DTIC ADA325671: Shaped Actuators and Sensors for Local Control of Intelligent Structures.

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4. TITLE AND SUBTITLE 5. FUNDING NUMBERS
Shaped Actuators and Sensors for Local Control of F49620-93-1--0537
Intelligent Structures
6. AUTHOR(S)
Amy Jean McCain
Edward F. Crawley AFOSR-TR-97
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)Q
Department of Aeronautics and Astronautics
MIT -- '
Cambridge, MA 02139
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'.ABSRACT The properties of the actuator to sensor transfer functions for various shaped strain actuator-
cj=ýsensor pairs on a Bernoulli-Euler beam are investigated. Analytical expressions for these transfer
Ci
functions and their associated dereverberated transfer functions are derived. It is shown that the
actuator-sensor pair can be designed such that its derevercerated transfer function will have a
desirable corner frequency and high frequency rolloff rate.
The analytical and dereverberated transfer functions of noncollocated actuator-sensor pairs are
compared to those of the collocated pairs.
C~)
Finite element models are constructed which add damping, finite actuator thickness, and finite
beam and actuator widths to the model. The actuator to sensor transfer functions are calculated to
(37)
these models, and the effects of the added factors are determined. It is shown that the transfer
bending modes of the three dimensional beam delay the rolloff of the actuator to sensor frequency
transfer function by at least two decades. Finally, experimental data confirms the results of the
three dimensional finite element model.
4. SUBJECT TERMS 115. NiU .13E RA PAGES
Actuators, collocated pairs, finite element model, rolloffIII
characteristics, Shaped actuators, sensors, transfer functions 16."PFCE CODE
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17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATI-ON OF ABSTRACT
Unclaasified Unclassified Unclasifie
%S%' 7540-01-280-5500 Stardard Porm 298 (Rev 2-89)
S'in Z;93*
Shaped Actuators and gensors for
Local Control of Intelligent Structures
Amy Jean McCain
Edward F. Crawley
Abstract
The properties of the actuator to sensor transfer functions for various shaped strain
actuator-sensor pairs on a Bernoulli-Euler beam are investigated. Analytical expres-
sions for these transfer functions and their associated dereverberated transfer func-
tions are derived. It is shown that the actuator-sensor pair can be designed such that
its dereverberated transfer function will have a desirable corner frequency and high
frequency rolloff rate.
The analytical and dereverberated transfer functions of noncollocated actuator-
sensor pairs are compared to those of the collocated pairs. General rules are found
which determine the frequency at which the transfer function no longer has an inter-
laced pole-zero pattern.
Finite element models are constructed which add damping, finite actuator thick-
ness, and finite beam and actuator widths to the model. The actuator to sensor
transfer functions are calculated for these models, and the effects of the added factors
are determined. It is shown that the transverse bending modes of the three dimen-
sional beam delay the rolloff of the actuator to sensor frequency transfer function
by at least two decades. Finally, experimental data confirms the results of the three
dimensional finite element model.
3
Table of Contents
1 Introduction 15
2 Analytical Models for Collocated Actuator-Sensor Pairs 19
2.1 Modelling of Point and Shaped Actuators and Sensors ......... ... 19
2.2 Wave Solution for Simple Transfer Functions ................... 25
2.2.1 Point Actuator and Sensor .......................... 26
2.2.2 Moment Actuator and Slope Sensor ................... 30
2.2.3 Rectangular Actuator and Sensor ...................... 32
2.2.4 Triangular Actuator and Sensor ...................... 34
2.3 Asymptotic Solution for Point and Shaped Actuator-Sensor Pairs 35
.
