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thesis entitled

USE OF CONCURRENT TASKS
FOR EFFECTIVE STRUCTURAL CONTROL MODELING

presented by

MARK NOLSCHON

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of the requirements for

M.S. degree in MECH ENG

 

 

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MSU Is An Affirmative Action/Equal Opportunity Institution
cw 3-9

mt

USE OF CONCURRENT TASKS
FOR EFFECTIVE STRUCTURAL CONTROL MODELING

By

Mark Edward Wolschon

A THESIS

Submitted to
Michigan State University
' in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1993

ABSTRACT

USE OF CONCURRENT TASKS
FOR EFFECTIVE STRUCTURAL CONTROL MODELING

By

Mark Edward Wolschon

Concurrent engineering is the simultaneous acquisition, dissemination, and application of
multi-disciplinary information during product design. Over the past decade it has been applied
successfully to almost every aspect of the product life cycle. However, its use in analysis,
testing, and mathematical modeling has been limited. In this study, a procedure based on
concurrent engineering principles is presented for developing and evaluating mathematical models
for actively controlled structures. One advantage of this procedure is that it exploits uncertain
linear systems analysis and the methods of robust control design. Specifically, it assumes and
treats the model not as a single, precise entity but rather as a set of models, inside of which the
actual system lies. As an example we apply the procedure to a cantilever beam with a
concentrated tip mass. Subsequently, the procedure is applied to a distributed-parameter plate

structure that incorporates a distributed-parameter actuator.

To my parents,
for all their inspiration and support,

iii

ACKNOWLEDGEMENTS

The author wishes to express his thanks Dr. Mukesh V. Gandhi, Dr. Philip M. FitzSimons,
and Dr. Brian Feeny for serving as his thesis committee and for their helpful considerations and
recommendations. Special thanks is extended to Dr. FitzSimons, whose countless hours of
consultation and information have proved invaluable for this and other projects. His help,
guidance, and inspiration has provided an environment where professional, as well as, academic
skills could be developed.

In addition, the author would like to thank the faculty and students in the Intelligent
Materials and Structures Laboratory at Michigan State University for providing the ideas and
resources necessary to complete portions of this thesis.

Finally, the finished version of this thesis would not have been possible without the help
and encouragement of Kelly S. Ulrich. Her patience, proof-reading and corrections, and

undivided moral support are gratefully and greatly appreciated.

iv

TABLE OF CONTENTS

List of Tables

List of Figures

Changed - INTRODUCTION
1.0 Motivation Behind Work
1.1 Thesis-Related Background Topics
/ 1. 1. l Concurrent engineering
1/ 1.1.2 Structural control
1. 1.3 Spatially-distributed actuators
1.2 Some Issues in Practical Structural Control Design
1.2.1 Obtaining reduced-order models
1.2.2 Actuator modeling and dynamics
1.3 Thesis Organization
Chapter} - A METHOD FOR CONCURRENT
STRUCTURAL CONTROL MODELING
2.0 Introduction
2.1 Large-Scale System Modeling

2.1.1 Modeling uncertainty in systems

2.1.2 Modeling the system in the presence of uncertainty

viii

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14
17
18
21

23

25

26

29

31

2.2 A Concurrent Modeling Method for Large-Scale Systems
2.2.1 The analysis problem
2.2.2 The synthesis problem
2.3 A Numerical Example
2.3.1 Analytical analysis formulation
2.3.2 Experimental analysis formulation
2.3.3 Synthesis problem formulation
2.4 Chapter Summary
21139121.} - EXPERIMENTAL ANALYSIS FORMULATION
FOR A DISTRIBUTED-PARAMETER STRUCTURE
3.0 Introduction
3.1 Electrorheological Fluids
3.1.1 Rheology of suspensions
3.1.2 The electrical behavior of ER fluids
3.1.3 The rheological behavior of ER fluids
3.2 Constitutive Model for the Actuator
3.2.1 Preliminary considerations
3.2.2 Identification of an electrorheological fluid
3.2.3 A model for the actuator
3.2.4 Numerical comparisons
3.3 Electrorheological Plate Identification
3.3.1 Experimental apparatus
3.3.2 Plate modal analysis
3.3.3 Input verification

3.3.4 Electrical and rheological dynamic coupling

vi

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43
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48
57
65
80

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82
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89
93
93
95
102
105
1 13
l 15
l 16
122

134

3.4 Chapter Summary
W - ANALYTICAL AND NUMERICAL ANALYSIS FORMULATIONS

FOR A DISTRIBUTED-PARAMETER STRUCTURE

4.0 Introduction

4.1 Structural Kinematics

4.2 Constitutive Equations

4.3 Governing Equations

4.4 Comparative Results
4.4.1 Static results
4.4.2 Dynamic results

4.5 Chapter Summary

m5 - CONCLUDING REMARKS AND
RECOMMENDATIONS FOR FUTURE WORK

BIBLIOGRAPHY

vii

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139
141
149
156
165
165
168

175

177

180

LIST OF TABLES

Table 2.1 Material and geometric properties
for the flexible beam configuration.

Table 2.2 Measured cantilevered beam natural frequencies.

Table 2.3 A comparison between measured and calculated
cantilever beam natural frequencies.

Table 2.4 Measured cantilever beam damping ratios.
Table 2.5 Physical bounds on model parameters.

Table 3.1 Dimensions for the two actuator models.

Table 3.2 Experimentally determined damping ratios at V = 0 kV.

Table 3.3 Experimentally determined plate natural frequencies.

Table 4.1 Material and geometric properties
for the plate configuration.

Table 4.2 Nondimensionalized deflections
for two three-layer sandwich plates.

Table 4.3 Comparison of the measured and computed
natural frequencies for the square plate.

Table 4.4 Comparison of the measured and computed
natural frequencies for the rectangular plate.

Table 4.5 Material Properties for the composite structural layers.

viii

54

59

8

63
108
121

123

167

I68

172

172

173

Figure 1.1
Figure 1.2
Figure 2.1

Figure 2.2

Figure 2.3
Figure 2.4

Figure 2.5
Figure 2.6
Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10
Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

LIST OF FIGURES

The electrorheological effect.
Principles of practical low-order control design.
A standard large-scale system modeling procedure.

Generalized uncertainty problem
where the actual dynamic system M’ E Z.

Common perturbation structures for representing uncertainty.
A general block-diagonal bounded perturbation diagram.

An example of a plant with
weighted multiplicative input and output uncertainty.

A hierarchical structure for constructing
and completing the analysis problem for large—scale systems.

A concurrent scheme for
evaluating dynamical system models and uncertainty.

A generalized nominal and uncertainty model optimization procedure.

An example of uncertainty regions
for a family 1‘ graphically shown in the Nyquist plane.

The configuration for the cantilever beam example.
The variation of A, with concentrated tip mass ratio.

First four normal mode shapes
for various concentrated tip mass ratios.

Step input response for various concentrated tip mass ratios.

Step input response for various beam bending stiffness.

ix

16

19

27

29

33

35

39

42

t

47

48

50

51
55

55

Figure 2.15
Figure 2.16
Figure 2.17
Figure 2.18
Figure 2.19
Figure 2.20
Figure 2.21

Figure 2.22

Figure 2.23

Figure 2.24
Figure 2.25

Figure 2.26

Figure 2.27
Figure 2.28
Figure 2.29
Figure 2.30
Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 3.5

Figure 3.6

Step input response for various values of B with a constant.
Step input response for various values of a with B constant.
Receptance response for various mass ratios.

Transient impulse response of the beam tip for two mass ratios.
Transient step response of the beam tip for two mass ratios.
Fuzzy Nyquist diagram associated with plant set.

Local fuzzy Nyquist diagram showing parametric uncertainty.

A pole-zero map outlining
global uncertainty regions included in the plant set.

Pole map showing averaged poles
and their associated uncertainty regions.

Bode plots for the three 4-mode nominal models.
Step response of the lower and upper parametric uncertainty bounds.

Multiplicative uncertainty for
both lower and upper perturbations on A.

A comparison of the SV and SSV representations for M.
SSV stability evaluation for the parametric uncertainty model.

SSV stability evaluation for the nonparametric uncertainty models.

Variation of additive nonparametric uncertainty and frequency bound.

A strain rate versus viscosity curve for a typical suspension.

Representative colloidal interaction
potential over particle surface separation.

Double-electric layer theory for ER fluids.

Changes in the structure as a result of
dielectric polarization of the suspension.

A schematic of observed shear stress/strain behavior of an ER fluid.

A figurative description of the
shear stress versus shear-rate Bingham model fit.

56
56
58
61

62

67

68

68
69

71

71
73
74
76
78

83

86

87

91

Figure 3.7
Figure 3.8

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12
Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16
Figure 3.17
Figure 3.18
Figure 3.19

Figure 3.20

Figure 3.21

Figure 3.22

Figure 3.23

Figure 3.24

Figure 3.25
Figure 3.26

Figure 3.27

Isothermal shear modulus for the insulate material.

Shear modulus curve fit for the insulate material data at 25 °C.

Experimentally determined non-Newtonian
viscosity for a hydrous ER fluid.

Experimentally determined shear stress for a hydrous ER fluid.

Curve fit comparison for
the non-Newtonian viscosity of the ER fluid.

Curve fit comparison for the shear stress of the ER fluid.
Equivalent transient electrical behavior for an ER fluid.

Actual voltage of the ER fluid medium
for an applied electric field of 2 IN.

A comparison between
experimental and simulated voltages for the ER fluid medium.

Geometric configuration of the actuator and its unit cell.
Geometric configuration and loading of the actuator models.

Discretized geometry for the two models.

Time response of the ER fluid model under periodic excitation.

Power spectral density of
the ER fluid model for various constant voltage inputs.

Steady-state comparison between
models 1 and 2 for zero voltage input.

Deformed configurations for:
(a) model 1, and (b) model 2 for zero voltage input.

Time response of the actuator model under periodic excitation.

Power spectral density of
the actuator model for various constant voltage inputs.

Schematic of the two plate specimens.
Experimental arrangement and instrumentation.

Experimental plate discretization.

xi

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96

97

97

98
99

100

100

102
104
106
107

109

111

112

112

114

114

115

117

117

Figure 3.28

Figure 3.29
Figure 3.30

Figure 3.31

Figure 3.32

Figure 3.33

Figure 3.34

Figure 3.35

Figure 3.36

Figure 3.37

Figure 3.38

Figure 3.39

Figure 3.40

Figure 3.41

Figure 3.42

Figure 3.43

Figure 3.44

Receptance for: (a)square plate,
and (b) rectangular plate at V = 0 kV.

Mobility frequency response for the square plate at V = 0 kV.

Mobility frequency response for the rectangular plate at V = 0 kV.

Mobility responses for:
the (a) square, and (b) rectangular plates at V = 0 kV.

Mode shape identification results for each plate
at V = 0 kV.

Transient response of the rectangular plate for
V= 2kVandfl =0.5Hz.

Transient response of the rectangular plate for
V= 2kVandfl=0.5Hz.

Transient response of the rectangular plate for
V= 2kVand0=0.5Hz.

Transient response of the square plate for
V=2kVand0=0.5Hz.

Frequency response of the rectangular plate
for a sinusoidal input of V = 2 kV.

Frequency response of the rectangular plate
for a sinusoidal input of V = 2 kV.

Frequency response of the recangular plate
for a square input of V = 2 kV.

Frequency response of the square plate
for a square input of V = 2 kV.

Frequency response of the recangular plate
for a pulse input of V = 2 kV.

Frequency response of the square plate for a
pulse input of V = 2 kV.

Frequency response of the recangular plate
for different voltage inputs at 0 = 1 Hz.

Frequency response of the square plate for
different voltage inputs at 0 = 1 Hz.

xii

119
119

120
120
121
124
125
125
125
127
127
128
128
129
129
130

131

Figure 3.45

Figure 3.46

Figure 3.47

Figure 3.48

Figure 3.49

Figure 3.50

Figure 4.1

Figure 4.2

Figure 4.3
Figure 4.4

Figure 4.5

Figure 4.6
Figure 4.7

Figure 4.8

Figure 4.9

Mode shape identification for the square plate activated with
a square pulse voltage input at various frequency.

Mode shape identification for the rectangular plate activated
with a square pulse voltage input at various frequency.

Reduction of the ER effect with time for a pulse length of
10 msec.

Reduction of the ER effect with time for a pulse length
of 500 msec.

Reduction of the ER effect with time for a pulse length
of 1 second.

Reduction of the ER effect with time for a pulse length of
5 seconds.

Cross-section in the x-z plane
of a distributed-parameter composite structure.

Cross-section in the x—z plane
of the displacement pattern for the structure.

A single structural layer fabricated as a composite laminate.
Configuration of the distributed-parameter plate.

Discretized geometry for a quarter of
the distributed-parameter rectangular plate.

Mode shape results for both plates at V = 0 kV.
Mode shapes for the clamped-edge square plate.

Natural frequencies for a multi-layered square clamped—edge
(a) aluminum plate, and (b) composite plate for V = 0 kV.

Natural frequencies for

a three-layer square clamped-edge aluminum plate
with varying actuator thickness and V = 0, l and 2 kV.

xiii

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143
149

157

169
170

173

174

174

Chem

INTRODUCTION

1.0 Motivation Behind Work

Competition on a global scale has driven every sector of American industry to search for greater
efficiency and productivity in the product life cycle. Striving to improve a product while
maintaining high quality, minimal cost, and shorter concept-to-market life is a challenging task.
Since performance is dependent on myriad conditions that enter all phases of product life, the
challenge to produce successful products can be met only with careful planning and systematic
execution. With an organized plan, a particular idea or concept can be designed so that major
tasks, operations, and requirements are considered in the initial stages of development.
Subsequent execution of the development plan must then include an ability to address problems
that occur during development in a quick and truly logical and rational manner. Systematic
execution possesses this capability and can produce solutions to these problems very quickly.
Clearly, product development strategies that are based on an organized and systematic plan and
that do not impose any restrictions on creativity and innovation are very useful to the design
engineer.“ ‘3 The ability to strive to improve and to guarantee the success of the product will
be directly related to these amenable characteristics of the product life cycle, and their use will

definiwa determine future global leaders in industry.

2

Strategies that have such salient features are known throughout the industrial community
as concurrent engineering, simultaneous engineering, or design-for-manufacture and assembly.
These development strategies are not new, but are very different from conventional industrial
practices. Success stories hrwe been mainly esoteric in knowledge. Unfortunately, a majority of
industries have relegated these methods to a level of thinking believed to be too difficult to
employ in existing technology. This expectation may be attributed to the large investments of
time and money deemed necessary to replace traditional product lines with concurrent ones.27
We find that significant product development changes in conventional practices have been very
slow. Therefore, in order to capitalize on the advantages that these methods may have, it is
essential to break traditional barriers and introduce new, innovative concepts, and products using
a concurrent approach.

Innovative design concepts and ideas that have potential commercial marketability must be
synthesized in a manner that guarantees their success. We find that many of the potential
problems associated with traditional product development are a direct result of an unsystematic,
inefi‘icient allocation of resources, and an inappropriate processing of information, experience,
and knowledge. For these reasons, conventional techniques needlessly make efficient and
efi’ective product development an insurmountable challenge.

In this thesis the preliminary stages of a particular concept that involves structural control
is investigated in an attempt to provide computer-based tools necessary for concurrent engineering
tasks. These preliminary stages will include research, theoretical studies, testing, experimental
development, and analysis, not necessarily in any specific order. The problems associated with
the preliminary stages of structural control design are problems that arise because of the
multidisciplinary nature of the field. Structural control involves a large number of engineering
disciplines, in particular, analytical dynamics, structural dynamics and testing analysis, materials

science, and control system theory. In the sense of concurrent engineering, the various

3
disciplinary functions might be planned for optimal execution in a parallel fashion. Such

functions will include obtaining a mathematical model of the structure and of all its components,
reducing the order of the model for practical considerations, verifying the reduced-order dynamic
model for acceptable accuracy, and Optimizing and implementing a control strategy that
accommodates the reduced-order dynamic model and that meets stability constraint and all
required levels of performance. It is possible to execute several of these functions concurrently
and with significant improvement in design quality and accuracy through the use of computer-
based tools. Therefore, we see that through extensive preliminary planning, concurrent functions
can be linked to tasks that allow the design engineer to address specific problems.

Problems arising in these stages of structural control modeling are attributable to the fact
that they are essentially distribumd-parameter systems represented by partial differential
equations. Hence their parameters vary explicitly with position. Models for distributed-parameter
systems have an infinite number of degrees-of-freedom and are characterized by nonrational
frequency transfer functions and infinite-order state space matrices. This characteristic causes
problems in structural control because much of control theory has been developed for lumped or
discrete-parameter systems, or systems of finite-order. These systems are typically represented
by linear ordinary differential equations. One finds that a practical control system for distribuwd-
parameter systems requires linear reduced-order models that are robust with respect to system
parameter changes and disturbances. Thus, the common practice in distributed-parameter control
is to obtain a full order model, either analytically or through identification, and to then synthesize
the controller using a reduced or finite-order model that captures the salient features of the system
dynamics. When this approach is used, it is necessary to discern resonant frequencies to within
5 percent accuracy, and mode shapes to within 20 percent accuracy, as required to enhance
stability and performance. " It is very important that the design engineer extract reduced-order

models that retain critical modes, yet account for neglected ones.

4

Other issues that are important to the design engineer in the modeling stage are the
dynamics and positions of the actuators and sensors. Such details are discussed very infrequently
in the technical literature, although their characteristics can have a significant effect on the
dynamics of the structure. This lack of discussion may be attributable to the various types of
sensors and actuators used and their particular application. Different results are observed for
collocated systems and for noncollocated systems. In the collocated system, the actuators and
sensors are coincidentally located on the structure, which is beneficial for reducing instability.
In the latter system, especially for a reduced-order model, instability and performance degradation
are possible unless precautions are taken before hand. In addition, certain actuators must be
accommodated for in the initial design of the structure, whereas other may be included later as
a possible add-on device. Hence, if the dynamics and location of the sensor and actuator have
a large impact on the structure’s dynamics, it becomes quite time consuming to develop general
governing equations for each different design generated.

The trend in structural control design has been the development of high fidelity reduced—
order models of the dynamics to improve design, analysis, and to aid in understanding system
behavior. With the increasing use of computer-based tools aimed at improving controlled
structural stability and performance through Optimization, it is plausible to conclude that accurate
dynamic models are quite complex. Model uncertainty reduces our ability to make precise and
significant statements concerning the dynamics of the system. The closer we examine the
physical behavior of the structure’s dynamics, the more apparent is the need to represent and
investigate the effects of this uncertainty. Since uncertainty is apparent in all phases of the
preliminary stages of the structural control design, it is unavoidable and must be dealt with in a
rational and systematic manner. Consequently, the design engineer must have tools that are
compatible with physiwa meaningful descriptions of uncertainty, and that can be mathematically

modeled to an accurate level for improved structural control design.

5

The problems discussed previously are investigated here for structures that incorporate
embedded electrorheological (ER) fluid actuators. These structures are represented by a specific
class of distributed-parameter systems, one that involves a spatially distributed actuator. Since
these structures or actuators are nonconventional, they enjoy several advantages over conventional
actuators. For example, they provide a uniformly distributed control input, all changes are
completely reversible and response times quick. They are fully contained within the structure
and are not exposed to the environment, they contain no moving parts, and their dynamic
response can be readily controlled using electric signals. These advantages make the ER fluid
structure a practical candidate for active structural control. Thus, when integrated with modern
sensors and control technologies, ER fluid structures can be essential components in a large,
flexible systems and industrial applications.

However, the dearth of adequate ER fluids and a lack of satisfactory understanding of their
behavior have been the primary barriers in the establishment of commercial applications)”
Accurate mathematical ER fluid models are very complex and difficult to develop, and current
models are usually limited to special cases. Much of this is due to the complex behavior of the
fluid, the significant effects of various parameters on the macroscopic properties of the fluid, the
reproducible manufacture of the fluid, and the dynamic interaction of the fluid and device
boundary. Clearly, these deficiencies preclude an effective commercial design of ER fiuid
devices and control systems.

It is the primary objective of this thesis to develop a framework for manipulating,
interpreting, and parallel processing analytical, numerical, and experimental data for ER fluid
structures in a logical and systematic manner. More specifically, this framework will be
established using concurrent engineering to formulate certain tasks, some of which will include
computer-aided engineering. These tasks will then be used to obtain good experimental and

theoretical correlations for full-order mathematical models of ER fluid structures, such as plates

6

and beams. These structures serve as useful examples for active control methods because of their
ability to be readily included as components of larger systems. Moreover, the application of
computer-aided engineering tools to structure development should prove that their role in design

is critical to the evolution of this diverse technology.
1.1 Thesis-Related Background Topics

Structural control is a subject that has been attracting more attention over the past few decades.
Much of the interest in this subject stems from the idea that the stability and performance of
certain structures can be improved by somehow enhancing the characteristic behavior of the
structure, either through the use of materials, devices, or control systems. Good performance
of the controlled structure demands a good interaction and understanding of all the disciplines and
areas involved. Certainly, trade-offs are necessary due to the innate conflicts that develop
between the different disciplines. Moreover, we see in structural control that there is a definite
boundary between what must be integrated within the design process and what must be added to
the structure after its completion. Accordingly, it may be reasoned that a concurrent engineering

approach towards structural control would be applicable and useful.
1.1.1 Concurrent engineering

W of all different phases during the product life cycle is commonly
referred to as product development, and the full process from conception-to-completion is
properly denoted the product cycle life. In conventional practices, product development is
subdivided into three major phases: conception, designtw
Typically, these three tasks are treated as discrete and separate entities whereby each is executed I}

in a sequential manner, and where all changes are performed in an iterative manner. In any

7

given phase, an engineer will be concerned mainly with a particular design or operation only
within his/her own discipline, so that discussion between engineers of different disciplines and
in diflerent phases is quite limited. As a consequence, the likelihood of not meeting specified
requirements would lead to increased production cycle time due to the repetitive cycling of design
changes and modifications.

Clearly, product development involves a broad spectrum of engineering disciplines that can
influence and overlap each other to a considerable extent. Problems initiated during conception
are likely to be exacerbated in the design phase. For instance, constraints that must be met by
the electrical component group may conflict with a completed prototype fabricated by the
mechanical design group because of a severe lack of multidisciplinary integration. Likewise,
parts that were optimally designed for easy assembly, for some reason, require a force fit in the
factory due to conflicting objectives stated in the design and manufacturing phases. In these
hypothetical cases, prospective solutions, which may have been overlooked in the initial phases
of design, need to be explored and executed. Obviously, conventional methods hinder further
advancement and show that, given the vast requirements imposed during development, serial-
based processes do not allow the design to stay abreast with emerging technology and knowledge.
Hence, we see that satisfying design objectives and stringent requirements from every aspect of
engineering, at every phase, is a major concern that cannot be efficiently resolved in traditional
product design methodologies.

Concurrent engineering permits all practical knowledge from the relevant multidisciplinary
fields to be combined and processed in a parallel manner. In such a large program, this is

accomplished through exceptional communication and collaboration, computer-aided engineering

. ._.—._,-_—.- -—~e

tools, and databases. Thus, a designer must establish close links with other personnel so that a

-"'-—---...... h _‘

good flow of: information and experience are equally shared.” The traditional "top-down," or

 

._.—-—._w___,_..—._..—-4

serial-based, approach to product design and development eventually could have organizational

J

8

repercussions that render inefficient activity. Therefore, close collaboration between different
engineers of different disciplines is necessary for a successful concurrently engineered product.
With the successful implementation of concurrent engineering, it follows that the difliculties f"

prevalent in current industrial practice are usually overcome when as much information as j

/

I

possible is integrated and interpreted throughout the development process.

1.1.2 Structural control

Structural control can be subdivided into two specific categories: passive and active control.
Passive control is used to improve the performance of the structure by enhancing its stiffness and
damping properties with the use of materials or devices. Passive structural control methods share
many similarities, the most common being their versatility. Not only can they be used in many
difl’erent applications, but these methods also have the capability of being implemented during
any phase of development. Required design modifications can be verified and made using
experimental or analytical optimization techniques, so that the structure’s response may be
tailored for specific frequency ranges, inputs, deflections, etc., without large delays in
production.“ In contrast, active structural control requires more intensive planning in the initial ‘ 1,
design stages because a closed-loop feedback system is used to control the structure. This closed- _)
loop feedback system incorporates actuators to provide the necessary forces to the structure, as
described by the sensors, which measure that response. The many advantages that active control
has over passive control are basically attributed to the design of the structure and the controller.
Active control permits a structure to be designed lighter than those without or with passive
control.” However, active control also dictates early consideration of the controller design
during the initial stages of the structural design.” Once the structure is completed, a control
system may not be viable and/or may be too costly to implement. One advantage of an early

consideration of the control system in the active control design is its permission of the use of a

9

variety of controller design and optimization techniques to stabilize the structure and achieve
required levels of performance. These advantages have caused a great increase in interest in
active structural control and its use in so many diverse applications.

A cursory review of the literature concludes that much of the motivation behind the work
in active structural control has evolved from the development of large, flexible structures intended
for space. Much of this directly relates to the Galileo spacecraft and the construction of the U.S.
Space Station scheduled to be launched in the near future”- '° Clearly, a structure intended for
space must be fabricated from materials with high stifl’ness- and strength-to-weight ratios. These
ratios must be large due to structural integrity and weight constraints, and also because of the
reliability and accurate positioning requirements for the structural appendages. Extra weight will
cause the rigid body dynamics to tightly couple with the flexible body motions.” Evidence has
shown that structures with these characteristics also tend to possess a low amount of damping.
As a result, they may sustain long lasting vibrations during maneuvers or due to disturbances that
can significantly degrade performance. For various other structures that depend on precise
tracking or positioning, i.e., robots, long boom antennas, telescopes, and solar arrays, this
problem is very important and critical to ensure an operation’s success.

Early work in structural control used models that were reduced to limit the full-order model
to include only its first few system modes. This is called Modal Control modeling.“ '3' ‘2' Modal
control was developed in these studies based on the assumption that the structure’s response is
dominated by its first few modes and that its residual modes can be neglected because they are
not directly affected by closing the loop. The modal control method has worked well when
applied to discrete-parameter structural systems, which amount to controlling the lumped structure
by controlling a finite number of modes.” As one of the original strategies for finite-dimensional
control, modal control has since been proven both theoretically and experimentally inadequate

for infinite-dimensional or distributed-parameter systems. The main difiiculty is that the reduced-

10

order model necessary for modal control is developed with only a finite number of modes, so
control over truncated modes is non-existent. Consequently, this implies that the control gains
must be of infinite-dimension to attain sufficient control using the modal control method. This
is truly impractical.

Other control theories that have been established for distributed-parameter systems have
been developed for both time and frequency domain analysis. Each control theory has its own
advantages, to be discussed here, however, all suffer a similar disadvantage manifest by the in-
finite-dimensionality of the structure. This disadvantage is one that requires the finite-
dimensional controller to satisfy certain criteria for realization and existence. Specifically, these
theories are based on modal approximations that must satisfy necessary and sufficient conditional
criteria for the selection of an appropriate number of critical modes and dynamic boundary
conditions. This criteria must be satisfied to have a stable control design.“

In a paper by Vidyasagar,m necessary and sufficient conditions were defined for a large
class of distributed-parameter transfer functions in the frequency domain under the assumption
that the dynamic models for these systems were known precisely. Lions” developed a general
mathematical theory for the control of distribumd-parameter systems that recognizes and details
three important steps in solving the optimal control problem. Convergence and stability criteria
for the modal control laws were considered by Gibson.” This study focussed attention on certain
essential criteria and on its direct correspondence with increasing system dimensionality. Lin“
surveyed the possible application of output feedback control using modal decoupling, pole
assignment, sub-optimal control, and optimal control to make some detailed conclusions about
assumptions and techniques associated with the theory of distributed-parameter control. Sakawa‘”
presented good results for the coupled bending and torsional vibration control of a cantilevered
beam using a hybrid feedback control scheme. Bruch” introduced an approach for orthotropic

plates with rotatory shear that minimizes two performance indices subject to the amount of

11

fiedback force that can be spent in the control process. Numerical examples of this open-closed
loop theory are given, however, with no discussion on how it is realistically feasible. Additional
control theories that have been developed in the manner described above include optimal
control," adaptive control,"- ‘°°' “‘ pole allocation,” “ and direct velocity feedback control
methods.”

The idea of introducing invariant natural modal coordinates to obtain a reduced-order model
for feedback control was introduced by Creedon,‘o Thkahashi,m and was further developed by
Meirovitch.“ " Although natural coordinates vary slightly from the general modal coordinate
descriptions aforementioned, they can be used to obtain a global and unique optimal closed-form
solution to the linear control problem for distributed-parameter systems in the time domain. This
theory has been termed the independent modal-space control theory” (IMSC), or natural control
theory,“ which is based on the notion that the stability of a linear system is determined purely
by the system eigenvalues and that no control efi’ort can be unnecessarily used. In particular,
IMSC makes use of a distributed control force to control the system’s eigenvalues, thus leaving
the system’s eigenfunctions unaltered. Both theories have been used in numerous applications
with adequate results. However, one specific problem involves finding a suitable control theory
to complement IMSC in order to determine the appropriate control gains for the system, since
a distributed actuator is necessary to provide the control force.

More recently, distributed-parameter optimal control theories have been modified for
combined controller and structure design. These modifications are made to include many
objectives that employ various constraints on the structure and controller’s performance. For
example, such objectives and/or constraints imposed on the determination of feedback gains,
sensor and actuator locations, structural natural frequencies and damping ratios, and structure

dimensions have been used in the optimization procedure. Typically, these objectives are

12

optimized for specific control/structure cost functions, such as weight, power consumption,
and/or vibrational response.

An integrating strategy for both structure and control system optimization was developed
by Khot, et at," who obtained a finite-element model of the structure and optimized the closed-
loop system for minimal weight. The constraint used to modify the structure was based on the
amount of internal damping, with a linear regulator used for the controller. Onoda105 combined
direct feedback and linear quadratic optimal control laws with a disturbance structural model to
establish total cost. The approach taken in this paper is similar to the large spectrum of literature
available on this subject, mainly considering a span of various costs posed in a single
control/structure objective function. Milman, er al,” employs a different method, one that
consists of a multi-objective approach for combined structure and control design, and uses a
family of weighted structural and control cost functions. The optimization problem here is posed
such that the combined approach is a natural extension of most well-established current
optimization strategies currently being used. Although the combined control and structure
optimization philosophy is sensible, its problems are very similar to most problems encountered
in optimization routines. Solutions are computationally demanding and require iterative
minimization algorithms which suffer from many difliculties, such as the choice of the starting
guesses, convergence, and multiple local minima.

The theories above present many fundamental mathematical principles for distributed-
parameter structures and their control designs in both the time and frequency domains. Many
of these theories are based on reduced-order modal coordinate characterizations and approxima-
tion methods in optimal and feedback control. However, very few of these methods can be
classified as robust since they are dependent on the assumption that the dynamic models are
precisely known, i.e. have no uncertainty. Control designs that account for uncertainty in the

dynamic system model are defined as robust control theories. Several specific theories central

13

to robust control include loop transfer recovery (LTR) techniques, linear quadratic gaussian
(LQG) methods, and H2 and H. optimal control. For distributed-parameter systems, robust
control has been employed by many investigators over the past two decades.” “' '3 These
methodologies suggest a controller design based on a nominal model with uncertainty described
by some perturbation from this model. After a model has been selected, robust control theory
guarantees the controller to perform satisfactorily under all possible deviations from the nominal
model as long as certain stability and performance criteria are satisfied.“ Hence, the modeling
of uncertainty is very crucial to how the final controller design rejects disturbances and follows
tracking inputs.

Most of these methods are linear time-invariant (LTI) frequency domain methods that
require very high bandwidth controllers. The need for high bandwidth controllers is manifest by
the large gain-phase margins necessary to make the structure robust to specified uncertainties.
However, with robust LTl systems, some inevitable trade-offs exist in the design process for
which accomodation must be made. Trade-offs such as unstable poles, nonminimum phase zeros,
and time- delays impose fundamental constraints on achievable performance, and imply
restrictions on the closed loop bandwidth of the system.”6 Generally, nonminimum phase zeros
place an upper bound on the system bandwidth. Since it is well known that distributed-parameter
transfer functions can have nonminimum phase zeros,” ‘°" ‘“ arbitrary truncation of the model
order may not include these zeros and give a false sense of full-order system stability. The
implications of this trade-off on robust control has been discussed by O’Young and Francis.”
Unstable poles impose a severe constraint on the system and can be clearly seen to worsen
structural control performance. Hence, it is very important that robust stability bounds are
satisfied. A model with a time-delay is actually a finite dimensional system of very high order,

so approximations are usually necessary. The use of a Padé approximation introduces modeling

14

error, which consequently limits achievable bandwidth, yet, the loss of performance is

insignificant compared to the robustness properties placed on stability and performance.“

1.1.3 Spatiallydistributed actuators

Essentially, actuators provide the motion, mobility, and changes that govern the response of a
controlled structure. Most conventional actuators accomplish this by using discretely located
forces or motion. A severe problem can occur when these discrete devices are placed at modal
nodes or lines. Tb observe and control the full continuum, the use of spatially-distributed
actuators is highly desirable.“ Certain materials can be used for these actuation purposes to
control damping, stiffness, and mass properties of or within the structure. Some of the more
common distributed-parameter actuator materials are shape-memory alloys, piezoelectric
materials, magnetostrictive materials, magnetorheological fluids and electrorheological fluids.
These materials all have unique innate characteristics that allow them to either change shape,
stifl’ness, position, energy dissipation, natural frequency, friction, and fluid flow rate in response
to changes in temperature, magnetic field, or electric field. Characteristics, applications, and the
history of these actuators are detailed by Gandhi and Thompson.’5

The concept of using a distributed-parameter actuator for control is not new. Applications
using distributed-parameter actuators date back to the 1950’s, where Olson'“ first used a
distributed-parameter actuator for electronic noise control. However, it was not until the 1970’s
that they were applied to mechanical structures.” Presently, the amount of research dedicated
to active structural control has dictated intensive interest in the use of distributed-parameter
actuators. Much of the work has been focussed on the application and control of structures using
piezoelectric and shape-memory materials.” Piezoelectric materials exert mechanical forces in
response to an applied voltage. These materials change shape when their electric dipoles

spontaneously align in the electric field, causing deformation within the material. Work in active

15

control using piezoelectric actuators has demonstrated their range of feasibility and applicability.
Examples are numerous, ranging from optical tracking devices to ink-jet printers, and in recent
years, acoustic attenuation, and active structural damping control.‘2 Shape-memory materials also
change shape. However, rather than having their dipolar properties change, they undergo a solid-
to-solid phase transformation in response to temperature changes. Under heating and cooling,
this phase transformation allows them to exhibit large, recoverable strains of up to eight
percent.” Applications for shape-memory materials include engine mounts, suspension systems,
and robotic actuators, which mimic the smooth motions of human muscles.”

