ACTIVE STRUCTURAL-ACOUSTIC CONTROL OF INTERIOR NOISE IN
VIBRO-ACOUSTIC CAVITIES
ASHOK KUMAR
DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY DELHI
JUNE 2016
©Indian Institute of Technology Delhi (IITD), New Delhi, 2016
ACTIVE STRUCTURAL-ACOUSTIC CONTROL OF INTERIOR NOISE IN
VIBRO-ACOUSTIC CAVITIES
by
ASHOK KUMAR
Department of Mechanical Engineering
Submitted
in fulfillment of the requirements of the degree of Doctor of Philosophy
to the
Indian Institute of Technology Delhi
June 2016
Dedicated to
My Supervisor Dr. S.V. Modak
&
My wife Sonika
My daughter Yashika Bagha
Certificate
This is to certify that the thesis entitled Active Structural-Acoustic Control of Interior Noise in Vibro-Acoustic Cavities being submitted by Mr. Ashok Kumar to the Indian Institute of Technology Delhi for the award of degree of Doctor of Philosophy is a record of bonafide research work carried out by him under my supervision and guidance. The thesis work, in my opinion has reached the requisite standard fulfilling the requirements for the degree of Doctor of Philosophy. The results contained in this thesis have not been submitted in part or in full, to any other University or Institute for the award of any Degree or Diploma.
Dr. S.V. Modak Associate Professor, Date: Department of Mechanical Engineering,
New Delhi Indian Institute of Technology Delhi
Acknowledgements
I wish to express my deepest gratitude to my supervisor Dr. S.V. Modak for guiding in my research endeavour. His constant support and guidance from choosing the right research topic to the writing of this manuscript has been very valuable. Without persistent encouragement and motivation provided by Dr. S.V. Modak this research work would not have been possible.
It was due to his initiatives and valuable instructions that this work got accomplished. It was his innovative ideas, wisdom, experience and goodness that kept this research going. I will forever remain indebted to you Sir for your encouragement, interest and supportive guidance throughout this research period.
I take this opportunity also to thank my other Student Research Committee (SRC) members- Prof. S.P. Singh and Prof. J.K. Dutt (of Mechanical Engineering Department) and Prof.
Santosh Kapuria (of Department of Applied Mechanics), for a critical review of this research work and for providing several constructive suggestions.
This is also time to express my sincere thanks to Mr. Ravi Sharma, Dr. Vineet Prabhakar, Prof. Rakesh Chandra, Dr. Subhash Chander for their guidance and support throughout my academic carrier.
I would also like to thank my fellow PhD scholars in no particular order Sky lab Bhore, Sharad K. Pradhan, Manoj Chouskey, Dipak V. Nehete, Dinesh Kumar, Sukesh Babu, Sachin, Faisal Rahmani and Amrita Puri who made every day easier by helping me in various ways. Special thanks to Sharad K. Pradhan for valuable scientific discussions and utterly selfless help.
I am thankful to Mr. S. Babu, Mr. D. Jaitly, Mr. K.N. Madasundaran, Mr. Ayodhya Prashad and Mr. Rishi Lal for their support and cooperation in the experimental work carried out in Design Research Lab.
I am thankful to my grandparents and parents for their encouragement and support to carry out this research work. I would like to thank my wife, Sonika, and my daughter, Yashika, for their patience, understanding, unconditional and immeasurable support during the course of this work. I am thankful to my wife who managed all other affairs, so that I may be free for the research analysis, and for sharing all my successes and disappointments over these years.
I can always count on her to make the best in any situation, without you it would not be possible for me to complete this or any other task of life.
I am thankful to IIT Delhi for providing a family accommodation in Nalanda Hostel.
Ashok Kumar
Abstract
Active noise control offers an effective alternative for control of low frequency interior noise in cavities like automobile passenger compartments, aerospace interiors, helicopters, marine vehicles, launch vehicles and enclosed spaces. Disturbances acting on the surrounding elastic structure are a major contributor to the generated noise. This thesis addresses the development of an active structural-acoustic control (ASAC) system using feedback control strategies. It also addresses development of methods for virtual sensing including incorporation of system identification in the control system design process. The feedback control strategies based on direct output feedback, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) control are explored for ASAC. Objective is also to achieve sensing and actuation through piezoelectric structural sensors and actuators. Numerical and experimental studies are performed on a 3-D rectangular box cavity with a flexible plate to evaluate and compare the performance of the proposed sensing and the control methods.