2.3.1 Methodology ........ ........................... 35
2.3.2 Correlation of Asymptotes using Modal Residue
Solution ........ .............................. 48
2.3.3 Asymptotes for Various End Conditions ................. 52
2.4 Discussion ......... ................................. 54
3 Analytical Models for Noncollocated Actuator-Sensor Pairs 57
3.1 Wave Solution for Simple Transfer Functions ................... 57
3.1.1 Point Actuator and Sensor .......................... 58
3.1.2 Transfer Function Relationship to a Pinned Boundary Condition 60
3.1.3 Moment Actuator and Slope Sensor .................... 63
3.2 Noncollocated Point Actuator-Sensor Pairs .................... 66
3.3 Noncollocated Shaped Actuator-Sensor Pairs ................... 71
3.3.1 Rectangular Actuator-Sensor Pairs ..... ............... 72
3.3.2 Triangular Actuator-Sensor Pairs ...... ................ 75
7
8 Contents
3.4 Discussion ......... ................................. 82
4 Collocated Shaped Actuator-Sensor Pairs in Realistic Geometries 85
4.1 Rectangular Actuator-Sensor Pair .......................... 85
4.1.1 Analytical Wave Model ............................. 86
4.1.2 Finite Element Models Using 2-D Beam Elements .......... 87
4.1.3 Finite Element Model Using 3D Brick Elements ........... 93
4.2 Triangular Actuator-Sensor Pair ........................... 96
4.2.1 Analytical Wave Model ............................. 96
4.2.2 Two-Dimensional Finite Element Models ................. 98
4.2.3 Three-Dimensional Finite Element Models ............... 102
4.2.4 Experimental Results ............................. 106
4.3 Discussion ........ ................................. 107
5 Conclusions 111
5.1 Analytical Models for Collocated Actuator-Sensor Pairs .......... 111
5.2 Analytical Models for Noncollocated Actuator-Sensor Pairs ...... .. 112
5.3 2D and 3D Solutions for Collocated Shaped Actuator-Sensor Pairs 115
References 117
A Transfer Functions for Collocated Actuator-Sensor Pairs 121
A.1 Transfer Functions for Point Actuator-Sensor Pairs .............. 121
A.2 Transfer Functions for Rectangular Actuator-Sensor Pairs ....... .. 123
A.3 Transfer Functions for Triangular Actuator-Sensor Pairs .......... 125
A.4 Transfer Function for Quadratic Actuator-Sensor Pair ............ 127
A.5 Cubic Actuator to Sensor Transfer Function ................... 128
List of Figures
2.1 Spatial Shapes of Distributed Actuators and Sensors: (a) Rectangular,
(b) Triangular, (c) Quadratic, (d) Cubic ...... ................ 20
2.2 Applied Forces and Moments Modelling Distributed Actuators: (a)
Rectangular, (b) Triangular, (c) Quadratic, (d) Cubic ............ 22
2.3 Displacement and Slope Measurements Modelling Distributed Sensors:
(a) Rectangular, (b) Triangular, (c) Quadratic, (d) Cubic .......... 23
2.4 Point Actuator-Sensor Pairs on (a) Infinite, (b) Semi-infinite (c) Finite
Beam .......... ................................... 27
2.5 Infinite Beam Broken into Two Beams at the Point of Actuation . . 27
2.6 Moment Actuator and Slope Sensor on (a) Infinite, (b) Semi-infinite,
(c) Finite beam ........ .............................. 31
2.7 Infinite Beams with (a) Rectangular and (b) Triangular Actuator-
Sensor Pairs ......... ............................... 33
2.8 Pinned-Pinned Beam with Point Actuator-Sensor Pair ............ 36
2.9 Transfer Function for Point Actuator-Sensor Pair Centered on a Pinned-
Pinned Beam ........ .............................. .38
2.10 Pinned-Pinned Beams with Collocated Actuator-Sensor Pairs (a) Rect-
angular (b) Triangular ........ .......................... 40
2.11 Transfer Function for Rectangular Actuator-Sensor Pair on a Pinned-
Pinned Beam, l1 = 6.67%1 ....... ........................ 41.