A rather detailed study of piezoelectric actuators applied to a one-dimensional beam analysis
is given by Crawley and de Luis.” They analyzed stresses, strains, and loads for a cantilever
beam with piezoelectric ceramic surface treatment. Bailey and I-lubbard,7 developed three
different control algorithms for cantilever beam configurations to control transient vibration.
These algorithms were implemented in an experimental analysis to verify theoretical equations
of motion and predicted response. Others contributing to this area have demonstrated the
potential of distributed-parameter actuators by simultaneously controlling a number of modes with
reduced spillover.” “‘- °° Active vibration control of plates using piezoelectric ceramics was
conducted by Dimitriadis, er 01.“ Conclusions drawn from this investigation include the ability
to excite the plate using distributed actuators, the observation that the input frequency of the
excitation markedly affects the modal response, and the location of the actuator strongly
influences excitation as well as spillover. Tzoum ‘35 introduced a theory for distributed actuators
that was discretized for isoparametric piezoelectric finite-element formulations. A distributed
modal analysis was examined and several case studies using Lyapunov control and constant gain
proportional feedback control were implemented.

The type of actuator used in this investigation is an electrorheological (ER) fluid. An ER

fluid is composed of fine conductive particles suspended in insulatory fluid. When an electrical

16

potential is applied to the ER fluid the conductive particles become polarized and structure
themselves to drastically alter the fluid’s stiffness and energy dissipation properties. Hence, the
ER fluid can be activated simply by placing it between electrodes of opposing polarity. However,
the magnitude and duration of the electrical potential determines the intensity and rate of these
changes. Thus, the capability of controlling the fluid is manifest by the electro-coagulation of
the particle. This phenomena is defined in the literature as the electrorheological effect (ERE)."'
“ It is the very nature of the ERE that permits it to be used as a spatially-distributed actuator
in a structural domain.

Throughout the past two decades, the focus on ER fluid technology and research has
become quite intensive.“ ‘3' m m "‘ ER activity and the ERE has merited extraordinary interest
due to the number of practical actuator applications. Detailed investigations have focussed on
the deformation, motion, and the rheological and electrical properties of ER fluids. Much of this

attention has been motivated by observed changes in the fluid’s appearance and constitutive

 

 

(a) No Electric Field (b) Electric Field Applied

Figure 1.1 - The electrorheological effect.

17

behavior. At zero electrical potential, the random dispersion of particles (see Fig. 1.1 (a)) in the
fluid medium has minimal influence on its rheological properties. In fact, its behavior can be
easily identified with that of typical Newtonian fluids. Although when it is subjected to an
electrical field, a detailed and unique formation of the particles (see Fig. 1.1 (b)) causes an
instantaneous transition in the fluid’s properties. These constitutive changes are comparable to
that of non-Newtonian fluid behavior. A marked distinction of this transition is the fluid’s ability
to support an increasing shear stress with increasing field intensity and without flow.2° Clearly,
without doubt, this characteristic of a yield stress is one possessed only by non-Newtonian
fluids.“

Nonetheless, as the overall behavior and capability of controlling an ER fluid has been
established, the actual controllability of the fluid depends on many inter-dependent parameters
that can have profound effects on the amount of ER activity. For example, the uniformity of
electrical field used, the rate of deformation of the disperse particles, the various additives and
the overall composition of the fluid,132 the surrounding temperature,“7 the dielectric characteristics
of the dispersion,“ and particle concentration all need to be considered. Such parametric
interaction has made constructing constitutive descriptions for general electrorheological fluids

a very complex task.

1.2 Some Issues in Practical Structural Control Design

A substantial number of problems in structural control can be attributed to the summary of steps
used in arriving at a practical reduced-order controller. Such a controller guarantees stability and
is capable of meeting acceptable performance constraints. Generally, constraints include both
frequency and time domain statements in these control system designs. The former usually
represents explicit performance criteria, while the latter reflects physical bounds such as actuator

limits, parameter ranges, rise time, overshoot, or sensor limits. Much of the difliculty with

18

ensuring that these constraints are not violated has to do with the representation of the modeled
structure. When the reduced-order model is linear, the problem is especially cumbersome, due
to the numerous assumptions and approximations that must be made about the actual plant.
Conventional modeling and controller design techniques frequently use the assumption that exact
knowledge of the system’s dynamics can be obtained. However, mathematical models and
control designs based on this assumption can be too restrictive and fail in reality. Conventional
control design leads to controllers that are too tight and may become unstable during operation.“
Moreover, in a real system, the internal and external disturbances that act on the uncertain system
cannot be excluded due to the complex interaction between the system and its environment.
Thus, all possible perturbations within the plant model must be bounded to guarantee stability and

requirements on performance in the face of uncertainty.

1.2.1 Obtaining reduced-order models

For practical distributed-parameter structural control design, simple linear controllers are
preferred over complex linear controllers for linear time-invariant plants. There are many
reasons for this, yet most result from the reduced amount of time and computational effort needed
to understand and operate them. Methods used to design low-order controllers for high-order
plants can be divided into two broad categories: (i) the direct method, where parameters defining
a low-order controller are computed by some optimization routine, and, (ii) the indirect method,
where a high-order controller or low-order plant model are first found, and a specific procedure
is used for further reduction.‘ These two approaches are shown graphically in Fig. 1.2 along
with a short listing of the different procedures used for system reduction. The fundamental
difference between model and controller reduction is that model reduction is based on open-loop
considerations. Controller reduction techniques must be based on mathematical existence

arguments that should, after all, preserve the closed-loop system stability and performance

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L , H, '
High Order Model QC # 13552:?
Modal CONTROLLER
kalman filter Direct REDUCTION
Design .
Luenberger MODEL Optimal
. . a1 (1 . .
0113;?” or er REDUCTION Frequency weighting
1 v
Direct output and Direct velocity feedback LOW Order
Low Order Model LQG’ H... H7. Decoupling control ’ Controller

 

 

 

 

 

 

 

 

 

Figure 1.2 - Principles of practical low-order control design.

The key objective in the direct method is to minimize a quadratic performance index subject
to the constraint that the controller be fixed degree, stabilizing, and linear time-invariant.“ ‘5
Direct methods have much promise and have been a research topic of interest in recent years due
to the ease of their implementation in control design packages. Nevertheless, a disadvantage with
the direct method is that it is unclear as to what mathematical basis is applicable for a large
variety of systems. For this reason, the remainder of this thesis will focus on indirect methods
of model reduction.

As for indirect methods, there are many different approaches, ranging from the very crude
modal approximations to the very sophisticated H. methods. There are at least two specific
methods for designing high-order controllers, and both are highly capable of using the large
amount of existing qualitative and conceptual knowledge that pertain to algorithms and
applications.‘ However, the procedures necessary for reducing high-order controllers to low-
order controllers are less developed. These procedures require precise, rational arguments in

order to retain the critical aspects of the system’s closed-loop dynamics. On the contrary, a very

20

crude approximation technique that can be performed to approximate the plant model in the time
domain is called modal reduction.“ This approach is quite similar to modal control, and when
used for distributed-parameter systems has similar drawbacks in the resulting low-order controller
design. A more sophisticated frequency domain model reduction method is the projection-and-
assembly method presented by Lee.‘0 In this method, modes that contribute significantly to the
control input and output are selected at the system level, and are then projected in a mathematical
sense onto the components of the system. The resulting reduced-order mathematical model of
the components are then assembled to yield a rather accurate model of the system dynamics for
only the selected modes. Reddy,“‘ proposed a method for minimizing the integral of the square
of the error between corresponding real and imaginary parts of original and assumed transfer
functions to evaluate coefficients of the approximate model. An advantage of his approach is that
it is computationally simple. Nonetheless, a few disadvantages are entailed by these frequency
domain methods, as with other time domain reduced-order methods, which have to do with
accuracy and mismatched transfer function zeros due to truncated or undesired modes.

In particular, the primary issue with the former indirect model reduction methods is the
inability to account for realistic actuators and sensors within the low-order control scheme. In
reality, actuators and sensors will not necessarily operate in a select frequency range, hence,
residual modes may become excited by the actuators and/or measured by the sensors. This
results in what is known as control and observation spillover, and both have been attributed to
instabilities and degradation of system performance.“ “- ‘° Over the past few decades, various
techniques have been established for minimizing the effects of spillover in discrete-parameter
systems. Curtain" developed estimates on spillover effects for all modes using a pole placement
method. Orthogonal and modal filters have also been used to better accommodate spillover by
several authors.“ "3 Filters were used to approximate and extract convergence properties from

distributed measurements over given time intervals rather than estimate the modal response of the

21

structure. Yang“2 eliminated instability by introducing a specific time delay in the control system
fiir certain mechanical structures. These techniques were initially developed for collocated and
noncollocated actuators and sensors. Only in the recent decade have they been applied to
distributed-parameter actuators and sensors. Balas9 used modal equations to obtain output
feedback control over a finite number of modes for a distributed-parameter elastic system,
specifically treating the problem of control spillover into the uncontrolled modes by instituting
spillover bounds. To alleviate spillover, Meirovitch and Baruh"2 proposed a technique that is
compatible with a limited amount of inherent damping to eliminate observation spillover

instability.

1.2.2 Actuator modeling and dynamics

In the former discussions, consideration of the issues pertaining to the physical effects that
actuators have on the mathematical model of a distributed-parameter system has been mentioned
briefly. Essentially, the type, location, and dynamics of the actuators are very critical when
implementing successful operations because most conventional actuators are discrete actuators.
They are discrete in the sense that they can only be located in select, finite regions of the full
structural system. Spatially distributed actuators will also suffer many of the same dilemmas,
because it would be impossible to actuate every member within a structure due to weight
constraints. Therefore, the consideration of the actuators must be sufliciently detailed to describe
their behavior on the system.

Actuators used in distributed-parameter systems not only require a detailed analysis of their
functions, but also detailed analysis for their dynamics and positioning. Spatially distributed
actuators have certain slight advantages here but are also inflicted with the same problem that
exists with all actuators, this being that they possess mass and inertia. In most cases, their mass

may be considered to be negligible if the structure is very large and heavy. However, for

22

vibration suppression of thin, lightweight structures, the distribution of the actuators will certainly
alter the modal response of the system significantly. Non-negligible actuator mass in structural
control design has been addressed only by a limited number of authors. Zimmerman‘“ focused
on required actuator masses for discrete actuators using optimal design techniques to minimize
control and position error subject to total mass and frequency constraints. This paper provides
an important inner-loop for the design of controlled structures with discrete actuators that
recognizes their critical duty for successful structural control. Such techniques have yet to be
applied to distribuwd-parameter actuators because of the myriad procedures for applying them
to the structure. However, this inevitable problem may lead to greater instability in the controlled
structure than what was initially encountered. Therefore, all actuated structures must be designed
to be sufliciently robust in order to account for variations in their inertial and dynamic properties.

In structural control design, the dynamics of the actuators must be taken into consideration
early in the simulation work, because their presence will have significant effects on the structure’s
performance.“ In a paper by Maghama and Joshi," the effect that actuator dynamics have on
their optimal actuator and sensor placement scheme vastly reduced controllability of their system.
Reduction became more pronounced as actuator bandwidth decreased or when it was near the
operational closed-loop bandwidth of the control system. The problem with these actuators is that
their bandwidth is finite, while that of structural system is infinite. Hence, extreme care must
be exercised so that control spillover of the low-frequency modes does not destabilize the
intermediate or higher frequency modes, and that critical modes of operation in the actuator are
never attained.”

Other minor issues such as reliability, functionability, and possible malfunction are very
important in the selection of actuators and control system design. Three possible motivations
usually exist for using active structural control: (i) to stabilize the structure at all time, thus, the

actuators are constantly in operation, (ii) to stabilize the structure over limited motion, thus, the

23

actuators must be dependable, especially during continuous on-off cycles, and (iii) to stabilize the
structure only during critical motion, hence, the actuators must be reliable when they have been
inactive for long durations. As illustrated, these motivations impose important design
requirements in the reliability and functionability of the actuators. However, a review of the

literature reveals minimal information on these important topics.

1.3 Organization of the Thesis

The organization of this thesis is based on the ideas introduced in Chapter 2. In Chapter 2, a
rather novel technique is presented for large-scale systems modeling. Specifically, this technique
addresses actively controlled large-scale structure modeling with controllers designed using robust
control theory. The framework of robust control theory is useful because it is very compatible
with concurrent engineering principles. In the past, concurrent engineering has been applied to
many aspecs of the product life cycle, but is use in the preliminary or initial stages of the design
process has been limited. Naturally, this is because system modeling is unique to the problem
at hand and no cookbook formula could exist that would be useful for the myriad problems
encountered. Therefore, these principles are used to develop a technique that deviates from
conventional structural control modeling procedures by taking into consideration uncertain linear
systems analysis. Take note that this modeling strategy is more of an outline that can be applied
to modeling than a strict standardized procedure.

The modeling procedure described above is applied to a simple example in the final section
of Chapter 2. For illustrative purposes, the example is a simple cantilevered beam. In this
example, analytical and experimental techniques are used to obtain an eflicient linearized model
and an associated uncertainty model. It will be shown that modeling uncertainty is a necessity
in any system because of is innate presence and is effect on the resuls of the mathematical

model. However, the goal here is not to model uncertainty where is mathematically convenient,

24

but where is actually occurs. Unfortunately, this objective is the most arduous task in the
modeling procedure, yet, as shown in this simple example, it yields the most benefits.

The final chapters specifically focus on the development of a linearized mathematical model
for distributed-parameter plates containing ER fluid actuators. Chapter 3 begins by giving a
detailed review of the current knowledge of ER fluids and the ER fluid effect. Both the
rheological and electrical aspecs of the ER fluids physical properties are considered.
Subsequently, models for the viscosity, shear stress, and electrical behavior are developed for a
specific ER fluid. These models are based on experimental identification techniques, as well as,
numerical and empirical methods.

Chapter 3 concludes with an experimental identification of ER fluid plates. Experimentally
determined information is obtained for two ER fluid plate configurations in an effort to
characterize their dynamics. This process is conducted using the structure’s frequency response
to set and identify bounds for constant and dynamic inpus. Unfortunately, this information is
presented in graphical form only due to the limitations of the data acquisition system. Finally,
the experimental resuls are compared to resuls generated from commercial finite element
analysis software. A unique characteristic of this software is that the nonlinear ER fluid model
can be incorporated via a subroutine written in Fortran.

In Chapter 4, an analytical model is developed for the ER fluid plate using assumptions
based on sandwich theory“ 9°: "° and constrained layer theory.“ The governing equations of
motion and their associated boundary conditions are obtained using Hamilton’s principle.
Subsequently, comparative resuls are generated for the static response of the plate using the
Naiver approach.“0 These resuls show good agreement with a previously obtained three-
dimensional elasticity solution,“ for the case of a simply supported three-layer sandwich plate.
Finally, conclusions are drawn from the above work and recommendations for future work are

discussed in Chapter 5.

Cheated

A METHOD FOR CONCURRENT
STRUCTURAL CONTROL MODELING

2.0 Introduction

Modeling is the basis for many types of system analysis and design. Mathematical models should
be expressive, easy to understand, should map all properties of interest, and should allow analysis
and decision-making concerning the real system.” In the case of large-scale dynamical systems,
however, there is a fundamental problem. These models sometimes tend to fail to describe the
behavior of the whole model appropriately because they are usually constructed by combining
various models of the system’s parts. During structural control modeling, this occurs as a result

of several important issues. Such issues pertain to uncertainty in all the approximations, ill-posed

component “109613. and/or inefficiently linked system component models. Generally, the

combined effects of all three issues that lend to deleterious effects on the dynamic mathematical
description of the system.

Models of large, controlled, structural systems can be either stochastic or deterministic.
They may also reflect the system’s behavior, the desired qualities of that system, or the
exogenous input conditions that may act on the system. There is a multitude of ways to represent
large-scale systems; however, there are three views of a system that need to be captured in the

model:”

25

/J

26

Structural viewpoint: accurately represents the structure of the system, its components,
and interactions between these components.

Functional viewpoint: reflects the ability of the modeled system to satisfy required needs
of the true system and to function as a whole unit.

Behavioral viewpoint: represents all the various states that the system goes through, and

the flow of processes.

A good modeling scheme is one that captures all three viewpoints of the system and
translates them to an appropriate working mathematical model. We find that there is no simple
method for doing this, because the components of most systems in practice are constantly active,
thus, every input change causes a state change in the system. Most structural control problems
definitely fit this category. For convenience it is useful to approximate the system by
characterizing it as a black box, in which every input is transformed into an output.21 Neverthe-
less, a complete modeling procedure must include not only what we know about the system, but
also what we do not know about the system. Only then can we capture the three viewpoins of
the approximate model and make significant statements about its dynamics.

It is the goal of this chapter to develop and outline a modeling method for representing
large scale systems. This representation will be based on a hierarchical procedure that integrates
the crucial aspecs of communication among the system components. Structural control systems
are active systems that must operate in a concurrent nature, hence the models used in design of

these systems must also reflect this concurrent methodology.

2.1 Large-Scale System Modeling

Modeling of systems can be carried out in many ways: by theoretical methods, by experiment,

and by any variety of combined theoretical and experimental techniques. In the case of large-

27

 

’ Collect known models of system components, I : ..

 

 

Supplement theoretical and/or experimental studies to create models for the unexplored partsi

 

 

[Simulate components and obtain good physical results. J _

 

 

I Study how the parts unite, establish interconnection equations] 3

 

 

Combine all components to form large-scale system. I

 

 

Simulate system and establish good physical correlation of system. I

 

 

Figure 2.1 - A standard large-scale system modeling procedure.

scale dynamic systems there is a standard modeling procedure that makes use of the results from
previous research in order to complete the analysis within a reasonable time period.” This
procedure, graphically outlined in Fig. 2.1, is indicative of typical serial—based, top-down
approaches because it is highly dependent on good representations of the system’s components
and their interactions. Hence, the steps in this procedure must be repeated until good correlation
of experimental and analytical results are obtained for the system’s components. Of course, it
would be natural to expect this standard methodology to yield accurate results for the system as
long as accurate results were obtained for its componens. However, this expectation is
sometimes unfounded even after many iterations. This is because mathematical models are only
approximations of the true physical process. When these models are combined model uncertainty

due to neglected, truncated, or approximated dynamics can cause substantial problems in system

28

behavior. Therefore, to make certain statements about a system and its components, it is

necessary to use precise, well-defined models that account for uncertainty.

2.1.1 Modeling uncertainty in systems

In all problems, small or large, linear or nonlinear, one has to evaluate uncertainty and the effects
of uncertainty on the dynamic model. An inherent trade-off in modeling occurs between fidelity
and simplicity. It is desirable to have models which closely match reality, yet which remain
easily analyzed. Obviously, solutions to this trade-off problem depend on the type of assumptions
made during the modeling process. Whether the model was arrived at by physical laws or perhaps
by measured data, these assumptions need to be evaluated and analyzed, as uncertainty is innate
to any modeling procedure.

This suggests a methodology which assumes and treats the system and its components not
as precise, single models, but rather as a set of models, inside of which the actual physical system
will lie. Hence, the dynamic system model can be viewed as a “fuzzy" description of the actual
process. This approach, referred to as set membership or unknown-but-bounded uncertainty,“
1“ has been described as a realistic technique for characterizing uncertainty. It has several
advantages, for example, it is less demanding than associated statistical descriptions because it
shows that the unknown process behavior cannot be modeled, and it is easier to use with
nonlinear systems and/or convert to a linear representation.” Thus, if we know that our nominal
model is closed and bounded by the set, then significant statements regarding stability and
performance can be made about that set. The mathematical framework for these two problem
will be discussed below.

The unknown-but-bounded uncertainty method adopted here falls into the deterministic
category,“ where uncertainty may be nonlinear (in terms of its influence on the system), and

time-varying (possibly fast), but its statistical information is never assumed. 11: In turn, all that

29

uncertain regions 2I

   

A i uncertain region

 

 

 

 

 

. ’
Dynamic System _/ Dynamic System
Parameter Space Mt Solution Space

Actual Dynamic System

Figure 2.2 - Generalized uncertainty problem where the actual dynamic system We Z.

is required is the knowledge of the set in which the uncertainty values lie, which is characterized
via certain deterministic properties such as compactness (closed and bounded). A graphical
interpretation of this framework is given in Fig. 2.2. Here we have all the dynamical system
parameters represented by A and the known or nominal parameters represented by 1,. We
suppose to know 1. not exactly, but have some information on it since it can be bounded and
nominal values can be identified. We are then interested in evaluating some model function MD.)
of this system to obtain an approximate solution to the dynamical system. If we let A represent
a normed rt-dimensional space and Z be a normed l—dimensional space, then given the operator

M, A is mapped into Z as follows:

MzA-DZ. (2.1)

We also have in this mapped region MA) = z E Z, and outside this region 2 E Z. Hence, our
‘ objective is to approximate an element MW of the space Z, knowing approximate information

about the elements X, or approximate information about an element 2 referenced from M0,).

30

Unless otherwise specified, these objectives are met by assuming that A and Z are equipped with

L, norms, and two infinity norms, L. and L.", where the latter can be written as
IIXIZ= mfx MINI. W.>0- (2.2)

Note that if an L." is used, component-wise bounds can be weighted with different values on
every element.

Figure 2.2 illustrates two particular types of uncertainty descriptions that can be classified
under the unknown-but-bounded formulation. These two type are known as: (i) parametric or
structured uncertainty, and (ii) nonparametric or unstructured uncertainty. To establish
conditions for these models, let us consider the sets of the following form to describe some
perturbation from the nominal or known model:

i) A finite perturbation set:

Z”-{z,, z,, z,,..., 2,} 6 Z (23)

ii) A parameterized perturbation set:

2' = {Mm | {xv 1,, x,,..., x3611} e z (2.4)

iii) A mixed perturbation set:

zws 2% 2' e z (25)

Eq. (2.3) is a nonparametric uncertainty description which is expressed in terms of specific single
perturbations. Nonparametric uncertainty descriptions are generally conservative, however, they
are useful for modeling small or large perturbations, unmodelled parasitic or neglected dynamics,
and system failure modes. In contrast, Eq. (2.4) represents the parametric uncertainty
description. This description corresponds to specified bounded uncertainty in the model

parameters, uncertainty with respect to the inpus. actuators, sensors, or outputs which are

31
represented directly and not lumped together as in Eq. (2.3). By assuming normed bounds on

this uncertainty description, it is possible to derive necessary and sufficient conditions, and very
non-conservative representations. Eq. (2.5) provides a characterization of uncertainty similar to
that graphically shown in Fig. 2.2. This representation is very powerful because it gives a
realistic portrayal of uncertainty by combining the advantages of the parametric and non-
parametric descriptions. In each of the perturbation sets listed above, a useful and meaningful
description will depend on its accuracy. By using the mechanistic arguments posed in this
mathematical formulation, we can norm-bound each perturbation to get 'tight" uncertainty

descriptions for the dynamic model.

2.1.2 Modeling the system in the presence of uncertainty

The importance of accurate uncertainty descriptions has been stressed in regard to uncertainty
model conservatism. We want our uncertainty descriptions to be as ”tight“ and as realistic as
possible, otherwise, mathematically-derived bounds may possibly give an invalid representation
of the real system.“ For this reason, necessary and sufficient conditions on robustness of the
uncertainty descriptions will be introduced here for a particular family of systems. These
conditions must be established to determine how uncertainty restricts the dynamic system model.
The term ”robustness“ is used in a context so to explicitly state that our uncertainty description
must be well-posed and maintain the true degree of system stability. Robustness conditions
require that even with the worst perturbed model substituted into the nominal model, the true
stability of the system will hold.

To begin let us consider the nonlinear, finite dimensional system

£0) " 115,143)
2‘0) = 80.9160 (2-6)

32

where at) is an n-dimensional state vector, u(t) is an r-dimensional input vector, and y(t) in an
m-dimensional output vector. The functions f and g are assumed to satisfy conditions for the
existence, uniqueness, and continuity of the solution of Eq. (2.6). The employment of a first-
order Taylor series expansion about an equilibrium point or a known solution of Eq. (2.6) yields

the following

x(:) = Am) + Bu(t) (2.7)
N) = Cx(t) + Du(t)

where A, B, C and D are matrices of appropriate dimension. Eq. (2.7) is a linear time-invariant,

finite-dimensional system which can be represented by a rational transfer function matrix

y(s) = P(s)u(s) (2.8)
P(s) = C(sI-A)"B +0 6 II

where P(s)E (5" and a member of the set II and I denotes the identity matrix.
Now, assuming that Eq. (2.7) represents an uncertain linear system such that in state-space

form, we have

A = A, + AA 60,, |[AA]U| sea,
B = Bo + AB 60,, ”A81“ 55b” (2.9)
C = C. + AC 600, |[AC]u| 55c”
D =- D. + AD eon, |[AD]UI std,

where all quantities with (I), refer to the nominal system model, 11,, 0,, 0c and 0,, denote
compact, bounded spaces, and 50 , 6b , 6c" and 6d,, restrict the additive uncertain perturbations
AA, AB, AC and AD to parallelpipeds within their appropriate spaces. We see that Eq. (2.9)
represents a generalized form for determining parametric uncertainty descriptions. By
substituting Eq. (2.9) into Eq. (2.8), we can postulate uncertainty descriptions for this system in
terms of a specific structure or as a relative uncertainty written as a rational Laplace transfer

matrix. To accomplish this, let us assume that each element pa in P is confined to a disk with

33

 

 

LE

 

 

 

 

 

 

   

Inverse Multiplicative
Additive Uncertainty Uncertainty

 

 

 

 

 

 

 

 

          

Multiplicative Output Multiplicative Input
Uncertainty Uncertainty

Figure 2.3 - Common perturbation structures for representing uncertainty.

radius a”(w) centered at the nominal [p,],, in the Nyquist plane

lpv-(pJ‘Il 5 a” (2.10)

or equivalently we can write

Irv-0M s 'g|(P)g| (2-11)
where a,(w) and road) are the additive and multiplicative (relative) uncertainty respectively. In
these two forms we are able to directly see how uncertainty affects the nominal model. Then
with this information we can select an appropriate uncertainty description.

If we wish to consider a structure for either a parametric and/or nonparametric uncertainty
description the following four perturbation structures are commonly used: additive (LA),
multiplicative output (L0), multiplicative input (L, , and inverse multiplicative output (Lg. These

four descriptions, which involve a perturbation L, are shown in Fig. 2.3. Initially, without

34

establishing the details between parametric and nonparametric let us consider L to be an actual
lumped perturbation. Now the magnitude of this perturbation may be measured in terms of a
bound on the maximum singular value of L, which is needed later to obtain the necessary

robustness conditions
3(L) 5 Kid) 8 max3(L), Vw (2.12)
PEI!
where the bound l(w) can also be interpreted as a scalar weight on a normalized perturbation A(s)

L(s) -- Imam, 3(A(iw)) s 1 v: = for. (2.13)
The four perturbation structures that correspond to Fig. 2.3 take on the following form
P - P. +L‘
2:215:17; (“4)
P - (I ~LE)“P,.
where the subscript (I), refers to the nominal plant model.

In particular, input and output uncertainty descriptions are useful for representing actuator
or sensor errors, neglected high frequency dynamics, or changing numbers of right half plane
zeros. The additive uncertainty description is applicable to plant errors, and the inverse output
uncertainty description is useful for representing low frequency errors and a changing number
of right half plane poles.“ Generally, the nonparametric uncertainty description l(s) will not
constitute a very tight description of the "real" uncertainty, since it reflects the relative
uncertainty r, , thus, P(s) will be larger than the actual set of all plants 11“” Roughly speaking,
Eq. (2.11) represents a better generalized form for identifying and using the nonparametric
uncertainty description because A is a full matrix. In order to achieve tighter bounds on the

uncertainty values in Eq. (2.10), we need to use a parametric uncertainty description with our

perturbation structure. Eq. (2.9), given in state-space form, is a parametric uncertainty

35

 

 

 

 

 

Figure 2.4 - A general block-diagonal bounded perturbation diagram.

description in the time domain. Numerically, these uncertainties can be identified directly, as
shown in Eq. (2.10), and written in terms of multiple individual perturbations L, of real or
rational transfer functions. If each individual perturbation L, satisfies Eq. (2. 13) then they can
be collected and placed into one large block diagonal perturbation matrix, where A = diag {A,,
A,, ..., A..}. If a perturbation structure is implemented, weighting matrices can be used to

normalize the uncertainty such that the actual perturbation L, (A,) is

L, = w,a,w,, 3(4) s 1 w. (2.15)
where W, and W2 are rational weighting matrices that must have no poles or zeros in the open
right half of the complex plane.“ ‘°

To analyze the degree of stability for both of these uncertainty descriptions, we construct
a block diagram for the system as shown in Fig. 2.4. The matrix M, called the interconnection
matrix, is the nominal system model as seen from the various uncertainties. It represents a block-

diagonal, bounded perturbation problem.‘9 The nonparametric uncertainty descriptions the

36

solution to this problem involves a generalization of the ordinary singular value decomposition
(SVD)." It is very important to determine the greatest singular value of a matrix, as it is
equivalent to the maximal ”gain“ of the matrix as the input vector varies over all possible
directions. The matrix value provides a reliable, nonconservative measure to determine whether
the robustness conditions of the model are satisfied. For the parametric uncertainty descriptions,
this measure is called the structured singular value (SSV), denoted by p, which also serves as an
essential analysis tool for verifying the robustness conditions.” “' “’- ‘°‘ Since all simultaneous
uncertainties can be put into block-diagonal form by merely constructing the associated M matrix,
SVD and SSV allow us to nonconservatively analyze the occurrence of uncertainty anywhere in
the model.

Now recall that a necessary and sufficient condition for asymptotic stability of a rational
transfer function is that all its poles lie in the open left half of the complex plane.” This
condition is easily verified for a transfer function, which has a finite number of asymptotic stable
terms, and can be extended to a transfer matrix via the Nyquist criterion.” "7 However, if our
plant is unstable, meaning it has poles in the right half plane (RI-1P), we want to retain this degree
of stability throughout the modeling process in order to improve performance in the control
design. For instance, if we are working with a reduced order model P(s), we make the
assumption that all the unstable terms of the full system are included in P(s). Hence, we want

to consider the degree of stability of a given transfer matrix to be defined by”
dim“) A max{81(rr)|a' is apole ofA}. (2.16)

Therefore, the degree of stability of the system must be equivalent for all models in the set. This
statement will be verified in the derivation of the robustness conditions on the M-A structure.
Consider L to be of the form given in Eq. (2.14), where the perturbation A satisfies Eq.

(2.13). Now if the nominal system model P,(s) is stable, M is stable and A is a perturbation

37

which can destabilize the system. However, if the nominal system model P,(s) is unstable then
M is unstable and A is a perturbation which can further destabilize the system. The following
theorem establishes conditions on M so that the degree of stability is maintained by A.
Theorem 2.1

Under the assumption that either all of the perturbations A are stable or all the members
P(s) of the set 11 of possible plants have the same number of RHP poles and if the absence
of encirclements of det(I - MA) about the origin of the N yquist D contour is necessary and
sufficient for robustness, the closed loop system given in Fig. (2.4) is stable and/or the
degree of stability is equivalent for all perturbations A if and only if the condition

Fauna» < 1, V0) (“7)
is satisfied.“ ‘9' ‘°‘
This theorem states that if Eq. (2. 17) is satisfied then no perturbation can destabilize the closed
loop M - A. Thus, the degree of stability is unaltered by uncertainty. From Theorem 2.1 we
can derive the necessary and sufficient conditions for robustness for the different uncertainty
perturbation structures given in Eq. (2.14). By comparing Fig. (2.3) with Fig. (2.4), we obtain:

1) Additive uncertainty perturbation

M = -lA(I+P°)" (2.18)

2) Multiplicative input uncertainty perturbation

M = -(I+P,)"P¢l, (2.19)

3) Multiplicative output uncertainty perturbation

M = -p,(z+p,)-v, (2.20)

4) Multiplicative inverse uncertainty perturbation

M = (um-11,, (2.21)

38

where for the nonparametric uncertainty descriptions, which follow from Eq. (2.13), all

perturbation quantities must satisfy

m.) s 1,, k = A, 1, o, E. (2.22)

When the perturbations A are restricted by a specific block diagonal structure, such as in
Eq. (2.15), Theorem 2.1 can be, in general, arbitrarily conservative. Therefore, to account for
the structure of the perturbations A, in order to achieve tight bounds on the uncertainty, let us
define the structured singular value (SSV) as an alternative to SVD. The function am), or SSV,
is defined such that p.” (M) is equal to the smallest maximum SVD of A needed to make (I - MA)
singular.“ m If no A exists such that det(I - MA) = 0, then MM) = 0. Hence, p"(M) is a
function of M and the structure of A. An interpretation of this definition in terms of the
generalized small gain theorem,“ similar to that given in Theorem 2.1, gives the SSV the tightest
possible bound on our system M.

