A New Adaptive Array of Vibration Sensors_Tesis

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A New Adaptive Array of Vibration Sensors

Hartono Sumali

Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in

partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Mechanical Engineering

Harley H. Cudney, ChairChris R. Fuller

Daniel J. Inman

Larry D. Mitchell

Alfred L. Wicks

July 1997

Blacksburg, Virginia

Keywords: Modal Analysis, Algorithm, Eigenvectors

Copyright 1997, Hartono Sumali


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A New Adaptive Array of Vibration Sensors

Hartono Sumali

(ABSTRACT)

The sensing technique described in this dissertation produces modal coordinates for monitoring

and active control of structural vibration. The sensor array is constructed from strain-sensing

segments. The segment outputs are transformed into modal coordinates by a sensor gain matrix.

An adaptive algorithm for computing the sensor gain matrix with minimal knowledge of the

structures modal properties is proposed. It is shown that the sensor gain matrix is the modal

matrix of the segment output correlation matrix. This modal matrix is computed using new

algorithms based on Jacobi rotations. The procedure is relatively simple and can be performedgradually to keep computation requirements low.

The sensor system can also identify the mode shapes of the structure in real time using Lagrange

polynomial interpolation formula.

An experiment is done with an array of piezoelectric polyvinylidene fluoride (PVDF) film

segments on a beam to obtain the segment outputs. The results from the experiment are used to

verify a computer simulation routine. Then a series of simulations are done to test the adaptive

modal sensing algorithms. Simulation results verify that the sensor gain matrix obtained by the

adaptive algorithm transforms the segment outputs into modal coordinates.


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ACKNOWLEDGEMENTS

I would like to express my gratitude to all who have contributed to this research endeavor,

especially the following individuals. I thank my major advisor Dr. Harley Cudney for his support

throughout my years as a graduate student, both financially and personally. I have learned so

much from him.

My special thanks are due to Dr. Larry Mitchell. I am grateful for the first-class education and

training I received from him, especially during the final years of my graduate studies. I thank Dr.

Dan Inman for his help, especially with my future career opportunities. I thank Dr. Al. Wicks for

his advise on signal processing and modal analysis. I am grateful that I had a chance to learn first-

hand information from Dr. Chris Fuller, a world-renowned expert in vibration and acoustics,

whose insight into the physical meaning of every mathematical expression never ceases to amaze

me.

I thank the agencies and companies which supported the various research projects I was involved

in: ONR, Westinghouse, DARPA, Cessna, and ARO. I thank many people who helped me with

experiments: Karsten Meissner, Rich Lomenzo, Ben Poe, and James Garcia. I thank those who

helped me with their knowledge, advice and discussion: Dr. Ricardo Burdisso, Chris Niezrecki,

Dr. Chul-Hue Park, Dr. Nesbitt Hagood, and many others. I thank my parents, friends, teachers,

and above all, I thank God.


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TABLE OF CONTENTS

1 Introduction ....................................................................................................................1

1.1 Modal Analysis and Modal Coordinates ....................................................................1

1.2 The Quest for Modal Coordinate Sensors ..................................................................2

1.2.1 Modal Filtering in Time Domain .......................................................................3

1.2.2 Modal Filtering in Spatial Domain ....................................................................3

1.2.3 Segmentation of Modal Filtering Sensors ......................................................... 7

1.2.4 Segmentation of Modal Filtering Sensors ......................................................... 7

1.2.5 Modal Filtering Using Adaptive Algorithms ......................................................8

1.3 Conventional System Characterization and Mode Shape Extraction ......................... 10

1.4 On-line Modal Analyzer: a Novel Concept .............................................................. 13

1.5 Overview of Dissertation ........................................................................................14

2 Simulation Method and Sensor Model ............................................................17

2.1 Model of Variable Host Structure ........................................................................... 17

2.1.1 Description of Structure .................................................................................17

2.1.2 Eigen-properties of Structure ......................................................................... 18

2.1.3 Equation of Motion and Its State-Space Form ................................................ 19

2.2 Simulation of Structure ...........................................................................................20

2.2.1 Sampling and Discrete-Time Model ................................................................ 20

2.2.2 Discrete-Time Model of One Mode ................................................................ 21

2.2.3 Discrete-Time Model of Multi-Mode Structures ............................................. 22

2.2.4 Selecting Sensor Configuration ...................................................................... 23

2.3 Segmented Sensor Model ........................................................................................ 25

2.3.1 Voltage Generated by Segment ...................................................................... 26

2.3.2 Array of Piezoelectric Film Segments on Beam ............................................... 28

2.3.3 Gain Matrix for Segmented Modal Sensor ...................................................... 31

2.4 Chapter Summary ...................................................................................................32

3 Numerical Simulation and a Proof-of-Concept Experiment ..................34

3.1 Verification of Digital Filter Model and Gain Matrix Formula .................................. 34

3.1.1 Verification of Digital Filter Model ................................................................. 34

3.1.2 Verification of Gain Matrix Formula ............................................................... 41

3.1.3 Modal Truncation and Spatial Aliasing .......................................................... 43

3.2 Proof-of-Concept Experiment .................................................................................47

3.2.1 Properties of Experimental Structure .............................................................. 47


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3.2.2 Experimental Setup ........................................................................................49

3.2.3 Experiment Results ........................................................................................51

3.2.4 Discussions on Experiment Results.................................................................. 55

3.3 Chapter Summary ...................................................................................................58

4 Adaptive Computation of Gain Matrices ........................................................59

4.1 Effects of Inaccurate Mode Shapes ......................................................................... 60

4.2 Adaptive Design of Modal Sensors ......................................................................... 62

4.2.1 Correlation between Modal Coordinates ......................................................... 62

4.2.2 Adaptive Computation of Sensor Gain Matrix ................................................ 65

4.3 Numerical Example .................................................................................................67

4.4 Chapter Summary ...................................................................................................75

5 Eigenvector Computing Algorithms .................................................................79

5.1 Jacobi Rotation Algorithm ......................................................................................79

5.2 Algorithm A, Convergence and Rotation Angle ....................................................... 85

5.3 Algorithm B............................................................................................................. 87

5.3.1 Development ..................................................................................................87

5.3.2 Numerical Example .........................................................................................91

5.4 Algorithm C............................................................................................................. 95

5.4.1 Development ..................................................................................................95

5.4.2 Numerical Example ...................................................................................... 100

5.4.3 Frequency-Domain Analysis ......................................................................... 102

5.5 Limitations.............................................................................................................112

5.6 Chapter Summary ................................................................................................. 113

6 A Mode Shape Sensing Technique ................................................................. 115

6.1 Sensor Gain Matrices and Mode Shapes ................................................................ 115

6.2 Lagrange Interpolation .......................................................................................... 117

6.3 A Numerical example ............................................................................................ 118

6.4 Chapter Summary ................................................................................................. 120

7 Conclusions and Future Direction ................................................................... 121

7.1 Conclusions .......................................................................................................... 121

7.2 Future Directions .................................................................................................. 122


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References ....................................................................................................................... 125

Appendix A LMS Computation of Gain Matrix ........................................... 130

Appendix B LMS Algorithm ................................................................................. 136

Appendix C A Control-Model Identification Procedure ........................... 138

Appendix D Stability of IIR Filters ..................................................................... 142

Vita ...................................................................................................................................... 146


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LIST OF FIGURES

1.1 Modal filtering in time domain ........................................................................................... 3

1.2 Modal filtering in spatial domain ........................................................................................ 4

1.3 Mobility magnitudes at 4 points .........................................................................................5

1.4 Modal mobility magnitudes for modally combined sensors ..................................................6

1.5 Modal sensor in continuous spatial domain ........................................................................6

1.6 Using an adaptive algorithm to create a modal sensor ........................................................8

1.7 Error signal history of adaptive modal filter .......................................................................8