2.12 Transfer Function for Triangular Actuator-Sensor Pair on a Pinned-
Pinned Beam, 1, = 6.67%1 . .............................. 42
2.13 Pinned-Pinned Beams with Collocated Actuator-Sensor Pairs
(a) Quadratic (b) Cubic ........ ....................... .. 43
2.14 Transfer Function for Quadratic Actuator-Sensor Pair on a Pinned-
Pinned Beam ......... ............................... 45
2.15 Transfer Function for Cubic Actuator-Sensor Pair on a Pinned-Pinned
Beam .......... ................................... 47
9
10 List of Figures
2.16 Modal Residue Solution vs. Dereverberated Wave Model for a Point
Actuator-Sensor Pair Centered on a Pinned-Pinned Beam ......... 49
2.17 Modal Residue Solution and Dereverberated Wave Solution where la =
6.67%1 on Pinned-Pinned Beam (a) Rectangular (b) Triangular . . . 50
2.18 Modal Residue Solution and Dereverberated Wave Solution where l1 =
6.67%1 on Pinned-Pinned Beam (a) Quadratic (b) Cubic .......... 51
3.1 Noncollocated Point Actuator-Sensor Pairs on (a) Infinite, (b) Semi-
infinite, (c) Finite beam ........ ......................... 58
3.2 Free-Free Beams with (a) Collocated, (b) Noncollocated Point Actuator-
Sensor Pairs .......... ............................... 61
3.3 Noncollocated Moment Actuator and Slope Sensor on (a) Infinite,
(b) Semi-infinite, (c) Finite beam ....... .................... 64
3.4 Noncollocated Point Actuator-Sensor Pair on a Pinned-Pinned Beam 66
3.5 Transfer Functions for Collocated and Noncollocated Point Actuator-
Sensor Pairs on a Pinned-Pinned Beam ...... ................ 69
3.6 Pinned-Pinned Modeshapes for Beam of Length l = 1 (a) 15th Mode
and (b) 17th Mode ........ ............................ 70
3.7 Analytical and Dereverberated Transfer Functions for Noncollocated
Point Actuator-Sensor Pair on a Pinned-Pinned Beam ......... ... 71
3.8 Noncollocated Rectangular Actuator-Sensor Pair (a) Sensor Shorter
than Actuator (b) Sensor Longer than Actuator ................. 72
3.9 Analytical and Dereverberated Transfer Functions for Noncollocated
Rectangular Actuator-Sensor Pair on a Pinned-Pinned Beam; l, = 0.51,
.(cid:127)= 0.051 ......... ................................. 74
3.10 Pole-Zero Spacing for Rectangular Actuator to Sensor Transfer Func-
tions; l = 0.2 ......... ............................... 75
3.11 Pole-Zero Spacing for Rectangular Actuator to Sensor Transfer Func-
tions; l1 = 0.4 ......... ............................... 76
3.12 Noncollocated Triangular Actuator-Sensor Pair (a) Sensor Shorter than
Actuator (b) Sensor Longer than Actuator .................... 76
List of Figures 11
3.13 Analytical and Dereverberated Transfer Functions for Noncollocated
Triangular Actuator-Sensor Pair on a Pinned-Pinned Beam ........ 78
3.14 Pole-Zero Spacing for Triangular Actuator to Sensor Transfer Func-
tions; 1a = 0.2 ......... ............................... 79
3.15 Collocated and Noncollocated Transfer Function for Triangular Actuator-
Sensor Pair on Pinned-Pinned Beam ......................... 80
3.16 Function for Triangular Sensor and Actuator Output ............. 81
4.1 Cantilevered Steel Beam with Rectangular Actuator-Sensor Pair of
Zero Thickness ......... .............................. 86
4.2 Analytical and Dereverberated Transfer Functions for Rectangular Actuator-
Sensor Pair on Cantilevered Beam ...... ................... 87
4.3 Transfer Functions for Rectangular Actuator-Sensor Pair on Cantilevered
Beam: Analytical with No Damping, and 2D FE Model with 1% Modal
Damping ......... ................................. 89
4.4 Dereverberated Rectangular Actuator to Sensor Transfer Functions for
2D FE Model of Cantilevered Beam with 1% Damping and Analytical
Model ......... ................................... 90
4.5 Cantilevered Steel Beam with Finite Thickness of Rectangular Actua-
tor .......... ..................................... 90
4.6 Transfer Functions for FE Model with Zero Rectangular Actuator Thick-
ness and with Rectangular Actuator Thickness of 7.5 mil on a Can-
tilevered Beam ......... .............................. 91
4.7 Analytical Asymptotes and Dereverberated Transfer Functions for Vary-
ing Rectangular Actuator Thickness on Cantilever Beam: 0 mil, 7.5 mil,
15 ril ......... ................................... 92
4.8 Finite Element Mesh of the Top View of Cantilevered, Steel Beam with
Rectangular Actuator ........ .......................... 93
4.9 Rectangular Actuator to Sensor Transfer Functions for 2D FE Model
with Finite Actuator Thickness and for 3D FE Model ............ 94