The following theorem establishes the necessary and sufficient robustness conditions for
the structured or parametric perturbations A.
Theorem 2.2 3" “' ‘9

Under the assumption that either all of the perturbations A are stable or all the members
P(s) of the set 11 of possible plants have the same number of RHP poles and if the absence
of encirclements of det(I - MA) about the origin of the Nyquist D contour is necessary and
sufficient for robustness, the closed loop system given in Fig. (2.4) is stable and/or the
degree of stability is equivalent for all perturbations A if and only if

#(MUwD < l. Vat. (2.23)

Much like Theorem 2.1, this Theorem 2.2 states that if Eq. (2.23) is satisfied then no
perturbation can destabilize the closed loop M - A and the degree of stability is unaltered by the

parametric uncertainty description. Take note that when the perturbations are replaced by a

39

 

 

 

 

r- W2I AI “’11 r W20 Ao W10

 

 

 

 

 

 

 

 

 

 

 

A1 - diagonal multiplicative input uncertainty perturbation
A0 - diagonal multiplicative output uncertainty perturbation

 

 

Figure 2.5 - An example of a plant with weighted multiplicative input and output uncertainty.

nonparametric or unstructured description, so that, A is a full matrix, then

MM) = 3(M) (2.24)

which follows from Theorem 2.1.“: ‘°‘

It has been shown that uncertainty can be characterized using parametric and non-
parametric uncertainty descriptions. These perturbations can then be applied directly where they
occur in the model. For example, if we consider a model that has both actuator and sensor error,
we can simultaneously apply weighted multiplicative input and output uncertainty perturbations
using a parametric uncertainty description. A schematic of such an uncertainty model is given
in Fig. 2.5.

In particular, we may combine these to form a more practical, mixed representation of

uncertainty. Where, for instance, unmodelled dynamics could be expressed as a nonparametric

4o

description and uncertainty in the pole locations could be expressed using a parametric uncertainty
description. An important point to remember, whether either a nonparametric or parametric
description is used, is that uncertainty must be modeled where it occurs, and not where it might

be mathematically convenient.

2.2 A Concurrent Modeling Method for Large-Scale Systems

An approach that includes the kind of uncertainty models described in the previous section
introduces two specific problems that need to be addressed during modeling: the analysis
problem, and the synthesis problem. The analysis problem is to determine if a given set
effectively portrays the system dynamics of the actual process. The evaluation of this problem
involves experimental testing, parameter identification, and model testing with all quantitative
examinations being limited to setting hard bounds. Specifically, such a procedure may involve
overshoot, undershoot, risetime and settling time evaluation, response envelope and input
bounding, and an examination of response effecs due to disturbances, noise, or actuator
operation. The object is to learn as much as possible about the system, is components, and their
interactions. The synthesis problem, on the other hand, entails finding a nominal model and an
associated uncertainty description that optimally and accurately fis the model set. To evaluate
this problem, the fundamental properties discussed in the previous section must be imposed and
evaluated. Of course, this usually requires a procedure that can efficiently access all the available
information acquired in the preceding problem step in order to limit the decisions in the modeling
procedure and the number of iterations. Hence, a concurrent engineering method would be a
very practical format in which to make these decisions during the modeling process.

In Figure 2.1, a standardized modeling procedure was presented for large-scale systems
analysis.m This procedure is characteristic of typical iterative top-down approaches that rely on

highly accurate representations for the system’s components. It was shown in the previous

41

section that such highly accurate representations are not always necessary. In fact, a highly
uncertain system can be modeled very efficiently. Innate system uncertainty can be non-
conservatively and accurately modeled.” Undoubtedly, it would be very effective if we could
take full advantage of this trade-off between what we know and what we do not know about the

system. Moreover, it would be very efficient if we could not only do this, but do it concurrently.

2.2.1 The analysis problem

For large-scale systems, a common difficulty arises with the need to efficiently represent the
components and their interconnections. A structure that can simplify these interactions should
be hierarchical so that systems can be built from the bottom up.“ Such a structure will permit
models and data, which describe a system or a collection of systems, to be incorporated into, and
extracted from, a single variable. For instance, such a variable might contain design,
performance, and/or input specifications, in addition to the plant model, controller or both.
Furthermore, these variables are flexible in that they allow functions to be represented in state-
space form, as a transfer function, as polynomials, etc., thereby vastly simplifying user
interaction. A general schematic for a controlled structure is given in Fig. 2.6. This structure
can be formed using commands in MATLAB’s Robust Control Toolbox. In Fig. 2.6, the
system’s components are broken into the respective parts and all their ancillary components,
bounds, and specifications are branched accordingly. Consequently, this simplified user
interaction allows us to make major or minor modifications and improvemens in the branches
such that results can be readily observed in the main system tree.

All the different techniques and ways used to obtain a model for the system and its
components can usually be classified in terms of either a forward or an inverse problem
evaluation. In particular, system identification methods can be categorized as an inverse problem

definition, whereas all others tend to fall into the forward problem solving class. Unlike the

42

LARGE-SCALE SYSTEM
Performance / \ Design

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Specs ; Specs
Combined Uncertainty
Com onent 1 --------------- in"- l ----r --------------- Co onent N
p System Controller ' mp

 

 

 

 

 

 

 

Uncertainty \\ Uncertainty -lge’scrcgp

 

 

 

 

 

 

 

 

 

 

 

 

Input Stability
\ Specs Specs
Actuator Sensor Performance Design Controller / Process Model /
Specs
1 l
errts Error V Performance Structure
Specs Specs

Figure 2.6 - A hierarchical structure for constructing and completing
the analysis problem for large-scale systems.
forward problem definition class, system identification techniques are very useful, because models
are derived directly from experimental data. These models are used for verifying predicted
dynamical models, setting bounds on inputs and parameters, and improving the fidelity of a
reduced order model. The details pertaining to the different types of system identification
techniques will not be discussed here. However, generally, system identification involves an
experimental apparatus of some kind in order to measure, acquire and process data. These data
possess measurement uncertainty, in addition to numerical uncertainty in the estimation algorithm,
and in conjunction with uncertainty apparent in the reduced or assumed model structure for the
system. Nonetheless, for all the uncertainty inherent in the procedure, system identification

provides the best methods known for verifying predicted models.” “‘ Thus, it is crucial to

43

couple the inverse problem definition process with the forward definition process in order to

obtain all available information and verify a model set.

2.2.2 The synthesis problem

The evaluation of all available information occurs in this portion of the modeling procedure. It
is here where models are rejected, and successive iterations may take place before a useful
nominal model and it associated uncertainty description are found. Therefore, assuming that the
work accomplished in the analysis problem is complete, we can use this information to
simultaneously characterize our given model. Regarding model reduction, fitting, and error
analysis, it is assumed that these will all take place when using a frequency description of the
model. The frequency domain is most useful because it provides the person modeling with some
measure of the model’s response over all frequency.“ “' This is very important in the steps
discussed here because most uncertainty models have limits that vary extensively with frequency
and models that must be based on frequency weighted bounds. These uncertainty models are very
common. The frequency domain also allows us to investigate frequency dependent parameters
and model them accordingly.

A general schematic showing how the synthesis problem may be conducted is given in
Fig. 2.7. This diagram takes into consideration the fact that information about a particular
component or system can be obtained analytically, numerically, and experimentally. In most
cases, analytical methods are highly limited and solutions are usually approximated using
numerical techniques. The key in the system identification portion of this procedure is to select
a model structure and its order to fit data using an appropriate estimator. Clearly, reduced-order
models are highly desirable, thus, the nominal model and identified models should reflect this.
Usually, from the identification procedure, enough data can be obtained to develop reasonable

nonparametric models. Since nonparametric models are necessarily lumped models, these

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Analytical Results
l Forward Problem
Numerical Results 1
i Parametric
. Uncertainty
Nommal Model .
1 Mixed Uncertainty
1 Model
Identification my “33533,?“
Actual Process , &

 

 

 

 

 

 

Inverse Problem

 

$ Nominal Model ’ Final Model

 

 

 

 

 

 

 

 

 

Figure 2.7 - Aconcugnt scheme for evaluating dynamical system models and uncertainty.

uncertainty descriptions can be readily constructed."

Some time after the nonparametric models have been formed, the task of making the
uncertainty model less conservative must be executed through a parametric identification. This
process is usually conducted by changing parameters in the numerical model and comparing these
results with the actual event. In some cases, it may be necessary to add the nonparametric model
may need to be added to the numerical model in order to complete this task. However, results
will be conservative if proper care is not taken. The difficulty with constructing a parametric
uncertainty representation is that a substantial amount of knowledge, understanding, and
information about the actual process must be obtained.

Finally, the two representations can be joined to form a mixed representation. Mixed

uncertainty representations can take on many different shapes and/or forms. They either can be

45

written out in a simultaneous manner or in a sequential manner. For example, in a sequential
type representation, we allow a parametric uncertainty description to degenerate into a
nonparametric description at a particular limiting frequency. This is based on the reasoning that
we will probably know more about the model and system behavior in the low frequency range
than we know in the high frequency range. After some experimental and/or numerical analysis,
a crossover region can be identified where all details about the system’s parameters become quite
fuzzy. It is this crossover region where the uncertainty model changes from parametric to
nonparametric to compensate for such things as unmodelled dynamics, neglected dynamics, etc.
The sequential mixed uncertainty representation provides us with the flexibility to tailor our
model to give the most nonconservative results.

Discussion about the nominal model has been brief partly because of the need for an
optimization procedure to be defined. A general diagram for a model fitting optimization
procedure is given in Fig. 2.8. This schematic closely resembles the procedure outlined in Fig.
2.7 with some additional decision making details. The kernel of the optimization procedure is
to fit uncertainty sets that have been obtained using numerical, analytical, and experimental
results to some predetermined nominal and uncertainty model. Such a method must be able to
search along all set boundaries and make very nonconservative fits. For most cases, it follows
that a search over all the given sets will involve first finding the center of the set, and second
verifying that it is a good representation of that set. Obviously, this center, which can be found
using a myriad of numerical techniques, represents the nominal model. Basically, the goal for
multiple uncertainty sets is to fit a continuous model through determined center points and
minimize conservatism in the uncertainty model given the robustness constraints discussed in the
previous sections. Clearly, the rigidity of the procedure outlined in Fig. 2.8 is kept to a
minimum because the general formulation of the nominal and uncertainty model is often isolated

from that of any common problem because several objectives will often need to be addressed and

 

 

 

Numerical Results .__/ o_____.,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Parametric
i Description
Actual Process
Identification u Determine Form of
Nominal Model
i v i
Search Boundaries Search Boundaries
F Obtain transfer + Uncertainty '
function fit estimation
Iterative Reweighting r
/ Model Checks
Stability
Nonparametric Robustness No
Description
Yes

 

 

 

 

 

 

Final Model ¢

 

 

 

 

Figure 2.8 - A generalized nominal and uncertainty model optimization procedure.

traded off in some manner. Therefore, requirements for an optimization procedure must enable
the model’s intentions to be expressed, yet be able to solve the problem and enter preferences into
a numerically tractable and realistic procedure.

In a graphical depiction, consider the family 1‘ of all possible plants represented by an
infinite number of uncertain regions in the Nyquist plane, as shown in Fig. 2.9(a). Since an
infinite number of uncertain regions are shown, we only see the uncertainty in terms of an upper
and lower bound. If we accept a nominal model to be the average of these two bounds,
optimization is unnecessary and this problem becomes rather trivial. However, if we have a
representation, as shown in Fig. 2.9(b), with a finite number of uncertain regions, the

optimization procedure becomes more difficult. This is because we must first determine a point

p:

0f

47

ilmaginary (C(00))

Lower bound ,.

 

>
Real (6(0))

Nominal model

 

Upper bound

 

 

A +
U Real (6(a))

Center of

Region Nominal model

 

Nonparametric
approximation

 

Figure 2.9 - An example of uncertainty regions for a family F graphically shown in
the N yquist plane.

inside each region that is at or is near its center, and secondly determine a continuous model that
passes through each of these points. Next we optimize the fit for a nonconservative uncertainty
model that varies with frequency. If we wanted to approximate each given region of uncertainty
in Fig. 2.9(b) with a normed-bound, then we would search for the smallest circle that contains
each actual uncertain region of interest and fit it with an additive uncertainty model. Although
this gives a very convenient uncertainty representation for numerical purposes, it is at the expense

of some conservatism, because it includes regions ouside the actual uncertainty.

48

2.3 A Numerical Example
2.3.1 Analytical analysis formulation

For an example of the technique proposed in the previous section let us consider a simple cantile-
ver beam with a concentrated mass attached to is free-end. We will assume elastic deformation
to be small and model the beam using Euler-Bemoulli beam theory. The configuration of the
beam and the corresponding notation are given in Fig. 2.10. Using variational principles, an

expression for the beam ’8 equation of motion can be written as:

El—af-w (x, t) + 8v [w(x, t)] +m—a—2-w (x, t) = —m—a—2w (x, t) (2.25)
3x4 5 3:2 3:2 °
and the corresponding boundary conditions
w(0, t) = 0 gym, t) = 0 (2.26)

deformed beam
H ............ ‘
... i niwat) x

.............. r I M >

2‘1

 

 

 

 

 

 

Walt)

 

 

 

 

//////////%

 

 

//

 

? “r

"///////

\

 

 

 

 

Figure 2.10 - The configuration for the cantilever beam example.

49

32 a3 32
El——2—(w(L, t)) = 0 Er—gwut, t)) = M—w (x, t)
8x 8):

3,2 (2.22)

where w(x,t) is the lateral displacement of an arbitrary point on the beam for any given time t, E1
is the beam bending stiffness, v is the damping operator, in is the mass per unit length of the
beam, M is the mass of the attached concentrated load, and WO( t) is a rigid-body external transla-
tion applied to the beam. The dimensions of the beam are given in terms of its length L, width W,

and height H. Also, M is regarded strictly as a point mass.

To begin the modeling procedure, we initially see that an analytical solution to Eq. (2.25)
can be obtained using the separation of variables method?5 Assume that the solution can be writ-
ten as

wow) = 2¢,(x)n,(t) (2.28)

It = 1
where 4), (x) are the normal mode shapes or eigenfunctions and 111t (t) are the time-dependent
generalized coordinates. By substituting Eq. (2.28) into Eq. (2.25), we obtain

1312 ¢;V(x)n,(t) + zvitgmmm +rfi2¢,(x)ii,(t) = 47'1“). (2.29)
k=l k=l

k=1

Now let us assume that the normal modes of the undamped system, as determined by the
eigenvalue problem, can be used to uncouple Eq. (2.29). To complete the eigenvalue problem, the
damping and the forcing terms in Eq. (2.25) are temporarily excluded and a general solution is

easily verified for the free vibration problem to be

41, (x) = A'sin (Akx) +B'cos (Akx) + C'sinh (Akx) +D'cosh (Akx), k = 1,23,...
(2.30)

where A', B', C', D' are constants of integration and it], are positive variables. By substituting Eq.

50

 

 

 

 

8
7r ....................................................................................... q.
k=3
8’ ..
LIE 5’ i
is k\ k“
NIX ‘> \ ______________________________________ "
" at -i
.s‘
r<
2* "
\ :1
1’ Rama—Eh k _,
o l L
O 2 4 8 8 10
M/(fizL)

Figure 2.11 - The variation of ink with concentrated tip mass ratio.

(2.30) into Eq. (2.25) and applying the boundary conditions. the following equation is yielded:

- C.
¢k “0 ' (cos (ML) + cosh (MU)
—(sin (ML) + sinh (kkL)) (cos (Akx) — cosh (lkx)) ]

 

[ (sin (Akx) - sinh (1,3)) (cos (AkL) + cosh (2th”)

k = 1,23,... (2.31)
where the infinite values for )4, must be determined numerically from

x 47} (cos (Apt) cosh (Akx) + 1)
k ' M(cos(AkL) sin/1(ka) - cosh(AkL)sin(AkL))

 

k = 1,2,3,... (2.32)

Once this equation has been solved for M, we can complete the eigenvalue problem by

determining the undamped natural frequencies

2 _ 2 El
mic - Mimi k= 12,3... (2.33)

The resulting values for k = I, 2 and 3 are given in Fig. 2.11 for a large range of tip masses. It is

obvious from the diagram that structural variation can be considered quite negligible for masses

fire
DUI:
ing
We 1
mix

“hit

 

 

 

 

 

O 0.5
Normalized Length (x/ L)

 

// \\.

-1 * '\ /
Mode 3

 

 

 

O 0.5
Normalized Length (x/ L)

M/ (52L) = 0,0 ——

Figure 2.12 - First four normal mode shapes for various concentrated tip mass ratio.

Under the assumption that the normal modes of the undamped system are unaffected by

51

=1.9——

[0,L] we can use the orthogonal relationship to form

 

Mode 2

 

 

 

0.5
Normalized Length (x/ L)

 

Mode 4

 

 

 

0 (is
Normalized Length (x/ L)

=3.9—0—

= 2.9 ..............

which acting in a manner proportional to the square of its natural frequency.

mm + 2v,,fi,.(r> mime) = Nth), k=12,3.-..

k=1

1

five times as great as the beam. It is important to determine whether the beam can actually sup-
port a mass of such magnitude and maintain the specified deflection constraints. The correspond-
ing mode shapes for various tip masses when k = 4 are given in Fig. 2.12. Clearly, from Fig. 2.12
we see that when no concentrated mass is added the mode shapes for the first four modes are at a

maximum. The effect of adding inertia at the end of the beam suppresses normal mode vibration,

damping, then by multiplying Eq. (2.29) by 41, (x) and integrating over the domain of the beam

tit:

[Lt

WI}

61;

Wire
SOIL}

L’lili.‘

Mitt

52

where

L
v", = [than [¢,(x)]dx, r,k=1,2,3,... (2.3411)
0

and the generalized modal forces are represented by
L
N, (t) = Jok (x) wodx =-= rtkwo, k = 1,2,3,... (2.34b)
0

Take notice that Eq. (2.34) represents an infinite set of coupled ordinary differential equa-
tions, so that, in general, damping produces coupling of the normal coordinates. In order to con-
tinue with this analytical solution and obtain values for the response of the beam the assumption
will be made that the damping operator is a linear combination of

64

v = orEI— + m 2.35
8x4 13 ( )

where or and B are constant coefficients. In this case, the equations can be easily decoupled and

expressed as

11,,(1) + 2gkwkfiku) + minkm = fikwo, k =1,23,... (2.36)

where E, denotes the modal damping factors of the cantilever beam defined such that g, < l. The
solution to Eq. (2.36) can be readily obtained by means of the Laplace transformation. With the

initial conditions set to zero, the nonrational transfer frmction from input to output is

co ..

G(s) = Z( "i J (2.37)

2 2
”I s +2§kmks+tok

 

After some algebraic manipulation of Eq. (2.37) the response of the beam becomes

t
1 —§kcok(t-r

n,(r) = ,0; mm ’sin(m,,,(t-r))dt. k=1,2,3,...
0 (2.38)

uh

“th T:

53

where
11,,(0) = 0 M0) = o

and 0),“, is the natural frequency associated with the damped system, which is simply

 

 

aw2+B 2
com: w,./<1—::> = m,/[1_[ 2;) j) (239)
k

The general response for the beam is obtained by inserting Eq. (2.38) into Eq. (2.28), with

the result

I

°° 1
(m) = ( )—JN I)
W k§1¢k x wkdo k( 2

“52m: (t—‘t

)sin (coma -1) )dt (2.40)

where ¢k(x) is given by Eq. (2.31) and N k( t) by Eq. (2.34b). Let us consider the case in which the
input has the form

W0 = "W0§(t) (2-41)

where W0 is a constant acceleration and g (t) the unit step function. Hence, we arrive at the step

response when Eq. (2.41) is substituted into Eq. (2.40) and integrated

( ) i —W0A’
w , t =
x k: la'mfi (cos (ka) + cosh (1,2))

(cos (ML) + cosh (KkL))—(sin (lkL) + sinh (MD) (cos(7th) — cash (2%)) ]

[1—__1__e-§‘m*tcos (mkdt - 9,9]

 

[ (sin (Akx) — sinh (Akx) )

/(l _ gi) (2.42)
where
g" 2 42
6k: atan [———-—) ( . a)
Jo -§i)

A’ = (f—k) (cosh (lkL) + l — cosh (ML) (cos (ML) - 1) ) (2.42b)

the com
“Wise
UP mass.
the bean

WIOC

54

Table 2.1 - Material and geometric properties for the flexible beam configuration.

 

Property Specifications
aterial Alurgnum Sheet

31.2 cm

2.60 cm

0.1143 cm

68.5 GPa

0.7133 g/ cm

0.00124

0.001

 

I

3

 

 

 

 

laugh

 

 

 

 

 

 

“5395!

 

 

L
B, . U¢§(x)dx+n7’:’z¢g(r.)] (2.42.)
0

Let us consider a beam with the material and geometric properties given in Table 2.1. With
the constant acceleration W0 set to 1 cm/ s2, we can use Eq. (2.42) to determine the transient
response of the bearn’s tip (x = L) for various conditions. These conditions will include various
tip masses, variations in the elastic bending stiffness, various damping factors, and variations in
the beam dimensions. It is anticipated that damping will be very small and vary slightly from
case to case and mode to mode. Therefore, the damping parameters will need to be bounded

CXperirnentally as determined in the proceeding formulation.

In Fig. 2.13, the analytically determined response of the cantilever beam for a step input,
via Eq. (2.42) and summed over k = 1 through 10, is given for various tip mass ratios of 0, 1, and
2- Clearly, the amplitude of the tip decreases with increasing tip mass, a trend which was evi-
denced in the eigenvalue results given in Fig. 2.11. Effectively, with infinitely increasing tip

mass, the response of our cantilever beam with the free-end boundary condition degenerates into

 

 

 

55

 

   

 

‘ _ a q 2 . 2

\
012 : #fld
new NNN
LLL \ MUM
W.IM\.IM\ .\ ..... 272
/// .‘ 0.0M
MMM = = __
.W ....... v BEE
_W 5 y .
\ :1. __m
r ........ __

..... I

v

..... .r.

 

1
Time (sec)

 

 

 

 

 

 

 

 

 

0.8

Figure 2.13 - Step input response for various concentrated tip mass ratios.

 
 

I‘
.U

AEEV “568335 AEEV EmEmumamB

Figure 2.14 - Step input response for various beam bending stiffness.

Figure 2.15 - Step input response for various values of B with a constant.

Displacement (mm)

Figure 2.16 - Step input response for various values of or with B constant.

Displacement (mm)

56

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.8 I l V T Y7 I

— B = 0.001

- - B = 0.05

m B = 0.10
0.6- a.=0.00124 ‘
0.4 -
0.2

o

-o.2 -
-o.4 -
'0“’o 0.5 1 3.5

2
Time (sec)

 

 

 

 

 

 

0.6L ' i i i i ‘ ‘ i ‘
0.4 ~ ‘ -
0.2 ~ _
0
oz
I
-o 4 — u =o.ooors
- — a = 0.0005 ! , y
“W a = 0.00124
0 a [5: 0.001 1 l
"’o 0.5 1.5

1
Time (sec)

 

57
one with a pinned boundary condition, thus tip displacement will approach zero. In Fig. 2.14,
‘ beam response is given for variations in the bending stiffness of up to :i: 20% of the value given in
Table 2.1. These results show this parameter significantly contributes to the magnitude of the
beam’s amplitude. Realistically speaking, 20% is a rather large variation, however, it serves to
prove how uncertainty in assumed constant values can alter final results substantially. Figures
2.15 and 2.16 are presentations of a step input response for variations in the constant damping
parameters or and B. It is assumed that for convenience, these remain constant for all modes and
are innate characteristics of the beam’s material. It is evident from Fig. 2.15 and 2.16 that or influ-
ence on decay is much greater than B but only over a limited range of values. This is partly
because or is proportional to the square of the modal natural frequency, yet it is limited because Ek
becomes imaginary at various modes when or is greater than 0.00124. We see that having two

parameters to adjust, which gives us more freedom when putting bounds on damping.

2.3.2 Experimental analysis formulation

The primary objective for this experimental portion of the analysis problem is to determine
bounds on the actual response of the beam. These bounds, in conjunction with the analytical
results, will be used to obtain a useful uncertainty model for our final reduced order, linearized
system. Accordingly, an Aluminum beam with the same dimensions as given in Table 2.1, and
configuration as shown in Fig. 2.10, was utilized in this example. To provide excitation a vibra-
tional shaker was employed. The beam was clamped onto a rigid mount that was attached to this
shaker. The concentrated masses used in the experiment were securely attached to the end of the
beam via rubber contact cement. It was important that a good bond was established between

these masses and the beam.

The experimental apparatus was set-up and the beam’s length was once again measured due
to variation in the clamping procedure. It was found that the measurement uncertainty was

approximately AL = :t 1 mm. This uncertainty was also consistent for the width measurement.

58

 

 

 

 

 

 

 

'13 7 — Mn'nL=o
_.. -28 MniL=tI9267
.2. --Mu§L=I.347I
E -43 . MImL: 2.16I2
"’ -58 "
an
'0 '-
g -73 t‘\ 3
a '88 3...”; \ 3' r ‘ ‘
g 403 .. «5:7
-|18 1.. ' ..-.v J . _.,_....~~ _~. __ ' . . \ .....
. i . .. _. Vr‘ "t'i‘dr W "" A tam-n . - . .. '
'3 I i F r r I
U 40 80 I 20 I 60 200 240 280 320 360 4“)
Frequency (Hz)
180
. _ ---MIniL=()
I35 ' S ': . 5 ._ M I iirL = 0.9267
J I 3 “Cm '- --MH7|L=|.347|
90 . : .1 1‘ ~~~~~ -M/riiL=2.16I2
3045*. El '2. .. It.“ .
: .- . . . ’. .3 .
z i“ war
90 ll i l I “Wt ”1““ '
-l35 K ' ”IL!“ 1‘ .JWww J .
I! i L-
"30 I I I T T I l I l l
0 40 80 I 20 I 60 200 240 280 320 360 4(

 

Frequency (Hz)

Figure 2.17 - Receptance response for various mass ratios.

Three small aluminum blocks of approximately 20.18 g, 29.98 g and 48.10 g were used for the
concentrated tip masses and the mass per unit length of the beam was 0.713 g/ cm. The measure-
ment cum in these readings were determined to be approximately AM = :t 10 mg. Additional
sources of uncertainty were also investigated. Uncertainty attributed to variations in the elastic
bending modulus over the length, a non-rigid tip mass attachment, excitations at the natural fre-
quencies of the mounting apparatus, beam warping, and fluctuating conditions of the boundary
conditions. Obviously, such uncertainties are not only small, as compared to measurement noise
and calibration error, but are difficult to quantify directly. Thus, they can be lumped together in

the form of a nonparametric model which will be determined later:

To measure the natural frequencies and damping factors for each mode of interest wideband
and nanowa periodic excitations were applied. Over a wideband range of 400 Hz the magni-

tude and phase of the receptance was measured. Receptance is a measure of the displacement of

59

Table 2.2 - Measured cantilever beam natural frequencies.

 

 

 

 

 

 

 

Frequency Concentrated Tip Mass Ratio

(Hz) 0 0.9267 1.3471 2.1612
001 9.25 4.25 3.85 2.8754
(02 58.625 45.91 38.875 44.25
(03 162.10 127.5 120.38 118.125
034 290.75 220.63 213.25 174.25
(05 473.25 347.13 325.21 225.75
(06 578.63 426.88 428.0 359.15

 

 

 

 

 

 

 

the structure under a constant input load normalized by that load.102 For various ratios of tip
mass to beam mass, the receptance of the beam atx = L is illustrated in Fig. 2.17. It is easy to see
from Fig. 2.17 that phase information contains extreme noise which greatly increases with addi-
tional mass. This noise is especially apparent in the very high and very low (rigid-body) fie-
quency regions. Wrth increasing tip mass the resonant peaks decrease and shift to the left as they
are dominated by strong regions of anti-resonance. This is also evident in the diagram for the
phase also. The greater the mass-dominance the greater the negative slope of the frequency
response magnitude.

For purposes of identifying these resonant peaks narrowband excitation was used. The first
six natural frequencies for the various tip masses are given in Table 2.2. These values are
observed to be in close agreement qualitatively with Fig. 2.11, where a general decrease in magni-
tude occurs with increasing tip mass. Table 2.3 presents a comparison between the calculated and
measured natural frequencies of the beam for M/ 51L = 0 and M/ 51L = 2.1612. As seen from
these results the analytical data provides an upper bound and the measured values provide the
lower bound on (on. The difference can be attributed to damping and energy losses through exci-

tation of the out-of-plane and torsional modes. Moreover, friction at the clamped boundary inter-

60

 

 

 

 

 

 

 

 

 

 

Frequency Ml rTrL = 0 MI rTrL = 2.1612
(Hz) Calculated Measured Calculated Measured
(01 10.132 9.25 3.023 2.8754
(1)2 63.50 58.625 45.621 44.25
(03 175.81 162.10 145.28 118.125
(04 347.43 290.75 301.74 174.25
(05 575.98 473.25 515.07 225.75
co6 860.42 578.63 785.29 359.15

 

 

 

 

 

Table 2.3 - A comparison between measured and calculated
cantilever beam natural frequencies.

 

face may add to these losses. Of course, this type of time-dependent boundary condition would

be very difficult to model.

The damping ratios were obtained from Fig. 2.17 using the half-power point method.102
This method is convenient and used here because it is valid for small damping. Results were
determined for various mass ratios and are given in Table 2.4. We see that damping tends to

remain fairiy consistent with increasing beam mass but decrease with increasing mode number.

Table 2.4 - Measured cantilever beam damping ratios.

 

 

 

 

 

 

 

 

 

 

 

Damping Concentrated Tip Mass Ratio

Rafi“ 0 0.9267 1.3471 2.1612
£1 0.0204 0.0204 0.0209 0.0205
{3,2 0.0173 0.0185 0.0193 0.0195
5,3 0.0096 0.0099 0.0102 0.0101
5.4 0.0088 0.0091 0.0095 N/A
£5 0.0071 0.0054 N/A N/A
£6 0.0008 N/A N/A N/A

 

 

 

61

 

 

 

 

 

 

 

 

 

 

— M I 5'11. = o
.. M / 51L = 2.l6l2
g
«f
3.). -t
T I F l l l l
2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
Time (sec)

8 —— M I 5L = 0

h _1 ' """"""""" M I 511. = 2.]6'2
- 4-i
g 0 —IL‘.-:;V- \ I A p ...... ‘
'73- -2 i
;
x .4 .-

6 -1

.g 7

T i T i i T l . l
I) 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
Time (sec)

Figure 2.18 - Transient impulse response of the beam tip for two mass ratios.

At the higher modes for the larger masses damping ratios were indeterminable. The results for the
modes greater than k = 3 are uncertain because the measurement signal was very weak. Hence,

further calculations will be based on the assumption that damping ratios remain constant fork 2 3.

Experimentally determined transient responses were obtained for both an impulse and step
input. These responses were completed for tip mass ratios of 0 and 2.16, respectively. The object
here was to set bounds on the overshoot, undershoot. and settling time for a pre-selected range of
tip masses. Figure 2.18 is a depiction of the transient response due to an impulse of approxi-
mately 30 N. The vast differences between the two cases are apparent, where tip displacement for
the 0 mass cases has an overshoot of about 6 mm and an undershoot of -2 mm. When a mass is
attached to the tip, displacement is greatly reduced. As seen in these results, since oscillations are
still occurring even after 8 seconds, damping must be very small. It took approximately 12 to 15

seconds for the tip to settle to its original position.

Amplitude (mm)

62

 

 

 

 

——M/fiiL=O
1.5.1 -------------- MlmL=2.l612
l-—4
.5 -L l I l I I
c ' . . t a . o o ' ' ' I O 0 . . ' . . o I
()_ . a. a a ' . _ u . . - ' . . ' . o a o o ' I o . 0 ° 1
~ .
. 5—1 % - ' ‘ I
[—1
-i,5 -—4
fl 1 l T I I T l l I I
It ()8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
Time (sec)

Figure 2.19 - Transient step response of the beam tip for two mass ratios.