1.8 Typical control-model identification ................................................................................. 12

1.9 Typical experimental modal analysis and modal filtering with conventional methods ......... 14

1.10 Modal analysis and modal filtering with the new modal analyzer ...................................... 15

2.1 Beam with pin and pin-with-torsion-spring boundary conditions ...................................... 172.2 Continuous-time and sampled (discrete time) systems ...................................................... 21

2.3 Representing a high-order structure with a parallel bank of second-order digital filters ..... 24

2.4 An infinitesimal piezoelectric element under strain ........................................................... 26

2.5 Strain in the film as a function of deflection of the beam ................................................... 27

2.6 Piezoelectric film and zero-impedance signal conditioner ................................................. 28

2.7 Spatial filters as modal sensors .........................................................................................30

2.8 Segment positions on beam ..............................................................................................31

3.1 Mode shapes of beam ......................................................................................................36

3.2 a)Z-plane poles, b)Denominator coefficients of the IIR-filter-equivalent of the beam ........ 37

3.3 Comparison between beams driving-point mobility and digital filters FRF ...................... 383.4 Time-domain comparison between second-order digital filters and ideal modal

coordinates: impulse response ..........................................................................................40

3.5 Contribution of each mode to segment outputs ................................................................ 41

3.6 Gain matrix for modal sensors .........................................................................................42

3.7 Modal sensor output compared to ideal modal coordinates: impulse responses ................. 44

3.8 Responses of a 10-mode filter to a 12-mode impulse excitation ........................................ 45

3.9 Modal sensor output compared to ideal modal coordinates: impulse responses of modes 8

and 9 ...............................................................................................................................46

3.10 Modal sensor output compared to ideal modal coordinates: FRF from force to sensor

output and modal coordinates ..........................................................................................46

3.11 Experiment setup .............................................................................................................493.12 Schematic picture of experiment setup ............................................................................. 50

3.13 FRF from force to segment outputs .................................................................................51

3.14 Sensor gain matrix W for transforming 20 segment outputs into 8 modal coordinates ....... 54

3.15 FRFs from force to sensor outputs ..................................................................................55

3.16 FRFs from force to sensor outputs, linear scale ............................................................... 57


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3.17 End connection to approximate simple support ................................................................ 58

4.1 Magnitudes of the responses of the sensor on the T* = 1 structure and on the T* = 10

structure...........................................................................................................................604.2 Phases of the responses of the sensor on the T* = 1 structure and on the T* = 10

structure ..........................................................................................................................61

4.3 Real parts of the responses of the sensor on the T* = 1 structure and on the T* = 10

structure ..........................................................................................................................62

4.4 Modal responses to random excitation .............................................................................63

4.5 Mode-1 coordinate. Mode-3 coordinate, product of mode-1 and mode-3 coordinates,

average of product of mode-1 and mode-3 coordinates .................................................... 64

4.6 Adjusting sensor gain matrix to diagonalize correlation matrix ......................................... 66

4.7 Sensor gain matrix adjustment using eigenvector matrix of segment output correlations ... 68

4.8 Sensor gain matrix computed with 16 sets of time data (W16) .........................................68

4.9 Ideal sensor gain matrix ...................................................................................................69

4.10 Sensor output correlation matrix usingW(16) .................................................................70

4.11 Sensor gain matrixW(256) ..............................................................................................71

4.12 Sensor output correlation matrix resulting fromW(256) ..................................................72

4.13 Sensor gain matrix calculated using 32768 time data points .............................................. 72

4.14 Sensor output correlation matrix resulting fromW(32768) ..............................................73

4.15 Sensor gain matrix calculated using 49152 data points ..................................................... 74

4.16 Sensor output correlation matrix resulting fromW(49152) ..............................................74

4.17 Magnitudes of adaptive sensor outputs and ideal modal coordinates ................................ 76

5.1 Jacobi rotation example ...................................................................................................835.2 Result of first sweep ........................................................................................................84

5.3 Result of second sweep ....................................................................................................84

5.4 Algorithm A ....................................................................................................................86

5.5 Typical rotation angle history of Algorithm A .................................................................. 87

5.6 Sensor output correlation matrix, Algorithm B, 256 time steps ........................................ 92

5.7 Sensor output correlation matrix, Algorithm B, 32768 time steps ..................................... 92

5.8 Rotation angle history, Algorithm B .................................................................................93

5.9 Sensor gain matrix (Algorithm B), 32768 time steps ........................................................ 93

5.10 Normalized magnitude responses of modal filter (Algorithm B) after 32768 time steps...... 93

5.11 Input connections to Algorithm C ....................................................................................99

5.12 Rotation angle history, Algorithm C................................................................................ 1005.13 Sensor gain matrix (Algorithm C), after 32768 time steps ............................................... 101

5.14 Sensor output correlation matrix (Algorithm C) after 32768 time steps........................... 101

5.15 Normalized magnitude response of modal filter with performance feedback: Mode 1 ..... 102





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5.23 Normalized magnitude response of modal filter with performance feedback: Mode 9 ..... 1065.24 Normalized magnitude response of modal filter with performance feedback: Mode 10 ... 107










5.34 Normalized magnitude response of modal filter with performance feedback: Mode 10 ... 112

6.1 Sensor gain matrixW .................................................................................................... 116

6.2 Mode shapes of beam, (x) ............................................................................................ 116

6.3 Beam with strain sensor segments .................................................................................. 117

6.4 Third row of sensor gain matrix .....................................................................................119

6.5 Reconstructed mode shape ............................................................................................ 120


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LIST OF TABLES

3.1 Physical properties of beam .............................................................................................34

3.2 Eigenvalues of beam ........................................................................................................35

3.3 Natural frequencies of beam .............................................................................................35

3.4 Gain matrix for modal sensor ...........................................................................................42

3.5 Physical properties of beam .............................................................................................47

3.6 Analytical natural frequencies of beam ............................................................................. 48


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CHAPTER 1

INTRODUCTION

Vibration is a very important phenomenon in machinery and structures. In some cases vibration

causes breakdown, malfunction or discomfort. In other cases vibration is the principal means of

operation. In many systems, from ships to musical instruments, quality and performance are

closely related to vibration. In those cases it is very important to understand and control vibration.

To understand and control vibration of a structure, first it is necessary to characterize the

vibrational properties of the structure, i.e. to have certain knowledge of how the parts of the

structure vibrate. Vibration characterization is essential in anticipating the vibration levels or

determining what actions to be taken if the vibration is to be controlled. However, the quest for

the ultimate control of vibration has advanced to the extent that active forces are now used tocounteract the vibration. This relatively new vibration control method is called active control of

vibration. This method requires sensing of the vibration in real time. Regardless of the active

control strategy, either feedback or feedforward, sensing is necessary.

This dissertation was conceived of an aspiration to contrive a system that performs both the

characterization and the sensing of vibration of machinery or structural components. This system

operates on the basis of adaptive processing of signals from distributed sensor arrays. This chapter

will give the reader an idea of the basic concepts, purpose, and expected results of the research

endeavor to develop the system.

1.1 Modal Analysis and Modal Coordinates

Vibration of a multi-degree-of-freedom system can be expressed in terms of the motion of the

systems along several coordinates. The equations governing the motion of the system can be

relatively simple or relatively complicated depending on the choice of the coordinate system.

Some coordinate systems result in coupled equations of motion. Coupling means that one cannot

solve any of the individual equations without involving the others.

The choice of coordinate system determines the degree of coupling among the equations. As a

rule, the more coupling exists among the equations, the more complicated the solutions are. Incontrolling the vibration of a multi-degree-of-freedom system, a coordinate system that leads to

no coupling among the equations also allows simple control schemes. In many cases, it is possible

to choose a coordinate system that results in no coupling among the equations of motion. The

coordinates in such a coordinate system are called the principal coordinates. These coordinates

are also called the natural coordinates.