A numerical model of the 3-D rectangular box cavity with the flexible plate (glued with piezoelectric patches) and with other five surfaces treated rigid is developed using finite element (FE) method. Collocated pairs of piezoelectric patches are used for sensing the vibrations and applying control forces on the structure. A finite element (FE) based numerical model of the piezo-structural-acoustic system is developed from which state space models for predicting the structural and the acoustic response in the physical as well
as in the modal domain are developed. Modal analysis of the cavity for single and multiple pairs of collocated piezoelectric patches is performed. The selection of optimal location of piezoelectric sensors and actuators based on observability and controllability grammians matrix is carried out.
The thesis proposes a new strategy for virtual sensing of the acoustic potential energy inside a vibro-acoustic cavity with the objective of achieving global active control of noise. It is based on the established concept of ‘radiation modes’ and does not add too many states to the order of the system. Sensing of radiation modes is carried out through two separate filters. A Kalman filter is used to sense the modal amplitudes of the radiation modes and a frequency-weighting filter is designed to model their radiation efficiencies.
The results of virtual sensing based on the proposed strategy are also compared with sensing based on the direct estimation of the structural modal velocities from the outputs of the piezo sensors and also with sensing based on an acoustic filter.
The thesis considered five feedback control strategies for ASAC. These are a) a control strategy based on direct output feedback (AVC-DOFB) b) a control strategy based on LQR to reduce structural vibrations (AVC-LQR) c) an LQR control strategy with a weighting scheme based on structural-acoustic coupling coefficients (ASAC-LQR) d) an LQG control strategy with an acoustic filter as an observer and e) an LQG control strategy with the proposed virtual sensing filter.
The first two strategies i.e., AVC-DOFB and AVC-LQR are indirect strategies in which noise reduction is achieved through active vibration control (AVC). The third strategy (ASAC-LQR) is an active structural-acoustic control (ASAC) strategy proposed in this work. This strategy is an LQR based optimal control strategy in which the information
about the coupling between the various structural and the acoustic modes is used to design the controller. The last two strategies are based on virtual sensing methods based on acoustic and radiation filters. The performance of these strategies is compared for active noise control in the 3-D rectangular box cavity.
An experimental evaluation of the ASAC system performance is carried out on a 3-D rectangular cavity with a flexible plate. The virtual sensing and the control system are implemented on a dSPACE controller board. A methodology to incorporate system identification in the development of LQG controllers is presented. System identification based on modal testing and finite element model updating is proposed to develop an accurate state observer in the form of a Kalman filter. An LQG controller is built by combining the Kalman filter based on the updated model and the LQR controller.
The results of the numerical and the experimental studies carried out using the feedback control strategies and the virtual sensing strategies show that the active structural- acoustic control system developed in this work is effective in suppressing the interior noise in cavities.