In Fig. 2.19 the transient response of the beam is given for a step input of magnitude equiv-
alent to W0 in the previous section. Results from this experiment are very close to the analytical
results presented in Fig. 2.13. By comparing the two cases using a linear systems approach, we
find that the 0 mass ratio case has a period of T8 = 0.1204 seconds and a damping ratio of approx-
imately 0.0087. The 2.16 mass ratio case has a period of T8 = 0.3704 seconds and a damping ratio
of approximately 0.0097. If we assume the first mode to be the primary mode that dictates the
beam’s response, we see that the damping factors obtained from Fig. 2.19 are much less than
those given in Table 2.4. However, these values are within a particular range when modes 1
through 3 are considered. This difference may be attributed to the averaging of the measurements
that was used to obtain the receptance results illustrated in Fig. 2.17. The period gives us some
indication of what response should be expected from a reduced order model. Also, since the natu-
ral frequencies and damping factors associated with each tip mass are different. the goal of the
synthesis problem will be to construct a nominal model that can be used to reflect the general
response of the model. We want to develop a nominal model that is able to accommodate uncer-

tainty in the tip mass. The motivation for this is similar to that used when formulating a model for

63

 

 

 

 

 

 

 

 

Natural hequency (Hz) Damping Ratio 7
Mode

1 0.008

2 44.0 64.0 0.008 0.020
3 115.0 175.0 0.008 0.015
4 174.0 300.0 0.008 0.015
5 220.0 475.0 0.008 0.015
6 355.0 560.0 0.008 0.015

 

 

 

 

 

 

 

Table 2.5 -Physical bounds on model parameters.

a single-link robot arm where the mass of the object being grasped is known only within a speci-

fied range.

The conclusion of the analysis formulation involves evaluating the above information.
After an examination of the analytical and experimental results we can set bounds that reflect the
actual dynamics of the beam. TWO specific sources of uncertainty need to be interpreted. These
relate to general uncertainty in the dynamic model and specific uncertainty in its parameters.
Essentially, because this example is rather simple, and it has been determined that the first mode is
the primary mode that dictates the beam ’s response, we do not have to account for neglected
dynamics. However, modeling errors in the high frequency dynamics cannot easily be neglected.
These errors, which are apparent in the analytical model, are attributed to the assumptions made
regarding small deflections. To accurately model higher frequency modes, we would have to
abandon Euler-Bemoulli beam theory for the more complex Timoshenko theory.119 Hence, it will

be necessary to fit our model with a frequency weighted lumped perturbation.

The second source of uncertainty occurs due to variations in the model parameters. Since
we identified several parameters earlier, such as tip mass uncertainty, measurement uncertainty,

and material specification uncertainty, these parameters will be considered. However, because the

64
modal natural frequencies and damping factors are explicitly governed by these parameters, we
can reduce the number of parameter variations to two where we allow the mass ratio to be
bounded between [0, 2.25]. Hence, the following bounds on parameter variations are given in
Table 2.5, where the superscripts L and U refer to the lower and upper bounds, respectively.
Obviously, from Table 2.5, the measured values serve as lower bounds whereas most of the calcu-
lated values serve as upper bounds. Also, since the measured values are included within the com-
pact set, we should be ensured that the actual system lies within this set. After evaluation in the

next section, these bounds could be reduced to attain a more conservative uncertainty model.

2.3.3 Synthesis problem formulation

Referring back to Eq. (2.28), Eq. (2.31), and Eq. (2.36) we find that our uncoupled set of govem-

ing equations can be transformed into a set of general equations written in state space form:

paw" 243)
1=CJ ('
where
2c = {110.710.111.08 ...... ,nknik} 25(0) = {250} (1433)
B = {We} (2.43b)
y: {W(L,t)} (2.430
and
' 1
0 1 0 0 0 0
0 0 0 1 0 0
0 0 40% ~2€1m1 0 0
0 0 0 0 0 0
A = (2.43d)
0 0 0 0 1
0 0 0 40% "251031:

 

 

65

(2.43e)

C=[L 0 0 (L) 0 ......... ¢k(L)0 ] (2.430

and go is a vector of initial conditions, and no, 1'10 represent rigid body motion of the beam.
Equation (2.43) resembles the general form of the LTI system given in Eq. (2.7) with the excep-
tion that it is not finite-dimensional because it contains an infinite number of modes, all of which
cannot be modeled. In order to continue this formulation, a truncated model will be derived using
only four of the modes identified in the analysis. It is assumed that this primary mode model
includes all characteristics in the residual modes of the beam’s dynamics. We then have a four
mode model a state vector containing 8 modal elements and two rigid-body elements. Also, take
note that this problem is single-input single-output (SISO), since we are only interested in tip dis-

placement for prescribed base motion.

If we assume that Eq. (2.9) is a valid representation for our uncertain system, then Eq.
(2.43) can be rewritten as a nominal model with bounded perturbations on its matrices. Let us
consider parametric model uncertainty. From Eq. (2.43e), we see that the tip mass directly affects
the system matrices both as a parameter and through the system’s eigenvalues. In order to model
this system so that all variations in tip mass are accounted for, we need to select some nominal
Operating conditions. It has been determined that tip mass ranges between a bounded mass ratio
of 0 and 2.25, therefore, calculated primary mode eigenvalues must range between [1.875,
1.0233]. For simplicity, we can determine a nominal operating condition by finding the mean of
this range and substituting it into Eq. (2.32) to obtain a nominal mass ratio. The mean for the
eigenvalue range was found to be 1.4491, which corresponded to a nominal mass ratio of 0.4421.
Therefore, after substituting this value into Eq. (2.43), we see that nominal matrices Bo and C0 are

easily established based solely on this nominal operating condition assumption.

66

 

 

Imaginary (0(in

 

 

 

 

Real (G(i(0))

Figure 2.20 - Fuzzy N yquist diagram associated with plant set.

Prior to constructing the nominal model, we transformed our state space model into a sum
of four transfer functions, as given in Eq. (2.37), for the nominal operating condition. The trans-
fer functions were used to get a physical understanding of the uncertainty region topology over a
desired wideband frequency range of 0 to 400 Hz. This was accomplished by mapping the uncer-
tain transfer functions to the Nyquist plane. The parametric in the modal natural frequencies and
damping ratios is represented by the fuzzy Nyquist diagram in Fig. 2.20. From Fig. 2.20 we see
that our plant set is open loop stable by the Nyquist criterion.29 It is also evident that the gain
margin for all possible plants and all modes is infinite, which is consistent with the experimental
data in Fig. 2.17. The resolution of Fig. 2.20 does not specifically show the parametric uncer-
tainty over the desired frequency range. To better observe the uncertainty regions, a local Nyquist
plot was constructed and is given in Fig. 2.21. This plot graphically shows the eflect of paramet-

ric uncertainty by nine individual fuzzy regions located at 0) = [.01, 400].

67

 

 

 

 

 

 

 

 

0.2 f
0.18 . ................ :E _
0.16 L ..................... ”:3: 4
2 0.14) i, ..~""..-j-.'-f~f‘°“ g .
.8 '-T '12-'31 3' 55 E
:— 0.12* ‘ 322.. ."'Z‘/"2" _ .53.. ' E «-
o ‘ L -,::_'~:.:i-: ’_..'- ‘~ ”...I: I E d
g ‘ /" ’/'- ' E; "z'
a? o 08 , ...... 1 .‘ '_) "::(r"-';.;.:."'-)f.,.’.::: ...... .gkiifi: ' .1
”tr.” ""3" “‘4 ......
E ...‘1' inn-q. --'.".:,...:Z ..... z,- ,,,,,,
0.08 t 1."."1' g ,,,,,,,,,,,, .5. ............ -I
0 ‘0‘ [ .3 _
0.02 $ _______ é ................. d
o l j l l 4
-0.1283 41.1282 -0.1281 -0.128 -0.1279 43.1278 -0.1277
Real (G(i0)))

Figure 2.21 - Local fuzzy Nyquist diagram showing parametric uncertainty.

Now looking closely at matrix A reveals that it is composed of the two uncertain parame-
ters. By focussing on the system matrix A the nominal model can be easily constructed by simply
using the averages of the upper and lower bounds given in Table 2.5. In most cases, however, an
assumption of this kind could lead to unwanted model error in critical frequency ranges. To avoid
introducing more error than what already exists, two additional models were constructed for com-
parative evaluation. The first nominal model is based on an average of the individual poles and
zeros in the model set transfer function. The second nominal model is an average of the individ-
ual elements in the A matrix. These two nominal models are refened to as 1 and 3, respectively,

and the parameter averaged model as nominal model 2.

Fig 2.22 is an illustration of the fuzzy pole-zero map for this system, which has four stable
poles and three zeros. The uncertainty regions associated with the zeros and poles are represented
by the dotted and dashed lines, respectively. It is clear from this map that our uncertain system
includes several nonminimum phase plants. Uncertainty descriptions that include RHP zeros

must be carefully examined and constructed.101 Figure 2.23 is a detailed representation of the

68

 

  

 

 

 

3000 . . . T .
. ' XPOIcS
m om
\ _
20001- \ H : <
m .
m1 .
_ __ _ ._ H ;_ _ _ _ __ _
1000+ m any 0..., 71 ~
.,_,_, .m@g____<m ________'-_.
x
<
m 0... ............................................................................................. ..
E .
- i&“.__‘ _‘ i’"——— ”‘1
4000+ I __ mm ______ _ __. _ _ JED- _J r
Uncertaintychions
w ‘ f forpole-zero
-2000L I p i -
/ , :
6&0 .
-300 1 1 1 1 1
- 50 -100 -50 0 50 100 150

Real Axis

Figure 2.22 - A pole-zero map outlining global uncertainty regions included in the plant set.

 

 

 

 

 

 

2500 1 1 . e r .
2000* ‘
Xr—f — —————— -X“
1500 \ “ \ x x \ V -
1000L *— — _ ‘ 7‘ -
2
500+ x k j x ‘
5; WF— at
E x
3 ° , ’4'“
E -500r x ”Hi—xx ‘
2(— -—/-ar
/ / 8 /
-1000~ ,4_ _ _ _,( ~
4500*- 3 J
-2000+ x x xActualPoles .1
EAvengePoles
-25 r r 1 r r
030 ~25 -20 -15 -10 -5 0 5
Real

Figure 2.23 - Pole map showing averaged poles and their associated uncertainty regions.

69

 

0' T V f l I T I

_ _ Nominal Modeu
Nominal Model 2 ‘

 

...,”... Nominal Model 3

dBUn)

 

~100r \

-1201 ~—

-140r

 

 

 

_1 l. I I L I I I
600 50 1 00 1 50 200 250 300 350 400
Frequency (Hz)

 

Figure 2.24 - Bode plots for the three 4 mode nominal models.

pole uncertainty and of the location of the averaged poles. Note that rigid-body mode causes one
open loop pole to exist at the origin. Bode plots of the tluee four mode nominal models are illus-
trated in Fig. 2.24. Close agreement between the three models is apparent in the range between
the first two primary modes. However, the difference between model 2 and models 1 and 3
becomes significant with increasing frequency. Models 1 and 3 are fairly consistent over the
Spectrum because they represent averages for the entire plant set. These results are also qualita-
tively consistent with the receptance plot given in Fig. 2.17. It follows from these results that
model 3 will be adopted for the 4th order nominal model.

Now that the plant set has been bounded and the nominal model defined. consider model-
ing parametric uncertainty in detail. To include parametric uncertainty in the model, we need to
formulate the plant using the book diagram in Fig. 2.4. Rewriting the uncertain equation (2.43) to

fit this structure requires the perturbation AA be placed into the following form

_ _ 2.44
A’ZwéJA-Aol _ AA _ WZAW1 ( )

70
where matrices W1 and W2 play the roles of placing the parametric uncertainty block A into the
diagonal structure given in Eq. (2.15). By substituting Eq. (2.44) into the uncertain version of Eq.
(2.43) we are able to write the following state space equations that are associated with the M-A

SU'UCIUI'CI

j = Aogc+Boy+W22

2=C2
z=AWg
A = diag {Awf’ A93, ...... , A2§1w1,A2§2(02’ ...... } (2.45)

where the subscript 0 refers to the nominal model. To obtain a diagonal matrix for A, the singular-
value decomposition (SVD) can be applied to AA. As a result, SVD provides two unitary matri-
ces which can be made equivalent to W1 and W2 such that the product of WZAWT is identically
AA. From Eq. (2.45), the overall augmented plant matrix and system matrix M written in state

space form becomes

AoBoWZ
G(A,s)scoo 0 =M (2-46)
wlo 0
T 00
A= A B=BW C= cw D:
1.] 1.1 [.1 [,0]

This form is a linear fractional transformation of Eq. (2.43). It is used to represent the transfer
function M “seen” by the uncertainty blocks as a transfer function from y to 2.36

Two step responses for an input of 1 cm /sec2 were conducted for the upper (maximum AA)
and lower (minimum AA) perturbations in the A matrix. The simulation results, illustrated in Fig.
2.25, are compared to the step response for the nominal model. The strong influence of the small
damping and high natural frequency in the lower bound response is similar to the experimental
results depicted in Fig. 2.19. In comparison to the nominal model and upper bound, we must also
note the large amplitude of oscillation about 1.4 mm in the upper bound. These large values inev-
itably show that the physical bounds on the uncertainty perturbations are large as compared to the

nominal model, possibly including more plant sets than are necessary. To determine the relative

71

 

 

 

 

3 T , r j
- -UpperParuneuicBounds
—"NominalModel
“‘mLoweerrneu-rc' Bonn
2.51:1 d:
l
I
Il 'I
I 1
II 1
2H1" '\ a
A I it H
E II ‘I It I“ \
g l: 'I ll ', 1‘ ”
0-0 I l \
=1.“I"" I 111! ‘
ms'HI'I‘IHIIH‘I‘In/ \\\ , .........
E I'lllll|' I"\’\/\JV\/”’\‘ q
0) II (I l' l\“ \I I
‘5} 1.11,“ Hi ‘1 1 3
._ H
O I
3.1:. ; .....
0'5 .5 .- 3 5: :': 37
0-2..-.‘.' ..... .4
0 1 2 3 4 5
Tlme(sec)

Figure 2.25 - Step response of the lower and upper parametric uncertainty bounds.

 

 

 

 

 

 

 

        

 

 

 

 

i --—UpperParameuicBounds

14g LowerParametricBounds
i
i

12% j
I
I
01101!
1’ Ii
2 ..
'E i!
8’ 81!
2 !I
to II
.2 I-
E 6%
q: l
C: i
i
4*.
i
i
21:

J. ............................................. ,‘ ........................

o' -’. . "-. - -
0 1 00 1 50 200 250 300 350 400

   

Frequency (Hz)

Figure 2.26 - Multiplicative uncertainty for both lower and upper perturbations on A.

72
magnitude of these additive perturbations the multiplicative form of uncertainty, given in Eq.
(2.11), was generated. Results for the upper and lower bounds are presented in Fig. 2.26. These
results verify the observations made in Fig. 2.25. Notice that both relative magnitudes exceed-
ingly cross unity at or near every mode past the first two modes. This is because our model
describes well the steady-state and low frequency behavior, but becomes highly inaccurate for
high frequency inputs. Furthermore, since we have included zeros crossing and located in the
RHP in the uncertainty description, we see Irl 2 1 at these modes, which implies that the phase is
completely unknown. The very sharp peak in the lower bound near the origin is caused by the
presence of the rigid-body mode. Since the rigid-body mode is located at the origin in the
Nyquist plane, it causes Irl = oo. This is unsuitable because rigid-body motion for this structure
is typically known, so that uncertainty at or near D.C. should be negligible. Consequently, these
results leave us two choices, to either go back and reformulate the physical parametric uncertainty
bounds, or to reduce the effect of the perturbation through the use of weighting matrices. The sec-

ond path is chosen for interest.

The results from Figures 2.25 and 2.26 suggest excessively large perturbations from the
nominal model. To evaluate stability robustness of the model due to these perturbations, we
employ the singular value method (SV or SVD) and the structured singular value method (SSV).
Figure 2.27 depicts upper SV and SSV bounds on M, normalized with beam length L and gener-
ated over a range of 0 to 400 Hz. It is evident from this diagram that the SV gives a slightly more
conservative upper bound than does the SSV. This is expected since SSV gives the tightest possi-
ble bound on M .47’ 101 Figure 2.27 also indicates an upper bound greater than 0 for both methods.
Thus, the magnitude of the perturbation causes the condition for stability robustness, as given in
Theorem 2.2, to not be satisfied for all A S 1. Earlier, it was determined that this model set is sta-
ble for all parametric variation, thus o““”““’ (G (s)) = 0. However, because the magnitude of
the perturbations include more plants in the set than were intended, we see that M is fully destabi-

lized by A, as determined by the conservative SV results and partially destabilized using SSV.

 

Theor
fippiie
were
thgi
dtpep

LAB,

73

 

100 . r r r

80 r ‘

Gain (dB)

 

 

 

'20 x Singular Values -
C Structured Singular Values
"‘0 so 100 1 50 200 250 300 350 400

Frequency (Hz)
Figure 2.27 - A comparison of the SV and SSV representations for M.

Since it is undesirable to make our system unstable by adding uncertainty, in order to satisfy
Theorem 2 .2 and to reduce the relative magnitude of the perturbations weighting matrices must be
applied to M. Three different weighting matrices were imposed on the uncertainty model. and
were compared. For convenience the same weight was applied to all A). Included in these three
weighting matrices were a constant valued model, a frequency dependent model, and a frequency
dependent multiplier scaled model that was obtained using the SSV subroutine provided by MAT-

LAB.36

The constant valued model was able to reduce the upper bound only for values below 10%
of the initial AA, as shown in Fig.2.28. Obviously, the use of this model would demand that either
the physical parametric bounds given in Table 2.5 be reformulated, or that the nominal model be
reduced to a lesser number of modes and complemented with a large additive nonparametric rep-

resentation. The frequency dependent model is given by the equation:

0.144s + 2880

D“): 2s+5600 ’

s = id) (2.47)

74

20 r - 1 f T r I
L - Constant Weight
1

 

 

—— Frequency Dependent Weight 4
— — Optimized Frequency Dependent Weight

Gain dB

-100

-120

 

-14O

 

 

4600 50 100 150 200 250 300 350 400

Frequency (Hz)

 

Figure 2.28 - SSV stability evaluation for the parametric uncertainty model.

This model imposes a weight of about 40% of the initial AA in the low frequency range (first two
modes), which gradually increases with frequency and rolls-ofl at approximately 65 Hz to reach a
final value of 65% of AA. As seen from Fig. 2.28 this model provides a much better uncertainty
representation because its reduces conservatism in the peaks. The weighting strategy allows us to
take advantage of what we know, such as the dynamics in the low frequency range. We see a
gradual increase in uncertainty of up to 200 Hz, approximately of the same frequency at which

experimental error caused poor results.

The third model is based on a multiplier method that uses convex optimal diagonal scaling

to accurately predict system robustness. Ideally, if M is diagonalized by a real positive mauix D

such that

u (M) s 6(DMD‘1) VD e Qd (2.48)

then "DMD-1”... can be made much smaller than II M II“, to provide very tight bounds and accu-

rate predictions of the system robustrress.36’ 101 From Fig. 2.28 it is apparent that this model

75
gives the tightest possible bound on parameter uncertainty, and not at the expense of physical
accuracy. However, one disadvantage with this method is that it is computationally demanding to

determine a new D at each frequency over the desired range.

Notice in Fig. 2.28 that the SSV for all three weighted cases decreases for frequencies past
225 Hz. This shows that uncertainty perturbations in this frequency range are highly attenuated
by the system dynamics. However, from our experimentally measured response we know that
two more natural frequencies exist between 220 and 400 Hz. Although these modes were not
included in the model, their presence cannot be overlooked. In addition, the uncertainty that was
identified as a result of the small deflection assumption, the assumed boundary conditions, and

other small errors, is investigated here using lumped uncertainty perturbations.

To accomplish this we employ a nonparametric uncertainty description. In Fig. 2.3, four
perturbation structures for representing nonparametric uncertainty are given. Here we will inves-
tigate and compare the additive and multiplicative output perturbation structures. The additive
structure is appropriate for modeling uncertainty due to unmodelled dynamics and uncertain RHP
zeros. On the other hand, the multiplicative structure is more appropriate for modeling neglected
high frequency dynamics and input or output errors. For additive uncertainty, the following

weighted perturbation was applied

5.5.sz+1408.1-1-137766
LA”) = 2 6
s +1001s+2.ll4e

 

A (s) , Vs = in) (2.49)

and for the multiplicative uncertainty the following weighted perturbation was applied

12.95 s + 0.7
4s2 + 3.4s + 8.34

 

LM(S) = NS). Vs = i0) (2.50)
such that A satisfied Eq. 2.13. Both uncertainty weights were determined by fitting continuous
stable minimum phase state-space realizations through selected data based on percentages of the

nominal model. For instance, in the low frequency region where only small variations are

76

 

0,‘ ................................................................................................ q

— — Nonparametric multiplicative uncertainty
Nonparametric additive uncertainty

 

-50

 

 

 

 

 

53-100 -
s
(U
o
4501' \\ / \-___,// \\ -1
-200r \-/”"?
-250 I l ' 1 1 L L
0 50 100 150 200 250 300 350 400

Frequency (Hz)

Figure 2.29 - SSV stability evaluation for the nonparametric uncertainty models.

expected. less weight is needed. In the high frequency region, error and uncertainty are typically
large, thus, system response can be relaxed using more conservative models. i.e., models that con-
tain more plants in the set. A suitable model was then fit so that uncertainty increased from 10%
to 65% in the crossover region between 100 and 200 Hz. Subsequently, the order of these

weighted models was reduced to the form given in Eq. (2.49) and Eq. (2.50).

The results from the nonparametric uncertainty evaluation are depicted in Fig. 2.29. It is
clear from these results that both models meet the stability bound given in Theorem 2.1. Accord-
ing to the multivariable stability margin,36 the primary resonance peak of the multiplicative
uncertainty model can tolerate a 29.2% perturbation before instability occurs, whereas the addi-
tive model can only be as large as 26.7%. This may occur because the weighted additive uncer-
tainty model greatly relaxes the response beyond the fundamental natural frequency. On the one
hand, it does this at the expense of increasing the order of the model from eight to ten. On the

other hand. the reason for the large conservatism in the multiplicative model is because uncer-

77
tainty is lumped together with perturbations that are relative to the nominal model. Although fre-
quency weighting slightly improves this situation, it cannot account for all the unnecesssary plants
admitted to the model set. The additive model directly affects nominal system response as
opposed to being relative to the nominal system. This idea follws the parametric uncertainty rep-
resentation where additive perturbations were applied to the system A matrix.

The results from the structured and unstructured uncertaiinty descriptions give som indica-
tion of the need for a mixed uncertainty model. We see that tight bounds are achieved in the low
and mid-frequency ranges using the parametric uncertainty description, which was formulated
from actual bounds on physical parameters. On the other hand, we find the nonparametric uncer-
tainty description advantageous for relaxing the nominal model in the high frequency range. By
combiniing these two models in a sequential manner we can construct a very realistic description
of the model uncertainty.

For the sake of completeness, let us consider a sequential approach for the mixed uncer-
tainty model instead of the simultaneous approach. For the sequential mixed uncertainty we must

impose a bound on the parametric description such that,

(A, s) Va) < (0‘

0(5) = G(s) ‘70)th s = i0) (2'51)

where 0), denotes the parametric bounding frequency. It is at this frequency where the parametric
uncertainty model fully transforms into the nonparametric uncertainty model. Prior to to, all RHP
zeros must be constrained to the LHP. This will ensure that all primary modes are properly
damped out. At (0, and beyond all RHP poles that exist in the parametric uncertainty description
must be constrained to the LHP. This ensures that our plant is stable at high frequency. For plants
with truncated or unmodeled dynamics this statement is important to the stability of the entire

model. It follows that at to, the nonparametric uncertainty description must be limited by

Amganll mlil’nmlG (A, s) I H a ‘35 (2.52)

78

0 '— l l 1 T I I
— - Transition Frequency Bound
— Additive Nonparametric Plant Variation

 

~20

d

fl—' "r—

-40

-60 .

-80

Gain (dB)

-100

-120

-140

-160

 

 

 

-1 80 I l l 4 I l L
0 50 100 1 50 200 250 300 350 400
Frequency (Hz)

 

Figure 2.30 - Variation of additive nonparametric uncertainty and frequency bound.

where k), is the high-frequency gain of the plant and e is the excess of poles to zeros of G(A,s).
Equation (2.52) signifies the lower limit of the parametric uncertainty description as a) approaches
infinity. Therefore, for G (s) e l" to smoothly transition from G (A, s) e I‘, it must be bounded

by the following relation

max

. k
A6 QFIIG(“D)II S —£;-

(0

Va) 2 0) (2.53)

t

 

 

where (21- is the set containing all nonparametric plants G(s).

In this example. k,,/0r approaches a gain of -37.47 dB /decade as 0) goes to infinity for a
pole-zero excess of unity. In Fig. 2.30, the variations of the perturbed nominal model for the addi-
tive nonparametric model is shown with the bound given by Eq. (2.53). We see that the final
intersection of the gain limit and the bound occurs at 237.8 Hz. If we select this frequency as the
transition frequency our additive nonparametric uncertainty model satisfies Eq. (2.5 3). Hence, the

transition is smooth between the parametric and nonparametric uncertainty descriptions. It is also

79
evident from Fig. 2.31 that if the bound on the right side of Eq. (2.53) was imposed a large
amount of conservatism would be included in the nonparametric model. As our nonparametric
model shows this large amount of conservatism is unnecessary and can be limited. Of course, in a
realistic problem we probably would move the transition frequency to the intersection after the
second primary mode. This would be most useful, as a practical problem would most likely be

reduced to include two primary modes instead of four.

2.4 Chapter Summary

In this chapter a procedure for modeling large-scale structural systems was introduced and
applied. This procedure is practical because it captures all aspects of the system while translating
them into an appropriate working model. We find that there is no one standard or simple way of
doing this because models are unique to the problem at hand. Thus, the method proposed here
assumes and treats the system as a set of models instead of a single system. This calls for a nomi-
nal model to describe the typical behavior of the system and an uncertainty model to describe per-
turbations from the nominal model. Modeling systems in the presence of uncertainty have been
shown through introduction of a general mathematical framework. Is is also shown that this pro-
cedure is very compatible with concurrent engineering. For example, during initial design,
parameters are very uncertain and imprecise. We can develop uncertainty models that gradually
improve as the design matures and the amount of imprecision and uncertainty subsides. In this
regard, a model is simultaneously developed through each stage of the design process. Finally,
our procedure is applied to a cantilever beam example. Using results from experimental and ana-
lytical analysis, a nominal model and parametric and nonparametric uncertainty models were
developed. These uncertainty models were then combined to form a single, mixed uncertainty
model. This example gives conclusive evidence of the usefulness and efliciency of mixed uncer-

tainty models and the concurrent design procedure.

EXPERINIENTAL ANALYSIS FORMULATION
FOR A DISTRIBUTE-PW STRUCTURE

3.0 Introduction

In the field of structural control, the need for good models to support robust control system
design has been indicated."- '2' “‘ Experimental system identification is one way to achieve this
goal. It can be used either to observe certain aspects of predicted dynamics or to verify the
reduced-order mathematical models needed to establish controller designs. Thus, through
carefully controlled experiments, reduced-order models of relatively high fidelity can be attained.
Such experiments would be designed to determine ranges for specific material, structural, or input
parameters, to estimate nonlinear system component bounds, or to verify fully-assembled
mathematical models. Therefore, we see that experimental system identification is a useful way
to provide the valuable information needed in the analysis problem of the concurrent modeling
procedure. However, system identification of distributed-parameter structures poses a unique
challenge for experiment design.” ”‘ The highly resonant dimensionality of the structure
imposes limits that are attributed to noise, deficiencies in model-order selection, and poor
correlation of higher frequency modes." All experimental techniques have some random
variation or error which must be assessed to establish the significance of the results. If the

models obtained from identification are to serve as descriptive references for simulations, it is

80

81

very important to minimize as well as investigate uncertainty. This is particularly important when
there is no general agreement on the physical mechanism of the observed behavior, nor on the
properties of the structure. In this case, poorly designed identification experiments will have little
efl’ect on any derived mathematical model and uncertainty may be too large to accurately quantify
or bound. On the one hand, if the problem is the selection of a constitutive equation that
accurately describes the relationship between a structure and its properties under certain
conditions, this can be resolved by fitting data to several different constitutive equations and by
making some comparative measurements." On the other hand, if the problem concerns the
interaction of the system’s components with its surroundings, many models need to be evaluated
and/or fit before accurate conclusions can be drawn and bounds can be determined.

For the electrorheological (ER) fluid structure considered in this study, a combination of
many aspects make any identification process cumbersome. The dearth of reliable and repeatable
testing methods for ER fluids has hindered attempts to formulate a collective mechanical property
database.m Since simple, quick, and convenient test methods are not currently available,
imperfections in the different models used for ER fluids make comparative conclusions difficult.
Thus, ER fluids must first be individually tested and fit to an appropriate rheological model in
most cases. Secondly, the integration of the ER fluid within the structure imposes certain
restrictions on the identification measurements. Assumptions must be made with care so that the
salient features of the system’s behavior are addressed in detail. It is plausible to assume that
only through an analytical and experimental identification, and a numerical correlation, will
critical sources of error be revealed.

The objective of this chapter is to begin the analysis problem in the mathematical modeling
procedure with an experimental formulation. This will entail an identification of the behavior of
free-edged plates composed of an ER fluid actuator core. In order to provide an accurate

description and bounds on parameters, a mathematical model for the ER fluid actuator is

82

developed. These descriptions will facilitate the analytical and numerical portions of this
investigation. A review of the ER fluid mechanism begins this chapter. Then a mathematical
model of the actuator is constructed to accommodate analytical and finite-element methods. A
numerical example of the actuator model is presented in order to demonstrate its capabilities.
Finally, a nonparametric experimental identification procedure is conducted for two ER fluid
actuated plates with rectangular and square geometries. Identification will involve an evaluation
of the dynamic response of the structure for zero electrical potential and while undergoing various
time-dependent voltage excitations. In conclusion, responses for the electrical and dynamic

behavior of the ER fluid are also considered.

3.1 Electrorheologlcal Fluids

3.1.1 Rheology of suspensions

When a large concentration of solid particles, called the solute, is dissolved in an insulatory liquid
solvent, and the size of the particles are much larger than the molecules of the liquid, the
admixture is defined as a sol suspension.” In a suspension, three kinds of forces coexist to
various degrees. These forces are of colloidal origin due to the electrical interaction between the
particles, the ever-present Brownian force, or (thermal) randomization of the particles, and the
local viscous forces acting on the particles." Thus, the presence of the solid phase alone
markedly affects the rheological properties of the original solvent to give the suspension unique
qualities of its own.

If we impose an identical deformation to the solvent and then the suspension, a significant
difference will be apparent between their measured stresses.” ‘3' 5' These differences are
attributed to the remarkable changes in viscosity, deformation, and rate of deformation as a result

of the solid-solid and solid-fluid phase interaction. A viscosity versus shear rate curve for a

83

Newtonian Behavior

 

Log Viscosity

non-Newtonian Behavior of a Suspension

 

 

Log strain rate

Figure 3.1 - A strain rate versus viscosity curve for a typical suspension.

typical suspension is given in Fig. 3.1. As compared to the common Newtonian fluid in the
schematic, the suspension shows a non-Newtonian decrease in effective viscosity which is known
as shear thinning. Shear thinning results from the aggregation and separation of solid particles
during steady shearing. At high and low shear rates, the suspension displays the Newtonian fluid
characteristics of the solvent as evidenced by the two plateaus. Of course, such factors as
concentration of the solute, temperature, pressure, particle size and particle surface quality all
have considerable effect on the suspension viscosity, as well as its response to deformation.

At rest, the suspension assumes a state of equilibrium where the particles tend to form a
structure that possesses minimal energy transfer. An innate characteristic of these fluid systems
is that the particles always arrange themselves so that the net forces are in mutual equilibrium."
This arrangement can vary according to the dominance of the three interacting forces. On the
one hand, if the particles are very small, forces acting on the particles are largely of Brownian
nature and the particles arrange themselves to form a thermodynamic equilibrium. On the other

hand, when the particles are large, forces are of colloidal form and the equilibrium arrangement

84

is either electrically attractive, or repulsive. Thus, when colloidal forces are electrically attractive
the structural arrangement can vary from near-spherical flocs to individual pearl-like strings.
When the colloidal forces are repulsive, the particles distance themselves from one another to
form a dispersion or a more structured pseudo-lattice dispersion.“ ‘3

Upon deformation, particle energy transfer in a dispersion increases as an uninterrupted
sequence of structure disassembly and arrangement occurs. Clearly, the rheological properties
of the suspension are strongly dependent on these considerations, as shown by Fig. 3.1.
Therefore, any modification of the suspension, either by adsorbtion of surfactant materials onto
the surface of the particles, presence of an activator phase, or the introduction of an electrical
potential across the fluid, will have direct consequences on the response of the particle structure,
and, the suspension’s rheological properties.