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The natural coordinates provide a basis on which to express mathematically the vibration of a

structure. On this basis, the vibration of a structure can be viewed as a summation of products of

a spatial function and a temporal function.

w x t x t m mm

M( , ) ( ) ( )=

=

1

, (1.1)

wherex is position and tis time. The spatial function is the mode shape of the structure, which

is a characteristic of the structure. The temporal function is called modal coordinate.

Real-time monitoring of modal coordinates is very important in active vibration control of

continuous structures. The use of modal coordinates in feedback control can prevent control

spillover, a phenomenon that results in degradation of performance or in instability (Balas, 1978).

Feedback control problems of continuous structures using modal coordinates can be viewed as a

problem of controlling single-degree-of freedom (SDOF) systems in parallel, with no interactionamong the systems (Meirovitch and Baruh, 1982). Decades ago, Porter and Crossley

(1972)

published a book dedicated to this control method, which is called modal control. Modal control

has been developed for several control applications such as vibration control of large space

structures (Davidson, 1990).

Several control theories have been developed using the modal control concept for various control

problems including LQG optimal control (Bai and Shieh, 1995). Positive Position Feedback (Baz

and Poh, 1996), and neural-network-based control (Chen et al., 1994). Modal control theory has

also advanced beyond linear structures (Slater and Inman, 1995). Modal coordinates are not just

useful in feedback control. Clark (1995) developed a feedforward control strategy that relies

heavily on the availability of modal coordinates. Modal control experiments have been done onvarious structures such as plates(Clark ,1991, Zhou, 1992, Gu et al., 1994, Miller et al., 1996),

cylinders (Sumali and Cudney, 1991, Finefield et al., 1992, Clark and Fuller, 1993), and highway

bridges (Shelley et al., 1991).

All of the above modal control methods require monitoring of modal coordinates. Sensing modal

coordinates in real time is so important that many researchers have developed a special area

within structural control dedicated to obtaining modal coordinates in real time (Meirovitch and

Baruh, 1985, Ouyang, 1987, Shelley, 1991). This area is called modal sensing. A short description

of some previous work in modal sensing is presented below.

1.2 The Quest for Modal Coordinate Sensors

Many researchers have developed techniques to create sensors that can produce modal

coordinates in real time. The proposed research work in this dissertation adopts the concepts of


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spatial filtering, segmentation, and adaptive signal processing. This section will mention a selected

sampling of previous work especially related to those concepts.

1.2.1 Modal Filtering in Time Domain

Monitoring modal coordinates of a vibrating structure can be done by processing signals from

sensors in time domain. Several researchers claimed that this processing can be done by filtering

sensor outputs with a bank of filters, each of which admits only certain frequency and filters out

other components of the signal. Balas (1978) introduced this modal filtering concept. Ouyang

(1987) developed a realization of this concept with a bank of special filters where each filter only

passes a single frequency that coincides with a natural frequency of the structure (See Fig. 1.1).

Davidson (1990) conceptually designed an electronic circuit that implements Ouyangs filters with

a set of phase-locked loops (PLLs) built with voltage-controlled oscillators (VCOs) that

generates pure sinusoidal signals. Davidson performed numerical simulation of a scenario where

his VCO-based modal filters are used to control many hundred modes of a large space structure.

Figure 1.1 Modal filtering in time domain.

1.2.2 Modal Filtering in Spatial Domain

Another method to obtain modal coordinates in real time from sensor outputs is by filtering the

sensor outputs in space. Basically this means assigning different weights to different sensoroutputs depending on which mode to be sensed. Sumali and Cudney (1991) performed some

experiments using this method. One set of weights produce one modal coordinate. This method

can be illustrated with the beam in Fig. 1.2. Assume that the exciting force is such that the

response is limited to combinations of the first four modes. In practice, this rather simplistic

assumption might be realized by several methods such as low-pass filtering the excitation. We

know that for the simple boundary conditions the mode shapes of the Euler-Bernoulli beam are

Point

sensor

Beam

v

f

( )

( )

v(t)VCO- based

filter

&$ ( )1 t

&$ ( )2 t

&$ ( )3 t

&$ ( )4 t


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sinusoidal. We use four point sensors to sense displacements at strategically assigned positions on

the beam, based on our knowledge of the mode shapes.

Figure 1.2 Modal filtering in spatial domain.

= Mode shape 1

= Mode shape 2

= Mode shape 3

= Mode shape 4

.414

1

1

.414

1

1

-1

-1

1

-.41

-.41

1

1

-1

1

-1

f

Point sensor

v

fj

1( )

( )

v

fj

2 ( )

( )

v

fj

3 ( )

( )

v

fj

4 ( )

( )

&$ ( )

( )

1

fj

v4(t)v3(t)v2(t)v1(t)

&$ ( )1 t

&$ ( )

( )

2

fj

&$ ( )

( )

3

fj

&$ ( )

( )

4

fj

&$ ( )2 t

&$ ( )3 t

&$ ( )

4

t


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This method can be described well with simulation results. The frequency response functions

(FRFs) from the force to the velocities at the four point sensor locations are shown in Fig. 1.3.

By combining the outputs of the four sensors with the right mixture as shown in Fig. 1.2, we can

obtain sensor outputs that are proportional to the individual modal coordinates, as shown in Fig.

1.4. The gains in Fig. 1.2 constitute a matrix that transforms the sensor coordinate system to themodal coordinate system. It will be shown later that these gains are closely related to the modal

matrix (or eigenvector matrix) of the structure.

If we increase the number of sensors, we get a higher spatial resolution. In the limit, for an infinite

number of sensors, the sensor gain matrix becomes continuous functions, each row representing a

mode. This modal sensing concept was invented by Lee (1987) for strain sensors such as

piezoelectric film. The width of the film is varied along the beam as a function of the modal sensor

weight. For a mode 3 sensor, such a sensor is shown in Fig. 1.5. Theoretically, this sensor is an

ideal mode 3 sensor, insensitive to any other mode. Lee and Moon (1990) successfully applied

this type of spatially distributed modal filter to vibration control. Many other researchers have

implemented Lees modal filters in various forms, for example: Structure-borne acoustic sensors

(Clark, 1992), one-dimensional modal sensors on plates (Zhou, 1992), one-dimensional modal

sensors on cylinders (Sumali, 1992; Clark and Fuller, 1993) sensors for feedforward modal

control of structures (Clark, 1995), acoustic sensing by volume-velocity (Guigou et al., 1995).

Burke and Hubbard(1990) explained the concept of spatial filtering sensors and applied the

techniques to control distributed parameter systems in general.

0 100 200 300 400 500 600 700 80010

-4

100

0 100 200 300 400 500 600 700 80010

-4

100

0 100 200 300 400 500 600 700 80010

-4

100

0 100 200 300 400 500 600 700 80010

-4

100

Frequency (Hz)

Yi1 ,m

Ns

Yi2 ,m

Ns

Yi3 ,m

Ns

Yi4 ,m

Ns

Figure 1.3 Mobility magnitude, Yv

fij

i

j

=( )

( )

, at four points.


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0 100 200 300 400 500 600 700 80010-4

100

0 100 200 300 400 500 600 700 80010

-5

100

0 100 200 300 400 500 600 700 80010

-5

100

0 100 200 300 400 500 600 700 80010

-5

100

Frequency (Hz)

&$

( )

1

fj

&$

( )

2

fj

&$

( )

3

fj

&$

( )

4

fj

Figure 1.4 Modal mobility magnitudes,&$ ( )

( )

i

jf, for modally combined sensors.

Figure 1.5 Modal sensor in continuous spatial domain.