Contents
Certificate ……….. i
Acknowledgements ………... ii
Abstract ………. iv
Contents ………. vii
List of Figures ……… xi
List of Tables ………. xvi
List of Symbols ……….. xvii
Chapter 1. Introduction and literature Survey ………... …. 1
1.1 Introduction ……… 1
1.2 Literature Survey ……….... 6
1.2.1 Analytical/ Numerical modeling of ASAC systems…...………. 8
1.2.2 Sensors and actuators for ASAC systems……….... 14
1.2.3 Optimal location of sensors and actuators……….... 16
1.2.4 Coupling between structural and acoustic modes………... 18
1.2.5 Radiation modes………... 19
1.2.6 Virtual sensing………... 20
1.2.7 Control approaches………... 22
1.2.7.1 Feedforward control...………... 24
1.2.7.2 Feedback control...………... 29
1.2.8 System identification………... 34
1.3 Concluding remarks about literature survey ……….. 35
1.4 Objectives of the thesis ……….. 37
1.5 Overview of the thesis ……… 37
Chapter 2. Numerical simulation of piezo-structural-acoustic cavity system for active structural-acoustic control…..….. 41
2.1 Introduction ………... 41
2.2 Numerical model of the piezo-structural-acoustic system……… 42
2.3 State space model of the structural domain of the cavity………. 51
2.3.1 State space model of the plant in physical domain……….. 52
2.3.2 State space model of the plant in modal domain...……….. 55
2.4 State space model of the acoustic domain of the cavity………... 57
2.5 Numerical study ……… 58
2.5.1 Details of the case study ……….. 58
2.5.2 Modal analysis of the piezo-structural and the acoustic systems...….. 66
2.5.3 Modal analysis of the piezo-structural system for other locations of piezoelectric patches...….... 72
2.5.4 Choice of location of the piezoelectric patches based on mode shapes. 76 2.5.5 Optimal location of the piezoelectric patches …………...…………... 79
2.6 Conclusions...……….... 82
Chapter 3. Feedback control strategies in active structural-acoustic control of a vibro-acoustic cavity……….. 83
3.1 Introduction ………... 83
3.2 Active vibration control using direct output feedback (AVC-DOFB)... 84
3.2.1 Open and closed loop system model...………… 85
3.2.2 Design of direct output feedback gain ………... 89
3.2.3 Results of closed loop control.……….. 92
3.3 Active vibration control using an LQR controller (AVC-LQR)……… 96
3.4 Active structural-acoustic control using an LQR controller (ASAC-LQR)... 100
3.5 Results of closed loop control using LQR controllers...………... 101
3.6 Comparative results...………... 109
3.7 Conclusions...………... 113
Chapter 4. Virtual sensing in a vibro-acoustic cavity………... 115
4.1 Introduction ………. 115
4.2 Virtual sensing using radiation filter... 117
4.2.1 Description of the radiation filter...………….. 117
4.2.2 Mathematical formulation of the radiation filter...……… 120
4.2.2.1 Frequency-weighting filter………... 120
4.2.2.2 Kalman filter………..………... 124
4.2.2.3 State space model of the radiation filter... 128
4.3 Virtual sensing using acoustic filter... 128
4.4 Virtual sensing using direct estimation... 131
4.5 Estimation of the exact acoustic potential energy... 133
4.6 Numerical study………... 133
4.6.1 Radiation modes of the flexible plate of the cavity...…….………….. 133
4.6.2 Results and discussion……….………. 137
4.6.2.1 Frequency response of the sensing system...………... 137
4.6.2.2 Sensing performance under impulsive and broadband disturbances... 143
4.6.2.3 Time response under impulsive and broadband disturbance... 147
4.7 Conclusions...………... 153
Chapter 5. Active structural-acoustic control using virtual sensing strategies ………... 155
5.1 Introduction ………. 155
5.2 Global active noise control using radiation filter...……….. 156
5.3 Global active noise control using acoustic filter...………... 160
5.4 Numerical study ………... 162
5.4.1 Frequency response of the control system...…………... 162
5.4.1.1 Using single pair of collocated piezoelectric patches for
sensing and actuation... 162
5.4.1.2 Using multiple pairs of collocated piezoelectric patches for sensing and actuation... 164
5.4.1.3 Effect on modal damping factors... 166
5.4.2 Control performance under broadband disturbances……….. 167
5.4.3 Time response of the control system...……….. 171
5.5 Conclusions...………... 178
Chapter 6. Experimental studies in active structural-acoustic control of a vibro-acoustic cavity …... 179
6.1 Introduction ………. 179
6.2 Active structural-acoustic control using an LQG controller incorporating system identification...…. 180
6.2.1 System identification...……… 181
6.2.1.1 FE model of the cavity structure...………... 181
6.2.1.2 Modal testing…………..………... 183
6.2.1.3 Finite element model updating of the plant... 184
6.2.2 State observer...……… 186
6.2.3 Linear quadratic gaussian (LQG) controller...……… 189
6.3 Experimental study...…………...………….. 193
6.4 Conclusions... ………... 213
Chapter 7. Conclusions and research contributions ………... 215
7.1 Concluding remarks ……… 215
7.2 Research contributions ……… 220
References ……… 221
Papers published and communicated ……….... 233
Bio-data ………. 234
List of figures
List of figures
Figure number and title Page no.