Essentially, ER fluids represent a class of sol suspensions that exhibit large modifications
in particulate structure when subjected to an electric potential. In general, ER fluids consist of
a particle constituent, a dielectric fluid component, a surfactant, and, in some cases, an activator
phase.” Observations of ER fluids at rest give indication of a dispersion, since repulsive
colloidal forces dominate particle interaction.“ 2" ”7 When an electrical field is applied to the
dispersion, the particles readily become attracted to one another as a result of polarization.
Simultaneously, as particles polarize and become charged dipoles, the dispersed solid phase re-
orients, re-structures or flocs to form column-like strings or bridges between the electrodes.
These strings define a new structure and equilibrium state that has very different and unique

rheological properties as compared to the original suspension.100

3.1.2 The electrical behavior of ER fluids

Conventional electrostatic theory indicates the existence of a uniform field between two electrodes

Separated by a dielectric medium, Yet as shown by Felici,’o when a dielectric suspension is

 

+

 

 

 

0

Total Potential ................

Interaction Potential (Volts)

van der Waals

 

 

 

Surface-Surface (distance/ particle radius)

Figure 3.2 - Representative colloidal interaction potential over particle surface separation.

inserted between two electrodes, the electric field is complicated by numerous local non-uni-
formities. Such local non-uniformities in the suspension affect particle polarimtion and
conduction, and cause field warping and very large particle motion. Furthermore, local non-
uniform field effects can continue for several minutes after the imposition of a steady-state global
voltage.“ We find that the local non-uniformities in the electric field contribute to the
mechanism that governs the behavior of an ER fluid and the formation pearl-like particle strings
that are held together by strong intenparticle forces.

Tb predict the behavior of the dispersion and its structure, accurate information about
particle interaction is required. TWO of the most important colloidal forces in a dispersion at rest
are the van der Waals potential, which is derived from the attractive molecular forces existing
at the atomic level, and the electrostatic potential. Comparatively, the former potential is strongly
attractive, whereas the later is effectively repulsive. In Fig. 3.2, the balance between these

interactive potentials is given for a dispersion at rest. This balance determines the apparent

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No Electric Field Electric Field Applied

Figure 3.3 - Double-electric layer theory for ER fluids.

pseudo-structure behavior of the ER fluid. For instance, a large repulsive electrostatic potential
causes the particles to form a stable-ordered lattice structure due to the disassociation of ionic
groups on the particle surfaces or the adsorption of ions from the solvent.“ "3' ‘3’ Conversely,
a large van der Walls potential stimulates agglomeration, and, in highly concentrated suspensions,
promotes aggregation to form a very disordered flocs.” In Fig. 3.2 the strong attractive
minimum of the total energy can lead to agglomeration, however, a repulsive maximum will
retard this attraction causing particles to separate. This gives qualitative indication of the
random, disordered and dispersed behavior of a dispersion and the properties of the ER effect.
It is generally accepted that the electrorheological phenomenon originates from particle
polarization induced by an electric field.” ‘°°’ no. "3 Currently, most models that describe particle
polarization can be categorized into three specific theories: i) a distorted double-layer theory, ii)

a water surfactant theory, and iii) an exclusive particle polarization theory. These three

87

interpretations have been summarized in a paper by Weiss, et at.139 Only for completeness, will
each theory be discussed here in brief.

The double-layer theory is graphically shown in Fig. 3.3. This theory involves the
distortion of an ionic layer surrounding each particle, causing each particle to align itself in a
direction of opposing charge. The water surfactant theory proposed by Stangroom,‘” suggests
that the minimum requirements needed by a material to exhibit the ER effect include a
hydrophobic liquid, hydrophilic and porous particle, and moisture. The moisture acts as a bridge
between particles, where ions in the water shift the amount of water surrounding the particle to
act as a 'glue".‘” Although water may be important for some ER fluid systems, recently, several
anhydrous fluid systems have been developed suggesting that the mechanism for ER activity may
be inherently attributed to the chemistry of the fluid constituents and not the apparent moisture
content of the solid phase. Treasurer, et 01)” suggests that this mechanism is dependent on the
intrinsic polyelectrolyte chemistry of the dispersed particle. The final theory is based on the
detailed theory of polarization5 to explain the five different stages of columbic particle attraction.

The importance of permittivity on the ER fluid dispersion subjected to an electric field
must be stressed. The very large difference apparent between the perrnittivities of the particles
and of the fluid solvent is responsible for polarization and structure formation within the ER
fluid)” Both structure and polarization are dependent on the electrostatic potential between
particles and on the non-uniformity of the electric field. Electrostatic potential is dependent on
the magnitude of the particles’ surface charge and dielectric constant of the solvent. The
adsorbtion of conducting surfactants on the nomconducting dispersion permit particle polarization.
Upon application of an electric field, the surface carries a charge introducing a formation of the
dipole moments of the disperse particles. For a structure and orientation of the dispersion to
exiSt, the particles must have a relative dielectric constant of sizable difference as compared to

the surrounding base fluid. This large difference between the two dielectric constants enables the

88

 

 

 

 

 

 

 

(c) orientation and
structure formation

(a) dielectrophoresis (b) spontaneous rotation

Figure 3.4 - Changes in the structure as a result of dielectric polarization of the suspension

particles to move towards regions of higher field strength, severely straining the interfacial layer
formed around the dispersed particles. The observed structures in the ER fluid originate due to
the induced separation of the charges on the surface of the structure. Origination is determined
by the interface of the necessary double electric, hydrate, or adsorbtion layers.“ ‘°°- ”9

At the macroscopic level, observations of an ER fluid in the presence of an electrical field
show the ability of the solid phase to create a strong structural skeleton or bridge between the
electrodes. This structural skeleton is manifest by particle electrocoagulation which is caused by
several varying mechanisms of particle migration. The polarization of the particles plays a very
important role in the structure formation and orientation as illustrated in Fig. 3.4. First, the
neutral particles are polarized. As a result, they experience torques while remaining stationary,

not being impelled to translate in a direction toward either electrode. Next, the dipoles tend to

b

d:

SE

T212
Cha
sis;
bah
0f If.

fluid

621;] ‘

89

align themselves to achieve a state of minimum energy in the electric field. This is called
dielectrophoresis. Dielectrophoresis, as shown in Fig. 3.4 (a), occurs as a result of field effects
on the translational motion of the neutral or weakly polarizable particles. ‘°° Simultaneously, as
more particles polarize, individual particle rotation is observed, as shown in Fig. 3.4 (b).
Finally, once the particle layer is polarized by ion adsorbtion, the strain on an individual particle
creates an electrostatic dipolar force between adjacent particles as they are forced to form the
aggregate bridge structure shown in Fig. 3.4 (c). It is this increase in the interacting dipolar
forces that results in an increase in the relative motion of the solid phase while new bridges are
being formed. Consequently, the continuous motion of the particles during aggregate bridge
destruction and reformation results in noticeable rheological property changes of the ER fluid
with changing electrical field. Rheological behavior of an ER fluid is discussed in the next

section.

3.1.3 The rheological behavior of ER fluids

An arbitrary material containing particle suspensions of colloidal dimensions, may, in some cases,
show viscoelastic behavior, and in other cases show viscoplastic behavior.” 7" '7 The differences
between the two materials are quite large, especially in terms of their respective mathematical
relations for stress, strain, strain rate, and viscosity. ER fluids, on the other hand, show
characteristic features of both types of behavior. They encompass linear and non-linear
viscoelasticity in certain areas of low electric field intensity and strain rate, and viscoplastic
behavior in regions of high stress and strain rate. It is because of this complex inter-dependency
of the rheological parameters on the external electric field that the mechanism by which an ER
fluid can be modeled is severely altered. Therefore, a complete rheological model will be
manifest by the range of the corresponding ER fluid’s application, and will depend on electric

field as well as deformation rate and magnitude, and thermodynamic effects.”

90

 

1"
2
H
In

Shear Stress

Increasing
Electric Field

is

Pre—Yield Post-Yield Shear Strain

 

 

 

 

 

Figure 3.5 - A schematic of observed shear stress/strain behavior of an ER fluid.

ER fluids behave in a slightly Newtonian manner only in the absence of an electric field.
When thus activated, the rheological properties of an ER fluid show non-Newtonian behavior
such as shear-dependent viscosity and a yield stress.“ ‘3' ‘5' ’7 Unlike Newtonian fluids, an ER
fluid can support a stress with no flow. Thus the minimum stress required to cause an ER fluid
to flow is termed the static yield stress. This maximum stress has been demonstrated in
numerous experiments clarifying that it must be a non-zero value“- 2" m' '27 In Fig. 3.5, a
hypothetical representation of the shear stress-strain curve is given for quasi-static defbrmation
of an ER fluid subjected to various uniform electric potentials. Illustrated here is the elastic-limit
yield stress, 1’, , the static shear stress, 1, , and the dynamic yield stress, r,I which is equivalently
the stress in the limit of zero shear rate.” 3' The dynamic yield stress is observed as a dynamic

equilibrium between the processes of structure destruction and formation. Wrth increasing

91

 

 

 

 

 

 

 

 

 

102—
14 ------- I ---------
T ~l -------- n ---------- a -------- u ------- I- -------- n ---------- lI---
. I .
. _____________
. . . ......... D" .0
A10"". 0 o o o O 0 . . . ............ f3" 0
«3 .............. C] o
n. .......... E]
T; -11 --------- '3" . E], 0
g -------- o
g o
100_ O—'—0 OVOltS
H 20Volts
Bingham Model H 100 Volts
' """" ' 600Volts
10-1 llllllrlllllllllllllllllml
10'l 10° 101 102

Shear rate (S'l)

Figure 3.6 - A figurative description of the shear stress versus shear-rate Bingham model

electrical field intensity these parameters also increase with the most noticeable change occurring
in the static and dynamic yield stresses. In solid mechanics we find that for a complete removal
of a stress exceeding 1, , the material never recovers and suffers a permanent strain, 7,, where
this strain signifies the elastic-plastic transition of the material. For ER fluids, this transition has
been defined as a 1% offset yield from an observed linear region.139 Evidence has been given
that this behavior is representative of most ER fluid materials,” m ‘27 however, it is possible that
these pre- and post-static yield stresses may not occur in this exact order if they exist at all.

An ER fluid behavior can be described by the two regimes of pre and post-yield. Over
the past few decades, the post-yield regime has received the majority of attention. A schematic
of this rheological constitutive behavior during post-yield is given in Fig. 3.6. The Bingham

model,” ” is the simplest model that provides a sufiicient description of this visc0plastic behavior

1 = 74(V) mi (3.1)

where u, is the Bingham viscosity or plastic viscosity of the fluid and V signifies the applied

sin
a ll
the
but
yiel
for

maj:
S % '1

00m

92
electric field. The Bingham model recognizes that the yield stress defining the onset of flow is

observed to change with an increase in electric field. Although, we cannot generalize the
behavior of all ER fluids by Eq. (3.1) since it is only an idealized representation, it has proven
very useful in the development of specific post-yield applications.” ”5

The response of ER fluids in the pre-yield regime has been a subject only of a limited
number of investigations.‘39 The pre-yield regime, depicted in Fig. 3.5, is defined by a yield
strain and static yield stress, 1, , and a complex shear modulus G. In Fig. 3.5, the pre-yield
regime is shown as a simple representation of a linear viscoelastic material. However, several
investigations have concluded that a viscoelastic representation of an ER fluid would not be
simple to model using linear viscoelasticity?“- ”°- “3 In accordance with linear viscoelastic theory,
a material is linear, provided its complex shear modulus is dependent on frequency alone and not
the shear amplitude.“ An ER fluid’s complex shear modulus is not only dependent on frequency
but also on strain amplitude, thus it acts like a highly nonlinear viscoelastic material during pre-
yield. Moreover, an ER fluid displays such behavior over a limited range of deformation, and
for a moderate range of frequencies.“ °°- 7°- '7' “3

The kind of applications that research has focussed on have shown that temperature and
moisture content play a large role in the rheological properties during ER activity. For a
majority of dispersions, the ER effect only occurs if a conductive additive, such as water, (over
5%) is adsorbed onto the surface of non-conductive solid particles.“ ”- '3‘ ER fluids of this
composition are hydrous mixtures and are usually termed wet fluids. For wet fluids, although
moisture plays an important role in the polarization and structure formation, it has been
recognized that a characteristic disadvantage of all hydrous fluid systems is their substantial
reduction in ER activity upon heating. Consequently, the principle motivation behind anhydrous
or dry fluid research has been requirements for higher temperature applications. The concepts

basic to wet fluids also apply to dry fluids, however, the present knowledge is quite premature.

93

3.2 Constitutive Model for the Actuator

One inherent physical limitation of an ER fluid is that its operating conditions dictate the use of
two embed electrodes of opposing polarity. This characteristic imposes important and obvious
restrictions on a prospective active control design by requiring that the structure have either negli-
gible electrical conductivity or some type of insulatory material containing the fluid between the
two electrodes. It follows from this observation that a structure with negligible electrical conduc-
tivity would be an ideal candidate for ER fluid embedment. However, for structures with electri-
cal conductivity, or structures composed of various compartmentalized fluid actuators, the
inclusion of insulation within the structural domain is necessary. Insulation would be used to con-
strain the fluid within or restrain the fluid from interacting with portions of the structure. Hence,
the insulation is an important material to be considered in the actuator design because it interacts
so closely with the fluid. It is not the goal of this section to develop a new constitutive model for
an electrorheological fluid, but to develop a mathematical model for the ER fluidfinsulatory actua-

tOI'.

3.2.1 Preliminary considerations

Unfortunately, the models developed in this thesis have some limitations that exclude them from
being totally general and applicable to all ER fluid structures. These limitations are a result of the
lack of knowledge associated with the complex behavior of the materials. Since we seek equa-
tions that accurately describe material behavior for specific conditions, and with a minimal num-
ber of parameters, empirical models will be developed to address these needs. Empirical models
are quite useful for this situation because materials can be significantly affected by such variables
as shear rate, temperature, pressure, concentration, and time of shearing. When specific variables
are fixed in a controlled experiment, the data collected should accurately reflect behavior for such
conditions. The disadvantage here is that the flmctionality and versatility of the mathematical

relation has been limited.

byl
fitl

ted

film.

qUtnc

94

The materials identified in this section will include an ER fluid and an insulatory material.
The insulatory material used throughout is a natural sheet rubber elastomer that was manufactured
by Dana Corporation. Elastomers possess a unique combination of physical, electrical, and per-
formance characteristics that make them good materials for a broad range of applications. In par-
ticular, they can be classified as nonconductors, since most have volume resitivities on the order
of 5x1010 ohms-cm.138 In this application, the electrical parameters were researched in order to
evaluate specific losses within the structure due to the high voltages applied across electrodes.
The dielectric constant for an operating frequency of 1kHz is 8, and it has a dielectric strength of
450 volts /mm.138 The possibility of power loss through capacitance was considered, however,
for the size of insulator materials included here these losses are quite negligible. Nevertheless, in
an actual structure these parameters may have long term effects such as dielectric breakdown.

heat-aging resistance, current-arc resistance, and hygroscopic induced electrical eflects.

The material properties of the elastomer are considered to be linear viscoelastic and readily
represented by a storage modulus. loss modulus, and loss factor. The most important effects on
the material properties of the elastomer are temperature, frequency, dynamic load, static preload,
and chemical contact.102 However, for this isothermal analysis, only the effects of frequency are
considered since the modulus of an elastomeric rubber always increases with increasing fre-
quency.” ‘33 Material data for the insulate in simple shear is given in Fig. 3.7. Normal stresses
in the rubber will be determined by a constant Young’s Modulus given by E = 1.225 MPa. An
effective range of temperature will assumed to exist at 20°-25° Celsius. Given this information, a
model can be empirically fit for the shear and loss modulus to yield an expression for the complex

shear modulus:
0(a)) = G’(m) (1+jn’(m)) (3.2)

where, the shear modulus G’ and the loss modulus n’ are,

6’ (0)) = 461.9129 + 8.468500 — 0.00210)2 (3.2a)

95

 

 

Y YVIVI l l rrrn

0’ (PSI)
q. ""3. r n
3 i
.._ 11’
32 '.

J
n
‘
Arnnm 4L LJAJA

 

 

 

 

 

 

23::
; :
”I a L 4 1 4 11A “'1 4 r L 1 LL i L
la la or ..r ..x e .2 u
Frequency (Hz) Frequency (Hz)
(a) Shear Modulus (b) Loss Factor

Figure 3.7 - Isothermal shear modulus for the insulate material.

n’((o) = &(22.571767m—O.3277812) (3.2b)

The least-square parameter fit in Eq. (3.2a) for the shear modulus data that was presented in Fig.

3.7 is given in Fig. 3.8. The density of the insulator, p in, is approximately 1376.62 kg/m3.
3.2.2 Identification of an electrorheological fluid

Of the diverse selection of ER fluids available, a model will be developed for a simple hydrous
fluid. This suspension comprises a weight-per-weight fraction of 45 percent Argo cornstarch and
55 percent vacuum silicone oil manufactured by Dow Coming and has a density, of, , of 970.0
kg/m3. Experimental rheological data for the ER fluid were recorded in a previous investigation
that employed a Rheometric 800 mechanical spectrometer that was outfitted for high voltage
applications.55 Steady shear measurements were conducted at temperatures ranging from 24 to 29
degrees Celsius for shear rates sparming 0.1% to 100%. The non-Newtonian viscosity of the fluid

is given in Fig. 3.9. These data effectively represent anticipated behavior of suspensions. In the

96

 

3.4 t f r

 

3.3» l + Measured data -
— Curve fit

3.2? A

 

 

3.1T _

2.9’ ‘

Shear Modulus (Log(PSI))
+

2.? _

2.7+ ,

—-

+

 

 

4L 1

 

2.8 l l 1 I
-1 41.6 0 0.5 1 1.5 2 2.5

Frequency (Log(Hz»

Figure 3.8 - Shear modulus curve fit for the insulate material data at 25 °C.

case of an ER fluid, this behavior is illustrated for a range of various measured electrical poten-
tials. The upper region at low shear rate is followed by a shear-thinning region, which is then fol-
lowed by a flattening-out region, where near constant values beyond 100% are expected. The
small range of shear rates demonstrates the limited scope of these previous experiments. Steady
shear measurements of the shear stress were also made for shear rates spanning 0.1% to 100:}.
These experimentally determined shear stress versus strain rate data are given in Fig. 3.10.
Clearly, the particles in the fluid become more difficult to separate with increasing voltage, as is
evidenced by the larger shear stresses. Moreover, at low shear rates, observations lead us to

believe that the stress of the fluid is independent of shear rate.55 This is indicative of the previous

discussions pertaining to the apparent yield stress possessed by the ER fluid.

A least-squares fit that minimized the L; norm of the difference between the experimental
data presented in Frg. 3.9 and Fig. 3.10 and the curve fits was completed. Over the given range

of shear rates the following functions for the viscosity and shear rate were determined

97

 

Viscosity (Log(Pa.s))

 

 

 

 

Shear-rate (Log(s‘1))

Figure 3.9 - Experimentally determined non-Newtonian viscosity for a hydrous ER fluid.

2.6

Shear Stress (Log(Pa))

 

: —— -'.— ~~ = = z s

   
 

 

 

 

I : . '
. ' ../
oooooooooooooooooooooooooooooooooooooooooooooooooooooooooo sol-c Inoooounc|IOIII0I no.0... 0d
.4 i
i c.
/
u
: /:
: v :
L ........................................................................ ............... -—l
'4
e '
) Z
O
3
O I
pa.......o-...--....-nou.I-o...o-cecon-oo-oo-oo-uadonnoooo can... noncong; oooooooooooooooooooooo --I
I l
i
I I
3 3
I I
i i
I O I
I ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo .coco-nun...:euoccoo-oua; oooooooooo
I I
. ; Ovolt/mmg
I I I
3 i '
I
l l l A l l

 

l .
'-2 -1 .s -1 -05 o 0.6 i 1.5 2

Shear-rate (Log(s“))

Figure 3.10 - Experimentally determined shear stress for a hydrous ER fluid.

98

n = Ale3 + Azéz + A3e' + A4é3V +A5é3V2 + A‘srlav3 + A713'2V3 +A8éV3 +

A962V+Aloész+AlléVz+A12éV+A13V+A14V2+A15V3

(3.3)

o = 518'3 +529? + 538' + s4e3v+sss3v2 + 566w3 + 37¢er3 +s,,.c:v3 + (3 4)
S9é2V+Swé2V2 + SuéV2+S12éV+ sumsmv2 + $15V3

where the A’s and S ’s denote constant parameters, if is the strain rate, and V represents the applied

voltage or electric field. The resulting fits for both ER fluid parameters are given in Fig. 3.11 and

Fig. 3.12. To note here, the fine mesh represents the curve fit and the coarse mesh denotes the

experimental data that is given in Fig. 3.9 and Fig. 3.10.

The transient electrical behavior of an ER fluid is very important for power consumption
analysis and time response estimates. Since ER fluids consist of conductive particles surrounded

by a dielectric medium, they essentially function as leaky capacitors}.39 Although, it is the leak-

viscosity 10"(Pa.s)

 

    

0.5
o -
voltage (RV) 2 strain rate 10"(1/s)

Figure 3.11 - Curve fit comparison for the non-Newtonian viscosity of the ER fluid.

99

250 \

200~

if

shear stress (Pa)
in 8
o O
1 I

o
1

 

_.r
"018
\L

100

    

80
0.5 40 60
20

0
voltage (W) 0 strain rate (1/s)

Figure 3.12 - Curve fit comparison for the shear mess of the ER fluid.

age of charge between particles that results in the measurement of an electrical current through the
ER material. To model this behavior, an assumed equivalent linearized lumped resistor-capacitor
circuit, such as the one illustrated in Fig. 3.13, can be used. This approach has been used by sev-
eral authors,121 to define a current-voltage law for the fluid using identification techniques. Here,
voltage measurements were taken using a 0.018 inch probe placed between two flat plate elec-
trodes located 1.36 mm apart. The hydrous ER fluid detailed in the previous section was con-
tained between the two electrodes. With one of the electrodes designated as a ground, a high
voltage electric field ranging from 1 to 2 kV was applied to the other. Voltage measurements were

then made in real-time using an HP dynamic signal analyzer.

Initially, a short experiment was conducted to determine if the assumed circuit representa-
tion provided a valid description for the ER fluid’s electrical behavior during dynamic excitation.

In Fig. 3.14, results for three different cases are presented: (i) voltage measurements for the fluid

l(X)

 

 

—— - Resistor Actual Mean
Applied V __ Capacrtor ER Fluid Voltage

Electric Field

 

 

 

 

Figure 3.13 - Equivalent transient electrical behavior for an ER fluid.

highly agitated with a dynamic excitation, (ii) the fluid thirty minutes after excitation, and (iii) the
fluid one hour after agitation. As shown in the illustration, there is not much difference between
the three cases, however, measurements taken prior to the experiment with the fluid at rest were
approximately ten time less than these data. The results give us evidence to conclude that the
equivalent ER fluid electrical circuit depicted in Fig. 3.13 is a valid representation for the transient

electrical response during and after dynamic excitation.

 

 

 

 

 

 

 

 

 

 

i , — During agitation
i l ......... 30 min. after agitation
4.5 _ g I ................ 60 minaftcr agitation
3 — I
g. l‘.
I
2»: 1.5— I
g I
a l
g 0 \
Q :
E .
< .
...s—
-3 s
l
-6
I I I I I I I 7
I
0 12.5 25 37.5 50 62.5 75 87.5 lOO i 12.5 125
Time (sec)

Figure 3.14 - Actual voltage of the ER fluid medium for an applied electric field of 2 kV.

101
To identify reasonable parameters for the resistor and the capacitor in the circuit, an itera-
tive identification routine was implemented using MATLAB. This routine employs a fourth and
fifth order Runge-Kutta ordinary differential equation solver to integrate the two following equa-

tions used to characterize the circuit given in Fig. 3.13:

RCg—V+V=Ri (3'5)
dr
dq q _

RE+E‘V am

where R and C denote the resistor and capacitor, respectively, 1: is the time variable, i is the cur-
rent, V is the voltage, and q is the Coulombic charge of the ER fluid. A comparison between the
simulated results and the experimental results are given in Fig. 3.15. Using values of R = 64.1 m
and C=10.3 nF, we see that the simulated results give a time constant of Tc = 11.56 msec. whereas
the experimental results give a time constant of approximately To = 10.38 msec.

The electrostatic distribution inside the ER fluid structure can be determined by considering
an electric field E as a potential problem in the three-dimensional dielectric medium 9. If we let
<I>(x,y,z,t) represent the time-dependent scalar electric potential, then the field is given as,

V d) = E (3.7)

and the governing equation for the three-dimensional domain Q is:

a_ 3 a_ if a 33 ._
8x(ePax)+ay(EPay)+az(€Paz)+0 — O (3.8)

together with the associated boundary conditions,

(I) = ¢0(Xi y. t) on r (3.93)

dd: do an: F, on r
e — + e — +e — =
Pax Pay Paz 7'

(3%)
where 8p (x, y, z) is the effective permittivity for the ER fluid medium, 0’ is the surface charge

density, F’ is the electrical flux or displacement, and I“ 1' 1‘2 are portions of the surface I‘ from the

domain I). For this analysis, 1‘1 represents the top and bottom surfaces of the fluid element and 1‘2

102

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mean Vollagc inside ER fluid Medium

4.5 "‘ I
A 3 .. i
> I
‘5 15.4 I
3
5 0 L
O.
E is .4

-3 d

Experimental
4.5“ -------- Simulation l-‘li
"" l l: l l l l I T r
0 20 .40 ,60 80 1.00 I20 I 40 l 60 I80 2.00
'l'llllc (sec)
.1 _—'—‘ *‘
Input Voltage Accross ILIL'CIHKICS
2 5 .4

.. 2 ‘4
>
:5 1.5 —~
0
3 I a
‘3
E 0.5"
<

0

-(l.5 "‘
" l l l l l I I l r
0 .20 .40 .60 .80 LOO 120 l.40 l.60 1.80 2.00

 

 

Tillie (sec)

Figure 3.15 - Comparison between experimental and simulated voltages for the ER fluid medium.

represents the four sides of the ER fluid element. The series solution for the problem given in Eq.

(3.8) can be written as

<I>(x,y, z, t) = 2 2 Amcos (malt) cos (Ebfl) (e-"wmz- emmz) (3.10)
m =- ln = I
where
b a
4 nity mm:
A... - EICOS(T)I¢0(X’W) CONTldxdy (3.10a)
0 0
and
2 2
m n
w’"" = — + — (3.10b)
a2 b2

For the small distances between electrodes, this electrostatic distribution will be almost negligi-

ble.

3.2.3 A model for the actuator

In this section we consider the result of incorporating the insulatory material and ER fluid into a

103

single functional unit of a composite actuator. The insulate, besides restraining the ER fluid and
insulating the structure, also has the function of transferring some of the applied loads. Thus, it is
of great importance to be able to predict the properties of the composite actuator, given the com-
ponent properties and their geometric arrangement. For this analysis, the rule of mixtures will be
utilized to accomplish this for a given generic geometrical configuration of the actuator. The rule
of mixtures provides a rough solution of the properties, because it considers the composite proper-
ties as volume-weighted averages of the component properties.” 68 Nevertheless, a very accu-
rate description is not necessary here because the ratios of insulate-to-ER fluid are anticipated to
be rather small.

Accordingly, the material model for the actuator will be based on the following assump-
tions:

Assumption (3.1)

1') There is no inter-conversion of mass and both constituents are chemically inert
to each other (no fluid to solid transmission),

ii) The insulatory solid is assumed to be linearly viscoelastic and incompressible,

iii) The electrorheological fluid is assumed to be ideal and incompressible,

iv) The actuator is homogeneous with all normal stresses in the fluid negligible.

Consider the unit cell geometry, taken from the infinitely large composite actuator, of thick-
ness d, depicted in Fig. 3.16. Fig. 3.16 illustrates a compartmentalized geometry for the actuator.
Such a geometry might be useful for limiting actuator effort to discrete portions of the structure,
or employing different ER fluids. Essentially, the unit cell is the smallest geometric entity that,
when repeated, will reconstruct the entire geometry. Here, the unit cell is idealized to be a paral-
lelepiped consisting of insulate material edges with an ER fluid core. Its orientation is defined by
an x-y-z Cartesian coordinate system. Use of the property of volume additivity allows us to define
the density of the actuator. Volume additivity is an intrinsic property that states that the total vol-
ume of the aggregate components is equivalent to the sum of the respective volumes of the super-

imposed continua at any time or deformation state.56 If we then consider the insulate mass min

104

Actuator Geometry J View A

 

 

 

 

 

 

 

 

 

 

 

Unit Cell Geometry l

 

 

 

 

 

 

Figure 3.16 - Geometric configuration of the actuator and its unit cell.

and volume Vin. and the ER fluid’s mass rm, and volume Vfl, the total mass and volume for the

actuator are given by the expressions,

mA =mfl +min (3.11)

VA =Vfl +Vin =bcd+2ad(2a+b+c) (3.12)

According to the rule of mixtures the actuator mass density p" is given by

“b = mfl+min = Vfl pfl+vin pin = P
A VA VA fl fl

 

+0," pin (3.13)

where vfl and Vin denote the volume fractions of the fluid and insulatory material, respectively.

Now, let us apply a small-amplitude oscillatory shear strain to the upper and lower surfaces

of the unit cell given in Fig. 3.16 in a direction parallel to the y axis. The nile of mixtures

105
approach will be used to determine the mechanical properties of the actuator. It has been assumed

that the insulate behaves like a linear viscoelastic material, such that in oscillatory shear, it obeys

the following isotropic constitutive equation for the shear stress and normal stress”: 5‘
r=G(to) (e+é) (3-143)
0 =50.» (e+e') (3.14b)

where the dotted quantity denotes differentiation with respect to time. However,, the ER fluid has
a constitutive behavior that is dependent on the strain rate and electric field potential as given in
Eq. (3.4). Now, if the two components adhere perfectly and have an identical Poisson ratio, each
will undergo an identical shear strain, cA .32, 63 This assumption holds since the ER fluid will take
on the shape of the enclosed cavity during deformation“:127 Therefore. according to the rule of

mixtures the constitutive law for the actuator assumes the form
0" = oinG’ (0)) e" + (oinG’ (to) n’ ((0) e'A + offlé", V) ) (3.15)

where the dynamic stress of ER fluid actuator is now characterized by a simplified nonlinear vis—
coelastic model that is dependent on frequency of oscillation, strain rate, and electric field. Tb
note, all pre-yield effects of the ER fluid are considered negligible because sufficient information

and data were not available to consider this phenomena.
3.2.4 Numerical comparisons

In order to verify the material model derived in the previous section, a simple numerical compara-
tive study was conducted. Tb verify the material model, a FORTRAN subroutine was written for
use with the finite element software ABAQUS.1 This enables us to physically model the nonlin-
ear material behavior of the actuator. The objective here is to compare and contrast the response
of the combined insulate/ER fluid actuator model given in Eq. (3.15) with experimental data, and
also with an actual unit cell geometry having separate insulate and ER fluid domains. A sche-
matic of the geometries for the two models with identical uniform shear loading conditions is

given in Fig. 3.17. As seen from Fig. 3.17, model (1) represents the actual unit cell geometry, and

 

 

 

l) Separated Model

 

 

 

 

 

 

l 2‘
D Fshear

Elevation for both d > y
f e\\\\\\\\\\\\\\\\\\\\\\\\\\‘\\\\\\\\‘\\\\\\\\‘

 

 

 

 

 

 

 

 

 

 

Ax
Actuator , ' _ T
_ . . 1’
2) Actuator Model . _ g ‘ I
<—-—— 2a+c ___,.

 

 

Figure 3.17 - Geometric configuration and loading of the actuator models.

model (2) represents the geometry associated with the actuator model. It is very important that
the orientation of the local Cartesian coordinate system shown in Fig. 3.17 is used for all element
and material axes. The actuator model and its mechanical properties are highly dependent on
geometry. Although the insulation is isotropic, the actuator acts orthotropic in the presence of an
electric potential, due to the ER fluid.. This observation holds only for parallel electrode geome-

tries.

For these two models, the geometry of the unit cell was discretized as shown in Fig. 3.18.

The associated dimensions for the unit cell are given in Table 3.1. The unit cell was discretized

107

Table 3.1 - Dimensions for the two actuator models.

 

Dimension Models

(cm)

 

a
O
\I
O
\l

 

 

 

b 3.6 3 6
c 8.6 8 6
d 0.16 0.16

 

 

 

 

 

1...!
1...!
1...!
1...!
1...!
1...!
1...!
1...!
1...!
1...!

Figure 3.18 - Discretized geometry for the two models.