+-

PVDF film

segment V1

Beam

Point

sensor

Mode 3

sensor

Frequency

Vofj

V

f

o

j

( )

( )

V

fj

1( )

( )


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1.2.3 Polyvinylidene Fluoride (PVDF) Film Sensors

Throughout the research work, the material used for vibration sensor is thin film of a piezoelectric

polymer polyvinylidene fluoride (PVDF). The use of this material in the structural dynamics

community was promoted mainly by Lee (1987). This material is particularly suited for thestructural modal sensing applications because it offers many advantages, including the following.

PVDF is lightweight. Compared to accelerometers (even very small ones), PVDF film has

negligible mass-loading effects, even on light structures. The polymer is also very compliant. This

feature enables us to neglect the changes in structural parameters due to stiffening by the film.

Another attractive feature of the PVDF film sensor is its strain-integrating nature. Unlike point

sensors, segments of piezoelectric film can be cut into shapes that convolve the strain on the

structures surface with a specified function (Lee and Moon, 1990, Collins et al., 1992). The

function can be selected such that the convolution results in a low-pass filtering effect in the wave

number domain. This effect results in the reduction of spatial aliasing, which is an important

property that we will address later in this dissertation.

PVDF film segments are much cheaper than accelerometers or other vibration transducers such as

strain gages, laser, or fiber optic transducers. The signal conditioning circuit for a PVDF sensor is

also much cheaper than the signal conditioning circuits for the other sensors. The simplicity to

attach large numbers of PVDF segments on structures is also very important in creating highly

distributed sensor arrays.

PVDF has a strong piezoelectric effect compared to most other piezoelectric materials (Lee,

1987). The material is also robust, both physically and chemically. It has been used also as

protective coatings against harsh environment, for instance, as vat liners for chemicals (Collins et

al., 1990). It endures time, temperature (up to about 120o C), and mechanical shock (up to several

hundred gs). The (mechanical) bandwidth of the film is very high (up to about 107

Hz). With

appropriate signal conditioning circuit, no dynamics is introduced by the sensor.

1.2.4 Segmentation of Modal Filtering Sensors

The idea of dividing the piezoelectric film layer into segments emerged primarily because a

piezoelectric sensor layer covering the whole host structure fails to detect anti-symmetric modes

(Cudney, 1992). Several researchers have investigated the use of segmentation in piezoelectricfilm sensors. Tzou and Fu (1992, 1992b) developed a theory for applying the segmented sensors

and actuators to vibration control of plates. Clark (1992) developed an adaptively-computed

sensor array gains to apply to segmented PVDF film sensors so that the sensor array emulates

vibration sensors and structural acoustic sensors. Sumali and Cudney (1993) developed a

technique to create modal sensor from an array of segmented piezoelectric film sensors. Callahan

and Baruh (1994) developed another method to create modal sensors from the same segment


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array configuration. Sullivan (1993) developed a special kind of distribution calculus to calculate

the responses of piezoelectric sensor patches of general shapes and to calculate the actuation of

arbitrarily shaped strain-induced actuators on multidimensional structures. Sullivan also developed

a sensor array that is weighted spatially according to a linearly varying function developed by

Burke and Hubbard (1990).

1.2.5 Modal Filtering Using Adaptive Algorithms

At least two methods have been developed to create modal sensors using adaptive signal

processing. The first method(Shelley et al., 1992) uses the sensor configuration shown in Fig. 1.6.

The outputs of the sensors are input to a linear combiner with some gain vector. The gain vector

is computed on-line using the LMS algorithm. The structure is excited with a random force

excitation. The output of the sensors are used to adjust the gain matrix iteratively until the output

of the linear combiner matches the output of a pre-programmed second-order digital filter. The

natural frequency and damping of the digital filter are pre-programmed based on the knowledge of

the natural frequency, modal damping, and residue of the desired mode. These parameters must

be known in advance. The advantage of this method is that knowledge of the mode shapes of the

structure is not required in computing the gain matrix. An example of the application of this

method is shown below. The details of a simulation of the adaptive modal filter are given in

Appendix A.

Figure 1.6 Using an adaptive algorithm to create a modal filter.

...Error

signal

(k)

Sensor output &$3(k)

-+

W3,1(k)

Adaptive

algorithm

V1(k)

BeamPVDF segments

f(k)

T*

Torsion spring

W3,2(k) W3,10(k)

Desired signal

d(k)

V2(k) V10(k)...


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This adaptive method seems to result in an impressive modal filtering effect. Figure 1.7 shows the

difference between the sensor output and the ideal modal coordinate. This error converging to

zero means that the sensor output converges to the desired modal coordinate.

0 0.01 0.02 0.03 0.04 0.05-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (sec)

Error(Volt)

Mode 3 sensor, T* = 1

Figure 1.7 Error signal history of adaptive modal filter.

Despite the impressive modal filtering effect, a more critical examination of Appendix A reveals

the fact that this technique is not likely to be of practical utility. Figures A.1 and A.2 in the

appendix show that the key element in this modal filter is the filter that generates the desired

signal d(k). This filter must be programmed with the natural frequency, modal damping, and

modal residue. Therefore, this method still requires complete knowledge of the natural

frequencies, modal damping, and residues of the modes of the structure. If one knows these

parameters and hence the filter coefficients, then there is no need for the adaptive linear combiner.

The filter alone can be used to generate the desired signal d(k), which is precisely the modal

coordinate. The real utility of this method would be in obtaining the mode shape.

The second method uses a parallel bank of second-order recursive digital. This method is based

on the modal decomposition of the multi-degree-of-freedom structure into a parallel bank of

single-degree-of-freedom systems as in Eq. (1) (Horvath, 1976). Wimmel and Melcher (1992)

used adaptive recursive filter algorithms developed by White (1975) and Hsia (1981). Part of thismethod, namely modeling a multi-degree-of-freedom (MDOF) system as a parallel bank of

second-order filters, will be adopted in this research work for the purpose of simulating a

structure with already-known modal properties with digital filters.


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10

1.3 Conventional System Characterization and Mode Shape Extraction

From the above discussion, we know that modal filtering requires knowledge of the characteristic

of the structure: either mode shapes or natural frequencies and damping ratios. In fact, the most

important step in virtually all vibration analyses is to obtain the natural frequency or eigenvalues

and mode shapes or eigenvectors of the system. The importance of eigenvalues and eigenvectors

can not be overemphasized. In all but the simplest structures, the modal properties of the system

must be obtained numerically or experimentally. The most popular methods to obtain the modal

properties of a structure by experiments can be easily classified into two broad categories. Each

category is different from the other in many ways: tradition and historical development,

mathematical foundation, and experimental knowledge base. Even the purposes of the different

categories are different. The first category is commonly called System Identification, the

second, Experimental Modal Analysis (EMA). In this section we will discuss the concepts and

efforts involved in computing mode shapes from experimental data using the two categories of

computation methods.

System identification, or more specifically Control-Model Identification (Juang, 1994) has

developed since the mid-sixties out of the necessity to control sophisticated systems such as

guidance and controls of aerospace structures. The main objective of this type of analysis is to

obtain the block diagram of the system to be controlled so that the control engineer can design the

controller, observer, sensors, and actuators. Modal properties of the structure can be obtained

easily from the resulting dynamic model of the structure. However, obtaining modal properties is

only a small part of system identification and often not an essential part. Much theoretical work

has been generated by many researchers on system identification (see for example the bibliography

of Juang, 1987).

EMA originated partly from such testing practices as Resonance Testing and Mechanical

Impedance Methods in the 1940s (Ewins, 1986). Development in electronics in the 1960s and

Cooley and Tukeys Fast Fourier Transform (FFT) algorithm created a revolution in signal

processing (Mitchell, 1986). Unlike Control-Model Identification, Modal Testing is especially

geared towards obtaining the modal properties of structures. The resulting estimates of modal

properties are used not mainly in active control of aerospace structures, but in various other tasks,

such as design modification and passive vibration control.