1.1. An active structural-acoustic control system 7
1.2. Active noise control system by Leug (1936) 23
1.3. Active noise control by Olson and May (1953) 24
1.4. Principle of feedforward control 25
1.5. Principle of feedback control 30
2.1. Flexible plate with two collocated piezoelectric patches (piezo-structural system)
43 2.2. One way coupling between piezo-structural-acoustic cavity systems 51 2.3. Short circuit electric boundary condition for sensor 52 2.4. Schematic representation of a rigid-wall rectangular box 3-D cavity with a flexible plate and a pair of collocated piezoelectric patches
59 2.5. FE mesh of the plate with single pair of collocated piezoelectric patches 62 2.6. FE mesh of the plate with multiple pair of collocated piezoelectric patches 63
2.7. FE mesh of the acoustic domain of the cavity 64
2.8. FE mesh of piezo-structural-acoustic domain of the vibro-acoustic cavity showing excitation point (node no. 103) on structure and acoustic response point (node 1941) inside the cavity
65
2.9. The plots of five structural modes of the plate with piezos in the range of 0- 400 Hz
70 2.10. The plots of first two acoustic modes of the cavity 71 2.11. Alternative location of piezoelectric patches for modal analysis 72 2.12. An alternative choice for the locations of multiple pairs of piezoelectric patches
75
List of figures
2.13. Mode shapes of structural modes 1, 2, 3 and 4 77
2.14. Mode shapes of structural modes 5, 6, 7 and 8 78
3.1. AVC system using direct output feedback control 85
3.2. Pole-zero plots at Gd = 0 and Gv = 0.08 (blue: open loop poles; red: closed loop poles)
90
3.3. Actuator voltage (Gd =0 and Gv = 0.08) 93
3.4. Open and closed loop sensor output voltage (Gd =0 and Gv = 0.08) 93 3.5. Open and closed loop structural displacement response at node number 103 94 3.6. Open and closed loop acoustic pressure at node number 1941 inside the cavity 95 3.7. Open and closed loop instantaneous SPL in dB at node number 1941 inside the cavity
96
3.8. AVC system using an LQR control strategy 97
3.9. Structural modal velocity (a) first mode (b) seventh mode 106 3.10. Acoustic nodal pressure with and without control at node number 1941 inside the cavity
107
3.11. Actuator voltage 108
3.12. Comparison of frequency response function of the acceleration of the structure in dB with and without control
110 3.13. Comparison of frequency response function (Pressure in Pa/ Force in N) of acoustic nodal pressure in dB with and without control
111 3.14. Comparison of frequency response function of the actuation voltage in dB 112 4.1. Virtual sensing of acoustic potential energy using radiation filter. Symbols used are: disturbance (1), control signal (2), sensor output (3), measurement noise (4), measured output (5), acoustic pressure (p), acoustic potential energy (Ep)
118
4.2. The Kalman filter model 125
4.3. Virtual sensing using direct estimation 132
4.4. The radiation efficiencies (or singular values) of the first four radiation modes of the structure
134 4.5. Radiation mode shapes of the flexible plate at 545 Hz (a) first mode (b) second mode (c) third mode (d) fourth mode
136 4.6. Frequency response of modal velocity of the first radiation mode estimated by 138
List of figures
the Kalman filter with a single pair of piezoelectric patches
4.7. Frequency response of sensing of global acoustic potential energy with a single pair of piezoelectric patches
139 4.8. Frequency response of modal velocity of first radiation mode estimated by the Kalman filter with multiple pairs of piezoelectric patches
141 4.9. Frequency response of sensing of global acoustic potential energy with multiple pairs of piezoelectric patches
142 4.10. Comparison of modal velocity of first structural mode estimated by the Kalman filter and estimated directly from the output of multiple pairs of piezoelectric patches in the presence of an impulsive disturbance
144
4.11. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and estimated directly from the output of multiple pairs of piezoelectric patches in the presence of an impulsive disturbance
145