108
into four layers of 50 hybrid solid elements. The hybrid element is especially useful for applica-
tions with incompressible or nearly incompressible materials as displacement and pressure are
represented as independent variables.1 The same discretized model was used for both cases,,and
as illustrated by the smaller elements included around the edges in Fig. 3.18, accommodations
were made for the separated model, or model 1.
The material model for the actuator was incorporated into ABAQUS as follows.1 The inte-

gration operator that was used for Eq. (3.15) is the simple, stable central diflerence operator:

_ A9
r+-l-At — E (3.16)

_ _, A9
Hg... ' 9: 7’ (3.17)

where g is some function, 9, is its value at the beginning of the increment, A; is the change in the
function over the increment, and At is the time increment. Applying this to the constitutive equa-

tion given in Eq. (3.15) gives

 

 

 

 

 

 

r— A -
2 A8
Aofz = Ki: L (AAt +3) Ag; + CAtf( Af, V) + At (Aefiz — 0:2)1: (3.18)
A 2 I A 892 A A T-
AO’yz = E _(AAt +B) Aeyz + CAtf F’ V + At (Aeyz — oyz) d (3.19)
where
A = oinG’ (to)
B = 0,,6’ (0011’ (m)
C = of,
Then the Jacobian matrix has the following terms
—a——(Ao" ) = —2— —(AA1: +3) + cat—340i; V)1 (330)
8A8; ‘2 A1: _ 3,38sz Ar ’ A
r' ABA
a A _ 2 d yz I
3A8;Z(Aoyz) - Ki (AAT'I’B) + CATWIK?’ V] (3.21)

109

 

 

 

 

700. . . . a
I \ A. "‘ '- 1 A
. a '8‘ .1: I, ll ll 1‘ I] II
GOOI’ I \ ’ 5‘ ‘ l l 1 l | l l I I I l ,1
"_ -—»......._ ._ -‘ / . )r I I I I ! I I 3 . I I I
" “w , 1". ..l’i
. . l l l l l
. , s l , I :I I
SOOI'I (1’ ‘I :l t? ,i'lll",:€
-' I' l I i It l l H 1
l I '
l‘ . , , . . I ‘.I., 1,1. ,I-‘- II '1 H I) I
840011. ‘. 4.4".“ ..~.""~,'.'4
,I l‘ u H ,‘ I
II
A \ a \ II I
:- I' _- ’\ x‘ ’\ ’\ I l ’ ll ’1 i
U) - - ---- "'_" \ ‘ ’I \ 1/ l \ \ I II '
200f’r \ / \ \J \/ \v/ I], \d Id (3
I
.l ;
1001aI I
I
II '
Il —V=0 I
OI W ......... v=10kv VJ
""V=l.5kV I
~V=2.0kV I
'IOOL . 1 r r j
0 0.5 1 1.5 . 2 2.5 3 3.5 4
Time(sec)

Figure 3.19 - Time response of the ER fluid model under periodic excitation.

since the normal and inplane stresses are neglected only these two components reside in this
matrix.

A periodic excitation which varied in frequency from 0.001 Hz to 120 Hz over a range of
10 seconds was imposed on the models. This excitation was in the form of a harmonic shear

e‘ = eocos (art) (322)

where so is the shearing amplitude and (0 is the circular frequency of oscillation. Results were
first obtained in the time domain and then mapped to the frequency domain for further analysis
using a discrete fast Fourier transform. Data in the frequency spectrum were passed through a
Hanning window and averaged over several simulations to improve resolution.

Initially, the ER fluid model was investigated singularly to compare its response with the
experimental results presented in the previous section. Figure 3.19 shows the response of the ER

fluid during harmonic shearing with an amplitude of so = 20% and for constant input voltages of

110
O, 1 kV, 1.5 kV. and 2 kV. respectively. It is evident from these results that ER fluid is capable of
supporting a definite yield stress which is indicative of the dynamic yield stress in ER fluids. This
increases with increasing voltage input, or until ER fluid breakdown occurs between the electrical
polarization of the particles, as shown in Fig. 3.5. It is also seen in Fig. 3.19 that the excitation
imposed on the ER fluid causes the stress to oscillate about this mean yield stress, with its magni-
tude dependent on the particular voltage input. The amplitude of these oscillations are highly
dependent on the frequency of oscillation, and is representative of the high stresses that are

encountered in the low shear-rate region of Fig. 3.12.

Effectively, limitations of the ER fluid model are apparent as a result of the approximations
made outside regions where experimental data were taken. Because the response is of a dynami-
cal nature, the innate inertial characteristics of the particles within the fluid and the fluid itself
must be considered at high frequency. Eventually, the motion of the fluid will be such that very
little aggregate structure formation will be possible. This observation puts large doubt in our

model at high frequency.

The results depicted in Fig. 3.19 were mapped to the frequency domain to obtain a discrete
representation of the power spectral density. Power spectral density is a measure of the power
transmitted at various frequencies prescribed by some IP (0)) |, where P((n) is the Fourier trans-
formed frequency response function These data are illustrated in Fig. 3.20 over a range of 0 to
100 Hz. Although the diagram is very unclear due to the limited number of discrete simulation
data, three distinct peaks are apparent. The peaks represent the amount of power entering the sys-
tem for all the frequency components of an excitation lying in an infinitesimal band between a)
and a) «I» do). For V = 0 these peaks are recognized at 24.7 Hz, 52.3 Hz, and 78.4 Hz, respectively.
Wrth increasing voltage, only the latter two peaks are apparent, with the middle becoming more
defined. This information tells us that between 50—56 Hz, and 75-82 Hz, we should expect mini-

mal power dissipation from the ER fluid model regardless of input voltage magnitude.

To make comparisons between model 1 and model 2, the same procedure above was

111

 

.I —V=0

ISOII --v=r.5kv -I
If. “V=2.0kV
III.
Isl
I f.
II“
.I"; ‘
II,\ II ‘
100 ”I?! ’5 \IIHIIII‘ 3‘." J \

  
   

I 'L"

I I II IIIlI I'll ~ ‘

I I I III} IrhIIIIIIIIIIIIIII I II IRI [III I - IN .‘I. M H I

I :III IIIIII IIKIIfllugII I I iii In?“ I II iIIII. ' II II l!

I I I ‘IfuiufliII a “I ”ZiLLI. II , ’I. IIIIII .IIII‘IM f I," N

I ' 'IIII III j IIIIIIIHIII‘II :HIIIIIII III: “I
IN ,,

   

Magnitude (dB(Pa))

 
  

I

- ' I .
I IIIIIWW .I/IIIII {III III. IIIII
, I I III
I

O

o F—_‘T“‘—

 

20 40 60 80 1 00
Frequency (Hz)

Figure 3.20 - Power spectral density of the ER fluid model for various constant voltage inputs.

repeated. Using a steady-state option provided by ABAQUS, the mean stress response for both
models was obtained over a frequency range of 0 to 100 Hz. Figure 3.21 shows a comparison
between these results for zero voltage input. Moreover, the results for the 2 kV case were qualita-
tively identical. With the addition of the insulate material, the stresses in the unit cell for both
model jumps by a magnitude of ten. It is evident that the mean stress decreases with increasing
frequency for both cases. However, the actuator model seems to under predict the actual geometry
by almost 100 Pa at low frequency, where in fact, the rule of mixtures model should be more
accurate. Intuitively, the close agreement in the high frequency range must be due to approxima-
tion error in the linear solver routine. This error may be attributed to the large differences
between the constitutive laws for the ER fluid and the insulate. Surely, a more accurate descrip-
tion of the unit cell would be necessary to compensate for these errors. Thus, the model obtained

through the rule of mixtures approach should be considered conservative for accurate modeling of

112

 

l

— Separate Model
"- Acrualor Model

40

 

100

80

60

Frequency (Hz)

20

0

 

annnl
~vuu

 

Figure 3.21 - Steady-state comparison between models 1 and 2 for zero voltage input.

 

(a)

Figure 3.22 - Deformed configurations for: (a) model 1, and (b) model 2 for zero voltage input.

113
the actuator. The associated deformed configurations for both models, which were determined at
a frequency of oscillation of 15 Hz, are given in Fig. 3.22. These two models are very similar in

appearance.

Figure 3.23 shows the response of the actuator during harmonic shearing. The actuator has
a unit cell geometry with the volume ratios of bin = 0.2808 and of, = 0.7192, which are
obtained using Eq. (3.12) and the data in Table 3.1. For a constant amplitude of so = 10%, and for
constant input voltages of 0, 2 kV, and 4 kV, these results show how voltage directly affects the
response of the actuator. Clearly, the small change in amplitude reflects the small effect that the
ER fluid will have on the dynamic response of a large structure. Hence, the ER fluid is severely
limited when drastic stiffness is necessary. The associated power density spectrum analysis for
the actuator model under various voltage inputs is given in Fig. 3.24. The results depicted in this
response spectrum demonstrate characteristics similar to those described in Fig. 3.20, however,
the larger peak near 80 Hz has disappeared. By using the solid zero voltage line as a reference we
see a small shift in the peaks is apparent for increasing voltage. Consequently, this observation
gives us some indication of what direction a structure’s frequency response will shift during exci-

tation.
3.3 Electrorheological Plate Identification

Here a nonparametric experimental identification procedure was conducted for the ER plate struc-
ture. The motivation has been to estimate the dynamic response of the structure in order to define
an uncertainty description. Another reason for identifying physical parameters of the structure is
to optimize vibration suppression qualities while maintaining a constraint on the maximum power
consumption of the ER fluid actuator. Stevems, et al,125 investigated time-dependent excitation
of an ER fluid clutch and concluded that it was possible to reduce power consumption by more
than 50% using excitation in the form of voltage pulses. However, the ER effect diminishes with

increasing frequency.136 Cleariy, there are advantages if BR fluid actuator excitation can be opti-

114

 

 

6.
O

   

0. 0 O.
3%: wuaig 8%

3.0kV
l’ = JOkV

i=1)
--\ =

he‘»

i

086

2.5

0.5

 

)

Time (sec

Figure 3.23 - Time response of the actuator model under periodic excitation.

 

  

0 .
= = -
VVV

100

80

60

40

20

 

 

 

.
0
A.»

23252 muEEmm—z

»
0
6

_
o
8

 

me
0

Frequency (Hz)

Figure 3.24 - Power spectral density of the actuator model for various constant voltage inputs.

115

| X 3 cm x 3 cm clamping fixture
I

 

 

<———12in. ——J

 

 

 

 

 

 

«— 12in. ———>

 

 

<— 12 in. ———n Rectangular Plate
Square Plate

21mm

$ /— Aluminum sheet 1 ¢
I \ . . I 0.019 in.
0.116 in. —— Silicone rubber edges 0.078 in.

ER fluid core

 

   

 

Figure 3.25 - Schematic of the two plate specimens.

mized using a voltage supply containing variable waveform, amplitude, frequcny, and phase char-
acteristics. Such optimization has been mentioned briefly in previous experoirnental

investigationsul’ 136

3.3.1 Experimental apparatus

A schematic of the two plate specimens employed in this analysis are depicted in Fig. 3.25. Each
plate was fabricated from 0.019 in. aluminum sheet and its inner cavity sealed with a 50 DA
durometer silicone rubber sheet, which was approximately 0.078 in. thick. 'Ib bond these compo-
ments, a silicone RTV rubber agent was applied to all surfaces and allowed to cure for a minimum
of 24 hours at room temperature. After cure, each plate was punctured with a needle through the

insulate and the cavity was injected with the ER fluid.

116

The experimental arrangement utilized in this study permitted a forced excitation and
dynamic response analysis of an Er fluid embedded plate with free-edge support conditions. The
corresponding instrumentation and experimental arrangement utlized in this investigation is
depicted in Fig. 3.26. Each plate was mounted to an aluminum fixture that was firmly attatched to
a Briiel & szer shaker unit via a small 3 cm square clamping fixture. The external forcing
imposed on the plate was supplied by an HP dynamic signal analyzer, and the time-dependent
voltage excitation was supplied by a programmable arbitrary function generator. In addition, a
Brfiel & szer force transducer was included to determine the mobility and receptance frequency
response measurements. The amplitude of excitation was maintained at minimal levels to reduce

nonlinear and other parasitic effects on each plate.

3.3.2 Plate modal analysis

Prior to determining the plate response for the voltage excitations, the first six resonant frequen-
cies and associated mode shapes of each plate were experimentally identified. In order to deter-
mine these dynamic characteristics, one quarter of each plate was discretized as illustrated in
Fig.3.27. In Fig. 3.27, each black dot signifies a point of measurement. By using the HP signal
analyzer as an excitation source and data acquisition system, receptance and mobility responses
were obtained. These responses were determined by imposing deterministic (sine-dwell) and ran-
dom (stochastic) forced excitations on each plate. For improved signal-to-noise ratio, the Han-

ning window was used for random excitation, and all data were exponentially averaged over time.

The plate resonant frequencies were identified from transfer and point receptance and
mobility frequency responses taken at discrete points. The receptance response for both the rect-
angular and square plates is depicted in Fig. 3.28. In Fig. 3.28 it is apparent that the frequency
response of each plate is relatively mass dominated, as evidenced by the large decrease with
increasing frequency. Mass dominance, as opposed to stiffness dominance, is also supported by

the existence of a resonance after each anti-resonance. Receptance response results were verified

117

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
      

 

 

 

 

 

        
 

    

     

 

    

 

 

 

 

 

Bn‘iel & Kjaer Type 8323 Tripp Lite
Laser Velocity Transducer 13.8 Volt D.C.
,, Power Supply
E Briiel & Kjaar Type 2635
Charge Amplifier
Plate .
- Bn’iel & Kjwr Type 8200
Specrmen -\ E i Force Transducer

l l '

HP 35660A Fixture
D ic Signal ............. . Trek 609C-6 .

: Analyzer High Voltage D.C. Amphfier

Function Source nie 18’ Voltage Input f(V,.0,l)
Shaker Unit

Briiel & Kjaer Wavetek Model 275

Power Amplifier Arbitrary
Type 2707 Function Generator
Figure 3.26 - Experimental arrangement and instrumentation.
I
c = I

4 8 12 16

30 7 O 11. 150

20 6. 100 140

1 5 913

——o—o——o—-o——

 

 

3 6 9' 12
20 5o 8. 110
4 7 1o

 

 

 

 

 

 

 

 

Figure 3.27 - Experimental plate discretization.

118
using a wide-band random excitation. This permitted an estimate of the frequency response func-
tion at the measurement point of interest in terms of both magnitude and phase. These results
were then divided by the input force frequency spectrum, as depicted in Fig. 3.29 and Fig. 3.30.
Phase plots depict the corresponding changes with resonance and anti-resonance and were used to

determine gain insight on the approximate resonsance values.

For purposes of identifying questionable resonant frequencies, this procedure was repeated
using narrowband random and pure sinusoidal excitations and repeated for several different points
of measurement. Multiple point measurements were averaged together using sinusoidal profiles
to ensure accuracy. For comparison purposes only, the magnitude of the mobility responses from
several additional points are presented in Fig. 3.31. As expected, those responses taken from the
square plate are in relatively close agreement due to symmetry. However, the responses at point
16 from the rectangular plate illustrates the beam-like influence of the longer side. Thus, as
expected, a resonance is noticed near every k (1),, along the frequency spectrum where k is the

mode number.

The damping ratios were obtained from Fig. 3.28-3.31 using the half-power point method
introduced in Chapter 2.102 Results from the various measurement points are given in Table 3.2.
We see that damping tends to remain fairly consistent in the domain of each plate, and increase
with increasing mode number. The large damping ratios in both plates for V = 0 kV can be attrib-
uted to both the insulate material that bonds the two aluminum electrodes together and the contri-
bution of the fluid inertia. It is evident that these damping ratios are much larger than the
structural damping ratios of aluminum alone. At the higher modes the results have large uncer-

tainty because the measurement signal was very weak.

Mode shapes for each plate were also identified visually by triggering the digital strobo-
scope laser velocity transducer simultaneously to measure the displacement of the plate at each
discretized point Data were then compiled, normalized with respect to the input, and graphically

plotted using a curve fitting routine. The results from this procedure are presented in Fig. 3.32

Magnitude («Hi [20 I.og(l mm/N)|)

Magnitude («Hi [20 11,2;(1 mm/N)|)

119

 

 

 

 

 

 

 

80 --«--—~- Point Receptance
70—1 — Transfer Receptance
60 "1'
50 -‘
40 -I
30 —‘
20 —
10 -‘

0 I I I I I F I I I

O 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(a)

('0 , ----~---- Point Receptance

60 Transfer Receptance

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 -«
'10 I I I I I I I I I
0 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(b)
Figure 3.28 - Receptance for: (a) square plate, and (b) rectangular plate at V = 0 kV.
G
$ 69 ~—~- Point Mobility or:
E 59‘i — Transfer Mobility .1
a
a
co
3
£
‘” I I I I I I I I I
- o 20 40 60 so 100 120 I40 160 180 200
Frequency (Hz)
a
.180—
A {560‘
g .540-
5 .720-
a:
D- mfi
4080-
.1260-
”‘0 T T I F I r I I I
o 20 4o 60 80 100 - 120 140 160 180 200

Plenum! (“1)

Figure 3.29 - Mobility frequency response for the square plate at V = 0 kV.

Magnitude (dB [20 Log(l mm/S°N)I)

Magnitude (dB [20 l-og(l mulls-N)”

120

by 1 - Point Mobility llU
59 4 —-- Transfer Mobility 0|

Magnitude (rill [20 | us” nrrrr/s-le)

 

 

 

Phase (deg)

 

 

‘ ' Y I ' ' I
I.) 2‘) 1.0 no so 100 no :40 .60 no 290

Frequency (Hz)

Figure 3.30 - Mobility frequency response for the rectangular plate at V = 0 kV.

 

60
mm Transfer Mobility at #1
50" — Transfer Mobility 111816

30—”.

 

 

 

 

I I I I I I I l I
0 20 40 6O 80 100 120 140 160 180 200

Frequency (Hz)

(a)

 

69
59—
49-
39 —
29 __..
r94

9—
-1 —I

-11

---- Transfer Mobility at #12
Transfer Mobility at #1

 

 

 

 

 

T I I T I I I l I
O 20 4O 60 80 100 l 20 140 160 180 200

Frequency (Hz)

(b)
Figure 3.31 - Mobility responses for the (a) square, and (b) rectangular plates at V = 0 kV.

121

 

 

Mode Shape Identification

Square Plate

 

Mode 2

 

Mode 3

 

Mode 4

 

Mode 5

 

 

Rectangular Plate

 

 

 

Mode 2

 

 

 

 

Mode4

 

 

 

 

Figure 3.32 - Mode shape identification results for both plates at V = 0 kV.

122

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Frequency Plate Damping Ratios
Response
Point 51 §2 £3 §4 §5 ‘56
f0 #1 0.01 0.015 0315 0.0I5 L001? 00191
0% g #13 0.009 0.011 0.013 0.0135 0.015 0.0194
#16 0.0112 0.0131 0.011 0.014 0.0156 0.021
:a #1 0.011 0.015 0.019 0.012 0.016 0.0185
g” is: #10 0.008 0.016 0.015 0.0146 0.0171 0.0187
g #12 0.009 0.012 0.011 0.0164 0.0167 0.019

 

Table 3.2 - Experimentally determined damping ratios for V = 0 kV.

3.3.3 Input verification

peaks the signal-to-noise ratio was too weak to identify further values.

and the first six resonant frequencies of each plate are tabulated in Table 3.3. Beyond six resonant

Each plate was subjected to simple voltage inputs and response measurements were taken in both
the time and frequency domains. The simple periodic voltage waveforms included alternating
(reversed polarity) sinusoidal, triangular, and square inputs. The frequency of the voltage input is

given by 9. To conduct the time domain measurements, a continuous narrowband periodic exci-

Table 3.3 - Experimentally determined plate natural frequencies.

 

 

 

 

 

 

 

Mode Square Plate Rectangular Plate
frequency (Hz)
1 19.50 15.52
2 37.87 31.36
3 64.56 53.15
4 118.3 76.35
5 138.7 111.5
6 181.8 155.6

 

 

 

123
tation was imposed on the plate and transient response was monitored. Results for the sinusoidal
and square inputs, taken at measurement point 1 for the rectangular plate, are depicted in Fig. 3.33
and 3.34. It is apparent in these diagrams that the ER fluid actuator has the largest effect on plate
vibration at peak voltages. The amplitude of the plate was greatly reduced for each voltage wave-
form, however, the greatest reduction is evidenced with the square voltage input. It is also impor-
tant to note the time lag associated with vibration suppression. This observation is important

when time-dependent voltages are considered and will be addressed later.

The final investigation included a square voltage pulse imposed on each plate. These
results are presented in Fig. 3.35 and Fig. 3.36, in which we see that the pulse input results in the
greatest reduction in plate amplitude, especially for low frequency excitation. The amplitude
response of the square plate is much less than the rectangular plate, and it is more stable.
Although, as illustrated in Fig. 3.31, the square plate is an inherently stiffer structure than the rect-

angular plate.

From these results, it is evident that the alternating voltage inputs have one advantage over
the pulse input in that they are better suited for low frequency steady-state excitations. For exam-
ple, in the case of suppressing a continuous amplitude disturbance, an in-phase sinusoidal voltage
input would be much better than a pulse input. This is because they apply the greatest amplitude
reduction over the excitation with the least consumption of power. The square voltage input
would be an exclusion because it is only undergoing a change in polarity rather than a change in
amplitude. Hence, the amount of power necessary to produce this potential is almost identical to
that of a constant voltage. Nevertheless, to compare the constant to the square voltage input in
terms of stiffness and damping improvement, the constant input would surpass because the alter-
nating square input tends to weaken dipolar bonding of the ER fluid particles.125 In transient con-
ditions, the most feasible of these four waveforms would be the pulse input. Since transient
conditions will contain a large spectrum of frequencies, the pulse input would provide the greatest

reduction in amplitude. Although, the reduction will come at the expense of power and a time lag

Voltage Input (kV)

Amplitude (rum)

Voltage Input (kV)

Amplitude tnrrnl

 

-» ~ E = 0 kV/mm
%- — E = 2 kV/mm. (1:0.5

 

 

 

 

I
(I ([4 (IX [.2 L6 2.0 3.4 2.8 3.2 3.6 4.0
Time (sec)

 

 

 

 

 

 

-|.5
‘3 I I I I I I I I I
t) 0.4 0.8 l.2 L6 2.0 2.4 2.8 3.2 3.6 4.0
Time (see)
Figure 3.33 - Transient response of the rectangular plate for V = 2 kV and Q = 0.5 Hz.

4 E = 0 kV/mm
3d ' — E = 2 kV/mm. 11 = 0.5
2-

 

 

 

 

 

 

IJ
L—4
!J
:21

I
(I (H UK I 3 I (I ll)

Time (sec)

 

 

 

 

 

I If I
0 0.4 0.8 1.2 L6 2.0 2.4 2.8 3.2 3.6 4.0
Time (sec)

Figure 3.34 - Transient response of the rectangular plate for V = 2 kV and Q = 0.5 Hz.

Voltage Input (kV)

Amplitude (mm)

Voltage Input (kV)

Amplitude (mm)

125

 

25-4
2-

1.5—
l—
.5 _t

 

 

 

-.5 "I
-1

eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee

°-----E=0kV
——E=2kV

 

 

 

 

 

 

Tune (sec)

 

4.0

 

1.5“
1—1

.5—

..5 -—
.1—
.15 _
-2

o—
k.“ .......

o I c
I I o
i I a .
I: c: O ' I
0 u a '
o . O . e
h :J.I:. ' i Q'- .0-
e. ...‘ 0 ~ 0
I. . . '.
- I
e . I t g ‘ .. ""l:n|I.
I ' i, 'I ‘
II . 1' . I.
' . I I I .-' \.
cl 0. . . "|‘ C
o. a. .' r
u '. ‘0'.
o c ' '
..U 'o '0
c e
. a: a.
I. O.
\ '0
D .I ‘
\
I

0. ii '{;""IaW"..
. 1' I '

..;:;;::;.::.
. i, l
flat-:9..- . - .-
u,'.' u‘.r.".
‘} .M’V.o‘|.

"333:.

O
M ”0.",
° ‘0‘ 2‘ ‘r-

'0 an ’ 'o b
’ "an ' n V ‘. .f

\M

 

 

1.6

T I
2.0 2.4

Tune (sec)

2.8 3.2 3.6

 

4.0

Figure 3.35 - Transient response of the rectangular plate for V = 2 kV and Q = 0.5 Hz.

 

2.5-4
2 _

1.5 -*

 

 

 

uuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuuu

-----E=0kV
—E=2kV

 

 

 

 

ooooooooooooooooooooooooooooooooo

 

 

 

 

I I
2.0 2.4

Tune (sec)

2.8 3.2 3.6

 

4.0

 

-.5 ""
-I -—t

91.5 —4

a
0‘.
\O
\
21'»
a
I
I.
'It
. II
c. I
I
‘.I::l..
.'cl'..f
s-.'v’,
-_.o."
..:I'
0".

' 3. ' ”ix-x.

 

 

I I
2.0 2.4

Time (see)

Figure 3.36 - Transient response of the square plate for V = 2 kV and Q = 0.5 Hz.

 

4.0

126

induced by the selection of an optimal pulse length and frequency.

To further investigate the effect of time-dependent voltages over a large spectrum of excita-
tions, frequency responses were generated for both plates. For convenience, the same waveforms
and measurement points were used. A random narrowband forced excitation was imposed on
each plate, and the corresponding magnitude and phase descriptions were recorded. These data
represent the plate frequency response for voltage input amplitudes of 2 kV and frequencies of 0,
l, 10, and 50 Hz, respectively. Square and rectangular plate response is illustrated in Figures
3.37-3.42 for sinusoidal, square, and pulse voltage inputs. The triangular input has not been
shown because of its similarity to the sinusoidal input We can conclude that the ER fluid effect
diminishes with increasing voltage frequency for all waveforms due to response consistency with
previous results. As expected, we see the largest reduction in amplitude over the largest fre-
quency range in the pulse input responses. It is also seen in this response that only small changes
in magnitude resulted from large changes in voltage input frequency. This is unlike the altemat-
ing input results, which show increases with increasing frequency. We must conclude that for
small pulse lengths, a dynamic electrical and rheological coupling takes place, induced by
dynamic agitation. This coupling must cause stiffness and damping to decay at a rate proportional
to the length of exposure time to the potential. Since the ER fluid changes from its liquid-to-solid
phase rather rapidly, it must take longer to return to a liquid, thereby retaining its salient proper-

ties for some time.

Using a constant frequency of 1 Hz, results were obtained for diflerent voltage input ampli-
tude. Magnitude results from these experiments are depicted in Fig. 3.43 and Fig. 3.44 for the
rectangular and square plate, respectively. For clarity, the phase representations corresponding to
each plot have been excluded, although each phase was identical to the results in Figures 3.37-
3.42. We see from Fig. 3.43 and Fig. 3.44 that a change in amplitude, as well as change in fre-
quency of the input voltage, has the capability of inducing a frequency response shift. It is clear

that the ER effect due to a 1 kV amplitude voltage input reduces overall plate vibration almost as

Magnitude (nu)

Phase (deg)

 

'” E = 0 kV/mm. r1 = 0 Hz
E=2kV/mm.fl= l Hz
E=2kVMm.fl= lOHz
E = 2 kV/mm. Q = 50 Hz

 

 

 

Magnitude (d1!)

Phase (deg)

 

 

 

 

 

 

 

 

Frequency (Hz)
Figure 3.37 - Frequency response of the rectangular plate for a sinusoidal input of V = 2 kV.

 

“’ W E=0lemm.fl=0Hz
10 E=2lemm.Q=le
. E=2lemm.Q=lOHz

E=2thnm.Q=50Hz

 

 

 

I I
0 IO 20 30 4O 50 60 70 80 90 mo
Frequency (Hz)

 

 

 

 

 

Frequency (Hz)

Figure 3.38 - Frequency response of the square plate for a sinusoidal input of V = 2 kV.

Magnitude (dB)

Phase (deg)

Magnitude (dB)

Phase (deg)

128

 

-50 —‘

 

 

......-.-. E = o kV/mm. Q = 0 Hz
5=2thnm.Q= l Hz
E=2kwmm.n= IOHz
E: 2 kV/mm. 0 = 50 Hz

 

T F |
0 IO 20 30 40 50 60
Frequency (Hz)

 

 

 

 

 

 

Freqmcy (Hz)

 

Figure 3.39 - Frequency response ofthe rectangular plan for a square input ofV= 2 kV.

 

 

._.......s E = o kV/mm. Q = 0 Hz
E=2kV/mm.Q=1HZ
E=2kvmim.n= IOHz
E: 2 kV/mm. Q: 50 Hz

 

 

 

 

 

0 IO 20 30 40 50 60 70
Frequency (Hz)

 

 

 

 

 

Frequency (Hz)

Figure 3.40 - Frequency response of the square‘plate for a square input of V: 2 kV.

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 E=0kwmm.r1=0Hz
01 E=2kV/mm.fl=IHz
. .. . E=2kV/mm.0=10Hz
S ' " ‘ " E=2kwmm.n=soaz
3
%
2
'a
:9
2
I I I I I
0 20 40 60 so [00 120 I40 160 ISO 200
Frequency (Hz)
U
480-4
A -360"
:0
g -540—
g '720_1 .......
5 900-1 Maw-:0 ..............................
4080-—
4260- ........
1.4—1'0 1 T I I I T I I .......... I .............
J 20 40 60 80 100 120 140 I60 180 2
Frequency (Hz)
Figure 3.41 - Frequency response of the rectangul'rrplm for a pulse input of V = 2 kV.
'” --------- E=0kV/mm.fl=0Hz
0- E=2kV/mm.Q=le
a E=2lernm.Q=SOI-lz
E
8
3
'e
if
2
73 I I I I I I I I ' I
0 20 40 60 80 100 120 I40 I60 180 2!!)
Frequency (Hz)
’31
3‘: W
at
2 .
a. ..................................
I I I I I I I I
0 20 40 60 80 100 120 I40 I60 180 21
Frequent (Hz)

Figure 3.42 - Frequency response of the square plate for a pulse input of V = 2 kV.

Magnitude (dB)

Magnitude (dB)

Magnitude (dB)

Magnitude (dB)

130

 

 

 

 

 

 

10
...__..~. E = 0 kV/mrn. Q = 0 Hz
0_ ------ E=1kV/rnm.Q=le
— E=2kV/rnnr,Q=ll-iz
-10"' '
.20— '-
-30 “
-40"I
-50—
.50 d
'70 I I I I I I I I I
0 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(a) sinusoidal input
10 -~--— E=0kV/mm,Q=OHz
01 . ...... E=lkVImm,Q=le
—— E=2kVIrnrn.Q=1Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 20 40 60 so 100 120 140 160 180 200
Frequency (HI)
‘ (b) triangular input
10 ---—-- E=0kV/rnm.QsOHz
Ofi cccccc E=lkV/mln.Q-=1Hz
. — E=2kV/rnrn.0=1Hz
o 20 40 6o 80 100 120 140 160 180 200
Frequencytflz)
(0) square input
6 —---- EsOkV/mesOI-lz
.10..q ...... E n 1 kV/rnrn. Q - 1H1
'. I: -— E=2kV/nrru,Q-1Hz
.20-
.30—
.40—
.50...
-60_
.704
.80 I I I I I I I I I
0 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(d) pulse rnput

Figure 3.43 - Frequency response of the rectangular plate for diflerent voltage inputs at Q = 1 Hz.

Magnitude (dB)

Magnitude (dB)

Magnitude (dB)

Magnitude (dB)

131

 

"...,... E=0kV/mm,Q=0HZ
...... E=lkV/mm,Qsle
— EsZkV/rnm,0=le

 

 

 

 

 

-70 I I I I I I I I T
0 20 40 6O 80 100 120 140 160 180 200
Frequency (Hz)
(a) sinusoidal input

10

~----—-— E = 0 Ithm. Q = 0 Hz
o— .".I ------ E=lkV/mm,Q=le
10‘ /‘ — E=2kV/mm,Q=ll-lz

 

 

 

 

 

I I I I I I I I I
0 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(b) triangular input
10 -~-—- E=0kV/mm.Q=0Hz
0_ ...... E:lkV/mtn,Q-1Hz

— E=2kVImm.Q=le

 

 

 

 

 

‘70 I I I I I I I I I
0 20 40 60 80 100 120 140 160 180 200
Frequency (Hz)
(c) square input
1° «W E=0kVImm, Q=0Hz
0_ ...... E=llemm,Q=ll-lz
x/ — E=2kV/rnrn,Q=le

 

 

 

I I
80 100

Frequency (HI)
((1) pulse input

120

 

200

Figure 3.44 - Frequency response of the square plate for different voltage inputs at Q = 1 Hz.

132
much as the ER effect resulting from a 2 kV amplitude voltage input for the alternating wave-
forms. This is evident in the rectangular plate response for sinusoidal and pulse inputs beyond the
first natural frequency. It is in this frequency range that the response from the two different
amplitudes becomes almost identical. However, the most notable response change in both plates
due to voltage amplitude occurs with the triangular voltage input. Here, use of a 2 kV amplitude
only reduced overall plate response by 10% for frequencies above 60 Hz, whereas a 1 kV ampli-
tude reduced overall plate response by over 18% beyond 85 Hz. Moreover, we see that the square

input contributes to the largest shift in the natural frequencies.

From the square plate response in Fig. 3.44, all three altemating voltage inputs had rela-
tively the same affect on the plate’s response. One possible reason for this is the larger innate
stiffness of the square plate. Observations conclude that the 1 kV amplitude performs much better
than the 2 kV amplitude for several voltage input cases. For instance, in the frequency range
between 40 and 80 Hz for a sinusoidal voltage input, plate vibration is reduced by over 12% due
to a 1 kV amplitude input and only 7% due to a 2 kV amplitude input. A possible explanation for
this exists with the attainable level of damping. When the ER fluid is charged with a 2 kV input,
the stiffness of the plate becomes greater, however, this effect does not guarantee that the level of
damping also becomes greater. In fact, at a level of 1 kV, we achieve almost the same level of
stiffness as we do for the 2 kV amplitude input if observe the slopes of the frequency response,
yet with an increase in damping, as depicted in Fig. 3.44 (c). This occurs around the second natu-
ral frequency in all cases. It must follow that the amount of random particle breaking and align-
ing at 1 kV is greater than at 2 kV. Consequently, this increases structural damping rather than

structural stiffening, which is more apparent in the fundamental natural frequency.