EMA is very commonly used in obtaining vibration modes to verify finite element or theoretical

models. Once the model is validated, it can be used to predict the responses to complex

excitations such as shock, or to proceed to more further stages of analysis. A validated model canbe used as a basis for further modeling and analysis. Modal Testing is often used to produce a

mathematical model of a component which may then be used in a structural assembly. Mode

shapes obtained from EMA can be used to modify the design of a component to improve its

vibrational characteristics, such as lower dynamic stresses, less acoustic radiation, relocation of

points of large vibration, etc.


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11

In terms of knowledge requirements, Modal Testing requires a thorough integration of 1)

Vibration theory 2) Accurate measurement, and 3) Signal processing. The area of Modal Testing

is very rich in experimental knowledge and intuitive rules that complement, sometimes even

circumvent, the mathematics. On the other hand, Control-Model Identification requires themathematics of modern control systems theory. In particular, most Control-Model Identification

algorithms rely on Singular-Value Decomposition (SVD). The state-space is the standard domain

of Control-Model identification.

Keeping the backgrounds of Control-Model Identification and Modal Testing in mind, we can

now make a reasonable judgment of the two categories of methods in terms of an important goal

in this dissertation: Computing mode shapes. Then we propose a new method of computing mode

shapes, which also results in a means of obtaining modal coordinates in real time. The advantages

of this new method over Control-Model Identification and the classical Modal Testing will be

discussed later.

For an explanation of the Control-Model Identification procedure, Juang (1994b) is an excellent

reference, on which the following paragraphs are based. The purpose of describing the Control-

Model identification procedure here is only to give an idea of the computational process. Figure

1.8 shows a typical sequence of variables to calculate in Control-Model identification of a

structure. Knowledge of SVD-based procedures such as one described in Appendix C is desirable

for further understanding of the rest of this section.

From the identification procedure in Appendix C, we learn that obtaining mode shapes from

experimental data using Control-Model Identification requires at least the following operations:

1. Fast Fourier Transform (FFT) to transform force and response signals into the frequency

domain. This process does not only require the application of Fourier transform algorithms,

but also other procedures to ensure good results, e.g. windowing, coherence computation and

checking, averaging, sometimes zooming, and so on.

2. FRF Computation from all input forces to all sensor outputs. This complex operation requires

multiplication and division. FRF computation must be done to all time data. In a highly

distributed sensor system with a high number of input channels, FRF computation requires

significant computing power.

3. Inverse FFT to transform the FRF back into time domain to obtain Markov parameters.

4. Singular value decomposition (SVD). Equations (C.4) through (C.7) in Appendix C show that

SVD is a key step in the identification process. This process is a lengthy sequence of matrixoperations (see, for example Golub and Van Loan, 1989).

5. Eigenvalue and eigenvector computation. This is another computationally extensive operation.

6. Transformation of eigenvalues from the discrete z-plane to the continuous s-plane (Eq.

(C.19)). This transformation requires evaluation of logarithms, a computation process that is

much less elementary than addition or multiplication. In the implementation of real-time

algorithms, this nonlinear operation may take many computation steps.


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12

7. Raising a matrix to the -1/2 power (Equation (C.9)). This process obviously needs extensive

computation.

8. Matrix inversion (Eq. (C.21)) and several multiplications. This is another computationally

extensive process.

From the above discussion, we can conclude that a typical Control-Model Identification process is

computationally extensive. If mode shapes are to be obtained and modal filtering are to be

performed in real time, this type of analysis must be done with tremendous computing power.

Less computing power may do the job if the computation is done with recursive methods.

Recursive algorithms are available mainly for single-input-single-output (SISO) systems. (Ljung

(1989) is an excellent reference.) The algorithms are mostly based on Recursive Least Squares

(RLS) or Kalman filter theories. These methods are not suitable for obtaining mode shapes

because they do not address the distributed nature of sensors required in mode shape

computation.

EMA is perhaps the best category of methods to obtain mode shapes. However, the current

procedures to compute mode shapes are not designed for on-line computation. Figure 1.9 shows

typical steps of modal testing and the computation steps to obtain mode shapes. The first steps in

EMA are identical to their counterparts in Control-Model Identification. The procedure shown in

the picture requires the first three operations performed in Control-Model Identification. EMA

programs use circle-fit, least-squares curve-fitting, and other averaging and error-minimizing

techniques. Like the processes in Control-Model Identification, those processes also require too

much computing power to apply in real time.

The foregoing discussion was not meant to argue that the current state of the art is full of

disadvantages and inefficiency. The argument is that the current practices of system identification

are not geared towards real-time application. Recent advances in adaptive algorithm have created

a trend towards streamlining the computational procedure by using recursive adaptive algorithms.

(See, for example, Ljung (1989b)). However, most of the current advances can be traced back to

SVD, least-squares curve-fitting including recursive least-squares error minimization and Kalman-

filtering-type algorithms, mainly because those are standard procedures. A typical system

identification expert is well-trained in, and feels comfortable with, those standard procedures.

Building on the current techniques can only improve the speed within the bounds inherent to the

nature of the underlying concepts.

The LMS algorithm (Widrow, 1985) is a notable exception to the above statements although the

underlying concept is least-squares error minimization. This algorithm is simple, versatile, andrequires very little computation, hence a lot of practical applications. The drawbacks of this

algorithm are, among others: Slow convergence, inability to take advantage of knowledge about

the system, lack of guaranteed stability. In terms of adaptive modal filtering, the application of this

algorithm is not very promising, mainly because it requires some kind of desired signal, as

discussed earlier.


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13

Figure 1.8 Typical Control-Model Identification. (Adapted from Juang, 1994b)

1.4 On-line Modal Analyzer: a Novel Concept

The proposed research work is to develop a structural vibration sensor system that obtains the

mode shapes of the structure in real time and performs modal filtering. This sensor system will be

called the modal analyzer. This novel system does not require discrete Fourier transform (DFT or

FFT), FRF computation, SVD, or matrix inversion. The modal analyzer works in time domain

using adaptive algorithms. This method is based on correlation and computation of eigenvectors.

Markov parameters

Hankel matrix, H(0)

Singular values

Time-shifted Hankelmatrix, H(1)

SVD

State Matrix

Eigenvalues =

Natural frequencies

and modal damping

Left singular vectors Right singular vectors

Output matrix Input matrix

Eigensolution

Mode shapes Modal amplitudes


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14

Therefore, it still needs eigenvector computation (step 5 in the Control-Model Identification

procedure described above.) This step is computationally extensive. However, the proposed

system performs this step recursively, improving the estimates of the eigenvectors gradually with

each iteration. Adaptive algorithms will be developed to perform this operation.

Comparison between Fig. 1.9 and Fig. 1.10 shows that the new modal analyzer bypasses many

computational steps between data acquisition and the computation of mode shapes and modal

coordinates. The modal analyzer does not compute natural frequencies and modal damping.

However, it performs modal filtering concurrently with mode shape computation. The

conventional methods require that the mode shapes be obtained first, and then the modal filter

constructed accordingly. If the structures parameters change, for example, due to temperature

change or drift in boundary conditions, then the fixed-parameter modal filter will no longer be

accurate. The modal analyzer, on the other hand, will track changes in structural parameters, and

adaptively adjust the modal filter so that the outputs are the correct modal coordinates.

1.5 Overview of Dissertation

Although a proof-of concept experiment will be conducted, most of the theory developed in this

dissertation will be tested on a simulated structure. This structure and its discrete-time simulation

will be described in chapter 2. Throughout the dissertation it is assumed that that the structure is

linear and self-adjoint, that the modes are real, and that the mode shapes are orthogonal. Most of

the theory developed here will apply only under those assumptions.

The sensor system is physically constructed from an array of piezoelectric polyvinylidene fluoride

(PVDF) film segments connected to electronic signal conditioning circuits. The design and

modeling of the sensor array are also described in chapter 2.

