K., "A Theoretical and Experimental
Study of the Response of Linkage Mechanisms Fabricated with
Composite Laminates," Journal of Sound and Vibration, Vol. 111, No.
2, 1986, pp. 399-428.

 

Thompson, B. S., "Composite Laminate Components for Robotic and
Machine Systems: Research Issues in Design,” Applied Mechanics
Review, Vol. 40, No. 11, 1987, pp. 1545-1552.

Tsai, S. W., and Hahn, H. T., Intgoduction to Composite Materials,
Technomic Publishing Co., Westport, Connecticut, 1980.

170

[T5] Thompson, B. S. and Gandhi, M. V., "The Optimally-tailored Composite
Laminate Design of High-performance Articulating Robotic Systems,"
ASM[ESD Advanced Composites: The Latest Developments, Conference
Proceedings, Dearborn, Michigan, November 1986, pp. 15-19.

 

[T6] Tauchert, T. R., and Adibhatla, S., "Optimum Elastic Design of a

Reinforced Beam," ournal of Composite Mategiala, Vol. 16, pp. 433-
445, 1982.

[T7] Tauchert, T. R., and Adibhatla, S., "Design of Laminated Plates for

Maximum Stiffness," ournal of Composite Materials, Vol. 18, pp. 58-
69, 1984.

[T8] Tauchert, T. R., Adibhatla, S., "Optimum Thermoelastic Design of
Laminated Plates," Journal of Thermal Stresses, Vol. 8, pp. 11-24,
1985.

[T9] Thomson, W. T., Theory of Vibration with Applications, Prentice-Hall
Inc., Englewood Cliffs, NJ, 1981.

[T10]Thompson, B.S., and Sung, C.K., "A Variational Formulation for the
Dynamic Viscoelastic Finite Element Analysis of Robotic Manipulators
Constructed from Composite Material," ASME Journal of Mechanisms,

Transmissions, and Automation Design, Vol. 106, 1986, pp. 183-190.

[T11]Turcic, D. A. and Midha, A., "Generalized Equations of Motion for
the Dynamic Analysis of Elastic Mechanism Systems," J. of Dypamic
Systems. Measureppnt. and Control, Vol. 106, 1984, pp. 243-248.

[T12]Turcic, D. A. and Midha, A., "Dynamic Analysis of Elastic
Mechanism Systems. Part I: Applications," J, of Dynamic Systems,
Measurement and Control, Vol. 106, 1984, pp. 249-254.

 

[T13]Turcic, D. A. and Midha, A., "Dynamic Analysis of Elastic Mechanism

Systems. Part II: Experimental Results," J, pi Dypamic Systems,
Measurement, and Control, Vol. 106, 1984, pp. 255-260.

[T14]Thompson, B. S., Zuccaro, D., Gamache, D., and Gandhi, M. V., "An
Experimental and Analytical Study of the Dynamic Response of a
Linkage Fabricated from a Unidirectional Fiber-reinforced Composite
Laminate," Transactions of the ASME, Vol. 105, 1983, pp. 526-533.

[T15]Thornton, W. A., Willmert, K. D., and Khan, M. R., "Mechanisms
Optimization via Optimality Criterion Techniques," ASME J, of
Mechanical Design, Vol. 101, 1979, pp. 392-397.

[V1] Vanderplaats, G. N., Numerical Optimization Techniques for
Engineering Design, McGraw-Hill, New York, 1984.

[V2] Venkayya, V.B., Structural Optimaiztion: A Review and Som

Recommendations, "lntl, J, Numr, Eeth, Engr,, Vol. 13, 1978, pp.
203-228.

[V3] Vanderplaats, G.N., "Structural Optimization - Past, Present and
Future," IAA Journal, Vol. 20, 1982, pp. 992-1000.

[V4]

[W1]

[W2]

[W3]

[W4]

[W5]

[Y1]

[Y2]

[le

[22]

[23}

171

Vanderplaats, G. N. and Miura, H., "Trends in Structural
Optimization: Some Considerations in Using Standard Finite Element
Software," SAE paper 860801.

Weaver, W. and Johnston, P. R., Structural amics by,Finite
Elements, Prentice-Hall Inc., Englewood Cliffs, 1987.

Whitney, J.M. and Husman, G.E., "Use of the Flexure Test for
Determining Environmental Behavior of Fibrous Composites,"
Experimental Mechanics, Vol. 8, No. 5, pp. 185-190, 1978.

Wolfe, P., "Methods of Nonlinear Programming,” Recent Advances in

Mathematipal Ppogramming, R.L. Graves and P. Wolfe(ads.), McGraw-
Hill, New York, 1963, pp. 76-77.

Wang, G. Y. and Wang, W. Q., "Fuzzy Optimum Design of Aseismic

Structures," Earthguake Engineering and Structural Dypamics, Vol.
13, 1985, pp. 827-837.

Wang, G. Y. and Wang, W. Q., "Fuzzy optimum Design of Structures,"
Engineering Optimization, Vol. 8, 1985, pp. 291-300.

Yang, P. C., Norris, C. H., and Stavsky, Y., "Elastic Wave
Propagation in Heterogeneous Plates," international Journal of
Solids and Structures, 2, pp. 665-684, 1966.

Yang, R. J., and Botkin, M. E., "Comparison Between the Variational
and Implicit Differentiation Approaches to Shape Design
Sensitivities," AIAA Journal, 24, 1986, pp. 1027-1032.

Zimmermann, H.J., Fuzzy Set Theory-and Its Applicaitons, Kluwer-
Nijhoff, Boston, 1985.

Zimmerman, H.J., and Zysno, P., "Decision and Evaluations by
Hierarchical Aggregation of Information," Fuzzy Sets and Systems, 10
(3), pp. 243-260, 1983.

Zadeh, L. A., "Fuzzy Sets," Information and Control, Vol. 8, 1965,
pp. 338-353.

 

   

"‘111111111111