Finally, it is obvious that the ER effect has negligible influence on rigid body motion or
vibration below the first natural frequency. It is also observed from the plate frequency responses
that significant reduction in plate vibration occurs in a limited narrowband range between the first

and second natural frequencies. From these results it can be concluded that the ERE is fairly con-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mode V-OkV V-2kV V-ZKV
“
.. ”00‘ I @
R/ k 2 R2
(rt-37.87141 o-‘lrflflz tn-39.9SHz
.. arts / \ J L
fl? \ fl Ir“
\V// \V/
2.< < 2.4
uh U m
5 DOC N/A Nut
m

 

 

 

 

 

 

 

 

 

Figure 3.45 - Mode shape identification for the square plate activated
with a square pulse voltage input at various frequency.

sistent over wideband frequencies for both plates when excited by a pulse voltage input. However,
the results from the other altemating voltage inputs give conclusive evidence of their narrowband
potential. It follows that a higher-power pulse voltage input would be acceptable to damp tran-
sient plate vibration, whereas the sinusoidal or triangular voltage inputs would be better for nar-

rowbarxi low-power steady-state plate vibration reduction.

To investigate the effect that time-dependent inputs have on the ER fluid structures, several
experiments were conducted using variable frequency and amplitude pulse voltage inputs to acti-
vate the ER fluid. In a manner similar to that performed earlier, the first five mode shapes were
identified for different values of pulse voltage input frequency. Figures 3.45 and 3.46 show the
results of the experimentally observed mode shapes for the rectangular and square plates, respec-
tively. Due to large damping and low measurement signals at high frequencies, several mode

shapes were too difficult to identify. In Figs. 3.45 and 3.46, the corresponding shift in the plate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mode V-OkV V-2kV V-2kV
0-13: 0.50!!!
1‘
(la-15.52": 0.34.53": III-3.9551:
2d
0.31.36“: 0.35.76": (II-34.21":
V"
3rd
/\ A A
0.53.15“! 0'55“": 0.54.111:
L.
4‘” fl@E g E
*1] [F 511 r
err-76.35": 0.135.! Ha m-ISISl-{z
5‘” > < NM NIA
0.11115":

 

 

 

 

 

 

Figrre 3.46 - Mode shape identification for the rectangular plate activated
with a square pulse voltage input at various frequency.
resonance peaks, as a result of the ERE, is evident. The most noticeable change is the apparent
modification of the mode shapes associated with the higher modes. For the square plate, this
modification occurs at the 3rd mode, and for the rectangular plate, this occurs at the 4th mode.
These changes exists as a result of the alternating stifl‘nesses, which is in turn due to the pulsing

effect which couples the zero voltage and nonzero voltage responses.
3.3.4 Electrical and rheological dynamic coupling

At some point during the design of ER fluid structures, it will be necessary to optimize the pulse
waveform for minimal power consumption. With the pulse waveform, we have the capability of
tailoring the pulse length and amplitude. As observed earlier, when a potential is imposed across
the plate, the response is relatively fast. However, at the termination of the potential, we see that
the dissipation of the ERE is very dependent on the length of the potential. From this reasoning, a

high frequency (long pulse length) voltage input would be able to maintain a fairly constant level

135

 

 

 

3 0.81 J
3 .
E e
g 0.6 -," —
_g- .
.5 .
”:3, 0.4M ~
E xmmx Model
2 o—-o Mode2

02+ .——t Mode3 ‘

+—-—l- Mode 4

 

 

 

 

 

A r r A

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
time (sec)

 

Figure 3.47 - Reduction of the ER effect with time for a pulse length of 10 msec.

 

 

 

g

:5

E q

z X—-—-—-X Model
0—0 Mode2

—3 Mode3 *
+—+ Mode4

 

 

 

 

 

 

Figure 3.48 - Reduction of the ER effect with time for a pulse length of 500 msec.

136

 

1" 0 ’0';~. OI.” .. 0 ° ‘
‘6’ e ’ 0

0.8

A

0.6

0.4

 

Normalized Displacement (pk to pk)

 

rt +——+ Mode4

 

 

 

 

 

o 4 L A A L A L

0 l 2 3 4 3 6 7 8

 

dun (as)

Figure 3.49 - Reduction of the ER effect with time for a pulse length of 1 second.

 

 

 

Normalized Displacement (pk to pk)

 

 

 

 

 

 

}- M Mode l
7‘ 0—0 Mode2
”f .—-t Mode3 ‘
+——+ Mode 4
0L a . a . .‘o 12 14

b (one)

Figure 3.50 - Reduction of the ER effect with time for a pulse length of 5 seconds.

137

of stiffness and damping over time, whereas the low frequency (short pulse length) would fluctu-
ate from a large reduction in vibration to large amplitudes, depending. Without a loss of general-
ity, we will call this phenomena the “capacitance efl’ect” of the ER fluid. This effect in not new ,
but has been indirectly acknowledged in several experimental investigations?“ ”3

For this identification experiment, a single pulse voltage input of 2 kV was imposed on the
rectangular plate as it underwent a forced sinusoidal excitation with an amplitude of 5 mm. After
imposing a voltage on the plate, the transient decay of the ER fluid efl’ect was monitored for
steady-state conditions until plate amplitude retumed to the original zero voltage value.

The results from this procedure are shown in Figs. 3.47-3.50. Peak-to-peak plate amplitude
was normalized using the zero voltage value for each associated mode. To identify values from
these results, data was fit using a Nelder-Meade simplex curve-fitting routine, as shown by the

solid line in Figures 3.47-3.50. The following equation was used to fit the data

5
MM) = 2C.(1—exp(—%)) (3.23)

i=1
where C,- are constants and (p,- are time constants. The routine was repeated to minimize the least-
square error. These results provide evidence for the conclusions made about the influence of the
ERE over the first few modes. We see that this influence is large and that the greatest vibration
reduction occurs in the first mode for each case. Figures 3.47-3.50 also resemble that of typical
first-order response.
When approaching the coupling between the electrical and rheological dynamics using a linear
systems perspective, it is clear that there is an increase in time constant for increased pulse length.
Hence, these results give conclusive evidence that an optimal pulse length exists to meet power

consumption constraints.
3.4 Chapter Summary

A brief discussion of ER fluids and the physical mechanism behind the ERE was presented. This

discussion included sections on the rheology of suspensions, the electrical, and the rheological

138
behavior of an ER fluid. Following this introduction, a constitutive model for the ER fluid actua-
tor was developed. It was shown that the ER fluid and actuator model, which were based on
empirical data, were comparative to available experimental data during low frequency shearing.
In steady-state, the actuator model provided marginal results of up to 25% difference between
actuator geometries at low frequency. This shows that estimated portions of the model need fur-
ther experimental verification. Finally, a detailed description of the experimental apparatus and
ER plate identification was given. Dynamic performance was evaluated for two plate configura-

tions. These results show the innate complexity of the ER fluid structure.

Chained

ANALYTICAL AND NUNIERICAL ANALYSIS FORMULATIONS
FOR A DISTRIBUTED-PARAMETER STRUCTURE

4.0 Introduction

The use of active control as a method for providing damping for large flexible structures has been
of particular significance in recent years. In most cases, these flexible structures are composed
of various subsystems that interact with the main structure to yield a very complex dynamic
system. Nonetheless, to provide effective control over the assembled main structure a sufficient
amount of knowledge about the interaction of the system’s ancillary components is required in
order to formulate a mathematical description of its dynamical behavior. Analyzing the main
structure as a complete unit is very difficult because of the computational intensity and time
necessary to determine and solve for the system response. One approach that can be applied to
these complex structures, and yield a more tractable analysis, is substructuring. Substructure
synthesis is a technique where the main structure is subdivided into various, discrete elastic
subsystems that are described separately with good accuracy, and reassembled to act as a single
structure by imposing certain geometric compatibility constraints on the individual subsystem’s
boundaries.“ 95.119 Since large flexible structures are actually a complex assembly of conven-
tional beams, plates, or thin shells, a mathematical model for the full structure can be constructed

of any combination of impedance, stiffness, and/or transfer matrices. Hence, each subsystem

139

140

model is an input-output relation that can be conveniently assembled to establish a linearized
dynamic model of the main structure. When numerical methods are applied to the linearized
model, e.g., finite-element analysis, relatively good approximations can be obtained for active
control design of the main structure.

This chapter is dedicated to the development and analysis of conventional plates that are
fabricated with embedded electrorheological (ER) fluid actuators. Since the behavior associated
with these distributed parameter systems is very complex, it would be extremely difficult to
mathematically describe them assembled in a large system. Actuated systems reduced to beams
and plates are ideal candidates to model since they are critical to the response of a flexible
controlled structure. Following, with our concurrent approach, it is the goal of this chapter to
obtain an analytical model for distributed parameter plates and beams that could be implemented
in a practical active control design by fitting certain parameters using numerical and/or system
identification methods.

The general configuration of the plate is one composed of alternating layers of ER actuator
material embedded between host layers of structural material. The constitutive material models
for the two structures are generalized to accommodate a diverse species of ER fluid actuators and
structural materials. The assumptions used to derive both models are based on fundamental
sandwich structure theory“ 9‘” “° and constrained layer theory,“ and the governing equations of
motion are derived using Hamilton’s principle. Sandwich structure analysis is useful here
because it gives us the freedom to include shearing effects and core displacements independently
but within a logical manner, where all interfacial boundary conditions are satisfied. Hence,
through-thethickness stresses are continuous and single—valued when the equilibrium equations
are used. Comparative results for the static and dynamic response of the plate is given. Our
objective here is to quantitatively contrast a simple case with previous published and experimental

results in order to accurately verify our model its applicability.

141

4.1 Structural Kinematics

The analytical model presented here focuses on multi-ply structures consisting of alternating lay-
ers of various structural and actuator materials. Since it is not the intent of this study to consider
the failure modes for these structures it will be assumed that perfect bonding exists between all
layers. This assumption is quite acceptable according to Kim and Hong,73 as long as bonding
stiffness does not drop below a certain critical level. To formulate the governing equations of
motion, a displacement approach is used where all strains and rotations are considered small as
compared to unity, and in-plane elastic deformation is assumed to occur about the cross-sectional
midplane of the structure.

Let us consider the structure composed of N layers of bonded actuator and structural layers
and of uniform height H, as shown in the cross-sectional schematic in Fig. 4.1. We see from Fig.
4.1 that N must be an odd number so that all actuator layers are structurally constrained. Properties

for an individual layer are identified by the subscript i, and the thickness of a layer is identified by

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

TN
7 _ . Cliffs”? L. .:, W I
T — h i T Layer N-I
”N ”M! .» - ' ~ " : ~=
Layeri
v v / _ ” *x e
H I / MIDPLANE / r H
—_ [13
h h _- H / 2
0 1 layer 3
J J -—h2 l Layerz
_

 

 

j Layer 1
T1

Figure 4.1 - Cross-section in the x-z plane of a distributed-parameter composite structure.

142
Ti. The coordinate system adopted for this model is a Cartesian system that consists of the longitu-
dinal coordinate x, the in-plane lateral coordinate y, and the transverse out-of-plane coordinate z,
where the x-y plane is coincident with the midplane surface of the structure, also shown in Fig. 4.1.
The displacement field chosen for this model is based on the higher-order shear deformation the-
ory proposed by Reddy.112 This theory accounts for the transverse shear strain and the stress-free

conditions at the surfaces. Given the following assumed displacement field:

u (x, y, z, t) = u” (x, y, 1:) + zyx (x, y, 1:) + zzgx (x, y, r) + z3§x (x, y, t)
v (x, y, z, 1:) = v" (x, y, t) + zyy (x, y, t) + 22C), (x, y, t) + 23% (x, y, t)

w(x, y, z, I) = WON’T) (4.1)

where u" and v0 represent the in-plane displacements of a point (x,y) on the midplane surface of
the structure, yr and 7y define the rotations of all normals to the midplane about the y and x axes,
Cx, C)» Ex and éy are the higher-order functions that will be determined using the imposed bound-
ary conditions, and r is the time variable. Eq. (4.1) yields the minimum number of variables nec-
essary to describe the field for a simple laminated composite. Thus, the following assumptions
are necessary to continue the kinematic description for the structure in Fig. 4.1:
Assumption (4.1)
i. In-plane deformation of the actuator layers vary linearly through-the-thicloress,
ii. In-plane deformation of the outer structural layers, i =1 and N, vary linearly through-
the-thickness, and their rotation is independent of the equivalent core,
iii. In-plane deformation of the inner structural layers, 1‘ =3 through N-2, vary linearly
through-the-thickness, however, their rotations are identical,
iv. Rotatory inertia in the structural layers is considered to be negligible,
v. Transverse displacement for the structure is independent of the z-coordinate,
vi. In-plane stiffnesses in the actuator layers will not contribute to the in-plane stiffness.
The application of this diSplacement field requires that the actual multi-layered configura-
tion in Fig. 4.1 be simplified according to Assumption (4.1). To accomplish this, we represent the

inner structural layers, i.e., between i=1 and N, by a lumped configuration that resembles the

143

 

 

 

 

 

 

 

MIDPLANE

 

 

 

 

 

 

 

Equivalent Core

 

Figure 4.2 - Cross-section in the x-z plane of the displacement pattern for the structure.

three—layered classical sandwich structure used in constrained layer theory.16 The simplified con-
figuration is illustrated in Fig. 4.2. Since the thickness of the inner structural layers is assumed to
be much smaller in magnitude than the actuator layers, it is possible to describe the rotation of the
inner structure by a single equivalent rotation. This simplification eliminates the need to represent
each irmer ith layer by an independent rotation, thus reducing the total number of rotations in the
model to three. In addition, the equivalent core permits the evaluation of the necessary transverse
stresses through-the-thickness of the structure, without imposing severe discontinuities in the ana-
lytical model. This is important because the inner structural layer’s contribution to the overall
structure’s stiffness in shear is much greater than their contribution in bending.

Equation (4.1) can be modified for the structure in Fig. 4.2 and written as:

u1 = 140+}:l (711: —yf) + 27; + 22C; + 2351
ac = u" + 215 + 22§f + 2%)? (4.2a)

u”: u°+hN_l(‘Yf—yiv) +zfi+22§f+z3§¥

144

v1: vo+hl (yd—y?) +zy;+zzi;;+z3§;
vC = v0 + 275 + 2%? + 239;? (4'2b)

vN= vo+hN_1(Y§—Y'yv) +zyg+zzcgl+z3§y

w1 = wC = w” = w
(4.2c)

where vi and 73 define the rotations of all core normals to the midplane about the y and x axes,
respectively, and the superscript C represents all quantities relating to the equivalent core. The dis-
placement field can be further simplified by applying the appropriate boundary conditions to each
of the layers. The boundary conditions imply that the transverse shear stresses must be continuous
through-the thickness and vanish on the surfaces of the structure, also the displacements must be
continuous across the interfaces. If we make the assumption that the structural layers are orthotro-

pic sheets in plane stress, and impose the following boundary conditions,

i: I N
i H ' H ’
oxz(x9yri‘2_) = 032 (xryrif) =0
612(X.y,hl) = 652(x,y,hl) G;z(x,y,hN_l) = 652(x’y’hN—l)
affix, y,h1) = 0;;(x, y,h1) 032(x,y,hN_l) = 0172“”, h~_l)

u‘(x.y,h,)=uC(x,y.h,) foam.-.) u”(x.y,hN_,)

v‘(x.y,hl) = vctx,y.h,) vctx.y,hN_,) = vN(x,y,hN_l)
(4.3)

we can form a set of twelve equations with twelve unknowns. The solutions for the twelve higher-

order functions in Eq. (4.2) are given as

1_ ShN_l(y}—fl) hN-1(7}"ch) HhN-IWJ-C‘Y})
1" 3H(2hN_l+I-l) 15h1(2hN_1+H) 4hi(2h~-r+H)
Shite-7,?) +3H(’Y}-Yf) + lief—7})
15h? (2hN_1 + H) 4hN_1(2h~_,+ H) (4.4a)

 

 

 

 

1
J

145

“if—m (Std-”+71” _ mtg—3’ _hi_1(4Hh,<yf—v})+15H2(v}—vf))
27H2(2hN_1+ H)2 45H2h§ 45H2hth_l (2hN_1+ H)
_ 20th (7749 37;) + 12th1 “1.5—7})
45H2h§(2hN_,+H)

 

 

 

 

(4.4b)

 

C _ h,2(,_1(—20h%(fl+y}) +4lel («ff-7}) +3H2(yf—y}))
J rszhihN_1(2hN_1+H)

—4h,,_l (Sywa) ”HWY—7f)

4.4c
12HhN-1 (2hN_l +I-l) ( )

 

 

C _ (60(2h,2,,_1+HhN_l)g—;)+20h,2v_l(5)dy+y}))(ij—yy)
éj - 135H2h~_1(2hN_1+H)2
_h,%,_,(4th(yf—y}) +3H2(y}—yf)) _
45H2h§h,,_l (2hN_1 + H)
_20hf(y;’—yf+3y}) +12Hh,(yf—y}) +9H2(yf—y})
45Hh§(2hN_,+H)

 

 

(4.4a)

{V = hN_, (zohfm; fly) +4lel mic-7}) +3H"-(yj€—y}))
’ 60H2h§ (2h,,_l + H)

20hN_,(57;V—yf) +15H(y;’—yjc)
6OHhN_1(2hN_1+I-I)

 

(4.4e)

 

146

a
(20h~_1 (517-11,?) —60h~_la—;T") (if-1;”)
135H2hN_1(2hN_1-I-H)2

h,2.,_1(4Hh1(ij—y}) + 3H2(y} 41.5)) 80hf()jY-yf+ 37]!) + 15Hh1 (yf—yjl)
45H2hth_l (2hN_1+H) 45th(2h,,_l +H)

§f=—

 

 

 

(4.40

where,
J' = x, y

Accordingly, by substituting Eq. (4.4) into Eq. (4.3), the strains can be evaluated for the above dis-

placement field and expressed as:

1_ 1 1 2 1 3 1
Ej—ej+zt<j+z}tj+zuj
85.: = e]? + 2K]? + 22).}: + z3uf j = x, y, xy (4.53.)
a” = e¢+2Ky+ zzly+ 2311?

I

and
1_ 1 1 2 1
Ejz—ejz+Zsz+z A);
c_ C c 2 C 0
8,-1- eszsz“ 1,12 1 =x,y (4.5b)
N_ N C 2 N
ejz—ejz+zt<jz+z 7.1.2
1_ c_ N-
ez — ez - £2 —0 (4.5c)
where,

e1_71+aW
1_ o l C l_ _
ej—ej+h1(1cj—1<j) ‘ ’ 3:)
C_ C...
cf -8;? elz—f+m
N_ o C_

1' = x, x U l = x, y (4.5a)

147

and the midplane strains are defined as,

80 _ a_u’0 avo 80 _ £920+8V0
‘ _ 8x Y 3y xy _ 3y 8y (4.56)

the curvatures for the structure are.

K; = — +— (4.50

i=I,C,N

and the hi ghcr-order terms,

. 3C . 3C . a? 3C . . . .
r _ x r = y r = x Y r = t l = t
_ ea: ._ at; .. __ awe
m-j w-a M-5 3 we
i= 1, C, N

To find the rate of traverse shear of deformation in the core we differentiate Eq. (4.5b) with

respect to the time variable to give
C C C
.c dexz = _in 51“" 61¢. 245. (4.6)

C , C C
éC = E"Z: d—yf-t-a—W+Zzi—y +322§y
Y’ dt dr 8y d“! dt (47)

where the dotted quantities, such as (ti) , denote with respect to time and the higher-order terms
become
dc

WC
2 2 2 2
dtf - hN_,(4Hh1+3H )af+4thhN_l—T -3th2

3'? 12th(HhN_,+2h§,_,)

if

6“ (4.6a)

148

2 dW 2 2 x
dgf 4(2hN—1+HhN-1)g; hN_1(4Hh1+3H )2;

_ 45Hh2 (HhN_ l+ 212,34)

dvfmjvf

hN_ l(20H112—12H2h —9H3)— )d

a? —h 3H(HhN_,+2h§,_,)
—"H2h2

 

 

 

 

45Hh,(Hh~_,+2hN_,) (4.6b)
and
(it; _ (013,4 (4Hh1+ 3H2) +4Hh$hN_ , - MHZ) )ciyf (4.7a)
dt 12th (HhN_ , + 2h§,_ 1) d1
2 dW 2 2 _y
£§=-4(2hN_1+HhN_1)Ey- _ hN._1(4Hh1+3H )d,‘
dz 3H(HhN_l+2h,2v_1) 45Hh%<HhN-1+2hi-1)
(fwd (onhf- 12H2h1— 9H3) -15H2h§)a‘Y,C
+ _
45th(H/1,,,_l + 2h,%,_ 1) C” (4.713)

Now differentiating Eq. (4.3) with respect to time yields the velocities for the ith structural layer

1 .450 )1de 1—9!-‘io—hd‘yf
dt ‘dr —d't 1dr
C C C C
nglf ‘35 szx 3d§x vc=flo+ fif+zzdy+zs y
dt dr dt d1: 1 d1 dr dr
uN=£lfo+h if ”-flo+h (hf
d’t N'1dt —d’t N'ldt
_d_w
_dT
(4.8)

where the time derivatives of the structural layer rotations and their higher-order terms in Eq. (4 2)

can be neglected based on Assumption (4 .1 .iv).

149

we

 

 

 

 

 

“s._.
3
A

 

 

V ,, MIDPLANE
. '1’ OF LAMINATE

 

 

 

1 m2

 

 

 

 

z}
1
\ \
h"!

 

 

Figure 4.3 -A single structural layer fabricated as a composite laminate.

4.2 Constitutive Equations

The constitutive relationship sought for this structure is derived to accommodate layers of isotro-
pic, orthotropic laminated unidirectional, laminated quasi-isotropic, or viscoelastic composite
materials. The assumptions made pertaining to the material model are listed below.

Assumption (4.2)

l. The material model for the structural layers are symmetric composite laminates
constructed from homogeneous and orthotropic lamina,

ii. For a viscoelastic structural material the moduli are modeled to follow linear
viscoelasticity and can be treated using the linear complex modulus approach.

Let us consider a single structural layer of flrickness T,- fabricated as a symmetric laminated
composite comprised of m stacked and rigidly bonded laminae, as shown in Fig. 4.3. Here, a lam-

ina is an orthotropic sheet in plane stress. In Fig. 4.3, we see that each kth lamina is located at a

150
vertical distance ti), from its midplane surface. Now consider a material coordinate system for a
lamina, with the principle axes 1, 2 and 3. The principle material axes for a lamina are oriented
such that the 1 axis is parallel to the fiber direction, the 2 axis is transverse to the fiber direction in
the surface of the lamina, and the 3 axis is normal to the lamina surface. Under Assumption
(4.21“), each lamina possesses a plane of elastic symmetry parallel to the 1-2 plane, so that the

constitutive equations for the kth lamina, of the ith structural layer, can be written as:

— —(i) —(i)~ -(i)

 

 

 

 

 

 

"a? as o
= as as o
_01‘5’) _ 0 0 (2:51“, 531
(t) (I) (i)
as = as o as
as) o as as)

(4.9)

where Eff) are the kth plane stress reduced elastic constants in the material axes of the kth lam-
ina, and the overbar signifies that the quantities are referenced to the material axes of that lamina.
In addition, 61 is the normal stress in the fiber direction, 62 is the normal stress perpendicular to
the fiber direction, and 012, 013 and 0'23 represent the shear stresses in their respective planes.
Similarly, the 8,-1- denote the strains in their respective directions. The correspondence between the

reduced elastic constants and the usual engineering constants is:

 

o E‘
u 1 ' v12"21
@- _ 152v12 Elv21
12 1“’12V21 1 ‘V12V21
o = E2
22 1 ‘V12V21

(4. 10a)

151

644 = 013
555 = 023
966 = G 12 (4.10b)

In Eq. (4.10), E) and E2 represent the Young’s modulus in directions 1 and 2, 012 , 023 , and C713
are the shear moduli for the lamina. and v21 and v12 denote the Poisson ratio when the lamina is
strained in either the l or 2 directions, respectively.

If the (x,y,z) system denotes the standard basis directions for the ith structural layer, then, to
construct each larnina’s contribution to the stiffness, the kth lamina must be rotated to this system.
With arbitrary fiber orientation, the variation of the kth reduced stiffness matrix can be obtained
by transforming the generally orthotropic [61km the specifically orthotropic reduced stiffness
matrix [Q]k. This transformation relates the mechanical properties of the kth lamina in the mate-
rial system to the geometrical (x,y,z) system through a simple rotation 6 that is positive counter-
clockwise. The stress-strain relation in Eq. (4.9) is transformed using the equation
(1)

(i) (i)

 

 

 

 

 

" T ' ‘ " W
(k) (k) (k)
0x 3 8x 8x
k ._ -1 ‘ (l k _ (i) k
0;) — [S] [Q1013 [S] 8;) "' [Q] (It) 8;)
C(15) 806) 80k)
_. xy_ _ xy_J _ ny

 

(k) (0 (k) (i) (k) (i)
C —1 — 8 8

x2 = S (i) 3 1:2 .__ (i) xz

m 11 [Q] m1] at) 1910,, a“)

GYZ Eyz yz

(4.11)

where [S] and [S] are the transformation matrices given by,

C’2 S2 CS

[S]: s2 c2 —cs

_ 2_ 2
2cs 2cs (c S ) (4,12a)

152

x C -S
[S] =
S C (4.12b)
and
C = cos (9)
S = sin (9) (4.12s)

To arrive at the constitutive relations for the structure, we must define the section force and
moment resultants per unit length in the geometric basis directions. Referring back to Assumption

(4.1) and Fig. 4.1, and employing the fundamental relationships for these quantities from classical

32. 68

plate theory, we can write for the first-order stress resultants,

H/2
_ (4.13)
(Nx,Ny,ny)— I (ox,oy,oxy)dz

—l-I/2
H/Z

(Mr My, Mxy) = I (are 0y, 010') zdz (4.14)

—H/2
”/2

(Q... Q,) = I (on, 0,942

—H/2

(4.15)

and write for the following higher-order terms,

H/Z
_ 2
(Px, Py,ny) _ I (ox,oy,oxy)z dz (4.16)

-H/2
”/2

3
I (6x, 0y, Oxy) 2 dz (4.17)
—H/2

(Y... Y, Y,,)

and,

H/2

Rx = I Oxzzdz (4.18)
-H/2

R = Joyzzdz (4.19)

S = I Oxzzzdz (4'20)
__ 2
3r I Cyz’ “’2 (4.21)

where Nx, Ny, ny are the stress resultants, M x, My, Mxy are the moment resultants, Q, Q), are the
section shear stress resultants, and P], Y}, Rj and S J- signify the appropriate higher-order contribu-
tions to these terms.

In Eq. (4.13), Eq. (4.14) and Eq. (4. 15) we see that the inplane stress resultants are found by
integrating over the structural layers only, therefore, when Eqs (4.13-4.17) are integrated over the

domain depicted in Fig. 4.2 we have

N- 3

 

1 _h21+1
2
(Nx,Ny,ny) =J.(o:,ol,o oxlymz+ z 5.; (o C,oC, 6C)dz+ 3' (0”, o”,ofy)d z
(4.22)
h hN—i
N- 3h
1 2 h2i+l N
Mxy) =J‘(ol,o‘, ox‘y)zdz+ 2 J( (5,0 oC o,C)zdz+ (of,o;’, o”)zdz
1111.211,“ (4.23)
l Bh2i+l N
(Px,Py,ny) =J;(O:.O')l,,ol y)22dz+ 2 J; (0C, 05,03) 22dz+ J. (afiofiofl'y) zzdz
h.-. (4.24)
_N_;__3_h:l:'1 hN
Yr Yy, ny) =J:(ol,6)l,,ol y)z3dz+ Z J; (o Qo§,ofy)z3dz+ I (ogofioivynsdz
hN-l

(4.25)

154
Now if we substitute the total inplane strains in Eq. (4.5) into the constitutive equation in (4.11),
and then substitute the resulting inplane stresses from Eq. (4.11) into the resultants in equations

(4.22-4.25) we obtain the following relationship for the ith structural layer

 

 

 

 

'- C (i) 1' 1(1)
{Nvay’ny}T _ C (i) {ex,ey,exy}T
{M M M }T A BDE {Kx’Ky’ny}T
1" y’ xy = D E F
T
{Pr Py’ PU} T F 1'; {33, Ay, 110'}

{Y Y Y }T ~Symm — We 11,, ”MT

_ x9 y! X), _1 l. ‘

 

 

(4.26)

where
i=I,C,N

and A ,2.) , 3(1) , D ,(3 , etc. are the structure’s stiffnesses for the ith structural layer. The relation in

In

Eq. (4.26) consists of two twelve element vectors and a twelve-by-twelve symmetric matrix. The

matrix is defined by the following expressions:

1
‘1:

1 l 1 l l 1 1 _ l 2 3 4 5 6
(A1", BI", DI", E1", Fln’Jln’ Kl") - Z J- [anI in) (192,2 92 ,Z 92 ,2 )d2 (427)
k=l
11-1

1
‘1:

N—3
2 m
C C C C C ' 2 3 4 5 6
(A1”,Bln,D,,,,E,n,Fln,JIC;,KlC;) = z 2![Q1n]::;+l)(l,z,z ,z ,z ,z ,2 )dz
i=lk=l

4 -1 (4.28)
4’
(Ali, 35%,, 01:, 127,, F11, 11,2, K53.) = 2 I [12,.) :2? (1, z, 2’, 23, 24, 25. 2") dz
k=1 (4.29)
191.1
I, n = 1, 2,6

To obtain the constitutive relations for the transverse portion of the structure we rewrite Eq.

155

(4.15) and Eqs. (4,184.20) as follows

 

—h2r—l —2—' hzj
(Q.,Qy) = 2 J (Gig—hog; l)arz+ 2 J- (0302542 (4.30)
i=1”2 j=lh2j—1
Nglh 1:1 I12}
(Rx,R ) = Z :I: (03: 1,02; l)zdz+ Z I (021‘, 021')de (4.31)
i=1}! i=1}!
N+l [v-1

h

— 2' “2—
(5.5): 2 I (oi; 1,oo;;-1)22dz+ 2

h
J (03,521);de (4.32)
i = 1h j= l h21-1

Equations (4.30-4.32) are equivalent to Eq. (4.15) and Eqs (4.18421), yet, they are divided into
their respective actuator and structural layer components. By substituting Eq. (4.5) into the con-
stitutive equations in (3.15) and (4.11), and, then substituting these resulting transverse stresses

into equations (4.30432) we establish the relations,

r- ‘1 (1) (i) (i)

 

 

 

 

 

 

{Qx,Qy}T A B D {exz’eyz }T
{R.,R,.}T = D E Us... 19le i=1, N
{Sx’ Sy} T _ symm F {1”, xyz} T
(4.33)
and for the core
— (2) (4) ‘
(6) (cs) (c, {Uh-”M2 + (hr-"3V 4 +...}
{Q} A B D {e} (hi—hfiv" (hi—h§)f"+
{R} = D E {K} + { 2 + 2 ..}
{S} ”m F m (hi-12M” (hi-him“
{ 3 + 3 +

 

 

(4.34)

156

where the superscript CS refers to the structural layers, and f is used to represent the function

f“) = f(ec,éc. co, V), 1': 2, 4, 6,... (4.35)

which is a generalized description of Eq. (3.15) for the ith actuator layer in the core. Then the

stiffnesses for the ith stnrctural layers are writtten as

l
m ‘k (4.36)
(4,1,1, 13,1", D)", E11”, F11") = 2 I [QM] 8.)) (1,2,22, 23, Z4)dz
k=l
‘k-t
1:3 {2:41
2 m
cscscscscs_ 2'1 234
(Aln’Bln’Dln ’Eln’Fln) " iakzl J [an] Eli)”- )(1,Z,Z 77' ,2 )dz (437)
4311
4’
m
(45,, 35,, D51, 55,. F?!) = 2 J [9,.) :15; (1,2,22,23,24wz (4.38)
k=1
’1-)
l, n = 4,5

Since Eq. (4.35) is dependent on strain, strain rate, voltage, and frequency of oscillation the actua-
tor layers stiffness is nonlinear and cannot be formulated in the same manner as Eq. (4.37). Thus,

it will be left as a function throughout the analysis and only dealt with for special cases.
4.3 Governing Equations

Here Hamilton’s principle is applied to derive the governing equations of motion for a plate and
beam. Since Hamilton’s principle is only applicable to conservative systems, the structure must
be initially treated as a fully elastic system, hence, the derivation of the equations of motion will
exclude damping. Once the energy functional is minimized for this provisional system, damping
in the structure can be introduced by invoking the correspondence principle.51 The notation
throughout this section will be such that the superscripts denote information pertaining to the indi-

vidual layers of the plate, respectively.

157

The configuration of the plate is depicted in Fig. 4.4. The plate is of uniform width W, total
length L, and uniform height H. Each layer is identified by the subscript i, and the thickness of an
individual layer is identified by ti. The total number of structural and actuator layers in the plate
N must also be odd so that the actuator layers are constrained. A similar Cartesian coordinate sys-
tem that consists of the global coordinates x, y and z is adopted for this model, where the x-y plane
of this system is referenced so that 2:0 at the midplane surface of the plate cross-section, as
shown in Fig. 4.4. For this analytical model of the plate we assume that Assumption (4.1) and

Assumption (4.2) holds true.