Numerical simulation of the structure-sensor system is presented in chapter 3. In this chapter we

also present an experiment to verify that the simulation procedure, indeed, represents the physical

system and to prove that the concepts used in the development of the adaptive sensor array are

physically realizable.

In chapter 4, we reveal the fundamental principle that we will use to develop a new algorithm to

perform adaptive mode shape calculation and modal filtering simultaneously. Numerical

simulation is presented to demonstrate the effectiveness of using this fundamental principle. The

development of the adaptive algorithms is presented in chapter 5. In chapter 6 we utilize a simple

formula to extract mode shapes from sensor gain matrices. Conclusions are presented in chapter

7. Also in this last chapter we discuss the possibility of constructing a hardware prototype of the

modal analyzer.


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Figure 1.9 Typical experimental modal analysis and modal filtering with conventional methods.

Force j

Time

Velocity1

FFT

FFT

FFT

( )( )

( )( )FRF

Velocty FFT Force

Force FFT Force FFT ij

ir jr

r

jr jr

r

=

FFT*

*

FFT

Velocity2

FRF1

FRF2

FRFi

FRFN

.

.

.

.

.

.

Modal

analysis

program

Natural

frequency

Damping

Mode shape 1

Mode shape 2

.

.

.

Modal filter

Modeshapes Velocity

Modal coordinates


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16

Figure 1.10 Modal analysis and modal filtering with the new modal analyzer.

Force

Time

VelocityN

Mode shape 1

Mode shape 2

.

.

.

Modal analyzer

Velocity1

Velocityi

Modal

coordinates

.

.

.


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CHAPTER 2

SIMULATION METHOD AND SENSOR MODEL

Throughout this dissertation we will use examples to help explain the concepts and theories that

we will develop. In the examples we will use numerical simulation of a structure. In the first half

of this chapter we will discuss the structure, modeling, and simulation, and give some insight into

the choice of the adaptive identification algorithm. In the second half of the chapter, we will

discuss the development of the theory of segmented sensors, the configuration of the sensor, and

its principle of operation.

2.1 Model of Variable Host Structure

2.1.1 Description of Structure

The structure we choose should be simple enough to model mathematically, but also allows for

variations in mechanical properties. For this purpose, we choose a simply-supported beam with

some torsional stiffness at one of its simple supports (Fig. 2.1).

Figure 2.1 Beam with pin and pin-with-torsion-spring boundary conditions.

The modal properties of the structure depends partly on the torsional spring stiffness T*. The

normalized torsion spring constant is

T TL EI * ( )= , (2.1)

where T* is the torsion spring constant in m-1

,L is the length of the beam in m, Eis the modulus

of elasticity in Pa, andIis the cross-section area moment of inertia in m4.

The following paragraphs explain the modeling and simulation of the structure. We assume a

linear Euler-Bernoulli beam with internal viscous damping. As in most vibrational analyses of

Forcef(t)

T* Torsion

spring


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18

structures, first of all we need to obtain the eigen-properties of the structure. Then we present the

equations of motion in the modal coordinates.

2.1.2 Eigen-Properties of Structure

The mth eigenvalue mof the beam is the mth non-trivial solution to the characteristic equation forthe undamped case is (Gorman, 1975).

( )cos sinh sin cosh sin sinh* =T 2 0 . (2.2)

The mth

natural frequency in rad/s, m, is

m mm

Lf L

EI

= =2

2

2 , (2.3)

wherefm is the natural frequency in Hz, m is the mth root of Eq. (2.2),L is the length of the beam,

EI is the product of the modulus of elasticity and the area moment of inertia, and L is the massper unit length. The mass-normalized mode shape is

m

mass normm

m

mmx

c x L x L( ) sin( / )

sin

sinhsinh( / )=

1, (2.4)

where the mass normalization constant cmass norm for each mode can be calculated by setting

L m

L

x x2

0

1( ) d = (2.5)

for all m. This equation results in a mass normalization constant for each mode shape.

( )

c Lmassnorm Lm

m

m

m

m

m

m

m

m mm m m m

= +

052

4 4 2

2

2

2

.sin sin

sinh

sinh

sin

sinh sin cosh cos sinh

(2.6)


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19

2.1.3 Equation of Motion and Its State Space Form

The velocity response of the structure to a point force excitation can be expressed in terms of

modal expansion as

& ( , ) ( ) & ( )w x t x t m mm

M

==

1

(2.7)

where m(x) is the mth

mass-normalized mode shape of the structure, and m(t) is the modalcoordinate, which is the solution to the equation of motion

&& ( ) & ( ) ( ) ( ) ( ) m m m m m m m f t t t x f t + + =22

(2.8)

where m(t) denotes modal coordinates. The dot above the variables denote time derivation. Thesummation limit M in Eq. (2.7) is theoretically infinity, but in practice must be limited to the

number of modes included in calculations; m2 is the eigenvalue; mdenotes the modal damping;

m(xf) is the value of the mass-normalized mode shape at the forcing point position.

In later chapters, we will need a state-space representation of the structure. The state-space model

is

& ( )

&&( )

( )

& ( )( )

t

t

t

tf t

=

+A B , (2.9)

where

AZn

=

0 I2

, (2.10)

is the matrix containing the natural frequencies and damping ratios, i.e.,

n2

12

2

=

0

0

O

M

, (2.11)

and


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20

=

2

2

1 1

0

0

O

M M

. (2.12)

0 and I areM-by-Mzero and identity matrices, respectively. In the second term, the input matrix

is

B =

0

0

1

M

M

( )

( )

x

x

f

M f

. (2.13)

2.2 Simulation of Structure

2.2.1 Sampling and Discrete-Time Model

Because data are normally acquired by sampling and processed by digital signal processingequipment, it is necessary in our analysis to model the input signal, the structure, and the output

signal as discrete-time processes.

Figure 2.2 shows a structure excited with a forcef(t). At this point we consider the point sensor.

The output signal is a velocity v(t) at some point. Because of sampling, the measured force and

velocity will be discrete functions of time step or sample number kinstead of continuous functions

of time. We call the sampled input f(k) and the sampled output v(k). We must represent the

frequency response function (FRF) of the structure by a discrete time FRF. This FRF must be

such that if the input f(k) is a sampled version of the force f(t), the output v(k) is a sampled

version of the velocity v(t).


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21

Figure 2.2 Continuous-time and sampled (discrete time) systems.

2.2.2 Discrete-Time Model of One Mode

To give a simple example of the process of representing a structure with a digital filter, first we

consider only mode-m of the structure. The frequency response of a single mode is like that of a

second-order system. According to Eq. (2.8), the continuous-time transfer function from force to

a modal coordinate rate & in the Laplace (s) domain is

& ( )( )

( )( )

( )

m m m f

m m m

sf s

s sf s

x s

s s= =

+ +2 22(2.14)

(Henceforth, what we denote by modal coordinate actually means the time rate of change of the

modal coordinate. This time rate is related to vibration velocity and is often more important than

the modal coordinate.) If we know the modal properties of a structure, i.e., the natural

frequencies, modal damping, mode shapes, and residues, we can synthesize the digital filter

representation using several methods. The digital filter is connected to the analog parts of the

system by A/D and D/A converters, usually with a zero-order hold. For this kind of signal

conversion, the discrete equivalent of the transfer function in Eq. (2.14) can be computed by

(Franklin et al., 1990)

H z zs

s s

f s( ) ( )

( )

( )=

111 Z

(2.15)

Equivalent discrete-time system

Sampler

Continuous-time system (ideal structure)

f(t) v(t)

f(k)

v(k)


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22

where Z{ } denotes thez-transform operator. It can be shown that substituting Eq. (2.14) into Eq.

