ributed sensor to
a distributed actuator is developed. The outputs of two sensors of different impedances are
combined electronically with the goal of increasing pole-zero spacing for improved per-
formance in low-authority structural control loops. The concept of a three-element actua-
tor-sensor module capable of adjusting the equivalent actuator and sensor impedances is
presented. The module consists of an actuator, and two sensors for measuring force and
strain. The output of the module is constructed by mixing the force and strain signals using
a mixing coefcient which can be used to tune the apparent sensor impedance for maxi-
mum performance. General shape of zero trajectories as a function of the mixing coef-
cient is derived. Mass-spring and beam models are used to further explore the behavior of
the zeroes of the mixed transfer function. Both an approximate beam model derived using
assumed mode method and the exact solution of the beam vibration equation are
employed. A practical implementation of the module is proposed. The design uses a piezo-
electric actuator with a collocated piezoelectric strain sensor and a novel piezoelectric
shear load cell. A test article was built, mounted on a cantilever aluminum beam, and
tested. Experiments veried the ability to increase pole-zero separation of a structural
transfer function by mixing the outputs of displacement and force sensors. At low frequen-
cies the overall shape of experimentally found zero trajectories compared well to the
results of beam models. Non-minimum phase zeroes encountered for certain values of the
mixing coefcient in both the models and the experiments limit the range in which the
mixed transfer function is attractive for feedback control.
4
5

ACKNOWLEDGMENTS

Funding for this research was provided by NASA Washington Space Engineering, under
contract NAGW-2014.
6
TABLE OF CONTENTS

7

TABLE OF CONTENTS

Abstract

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3
Acknowledgments

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
Table of Contents

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7
List of Figures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9
List of Tables

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
Nomenclature

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
Chapter 1.
Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17
Chapter 2.
Modeling systems with force and Strain Sensors

. . . . . . . . . .

25

2.1 Actuator-Sensor Module . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2 Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.3 Zero Trajectory Plot . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Lumped Parameter System . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.3 Finite-Dimensional Models of Beams . . . . . . . . . . . . . . . . . . . . 60
2.3.1 Application of the Assumed-Mode Method to Beam Vibrations . . . 61
2.3.2 Fixed-Free Beam with an ASM as a Force Actuator . . . . . . . . . . 64
2.3.3 Fixed-Free Beam with ASM as a Moment Actuator . . . . . . . . . . 71
2.4 Innite-Dimensional Models of Beams . . . . . . . . . . . . . . . . . . . . 77
2.4.1 Solution of the Beam Equation . . . . . . . . . . . . . . . . . . . . 78
2.4.2 Fixed-Free Beam with an ASM as a Force Actuator . . . . . . . . . . 81
2.4.3 Fixed-Free Beam with an ASM as a Moment Actuator . . . . . . . . 88
2.5 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Chapter 3.
Experiment Design

. . . . . . . . . . . . . . . . . . . . . . . . . .

99

3.1 Conceptual Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 Component Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
3.2.1 Actuator and Strain Sensor package: QuickPack . . . . . . . . . . . 104
8

TABLE OF CONTENTS

3.2.2 Force Sensor: a Shear Load Cell . . . . . . . . . . . . . . . . . . . . 106
3.3 Component Integration and Manufacturing . . . . . . . . . . . . . . . . . . 109

Chapter 4.
Experimental Results

. . . . . . . . . . . . . . . . . . . . . . . . .

121

4.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2 Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3 Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

Chapter 5.
Conclusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141
References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147
LIST OF FIGURES

9

LIST OF FIGURES

1.1
Controlled Structures Technologies (CST) framework. . . . . . . . . . . . . . 18
1.2
Actuator and sensor spectra [Fleming, 1990]. Sensor impedance is dened as the
output signal content relative to two extremes: a generalized force and a generalized
displacement sensors; also shown are special actuator-sensor pairs: complementary
extremes (arrows), positive compliments (circles), negative compliments (squares),
positive non-compliments (crosses) . . . . . . . . . . . . . . . . . . . . . . . 19
2.1
Typical sensor applications: (a) strain sensor placed in parallel with a structural stiff-
ness and, in this case, an actuator; and (b) force sensor placed in series with an
actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2
A conceptual representation of a three-element actuator-sensor module (ASM) does
not imply any modeling technique or practical implementation. . . . . . . . . 28
2.3
Static lumped-parameter model of an actuator-sensor module (ASM): (a) diagram;
(b) force balance at spring juncture; (c) force balance at the juncture between the
load cell and the structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4
ASM/structure integration can be cast as a feedback problem. . . . . . . . . . 42
2.5
An illustration of the relationship between the poles of the original uninstrumented
structure, the zeroes of the force transfer function and the travel range for the zeroes
of the mixed transfer function. . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6
Expected shape of the zero trajectory for one mode of the mixed transfer function,
monotonic change in zero frequency is assumed; also shown are the interpretations
of the special values of the mixing coefcient in terms of the mixed transfer func-
tions they produce. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.7
Mass-spring system with an actuator-sensor module (ASM): (a) schematic; (b) free-
body diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.8
Sample strain (solid) and force (dashed) transfer functions for an ASM connected to
a mass-spring system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.9
A typical zero trajectory plot for a mass-spring system with an in-line ASM. . 53
2.10 Step response of the mass-spring system with
(solid),
(dashed), and
(dash-dot) demonstrates NMP behavior of the system with the mixing coef-
cient
between
and
. Steady state values are shown as horizontal lines
with the same line styles as the corresponding time responses. . . . . . . . . . 54
2.11 Zero locus of the mixed transfer function of an ASM connected to a mass-spring
system; also shown are the system poles (crosses), strain sensor zeroes (circles), and
force sensor zeroes (diamonds). . . . . . . . . . . . . . . . . . . . . . . . . . 55 0.5 = 1.0 = 2.8 = 1.0 = R S
10

LIST OF FIGURES

2.12 Pole-zero separation in the strain sensor transfer function as a function of strain sen-
sor and load cell stiffnesses for an ASM attached to a mass-spring system. . . 56
2.13 Mixed transfer function zero dependence on the mixing coefcient and the relative
load cell stiffness plotted normalized between 0 and 1 within the bounds imposed
by the zeroes of the fo