4.12. Sensing of global acoustic potential energy with multiple pairs of piezoelectric patches for a broadband disturbance with measurement noise SNR of 60dB
146
4.13. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with single pair of piezoelectric patches
148 4.14. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with multiple pairs of piezoelectric patches
149 4.15. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with single pair of piezoelectric patches at SNR of 60dB
150
4.16. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with multiple pairs of piezoelectric patches at SNR of 60dB
151
4.17. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with single pair of piezoelectric patches at SNR of 6dB
152
4.18. Comparison of modal velocity of first radiation mode estimated by the Kalman filter and its true value with multiple pairs of piezoelectric patches at SNR of 6dB
153
5.1. Global active structural-acoustic control using radiation filter. Symbols used are: disturbance (1), control signal (2), sensor output (3), measurement noise (4), measured output (5), acoustic pressure (p), acoustic potential energy (Ep)
157
5.2. Global active structural-acoustic control using acoustic filter. Symbols used are: disturbance (1), control signal (2), sensor output (3), measurement noise (4),
160
List of figures
measured output (5), acoustic pressure (p), acoustic potential energy (Ep)
5.3. Global acoustic potential energy with and without control with a single pair of piezoelectric patches
163 5.4. Open and close loop frequency response of global acoustic potential energy with multiple pairs of piezoelectric patches
165
5.5. Random disturbance acting on the structure 168
5.6. Control of global acoustic potential energy with multiple pairs of piezoelectric patches for a broadband disturbance with measurement noise SNR of 60dB
169
5.7. Control of global acoustic potential energy with multiple pairs of piezoelectric patches for a broadband disturbance with measurement noise SNR of 6dB
170 5.8. Control of acoustic nodal pressure at node number 1941 with single pair of piezoelectric patches
171 5.9. Control of acoustic nodal pressure at node number 1941 with multiple pairs of piezoelectric patches
172 5.10. Control of acoustic nodal pressure at node number 1941 with single pair of piezoelectric patches at SNR of 60dB
173 5.11. Control of acoustic nodal pressure at node number 1941 with multiple pairs of piezoelectric patches at SNR of 60dB
174 5.12. Control of acoustic nodal pressure at node number 1941 with single pair of piezoelectric patches at SNR of 6dB
175 5.13. Control of acoustic nodal pressure at node number 1941 with multiple pairs of piezoelectric patches at SNR of 6dB
176
6.1. A stochastic state observer 187
6.2. Linear quadratic Gaussian (LQG) controller for ASAC 190
6.3. Experimental vibro-acoustic cavity 194
6.4. Experimental set-up for modal analysis (a) Modal hammer (b) FFT analyzer (c) Accelerometer (d) PZT patch (e) Flexible plate (f) Clamping screws (g) Acoustic cavity (h) Microphones (i) Sound pressure measuring holes
196
6.5. Torsional spring stiffness updating parameters for the plate 200 6.6. Comparison of the overlays of inertance FRF-H161, 84 using IESM with the measured FRF for FE model (a) before updating (b) after updating
202
6.7. Experimental setup used for ANC (a) vibro-acoustic cavity (b) ENDEVCO signal conditioner (c) dSPACE controller board (d) analog low pass filter (e) Controller (f) Piezoelectric voltage amplifier (g) CRO
205
List of figures
6.8. Schematic of the experimental setup used for ANC 206
6.9. Kalman filter performance evaluation at L1 and L2 location 208 6.10. Comparison of the estimated acceleration with the measured acceleration at location L1
209 6.11. Comparison of the estimated acceleration (L2) with the measured acceleration at location L1
210 6.12. Frequency response function with and without control at a microphone position in the cavity
211 6.13. Open and closed loop frequency response function at a node inside the vibro- acoustic cavity
212
List of tables
List of tables
Table number and title Page no.