(2.15) results in the discrete transfer function

& ( )

( )

( )m m

m m

z

f z

b z z

a z a z=+

+ +

1 2

11

22

1(2.16)

where the filter coefficients are

b x Tm m fm m

m m

m m=

( )

sin( )exp( )

1

1

2

2, (2.17)

a Tm m m m m12

2 1= exp( ) cos( ) (2.18)

a Tm m m2 2= exp( ) (2.19)

In discrete-time domain, the second-order filter representation of the single-mode structure is

& ( ) & ( ) & ( ) ( ) ( ) m m m m m m mk a k a k b f k b f k = + + 1 21 2 1 2 (2.20)

What we have done so far is:

1. Modeled the structure with its modal properties, i.e., m, m, and m(x0)2. Transformed the equations of motion in time domain into transfer functions in the s-domain3. Transformed the continuous transfer functions in the s-domain into discrete-time transfer

functions in thez-domain

4. Transformed the discrete-time transfer functions into digital filter coefficients

2.2.3 Discrete-Time Model of Multi-Mode Structure

Continuous structures, such as the example structure we use in this research work, have an

infinite number of modes. In practical calculation and numerical analysis we can only include a

limited number of modes. Each mode increases the order of the transfer function by two. Ingeneral, anM-mode structure has a 2Mth order transfer function of the form

& ( )

( )

.. .

.. .

w s

f s

s s s

s s s

MM

MM

=+ + + +

+ + + +

b b b b

a a a

0 1 22

22

1 22

22

1(2.21)


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23

The above transfer function can be represented by a 2Mth

order digital filter of the form

H zv z

f z

z z z

z z z

MM

M

M( )

( )

( )

...

...

= =+ + + +

+ + + +

0 11

22

22

1

1

2

2

2

2

1

(2.22)

To decompose the multi-mode transfer function into several single-mode transfer functions, we

can decompose Eq. (2.22) by partial fractioning into

H zb b z b z

a z a z

m m m

m mm

M

( ) =+ +

+ +

=

0 11

22

11

22

1 1(2.23)

The last equation can be realized as a bank of second-order infinite-impulse-response (IIR) digital

filters connected in parallel with a common input. The outputs of the filters are summed up to

produce the total output of the system. This digital-filter representation of the structure is shownin Fig. 2.3.

The representation of the structure as a parallel bank of second order digital filters lends to modal

analysis because the transfer function of the parallel filters is exactly the same as the modal

expansion of the structures transfer function. Each second-order filter has three coefficients that

are the results of transformations from the natural frequency, modal damping, and modal residues.

Even modal analysis could be done adaptively if we find a way to compute the filter coefficients

recursively.

2.2.4 Selecting Sensor Configuration

The above conclusion seems very promising because it shows the possibility of doing on-line,

adaptive modal analysis. The most difficult part of realizing this concept is the adaptive algorithm

that would be necessary to adjust the filter coefficients without requiring a high computational

load that precludes the on-line signal processing. There is however another important problem

with the parallel IIR filter configuration, namely, stability. Basically, stability requires that


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24

Figure 2.3 Representing a high-order structure with a parallel bank of second-order digital filters.

This issue is discussed separately in appendix D. The conclusions we can draw from the

discussion is given below.

If the system to be identified is a high order system such as continuous structures, the order of the

acceptable auto-regressive-moving-average (ARMA) model in Eq. (2.22) is very high. With high

order models, a slight error or variation from the true values can very easily drift the digital filter

poles out of the stability disc. Moreover, the algorithm used to compute the coefficients may

fBeamForce

transducer

v

Velocity

sensor

b z z

a z a z

1 11 2

1 1

1

2 1

21

,

, ,

( )

+ +

b z z

a z a z

m

m m

11 2

11

22

1

,

, ,

( )

+ +

b z z

a z a z

M

M M

11 2

1 1 2 21

,

, ,

( )

+ +

M

M

y

f

1( )

( )

y1

ym

yM

v^

T* Torsion

spring

$( )

( )

v

f

v

f

( )

( )


35/156


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26

layer of sensor film for each mode. The most important advantage of our sensor is that it can be

made adaptive.

2.3.1 Voltage Generated by Segment

Within its linear region, piezoelectric film generates electric charge proportional to strain and to

the area of the film. Figure 2.4 shows an infinitesimal element of original length dx and width b.

The three orthogonal directions for an electrically anisotropic piezoelectric material are shown by

the arrows. The electrodes are on the 1-surfaces (top and bottom) of the element. The element is

stretched in the 3-direction of the piezoelectric material. The charge generated by this element

under strain is

dq e dx= 31 , (2.26)

where e31 is a piezoelectric charge-density-per-strain constant in the 3-direction.

Figure 2.4 An infinitesimal piezoelectric element under strain.

Figure 2.5 shows a segment of a beam with a rectangular piezoelectric film patch attached on the

top surface. The left end of the film is at x = x1, and the right end is at x = x2. The beams

deflection as a function of position x is denoted by w(x). The piezoelectric film is so compliant

compared to the beam that we can assume that the film does not affect the bending of the beam

with any loading. Assuming Euler-Bernoulli beam theory, we can calculate the strain in the film,

based on how the film is stretched or compressed as a function of the flexure of the beam.

dx

b

dx

1

2

3

Directions


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27

Figure 2.5 Strain in the film as a function of deflection of the beam.

Neglecting the small thickness of the piezoelectric film, the strain on the piezoelectric film can be

considered the same as the strain on the surface of the beam, which is

( , )

( , )x t

h w x t

x=

2

2

2, (2.27)

where h is the thickness of the beam. We obtain the total charge generated in the piezoelectric

film by substituting Eq. (2.26) into Eq. (2.27) and integrating over the length of the film to yield

q t e bh w x t

xdx e b

h w

x

w

xx

x

x x

( )( , )

= =

31

2

2 312 2

1

2

2 1

. (2.28)

Thus, the charge generated by the film is proportional to the difference between the slope at the

right end and the slope at the left end of the piezoelectric film.

The electrical circuit model of the piezoelectric film is a charge generator in parallel with the

capacitance of the film (Kynar Piezofilm Technical Manual, 1987). The voltage generated by the

segment depends also on the signal conditioning circuit connected to the piezoelectric film. The

signal conditioning circuit that we will use is an Op-Amp circuit with theoretically zero input

impedance. The circuit representation of the film and the signal conditioner is shown in Fig. 2.6.

Deflection

w(x) x

x1 x2

h

Piezoelectric

film

Neutralaxis

w(n-1)w(n)


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28

Figure 2.6 Piezoelectric film and zero-impedance signal conditioner.

Because we connect the film to a zero-impedance signal conditioner, the output terminals of the

film are effectively shorted together. The capacitor Cdoes not draw any current and therefore can

be ignored. In this case, the output current is simply the time derivative of the charge generated by

the film. For the signal conditioning circuit, the output voltage is the product of the current i(t)

and the feedback resistance , thus,

V tdq t

dt( )

( )= (2.29)

Substituting Eq. (2.28) into Eq. (2.29), we obtain the output voltage of the piezoelectric film

segment as a function of the deflection of the beam.

V t e bh w x t

x

w x t

xx x

( )& ( , ) & ( , )

=

31 2

2 1

(2.30)

2.3.2 Array of Piezoelectric Film Segments on Beam

As shown in Fig. 2.7, N piezoelectric sensor segments are mounted on the beam. We willconstruct modal filters, i.e., linear combiners whose output emulates modal coordinates of the

vibrating beam. Each modal filter consists of an array of weights (gains) that multiply the outputs

of the segments. The mode-m filter is an array of gains Wnm, where m denotes the mode and n

denotes the segment number to which the gain is connected. The output of each segment is

denoted by Vn. The output of the mode-m sensor, denoted by &$ ( ) m t , is the weighted sum of the

segment outputs,

q(t) CCharge

generator

Piezoelectric film

Op-Amp

+

-

Output

voltage

i(t

V(t)

Signal conditioner


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29

&$ ( ) ( )m t tmn nn

N

==

W V1

. (2.31)

If we haveMmodal sensors, we can write the outputs of all of them as

&$ = WV (2.32)

where &$ is anM-vector of sensor output, W is an (MxN) matrix of gains, and V is anN-vector

of segment outputs [V1, V2, ..., VN]T. Our task now is to determine the gain matrix W so that the

modal filter outputs &$ emulates modal coordinates.