We turn toward the evaluation of the various energies involved in order to be able to
express the Lagrangian function in terms of our plate variables. From Eq. (4.2), there are a total

of seven displacements which are functions of three midplane deformations and six rotations.

ll

2

- 67

 

 

 

 

 

 

 

z - ' -A
Structural
La er
I t” y Actuator Layer flr

H

i—
3— w i

  

 

i
_T

Figure 4.4 - Configuration of the distributed-parameter plate.

158

When we consider the kinetic energy T we have:

T =-. %ijJ[(121)24-(12C)2+(d~)2+()31)2+(vC)2+(vN)2+w2]dsz
R H

(4.39)
where fir is the mass per unit area of the plate and is given by
N
” = Zp'wi—hm) (4.40)
i=1
Substituting Eq. (4.8) into Eq. (4.39) and integrating with respect to 2 we get
1 ~ .0 2 .0 2 ' 2 H2 . .
T= iflmmmu) + (v) + Q) ) +(h%+-17 +12%.-.) [(79% (1,621
R
.0 . 2 .0 . 2
+2(h~_1-h1) [u (YE) +v (75) mm (4.41)
The elastic strain energy for the structure is given by
l
U = 5.111 {o 8 +0 ey +0 8 z+oxyexy+oxzexz+oyzeyz} dsz (442)

Here we see that the triple integration is performed over the volume of the structure, where dA
denotes a small surface element of the structure in the x—y plane. Now, if we take into account the
assumptions made in the previous section and substitute Eq. (4.11) into Eq. (4.42), we find the

strain energy for the ith layer is

(1(1): %RJ' (Q3) (8(1))2 +Q(i) (8(1))2 +Q6(5i) (8(i))2 +Q(i) (8;?)2

+QU) (8(1))2:2Q1(;) 8:08;!) +2Q16) 8:1')8(i)+2Q2(i)€ (i) €(i)+2Q(,5')€ (1') 8(1)

8y: 8x2

+Qi‘7eg)+R§i)K§:)+S,fn7t£i)+ Qwez (i) +R‘0K (i) +£03.27 }dsz

i=1,C,N (4.43)

159
Subsequently, when the strain-displacement relationship in (4.2) and the constitutive relationships
in (3.15), (4.26), (4.33) and (4.34) are substituted into Eq. (4.43), integrated with respect to z, and

then summed, the following equation can be written for the total strain energy of the plate.

1 C N 1 N 3V0 alto 1 N
(1:!!{3‘5 (N +N +N )+—— (N +NC+N ”(5; +a_y'1(ny+ny+ny1 +

g—f(thN_l—Nihl)+311€(NN}1N_1—N;h1)+(g—YS+:}§)(nyhN_l—Nih)+
l l l l
h,(N;::+N1:: +N1 (gy— +311))—h,,_ l(NNN3:v+NN::N+NN (g: 9:1)H
3W 1 C N 3W 1 C N 1 BYC+MNCW+A£3171+ 816+
5;(QM+Qx+Qx)+-(.E(Qy+Qy+Qy)+M—“1‘+MCE"‘+ METx _
l l
11411:;1—NN+114,1,y(::11+a J) +Mf (371+ 52”) +MQ’ (311.4% +Qlyl+Q;yl+
a l a l 1 ICE) C C
vaf+Q§1F+ QW+QWJ+P18— C" "+P1aC+P i,‘(3—i+ 3:111?) a: +PC3—:+
acC+—C:C IVS—EN a:acN+P~acN+:—CN Yliilaxla§1+
P191154“):+1 ”WP ay ’fivy‘afl xy+) ‘Wyzr—yy+
351351 3&5 3§C 3&5 _égC YNaéN 3%” NBEN 3%”
Yiy($+a —x’+) +YC— +Yf$y+ny(a—;+ x’+) +YNayy+ ny(-a—x "+—’)+
2(R:§;+R;§:) +2(Rfcf+Rf§yC) +2(Rf§f+x1:’;§g’) +3(s;§;+s;§;) +
3(sf§f+sf§§) +3(s§’§fj+sg’§g) }dA (4.44)

where the nonlinear section shear stress resultants for the core are

C C - 4.44a
Q. = Q, (80.2% m. V) 1 1
Rf = Rf (2?, 15C, to, V) (4.4%)
sf = sf (eC, éC, a), V)

(4.444;)

Equation (4.44) is more tractable when only symmetric composite laminates are used. Hence, the
terms 8,-], 13,-], and Jij are identically zero. Moreover, when the plate structure is symmetric the By,

Eij terms for the core layers are also zero.

160
The total potential energy for the plate can be formed by considering transverse and inplane
loading. For transverse bending, we are concemed with loads generated by normal tractions

applied to the top and bottom surface of the plate which leads to the following expression for the

potential energy
W = -”qwdA (4.45)
R
where q is equivalent to
,- H ,- H
q = Oz 5) ’°z(-§) (4.46)
For inplane loading,
1 3w 2 8w 2 awaw
V — -2-‘”[Fx($) +FY(§_)7) +2F‘Y(B—x§y_)]dA (4.47)
R

which have been taken directly from Whitney.140 In Eq. (4.47), Fx, Fy and ny denotes the initial

inplane force resultants applied to the pre-buckled state.

From Hamilton’s principle the goveming equations of motion and the proper boundary con-
ditions for the plate are determined by substituting (4.41), (4.44), (4.45), and (4.47) into the varia-

tional equation and integrating over an arbitrary time interval

1:1

51 = I(5T—6U—5V—8W)dt = o (4.48)

10

Integrating the expressions in Eq. (4.48) by parts, and collecting the coefficients of SuC, 8v”, 5w,

8y}, 87f, Sny, My] , Myc, and My”, we obtain the following equilibrium equations:

a 1 C N a 1 N
—(Nx+Nx +Nx) +$(ny+ny+ny)

8x
dzu" (177::

2
2 2 H
a; “E (h~_1+h1+-i§ ] (4.49a)

161

a 1 N a 1 N
B—JC_(NNN+N‘SN+NXN) +§§(Ny+Nf+Ny)

20 dzyc H2
_mPH[(hl—h,,_,)dT Tr (h~_1+h1+12)] (4.491))

5v”:

5w:
(—a—2(Y'+YC+YN) +—a_zr1+1’C+YN+4iZ(Y1 +YC +1’N)
8X2 x x x ayz )1 y y axay xy x)’ xy

_ a 1 C N__ a 1 C N _4_
35(sx+sx+sx) 337(Sy+5y+5y))(3H2)+

2 2 2
3w 32w 3w +q=2Hdw

_{9_ 1 C N A 1 C N _ _ __
ax(Q1‘ + Q + Q) + ay(Qy + Qy + Q) + anx2 +Fyay2 + ZFJ’Bxay drz
and the curvatures about the y axis,
5? 1:
ayf arc

2 2 N l N 3
hth_1(4H—20) (333(Y1+YE+Y,)+§7(Y,,+Yf,+Y,,))+9H hN_l(§-£ +5;
90H1hfh},_ 1 + 451113sziz,,,_l

2 2 N
hN_, (12H 1:, — 60Hh1) (5170'; + Yf)+§—y(1';y+ 1(3)) + 3H2h§_1§—y(ny-5Y1y+ Y”)
+
90H1hfih,1;,_, + 45H3hth_ 1

31112hf,,_l (§;(Yf+ Yf—SY;)) +h,2,,_,(s;+sf+SN) (60Hh1—12/fi)
+

9011211112?M + 4517311th 1
3H2h§,_,(ss;-sf—sf) + (s;+sf) (180Hh¥hN_1-36H2h1hN_l) —27H3h,,_,sf
90112hih124-1 + 45111111191,“l

3N1 2)le a_M; a_M;y_O

_- l _x _ _
Qx+hl(ax +ay )+ax +3y
(4.49d)

162

and,

61f:

aYC arc
”11%— 1 “1H”: + 311553;”? Yf+ Y?) +53—y-(Yl,+ Yf,+ 1’21,» - 91:113/z,,,_l (a; +5?”

 

2 2 2 3 2
9011 hth_l+45H hth_,
2 2 l l 2 2 l N
hN_, (15H 111—601111,) (53}(Yx+ Yf) 448370;” 1%)) + 15H h~_1§—y(ny—ny+ny)

 

 

 

 

+
9orvzhfihf(,_l + 4511i3h%hN_l
15H2h§(§—x(1’f— Yf— Yb) +h,2(,_1 (51+sf+s,’;’y) (36Hh1—9hf)
2 2 2 3 2 +
90H hth_l+45H hth_l
3H2h12v-1 (55:43-3?) + (s;+sf) (4517IIz§/1,,,_l —60H2h,h,,_,) —27H3h~_1Sf+
2 2 2 3 2
90H hth_l+45H hth__1
3N1 3N1 8N” 3N” aMC BMC d2 0
_ C _xy__x _xy__x _x _xy= _ _u
Qx+h1(3y 8x )+hN’l(8y 8x )+3x +3y 2H(h~’l hl)d’c
(4.4%)
516:”:
11th (2011—120) (i(Y1+YC-1-YN) +i(r1 +YC +YN))+45H3h (a—" +33")
1 -1 ax x x x a), x)’ J‘)’ 1‘31 N'l ax 8y +
90H1hfhi,_, + 4511311111,“
2 N
hN_1 (4H2h1—30th) (£03111 YE) +%(Y;y+ 13%)) +15H2hN_1§—y( xy—sriyuxy) +

 

90H2hfhi,_, + 45 H3h§hN_,

4511211,?“l (53;(Yf+ Yf-SYD) + 11,2“ (s;+sf+sf,) (3Hh1—45hi)
+

 

90112111113“ + 45H3h§hN_,
4H1h§,_, (2551-35—31!) + (5}, +55) (135Hh21’hN_1 —72H2hth_ l) — 9H3hN_ 15,?
9011211112; 1 + 45H3h§hN_1

1 h 3N}r 3N},y 3M1, aM;y_O

 

(4.490

163

and the curvatures about the x axis,

64,1:

an arc

 

+
90H2h§hfv_ 1 + 45113Ia§11,(,_l

hN_, (121121;, - 60th) (g—xwifi ny) +%(Y; + 25)) + 3H2h,2(,_1:%-(Yf-5Y;+ Y;’) +

 

901211111934 + 451L111h§hN_l
22 a C N 1 2 1 C N 2

 

+
90H2h§h§_ 1 + 45H3hfh,,_l

3H2h,2(,_1 (551—5541;) + (s;+sf) (180Hh‘l’-h,,,_l -36H2h,hN_,) —27H3hN_,syC
2 2 2 3 2
90H hth_1+45H 121/1%,
1 1 l 1
5w aNy 3M”, aMy

_ 1 _xy _ _ _ =
Qy+hl(3x +832 )+3x +3y O (4.49g)

 

and,
57f:
arf arc
‘1?“ 14””11‘3‘”) <%<Yiy+ny+Y¥y> +57%”? +41» -9H3h~—1 <;; ”551) +

 

90H2h§hfv_ , + 45113hth_l

h,(,_l (15111211140th (5)7":2’" ny) «hiya/)1, + if» + 15H2h12v_1%(l’; — Yf+ Y?) +

 

9017211th 1 + 45H3h71'hN_l

15H2hi<§7(1’fy' Yf,- 1310))+h§,_,(s;+sf+s;1)(36Hh1—9hfi)
+

 

90112119; 1 + 45H3hfihN_1
3H2h§_1(55;_55_5g) + ($4155) (45thhN_,-60H2h,h,,_l) -271=13h,,,_lsf+
90H2hfhfi,_ 1 + 451113Iz§12,,,_l
3N1 3N1 3N1; 9:11;; BMC aMC dzvo

_ C _Y___xy _ _ _xy _y= _ _
Qy+hl(ay ax )+hN‘1(3y ax +3): +ay 2”“‘N-1 ”1141

 

(4.49h)

164

and finally,

My”:

ayf BYC
h}h§,_1(20H— 120) (515131” 1131+ Y§1)+afiy-(Y;+ Yf+ 111111)) +45H3hN‘1(_8-3c_ +5?)

 

901112hfhfv_1 + 45H3hfh1,_1

hN_1 (4112111 - 3011/11) (§;(Y1111+ ny) +%(Y;+ 1f» + 15H2h121,_1aa—y(1’f—5Y111 + Y’j) +

 

90H2hfh121,_1 + 45 11312121114
45H2h,21_ 1 (1137(ny+ 11211—513111»+h121_1(s;+s§+s§’)(312111-4512?)

 

+
9011211112121,_1 + 45H3h§h111_ 1

4H2h§,_ 1 (2551, -S§ —s§’) + (51114-55) (135Hh111h1,_1-72H2h1h1,_1) —9H3hN_1SyC
90H2h‘11h111,_ 1 + 45H3hfh1,_1

8N1 8N1 3M1 3M1
—Q;+h1(§-i”+—y)+—N’+—y=0

 

 

 

 

8y 8x 3y
(4.49i)
The boundary conditions are of the following form:
“(1') nx+ v“) n), or N1?) ni+Nym ni+2N$ nxny
—umnx+ vmny or (N;0 -N1E0)nxny+N1f;) (113-n3)
. . . . 4H2h2_ —3H2h h _
(9,514,515,114.(go—Ryan,— 2 2’21 :2” ‘
w 0, 90H h1hN_1+45H h1hN_1
((nx%—ny§a;) ((1'11‘1 — rfi‘1)n1n1+r1f;1(n§—n§)))
183—2)“ + 3;"? 0r Y1?) n: + Yy“) n: + 2115511111,
2 2
[M _ 12Hh~_l—H hth-l Y(i))n2
" 90H2h§h§1_1+45H3h§hN_1 " "
SHh _ -20H2h h2_ .
‘Yionx-l-‘Ymny or + ([My- 2 £21 ; 12v 1 Y;‘))n§+nxny]
1 90H h1hN_1+45H h1hN_1

2 2
HhN-r'H hth-l (i)
2 Mn" 2 2 2 3 2 ny
90H h1hN_1+45H h1h1,_1

 

(4.50)

i=I,C,N

165

4.4 Comparative Results
4.4.1 Static results

To demonstrate how this distributed-parameter plate theory compares to existing results, let us
consider the plate given in Fig. 4.4 to be simply supported and subjected to the following static
surface load

nx y

4 = q.sin(W)sin(’—‘17) (4.51)

Accordingly, with the plate being simply supported, the boundary conditions are

—L L —W W
w(x,—2—)—w(x,-2—)—w(——2—-,y)—w(-2—,y)-0
(i) -L _ (i) L _ (i) -W _ (o W _
m -L _ (i) L _ (o -W _ (i) W _
My (1,7)-My (x,§)*Mx (TOO-M; (5")0—0

i —L i L r ’W i W
71110,?) #1111045) 41(7)) =v§’(§.y) =0
(4.52)

i=1,C,N.

Use of the N aiver approach described by Whitney140 permits us to solve Eq. (4.49). For simplic-
ity, consider the plate to be composed of three layers, i.e. N = 3, where the actuator layer is sand-
wiched between two identical face aluminum sheets with constant thickness. Table 4.1 provides

the assumed properties for the plate.

For this case, Eq. (4.49) is greatly reduced because all time derivatives are set to zero and
several of the plate stiffnesses are identically zero. From Table 4.1 the following stiffnesses fall
into this category

B11=E11=J11=0 I,k=1,2,4,5,6

(i)_ (0_ (0__ (i)_ (i)_ (i)_ (i)_ (0..
Are-A26 "Dre- 26-F16—F26-K16- 26-0 (4.53)

166

Table 4.1 - Material and geometric properties for the plate configuration.

 

 

 

  
 

  

  

 

 

 

 

 

 

 

 

 

 

Structural Layers Actuator Layers
Property (1' = 1,3) Specifications Property (i = 2) Specifications
Material Aluminum Sheet 6’ 1.225 MPa
L 31.2 cm v (Poisson) 0.45
W 31.2 cm V111 (volume) 0.15
H 0.1143 cm Vfl( volume) 0.85
511) 68.5 GPa p 1224.1 kg /m3
p 2714.3 kg /m3 E( 1) 1.225 MPa
v(i)(Poisson) 0.334 qo 1 N
h( 1) 0.50 mm w 31.2 cm

 

 

 

(1') _ (1)17“)
A45 “D45 = F45 =0
1:1, C, N

and using Eq. (3.15) the core stiffness reduces to

C _ 1 C
oxz—uinG (0)8“
C _ I C
oyz—uinG (0)13z

Following the Naiver procedure, we assume a solution which satisfies Eq. (3.52) and is of the fol-

lowing form

2:, W '1’”) ' (“yr
msm W sm T

m,n:1

701:2 X('21cos(—x)sin(—y)

m,n=l

2 Y“) sin(1—c£) cos(n—y—)
1 "1" W L
mm:

i=l,C,N

(4.5 3con’t)

(4.5421)

(4.54b)

(4.55)

167
where WW1, X11511,1 and }’("),,1,1 are unknown constants to be determined. By substituting Eq. (4.55)

into Eq. (4.49), these seven constants can be solved for algebraically. For brevity the equations

are not shown.

Results for the maximum nondimensionalized deflections, W, of the plate are given in Table
4.2 for various values of W /H, hence various actuator thicknesses. Note that the length—to—thick-
ness ratios used in this section would not be practical for this type of structure, but are only used
for comparison purposes. Below, these results are compared to an exact three-dimensional elas-
ticity solution obtained by Pagano.108 It shows fiorn these results that good agreement between
this theory and exact solutions are obtained over a wide range of aspect ratios. However, for large
aspect ratios, ER plate deflection is slightly higher than the elasticity solution. This is expected
because the three dimensional effects occurring in this model have been neglected. Also, because
this theory has been based on sandwich plate theory and constrained layer theory, the in-plane
stresses in the core have been assumed to vary linearly through-the-thickness. This means that for
all plate aspect ratios, this theory will greatly over-predict the true shearing stresses unless shear

correction factors are used.112 It would be highly effective to use shear correction factors, espe-

 

 

 

 

 

 

 

 

 

 

 

Nondimensionalized Deflections
W m Paganolog ER Plate
4 0.682 0.49128
10 0.2191 0.16537
50 0.9761 1.05721
100 0.0951 0.09684

3 (l)
w = 10(3’LL) w(O, 0)
(10W4

 

 

 

Table 4.2 - Nondimensionalized deflections for two three-layer sandwich plates.

168
cially when integrating the shear stresses for this plate, because good predictions are necessary

due to the nature of the actuator and its dependence on the shear rate.
4.4.2 Dynamic results

A finite element analysis of the distributed-parameter plate was conducted to obtain results for its
dynamic behavior. Square and rectangular plate configurations were investigated, both had the
same dimensions and geometries as the plates given in Fig. 3.25. The nonlinear nrle-of-mixtures
material model developed in Chapter 3 was implemented for the actuator. For simplicity, the
same actuator material is always sandwiched between structural faces of identical geometry and
material properties. Furthermore, all structures are symmetric about their midplanes. An exami-
nation of the frequency response for steady-state excitations was conducted for free-edge and
fixed-edge boundary conditions in order to compare with experimental results in Chapter 3.
Modal natural frequencies and damping factors were estimated for both cases and for various con-

stant voltages to set bounds on physical parameter uncertainty.

The plates were discretized using 200 reduced integration, eight-node, second order shell
elements with thick shell control. In ABAQUS, the shell element has several unique qualities that
make it a good candidate for these structures. Its shell elements are provided with a composite
material option that allows us to select the number of integration points through-the-thickness.
The number was set to 3 integration points foe the stnrctural layers and 5 for the actuator layers.
The shell element also includes a transverse shear option that permits the use of shear correction
factors. For the sandwich structures considered here these factors are evident in the following

equation
111/2

(Q.,Q1) = I 05611165019412 (4.56)
-H/2

where k5 and k6 are the shear correction factors about the y and x axes, respectively. For these

structures k5 = 0.4098 and k6 = 0.6915, which are reasonable for sandwich structures.140 Finally,

169

-----‘\

\
\\\\\\\\\K\
\\\\ \\\\\

 

Figure 4.5 - Discretized geometry for a quarter of the distributed-parameter rectangular plate.

the thick shell element used here is able to accommodate full six degree-of-freedom models,
whereas the solid brick element would be limited to the three displacements, and would not
include the necessary nodal rotations experienced by such a low stifl‘ness structure. An illustra-
tion of one quarter of the discretized rectangular plate, shown as a solid model, is presented in
Fig. 4.5. Notice that the aspect ratio around the edges was reduced to obtain better solution con-

vergence due to three dimensional free-edge effects.

Figure 4.6 is an illustration of several zero voltage displaced mode shapes for the identical
three-layered structures used in the experimental analysis. These mode shapes were computed
using a linear subspace routine available in ABAQUS.1 The flat area in the center of both plates
represents the 3 cm square clamping fixture used to attach the plate to the shaker. Due to the mag-
nification of the modes, its appearance seem to have a large effect on the results. We see many
similarities between the experimentally observed results in Flg. 3.33 and these results. However,

one disadvantage in making the experimental observations is that very close mode shapes of low

 

Square Plate

 
 

170

Q

.
. .0
.o”

       
  

.9

    

O
:“.

0“

'3'.
O 9
O O
.0
o:.

O

'0
.0

‘0

..‘...o
O O.
O
O

o“...

9’...

O
O

   

9
0’9

0

o
o

 

o

0..

0..
’

   

      

’

 

O
9’”.
Q...
9..

 

Mode Shape Identification

 

Mode 1

 

 

Rectangular Plate

I

  

 

 

 

 

 

 

     

 

  

 

  

 

 

 

        

 

 

   

    

 

0 kV.

 

6 - Mode shape results for both plates at V

4

Figure

 

 

171

Table 4.3 - Comparison of measured and computed natural frequencies for the square plate.

 

Square Plate Response for Constant Voltage Inputs

 

 

 

 

 

 

 

 

 

 

 

 

Natural V=0kV V=lkV V=2kV
Frequency
(Hz) measured computed measured computed measured computed
(1)1 19.50 19.699 22.06 21.53 23.94 22.96
002 25.525 28.25 29.923
033 25.525 28.25 29.923
(04 37.87 41.556 41.47 44.76 63.50 47.036
(05 64.56 60.585 71.15 65.58 72.28 69.92
006 70.821 78.072 82.821
(1)7 70.821 78.072 82.821
(03 118.3 93.574 126.35 105.48 129.065 113.54

 

 

 

 

 

 

 

Table 4.3 - Comparison of measured and computed natural frequencies for the rectangular plate.

 

Rectangular Plate Response for Constant Voltage Inputs

 

 

 

 

 

 

 

 

 

 

 

 

Natural V=0kV V=lkV V=2kV
Frequency
(Hz) measured computed measured computed measured computed
0)] 15.52 15.073 19.76 22.45 24.53 25.062
(02 17.9 24.85 28.595
(03 31.36 29.21 33.21 36.763 35.76 45.763
004 53.15 53.34 54.3 55.914 55.45 56.020
(05 68.095 70.577 74.971
(06 83.223 83.223 88.663
(07 76.35 87.576 132.12 102.76 135.11 111.15
(03 104.25 111.025 127.29

 

 

 

 

 

 

 

 

 

172
magnitude were indeterminable. For this reason, several modes have been omitted in order to
compare the two sets of analysis results. It appears that ABAQUS has no problem computing
mode shapes that reside only fractions of a frequency apart. In Tables 4.3 and 4.4, natural fre-
quencies for both plates are given for the identified mode shapes in Fig. 3.3 and Fig. 4.6. These
mode shapes were matched up and compared to computed shapes then their associated natural fre-
quencies were tabulated. The blank entries in Table 4.3 and 4.4 signify computed values for
modes that were not observed in the experimental results. For instance, the results for modes 2
and 3 for the square plate occur at the same frequency and are very small as compared to the fun-
damental and second observed mode shapes. It is evident here that ABAQUS provides good
results for the fundamental and second observed modes for all input voltage cases considered.
Although, beyond two modes it was very difficult to determine a match for the experimental val-
ues because the natural frequencies are very close to one another. This situation was not encoun-

tered experimentally, where all mode shapes seemed to be separated over larger ranges.

Table 4.5 - Material properties for the composite structural layers.

 

 

 

 

 

 

 

 

 

 

 

 

 

Composite Structural Layers

Property (k = 1,2) Specifications
Material Graphite / Epoxy
1,11 1) 0.025 mm
151,“) 135 kN /mm2
522‘ 1) 13 kN innn2
G 12‘ 1') 0.641 kN /inm2
G 1 3‘ 1) 0.641 kN /mm2
023(1) 0.434 kN /nnn2
V12( 1) 0.38
V211 1) 0.38
p 1553.98 kg /m3

 

 

 

173

 

 

 

 

 

 

 

 

Figure 4.7 - Modes shapes for the clamped-edge square plate.

Good results for the mode shapes and natural frequencies of the first two observed mode
shapes permitted an investigation of the variation of natural frequency with the clamped edge
boundary condition. Results were obtained for a square clamped-edge plate with the same width
and length geometry as given in Table 4.1. However, two specific structural material cases were
investigated for V = 0 kV. The first case was a plate composed of aluminum structural layers and
the second case was a plate composed of a cross-ply, [0°/ 90°], graphite/epoxy composite. Mate-
rial data for the plate with the composite structural layers is given in Table 4.5. The mode shapes
associated with the first six modes of both plates are given in Fig. 4.7. These mode shapes were
identical for both clamped edge cases. It is evident from Fig. 4.7 that modes 2 and 3 must occur
at the same natural frequency because mode 3 is just a generalized coordinate shift of mode 2.

We also see that the number of node lines increases with 2n for this boundary condition

174

.

”8888838853

883888

   

Freauencv (Hz)
Frequency (Hz)

  

3 4
Mode Number
(a)

(b)

 

Figure 4.8 -Natural frequencies for a multi-layered square clamped-edge (a) aluminum plate,
and (b) composite plate for V = 0 kV.

LII
Ill

 

Frequency (Hz)

 

 

 

r l l l l

a 4 5 e 7
Thickness (mm)

 

0"
-e
N

Figure 4.9 -Natural frequencies for a three-layer square clamped-edge aluminum plate
with varying actuator layer thickness and V = 0, 1 and 2 kV.

175

Results given in Fig. 4.8 are from an investigation of the variation of natural frequencies
with different total number of layers. Values were obtained for a 3, 5 and 7 layered clamped-edge
composite and aluminum plate. In each case, the thickness of the structural layers was constant so
that the total thickness H changed. For the 5 layer composite plate, the 3rd layer, i = 3, was
divided into two cross-ply lamina. k = 4, in order to maintain symmetry through-the-thickness.
As observed in Fig. 4.8, the natural frequency increases with an increasing number of layers. This
increase is most significant in Fig. 4.8 (a) for the aluminum plate, although the natural frequencies
for the composite plate are larger. In Fig. 4.9, results are shown for a three layer clamped edge
aluminum plate with different actuator thicknesses and different voltages. It is apparent that the
natural frequency increases with increasing voltage, which is in agreement with experimentally
observed results. However, as 11(2) increases the natural frequencies begin to approach an asymp-
totic value. This occurs because bending stiffness is not increasing as rapidly as total plate thick-
ness since structural layer thickness is kept constant In addition, we see the natural frequencies
increasing as ha) decreases for the converse reason. These values approach the natural fiequency

of the plate when 11(2) = 0.
4.5 Chapter Summary

Both analytical and numerical analysis formulations were conducted in this chapter. An analytical
model was derived based on a higher-order plate theory. This was further complemented with
assumptions from classical sandwich theory and constrained layer theory. Next, the Hamiltonian
was used to arrive at the governing equations of motion and associated boundary conditions. It
was shown for static loading conditions, using the Navier approach, that close agreement with
previously published results was possible. Time constraints did not permit a full dynamic investi-
gation of the analytical model. Numerical results were presented with the use of a commercial
finite-element code. Frequency response results were generated for the two plate configurations

considered in Chapter 3. Once again time did not permit a full time-dependent dynamic input

176
voltage investigation. These computed results showed many similarities between experimentally
determined results. Natural frequencies for the first two modes for all voltage inputs are in close
agreement despite some difficulty determining whether observed natural frequencies matched
with computed values. With the actuator model and structure partially verified, the response of
the square plate was examined for various materials, actuator thicknesses, and total number of
layers. These results were then compared and contrasted.

Due to the large uncertainty in this analysis is was too difficult to accurately set physical
bounds for the distributed—parameter plate considered in this analysis formulation. Uncertainty
exists as a result of numerous sources, including: i) inertial damping of the fluid, ii) non-rigid
bonding conditions between layers, and iii) the replication of the experimental plate boundary
conditions. In fact, uncertainty in this case can be a result of objective and subjective causes. In a
subjective manner, uncertainty is introduced due to the extreme variations in the operating condi-
tions and the actuator material. Uncertainty introduced objectively is a result of the inability to
replicate the boundary conditions in the experiment to the computer model. These two sources of
uncertainty must be reduced in order to set bounds on the structure. One possible way of reducing
subjective uncertainty is to use narrowband excitation and determine experimental values for the
constitutive model of the actuator. This would allow us to experimentally identify a material
model for better comparative purposes. Secondly, to reduce objective uncertainty, we could use
simply-supported or clamped-edge boundary conditions instead of the free-edge conditions.
Although simply-supported conditions are difficult to completely replicate, they would allow a

much better comparison than the free-edge boundary conditions that were used.

ChapteLS

CONCLUDING REMARKS AND
RECOMMENDATIONS FOR FUTURE WORK

A generalized outline for a mathematical modeling procedure has been established in the spirit
of concurrent engineering. Essentially, this procedure has been developed for evaluating a
variety of large-scale system modeling problems that exist at the preliminary and initial stages
of design. However, in this study, only a certain class of problems involving smaller scale
controlled flexible structures has been considered. The reason being that a large-scale assembly
can often be decomposed into conventional beams, plates, and shell structures. These problems
in structural control have been addressed by exploiting the highly acclaimed methods of robust
control theory and uncertain linear systems analysis and combining them with the salient features
of concurrent or simultaneous engineering design and development.

Specifically, this concurrent modeling procedure assumes and treats a mathematical model
not as a single and precise model but rather as a set, inside of which the actual system lies. This
framework calls for two specific models: (i) a nominal model, and (ii) an uncertainty model. In
this procedure, it is taken into consideration that tight time constraints would typically limit the
construction of two models during the design process. However, instead of deriving each model
in series, both models can be derived concurrently as long as applicable physical laws,

assumptions, and equations are available to effectively describe the actual process. Thus, by

177

178

using this procedure in the design process, one is able to improve the accuracy of the
mathematical model simply by improving the accuracy of the uncertainty model. Therefore, over
the course of the product or design development program, and as parameters become more
precise, uncertainty models should gradually mature with the structural design and its control
system.

In Chapter 2, discussion focussed on how the modeling procedure can be divided into two
specific problems: (i) the analysis problem, and (ii) the synthesis problem. In this study, the
modeling procedure has been applied to a cantilever beam example and plates containing a
distributed-parameter actuator. Both analysis and synthesis problems have been investigated in
detail for the cantilever beam, however, only the analysis problem has been completed for the
plates. In particular, our results show that this procedure provides models that can be very useful
for locating and identifying bounds where the actual system lies. This procedure does have some
drawbacks due to its ”very" generalized form. Experience plays a large role in selecting
uncertainty models, extracting correct analytical and experimental information, and fitting this
information to an appropriate model in a judicious manner. For example, throughout the
cantilever beam example, it is evident that neither parametric nor nonparametric models can fully
satisfy full model uncertainty. Clearly, the very conservative nature of the parametric model at
high frequencies severely limits its applicability. To include a parametric model at these
frequencies will cause higher modes to become unstable for high controller gains. For this
reason, a sequential mixed uncertainty model was chosen because it effectively represents the
uncertainty over the selected frequency spectrum. Although, as a consequence, much of this
decision-making for choosing a transition frequency was based on prior experience.

Now, the actuator used in the plate analysis formulation is an electrorheological (ER)
fluid. Its electrical and rheological properties have been investigated in order to formulate pheno-

menological equations that would be useful in numerical and analytical analysis. Due to its innate

179

complexity, an empirical model was developed to describe its rheological properties. This model
accounts for the global damping and stiffness properties under the influence of electrical
potentials. It has been shown through experimental, numerical, and analytical formulations, that
extreme coupling between the electrical and rheological properties produces a "dynamic
capacitance” effect that is very difficult to model. Consequently, this characteristic excluded
consideration of time-dependent voltage inputs and transient vibration. However, steady-state
vibration was investigated and good correlations between the experimental and numerical results
were obtained for the first two observed natural frequencies at all voltage levels. Damping was
not computed nor looked at in detail because of the limitations of the numerical software.
Therefore, uncertainty bounds could not be established for this analysis.

In conclusion, this thesis has been an exposition of a concurrent modeling procedure.
Two useful applications have been considered with varying degrees of difficulty. Results from
these two models have shown that the procedure is highly dependent on the experience of the
person modeling and the complexity of the problem at hand. Despite these disadvantages the
framework of this modeling procedure, which is based on uncertain linear systems analysis and
robust control theory, yields good practical controllers. Thus, future work will include the full
integration of this procedure within a process that can accommodate simultaneous structure and

controller design and optimization.

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