4.10 Three-Dimensional Modeshapes near 16000 Hz of Cantilever Beam
with Rectangular Actuator-Sensor Pair ...................... 95
12 List of Figures
4.11 Cantilevered Steel Beam with Triangular Actuator-Sensor Pair of Zero
Thickness ......... ................................. 97
4.12 Analytical and Dereverberated Transfer Functions for Triangular Actuator-
Sensor Pair on a Cantilevered Beam with Realistic Geometry . . . . 97
4.13 Transfer Functions for Triangular Actuator-Sensor Pair on Cantilevered
Beam: Analytical with No Damping, and 2D FE Model with 1% Damp-
ing .......... ..................................... 99
4.14 Dereverberated Triangular Actuator to Sensor Transfer Functions for
2D FE Model with 1% Damping and Analytical Model ........... 100
4.15 Cantilevered Steel Beam with Finite Thickness of Rectangular Actua-
tor ......... ..................................... 100
4.16 Transfer Functions for FE Model with Zero Triangular Actuator Thick-
ness and with Triangular Actuator Thickness of 7.5 mil ... ....... 101
4.17 Analytical Asymptotes and Dereverberated Transfer Functions for Vary-
ing Actuator Thickness on a Cantilever Beam: 0 mil, 7.5 mil, 15 mil 102
4.18 Top and End View of Three-Dimensional Steel Beam with Triangular
Actuator-Sensor Pairs ........ .......................... 102
4.19 Three-Dimensional Perspective of Steel Beam with Triangular Actuator-
Sensor Pairs .......... ............................... 103
4.20 Triangular Actuator to Sensor Transfer Functions for 2D FE Model
with Finite Actuator Thickness and for 3D FE Model ............ 104
4.21 Modeshapes of Cantilever Beam with Triangular Actuator-Sensor Pair
Showing First and Third Mode Transverse Bending ............. 105
4.22 Experimental Transfer Function and FE Model Transfer Function . . 106
4.23 Transfer Function for FE Model of Rectangular Actuator-Sensor Pair
with d is Active and Inactive ....... ..................... 108
32
4.24 Transfer Function for FE Model of Triangular Actuator-Sensor Pair
with d is Active and Inactive ...... ..................... 109
32
List of Tables
2.1 Low and High Frequency Asymptotes for Point Actuator-Sensor Pair
at Different Locations on a Pinned-Pinned Beam ............... 38
2.2 Low and High Frequency Asymptotes for a Pinned-Pinned Beam 53
2.3 Low and High Frequency Asymptotes for a Cantilevered Beam 53
.
2.4 Low and High Frequency Asymptotes for a Free-Free Beam ........ 53
3.1 Values of k for Poles of Beams with Pinned Boundaries and Zeros of
Noncollocated Point Actuator-Sensor Transfer Function ........... 63
3.2 Low and High Frequency Asymptotes for Different Placement of Point
Actuator and Noncollocated Point Sensor; x, Is Positive ........... 68
3.3 Low and High Frequency Asymptotes for Different Placement of Point
Actuator and Noncollocated Point Sensor; zx Is Negative .......... 68
3.4 Low and High Frequency Asymptotes for Noncollocated Rectangular
Actuator-Sensor Pairs on a Pinned-Pinned Beam ............... 72
3.5 Low and High Frequency Asymptotes for Noncollocated Triangular
Actuator-Sensor Pairs on a Pinned-Pinned Beam ............... 77
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Chapter 1
Introduction
Modern controlled structures are being designed for high performance, precise appli-
cations requiring pointing accuracy and vibration suppression. These specifications
typically require control systems, which often include many sensors and actuators, to
meet these performance objectives.
Aubrun [3], and Hall, et al. [12] have discussed the use of hierarchic control in
flexible structures. This included a global control loop, which is a multiple input,
multiple output (MIMO) system, whose transfer function includes the overall perfor-
mance objective. The information in the global control loop is obtained from several
single input, single output (SISO) local control loops. It is generally true that if
damping can be added to the local control loop, the robustness of the global control
loop will be improved.
This work focuses on the design of the strain actuator to strain sensor transfer
function of the local control loop. Collocated and dual actuator-sensor pairs will
produce a transfer function with an alternating pole-zero pattern and no associated
phase loss. Damping can always be added to the plant transfer function with rate
feedback. If rate feedback or any positive real compensator is used on the transfer
function of a collocated and dual actuator-sensor pair, stability is guaranteed [5].
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