As shown in Fig. 2.8, the left and right end positions of the nth segment are denoted by n-1 and n,

respectively. We connect each segment to signal conditioning circuit with zero input impedance.For the n

thsegment, the circuit translates the current into voltage Vn, which can be obtained by

Eq. (2.30).

( )Vn n nt w t w t ( ) & ( , ) & ( , )= 1 , (2.33)

where

=h

e b2

31 , (2.34)

the dot above the variable w denotes time derivative and prime denotes spatial derivative.


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Figure 2.7 Spatial filters as modal sensors

Vn VNV1

BeamPVDF

segmentsForce

transducer

f

Mode-1

sensor&$ ( )

( )

1

f

&$ ( )

( )

Mf

.

.

.

.

.

.

Mode-m

coordinate

W1,1

Wm,N

Wm,1

WM,N

W1,n

Wm,n

WM,n

WM,1

W1,N

. . . . . .


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31

Figure 2.8 Segment positions on beam.

2.3.3 Gain Matrix for Segmented Modal Sensor

The time rate of change of a deflection slope can be expanded into its modal componentsaccording to Eq. (1.1),

& ( , ) ( ) & ( ) = =

w x t x t mm

M

m 1

, (2.35)

where 'm(x) is the slope of the mth mode shape, and m(t) is the m

th modal coordinate.

Substituting Eq. (2.35) into Eq. (2.33), we obtain the segment outputs Vn, n = 1, ..., N, as

V = CS & . (2.36)

where the (n,m) component of the segment output matrixCS is defined as

( )CS ( , ) ( ) ( ) ; ,..., ; ,...,n m n N m M m n m n= = = 1 1 1 . (2.37)

and & is the modal coordinate vector

&

& ( )

&( )

=

1 t

tM

M . (2.38)

The segment output voltages in Eq. (2.36) are input to the linear combiners (i.e., multiplied by the

gain matrix W) to produce the sensor outputs in Eq. (2.32). The sensor output vector &$ and the

gain matrix W contain as many rows as there are linear combiners. One linear combiner emulates

one modal filter.

PVDF segments

f(t)Simple

Support

x

n-1 n

Vn

xfBeam


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32

Substituting Eq. (2.36) into Eq.(2.32), we can express the outputs of the linear combiners as

functions of the gain matrix, the segment output matrix, and the modal coordinates,

&$ & = WCs . (2.39)

For the segment array to behave as modal coordinate sensors, the segment outputs V must be

multiplied by correct gains. The gain matrix W is the matrix that ideally will transform the

segment outputs V into the modal coordinates & . Equation (2.39) shows that the estimated modal

coordinates &$ will be equal to the ideal modal coordinates &only if the gain matrix is

W CS1= (2.40)

We will use the above equation to obtain the ideal gain matrices of linear combiners.

If we substitute the above weight matrix into Eq. (2.39), at the first glance it seems that the

estimated modal coordinates are equal to the ideal modal coordinates. However, this is not

exactly the case. Note that the gain matrix W and the segment output matrix CS must be square.

From Eq. (2.40), it is clear that to create a sensor to sense M modes we need at least N = M

segments. Theoretically, the structure has an infinite number of modes, therefore, the size of the

segment output matrix CS should be infinity. In practice, we can only use a finite number of

segments. Therefore, the segment output matrix CS is actually a truncated version of the ideal,

and the weight matrix obtained by Eq. (2.40) is only an approximation of the ideal weights for

modal filters. This approximation will result in a non-ideal estimate for the modal coordinates, as

will be shown in a later chapter.

2.4 Chapter Summary

In this chapter we reviewed and developed some concepts that are required to create a sensor-

and-signal-processor system that functions as a modal coordinate sensor. This system will receive

its input from the mechanical vibration of a structure and produce outputs that are proportional to

the modal coordinates of the structural vibration. As a transducer to convert mechanical vibration

into electrical signal, we choose an array of segments of piezoelectric film. The signal-processing

part of the system consists of linear combiners that act as modal filters. Each modal filter shall besensitive to only one mode, and filter out any other modes. This chapter reviewed and developed

some concepts necessary to design the modal filters.

To facilitate understanding of the concepts, we developed a structural model to use as examples

in analytical development and numerical simulation. The model we have chosen is a simply


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supported beam with an additional torsion spring at one end. The excitation is a point force. We

derived the natural frequencies, mode shape, and equation of motion for the structure.

We developed a discrete-time model for the structure. Mechanical structures can be modeled by

equivalent electrical circuits. When the signals are sampled, both can in turn be modeled withdigital filters. The modal-expansion form of the response of the mechanical structure is a

summation of second-order-type responses. Therefore, we derived a method to model a

mechanical structure with a filter composed of parallel second-order digital filters. This method

transforms the modal properties of the structure (natural frequency, damping, and residue) into

filter coefficients. The resulting discrete-time model is not only useful in simulating the structure

with a digital computer, but also useful in further analysis of the characteristics of the structure.

The relationship between the filter coefficients and the modal properties is known precisely.

Therefore, if we can adaptively adjust the filter to match the structure, we shall be able to extract

the modal properties of the structure from the filter coefficients. Thus, the parallel-second-order-

filter model of the structure could be used as a basis for developing an adaptive algorithm that can

be used in on-line modal parameter estimation. A problem with this approach is stability. Many

metal structures have very low damping that cause the z-plane poles of the structures to be very

close to the edge of the unit circle. Adaptive algorithms for identifying these poles can easily come

up with an unstable pole in the adaptation process, throwing the whole adaptive identification

process into instability. Stability problems force us to consider other filter configuration for

system identification. We chose a simple linear combiner configuration to do the filtering in space

rather than time.


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CHAPTER 3

NUMERICAL SIMULATION AND PROOF-OF-

CONCEPT EXPERIMENT

The work discussed in the chapters subsequent to this will rely heavily on numerical simulations.

In this chapter we describe an experiment to ensure that our simulation closely represents real

structures which it is designed to simulate. Towards the end of this chapter, we will also

understand the need for adaptive modal filters.

3.1 Verification of Digital Filter Model and Gain Matrix Formula

3.1.1 Verification of Digital Filter Model

To verify the method we use to represent the structure with digital filters, we shall present a

numerical example. As an example, we simulate a beam with boundary conditions as described in

Fig. 2.1, which are a pin at x = 0 and a pin and a torsion spring at x = L. The properties of the

rectangular beam are given in Table 3.1.

Table 3.1 Physical properties of beam

With the above physical properties, the parameter12L

EI

L= 23.9596 s-1. The eigenvalues can be

computed by solving Eq. (2.2). For several nondimensional torsion spring constant T* (defined in

Eq. (2.1)), these eigenvalues are as shown in Table 3.2. The T* = 0 case is the case of simplysupported beam without a torsion spring and is included in the table for comparison.

Property Value Unit Property Value UnitLength (L) 0.4382 m Density () 2700 kg/m

3

Width (b) 0.0381 m Youngs modulus (E) 68(10)9 Pa

Thickness 3.175 mm Poissons ratio 0.3


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Table 3.2 Eigenvalues of beam

The natural frequencies can be computed by Eq. (2.3). For several nondimensional torsion spring

constant T*, the natural frequencies of the beam are as shown in Table 3.3.

Table 3.3 Natural frequencie

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A New Adaptive Array of Vibration Sensors_Tesis