Optimal location and gains of sensors and actuators for feedback vibroacoustic control of plates

130 Pages·2014·5.7 MB·English

POLITECNICO DI MILANO
Scuola di Ingegneria Industriale e dell’Informazione
Corso di Laurea Magistrale in Ingegneria Spaziale
Optimal location and gains
of sensors and actuators for feedback
vibroacoustic control of plates
Relatore:
Prof. Lorenzo Dozio
Tesi di laurea di:
Silvio Patruno
Matr. 771437
Anno accademico 2013/2014

"Scientists discover the world that exists;
engineers create the world that never was."
Theodore von KÆrmÆn

Acknowledgments
My (cid:28)rst and dutiful thanks go to Prof. Lorenzo Dozio, for guiding me in the
writing of this thesis, masterfully, with great willingness and con(cid:28)dence; and
for having instilled in me a passion for the control of structures through his
teachings and advice.
My thanks also go to all the teachers met during these years of study at
Politecnico di Milano. This thesis is the result of all their precious teachings
and passion that they managed to send me. I owe them a message that I
managed to catch in recent years: Engineering is not a way to do, but a way
of thinking and seeing things.
I would also like to remind all my fellow travelers. Years have been chal-
lenging, full of sacri(cid:28)ce and hard times. A big "thank you" goes out to all
my friends and colleagues with whom I shared and overcome these moments.
I would like to mention the unforgettable members of Laputa Team, a group
of fantastic people, old and new college mates, and a distant but close com-
panion of life, Federico Secondi.
Finally, a special thanks goes to my little scricc, my family and my old
friends, for the patience, the support, and the encouragement during this
whole period of studies.

Contents
Abstract XI
1 Introduction 1
1.1 Problem understanding . . . . . . . . . . . . . . . . . . . . . . 3
2 State of the Art 5
2.1 Optimal locations on a membrane - K. Xu, P. Warnitchai, T.
Igusa - 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Formulation of the problem . . . . . . . . . . . . . . . 6
2.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Optimal discrete locations on a membrane - M. Abdullah - 1998 10
2.2.1 Discrete location programming technique . . . . . . . . 11
2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Problem statement 19
3.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1 Ritz formulation . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 Dynamic Equations . . . . . . . . . . . . . . . . . . . . 23
3.1.3 State space representation of the structural model . . . 26
3.2 Radiation model . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Sound Radiation formulation . . . . . . . . . . . . . . 27
3.2.2 Formulation in terms of Elementary Radiators . . . . . 28
3.2.3 Radiation modal expansion (RME) . . . . . . . . . . . 31
3.2.4 State space representation of the acoustic model . . . . 35
3.3 Global state space representation . . . . . . . . . . . . . . . . 37
4 Optimization problem 39
4.1 Gradient-based numerical optimization . . . . . . . . . . . . . 43
7
CONTENTS I
5 Convergence study 45
5.1 Convergence of the structural model . . . . . . . . . . . . . . 46
5.2 Convergence of the acoustic model . . . . . . . . . . . . . . . 49
6 Analysis and Results 53
6.1 Optimization of gains with a (cid:28)xed grid of controllers (Engels
[4]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.1.1 Emprovement of the grid arrangement . . . . . . . . . 57
6.2 Di(cid:27)erences in minimizing Kinetic energy or Radiated sound
power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 Comparison between (cid:28)xed locations and optimized ones . . . . 62
6.3.1 Optimal placement and gains for a single controller -
Kinetic energy optimization . . . . . . . . . . . . . . . 62
6.3.2 Optimal placement and gains for a single controller -
Acoustic power optimization . . . . . . . . . . . . . . . 69
6.3.3 Optimal placement and gains for 16 controllers. Ki-
netic energy and acoustic optimization . . . . . . . . . 75
6.3.4 Optimal placement and gains for 100 controllers. Ki-
netic energy and acoustic optimization . . . . . . . . . 82
6.4 Results for di(cid:27)erent boundary conditions . . . . . . . . . . . . 86
6.4.1 Fully clamped plate (CCCC) . . . . . . . . . . . . . . 86
6.4.2 Half clamped, half simply supported plate (CCSS) . . . 89
6.4.3 Clamped free plate (CCCF) . . . . . . . . . . . . . . . 91
6.5 Results with the addition of complicating e(cid:27)ects . . . . . . . . 93
6.5.1 Elastic point supports . . . . . . . . . . . . . . . . . . 93
6.5.2 Elastic line supports . . . . . . . . . . . . . . . . . . . 95
6.5.3 In-plane loaded plate . . . . . . . . . . . . . . . . . . . 97
6.5.4 Rigid concentrated masses . . . . . . . . . . . . . . . . 99
6.5.5 Sensors and actuators with mass . . . . . . . . . . . . 101
7 Conclusions 103
7.1 Future developments . . . . . . . . . . . . . . . . . . . . . . . 104
A Input parameters 105

List of Figures
1.1 Rectangular Kircho(cid:27) plate. . . . . . . . . . . . . . . . . . . . . 3
2.1 Rectangular membrane. . . . . . . . . . . . . . . . . . . . . . 6
2.2 Placements for the single controller case. . . . . . . . . . . . . 8
2.3 Superimposed placements for the single controller case - nodal
lines for the open loop modes. . . . . . . . . . . . . . . . . . . 8
2.4 Placements for 11 controllers case. . . . . . . . . . . . . . . . . 9
2.5 Superimposed placements for the 11 controllers case - nodal
lines for the open loop modes. . . . . . . . . . . . . . . . . . . 10
2.6 Placements of 100 controllers. . . . . . . . . . . . . . . . . . . 11
2.7 Transformation of variables. . . . . . . . . . . . . . . . . . . . 13
2.8 Cost function J with respect to single actuator/sensor place-
ment, 1 mode, gain = 0.160. . . . . . . . . . . . . . . . . . . . 15
2.9 Actuator/sensorplacementforonemode: •,J(0.6,0.6)=3.564
and gain = 0.106; ×, J(0.5,0.5) = 3.244 and gain = 0.125. . . 15
2.10 Cost function J with respect to single actuator/sensor place-
ment, 11 mode, gain = 0.160. . . . . . . . . . . . . . . . . . . 16
2.11 Actuator/sensorplacementforonemode: •,J(0.8,0.6)=45.775
and gain = 0.279; ×, J(0.790,0.825) = 45.468 and gain = 0.264. 16
2.12 Optimalplacementfor11actuators/sensorsfor11modeswhere
J =5.873andgains={0.190,0.192,0.192,0.192,0.192,0.192,
0.195, 0.195, 0.199, 0.199, 0.208}. . . . . . . . . . . . . . . . . 18
2.13 Optimalplacementfor11actuators/sensorsfor11modeswhere
J =7.614andgains={0.168,0.178,0.194,0.206,0.215,0.221,
0.221, 0.227, 0.246, 0.275, 0.281}. . . . . . . . . . . . . . . . . 18
3.1 Rectangular plate lying in the (x;y) plane. . . . . . . . . . . . 21
3.2 Subdivision of a panel into elementary radiators. . . . . . . . . 29
III