2.1. Natural frequencies of the steel plate and plant and the acoustic cavity 66 2.2. Natural frequencies of the steel plate with single pair of piezoelectric patches
74
2.3. Natural frequencies of the steel plate with multiple pair of piezoelectric patches
75 2.4. Observability grammians eigenvalues for first few modes of structure 81 3.1. Characteristics of various closed loop poles as a function of Gv 91 3.2. Modal structural-acoustic coupling coefficients (Cij) 104 3.3. Modal damping factors with different control strategies 113 5.1. Modal damping factors with single pair of collocated piezoelectric patches 166 5.2. Modal damping factors with multiple pairs of collocated piezoelectric
patches
167
5.3. Comparison of RMS value of acoustic nodal pressure (in dB) in case of broadband disturbance. The values inside the brackets representing reductions after the control
177
6.1. Experimental natural frequencies and damping factors of the cavity structure and the acoustic cavity
197 6.2. MAC values and FE model and experimental natural frequencies 198 6.3. The intial and the values of the updating parameters after updating 201 6.4. Comparison of the correlation between the measured and the updated natural
frequencies and the mode shapes
201 6.5. Modal structural-acoustic coupling coefficients (Cij) 204
List of symbols and abbreviations
List of Symbols and Abbreviations
Nomenclature
a Modal amplitude of radiation modes
ˆa Estimated modal velocity amplitudes of radiation modes ACL Closed loop state matrix for LQR controller
A,Ba, Bg C,D State space matrices of the plant
AA, BA, CA, DA State space matrices for the acoustic filter
Af ,Bf , Cf,Df State space matrices for the frequency-weighting filter AK,BaK, BgK , CK, DK State space matrices for the Kalman filter
AR, BR, CR, DR State space matrices for the radiation filter
AZ, BZ, CZ, DZ State space matrices for the frequency-domain filter A ,u B , ua B ,gu Cu, D u State space matrices of the updated Kalman filter model b State vector of frequency-weighting filter
C Capaciatnce of the charge amplifier
List of symbols and abbreviations
CA Acoustic viscous damping matrix
CAS Structural-acoustic coupling matrix in the modal domain CE Elastic constant matrix at constant electric field
Cnew Modified damping matrix due to effect of control damping matrix
CS Structural viscous damping matrix
u
CT Updated damping matrix of the cavity structure d State vector of the frequency-domain filter
D Electrical displacement
e Piezoelectric stress coefficients
E Electrical field vector
e
Ep Acoustic potential energy for an acoustic element
Ep True global acoustic potential energy in frequency domain ˆ '
Ep,R Square root of the acoustic potential energy estimated by the radiation filter
'
ˆ A
Ep, Square root of the acoustic potential energy estimated by the acoustic filter
ˆp
E Estimated global acoustic potential energy
g Vector of the random disturbances acting on the cavity Gd Displacement gain correponding to the sensor output voltage Gv Velocity feedback gain correponding to the rate of the sensor
output voltage
List of symbols and abbreviations
G LQR gain matrix due to structural modal velocities and radiation filter states
GA LQR gain matrix due to structural modal velocities and acoustic filter states
I Identity matrix
KT Combined structural and piezoelectric stiffness matrix
a
Kwφ Electro-mechanical coupling matrices between the structure and the actuators
s
Kwφ Electro-mechanical coupling matrices between the structure and the sensors
KA Acoustic stiffness matrix
Kaφφ Electric capacitance matrix for actuators Ksφφ Electric capacitance matrix for sensors Kstr Structural stiffness matrix
Ksen Piezoelectric sensor stiffness matrix Kact Piezoelectric actuator stiffness matrix
K Full state feedback LQR gain matrix
Knew Modified stiffness matrix due to effect of control stiffness matrix
u
ΚT Updated stiffness matrix of the cavity structure
L Steady state Kalman gain matrix
L u Updated Kalman filter gain
MT Combined structural and piezoelectric mass/inertia matrix
List of symbols and abbreviations
MA Acoustic mass/inertia matrix
Mact Piezoelectric actuator mass/inertia matrix Msen Piezoelectric sensor mass/inertia matrix
Mstr Structural mass matrix
u
ΜT
Μ Μ
Μ Updated mass matrix of the cavity structure
na Number of acoustic modes
ns Number of structural modes
p Vector of the nodal acoustic pressures
p Acoustic pressure inside the element
pe Vector of elemental nodal pressures
qa Vector of the electric charges at the actuator electrodes qs Vector of the electric charges at the sensor electrodes qd Modal displacement gain for LQR controller
qv Modal velocity gain for LQR controller QVib Weighting matrix for structural states r State vector of the radiation filter rA State vector of the acoustic filter
R Weighting matrix for the control input
g, v
s s Process and measurement noise covariance respectively
S Structural-acoustic coupling matrix
Sca Gain of the charge amplifier in mV/pC
List of symbols and abbreviations
U Orthonormal matrix
v Measurement noise vector
V Acoustic volume matrix
V m Diagonal acoustic modal volume matrix
w Vector of structural transverse and rotational degrees of freedom
wɺɺ Vector of physical acceleration of the cavity structure ˆu
wɺɺ Estimate of the physical nodal acceleration
Wo Diagonal matrix of controllability grammians eigenvalues ys Sensor output voltage corrupted with measurement noise y Measured acceleration
u
ym Estimate of the measured acceleration
Za Structure to acoustic modal frequency response function matrix
Greek Symbols
ζS Piezoelectric dielectric matrix at constant mechanical strain φa Vector of voltages on the piezoelectric patches used as the
actuators
φs Vector of voltages on the piezoelectric patches used as the sensors
ρact density of the piezoelectric actuator
List of symbols and abbreviations
ρstr density of the structure
ρsen density of the piezoelectric sensor
ψS
ψ ψ
ψ Mass normalized in-vacuo eigenvectors of the structural-piezo system
ξS Viscous modal damping factors of the structure
λS Square root of the eigen values of the structure-piezo system
ψA
ψ ψ
ψ Mass normalized eigenvectors of the acoustic system
ξA Viscous modal damping factors of the acoustic domain λA Square root of the eigen values of the acoustic system
2
λλλλS Eigenvalue matrix of the structure-piezo system ΛS
ΛΛ
Λ Diagonal modal damping matrix of structure-piezo system representing ΛSii= 2= 2= 2= 2ξSiλSi
2
λλλλS Eigenvalue matrix of the acoustic system ΛA
ΛΛ
Λ Diagonal modal damping matrix of acoustic system
representing ΛAii = 2= 2= 2= 2ξAiλAi
ηηηηS Vector of structural modal displacements ɺS
ηηηη Vector of structural modal velocities ɺɺηηηηS Vector of structural modal accelerations β1 Vector of structural modal displacements β2 Vector of structural modal velocities
β State vector of structural modal displacement and velocities ηA Vector of acoustic modal pressure amplitudes
List of symbols and abbreviations
αA Acoustic frequency response function matrix in the modal domain
σ Matrix of singular values
ˆK
β Estimated structural modal vector by the Kalman filter
ˆ2K
βɺ Estimated structural modal velocity amplitudes by the Kalman filter
ψR
ψψ
ψ Radiation mode shape matrix
φca Sensor output voltage passing through charge amplifier ω Excitation frequency
α Weight on the acoustic potential energy
λλλλX Experimental eigenvalues of the cavity structure
ψψψψX Experimental eigenvectors of the cavity structure
ξX Experimental viscous modal damping factors of the cavity structure
u
λλλλS Updated eigenvalues of the cavity structure
u
ψS
ψ ψ
ψ Updated eigenvectors of the cavity structure
u
ΛS
ΛΛ
Λ Updated modal viscous damping matrix of the cavity structure
List of symbols and abbreviations
Subscripts
a, act Actuator
A Acoustic
ca Charge amplifier
CL Closed
f, z Filter
K Kalman
L Laminate
N-L Non-laminate
nu Number of updating parameters
R Radiation
s, sen Sensor
S, str Structure
T Total
Superscripts
a Actuator
E Constant electric field
e Element
S Constant strain
s Sensor
List of symbols and abbreviations
T Transpose
u Updated
Abbreviations
ANC Active Noise Control
ASAC Active Structural-Acoustic Control
AVC Active Vibration Control
DOFB Direct Output Feedback
DOFs Degrees of freedom
FE Finite element
FRFs Frequency Response Functions IESM Inverse Eigen-Sensitivity Method
LMS Least Mean Square
LQG Linear Quadratic Gaussian
LQR Linear Quadratic Regulator
MAC Modal Assurance Criterion
PVDF Polyvinylidene Fluoride
SNR Signal to noise ratio
