January 2014

NASA/CR–2014-218147

Active Vibration Control for Helicopter Interior 

Noise Reduction Using Power Minimization

J. Mendoza, K. Chevva, F. Sun, A. Blanc, and S. B. Kim

United Technologies Research Center, East Hartford, Connecticut
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January 2014

NASA/CR–2014-218147

Active Vibration Control for Helicopter Interior 

Noise Reduction Using Power Minimization

J. Mendoza, K. Chevva, F. Sun, A. Blanc, and S. B. Kim

United Technologies Research Center, East Hartford, Connecticut
Available from:

NASA Center for AeroSpace Information

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Hanover, MD 21076-1320

443-757-5802

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and Space Administration.
Abstract

This report describes work performed by United Technologies Research Center (UTRC) for

NASA Langley Research Center (LaRC) under Contract NNL11AA06C. The objective of

this program is to develop technology to reduce helicopter interior noise resulting from multiple gear meshing frequencies. A novel active vibration control approach called Minimum

Actuation Power (MAP) is developed. MAP is an optimal control strategy that minimizes

the total input power into a structure by monitoring and varying the input power of controlling sources. MAP control was implemented without explicit knowledge of the phasing

and magnitude of the excitation sources by driving the real part of the input power from

the controlling sources to zero. It is shown that this occurs when the total mechanical input

power from the excitation and controlling sources is a minimum. MAP theory is developed

for multiple excitation sources with arbitrary relative phasing for single or multiple discrete

frequencies and controlled by a single or multiple controlling sources. Simulations and experimental results demonstrate the feasibility of MAP for structural vibration reduction of

a realistic rotorcraft interior structure. MAP control resulted in significant average global

vibration reduction of a single frequency and multiple frequency excitations with one controlling actuator. Simulations also demonstrate the potential effectiveness of the observed

vibration reductions on interior radiated noise.

1
Acknowledgements

The authors would like to acknowledge the technical support and leadership of Ran Cabell

and the project support of Susan Gorton from the NASA Langley Research Center. This

work was sponsored by the NASA Subsonic Rotary Wing Project under the Fundamental

Aeronautics Program. The authors would also like to acknowledge Paul Braunwart and

Kenji Homma from UTRC for their help with the experiments and thank Bill Welsh and

Tom Millott of Sikorsky Aircraft Corporation for their active participation in the program

providing insights into practical helicopter interiors, dynamics, and controls.

2
Contents

1 Executive Summary 25

2 Introduction 29

2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Program goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4 Proposed Active Vibration Control Approach . . . . . . . . . . . . . . . . . 38

2.5 Report Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 MAP Control for Single Primary and Single Secondary 42

3.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3
3.2 Theory of Minimum Actuation Power (MAP) . . . . . . . . . . . . . . . . . 45

3.2.1 Input power minimization . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.2 MAP theory for a simply supported plate actuated by two piezoelectric

patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.3 MAP control for multifrequency excitation . . . . . . . . . . . . . . 53

3.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.2 Vibration response under primary excitation . . . . . . . . . . . . . . 57

3.3.3 Vibration response with MAP control . . . . . . . . . . . . . . . . . . 58

3.3.4 MAP control at resonant and off-resonant frequencies . . . . . . . . . 59

3.3.5 Optimal secondary location for a frequency band . . . . . . . . . . . 64

3.3.6 MAP control as a function of separation between primary and secondary patch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.3.7 Multifrequency excitation . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4 MAP Implementation with Electric Actuator . . . . . . . . . . . . . . . . . . 73

3.4.1 Electromechanical coupling between the PZT actuator and the structure 73

4
3.4.2 Real-time control algorithm to implement MAP for a single secondary

actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Experimental Demonstration of MAP . . . . . . . . . . . . . . . . . . . . . . 79

3.5.1 Overview of experimental verification . . . . . . . . . . . . . . . . . . 80

3.5.2 Test instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.5.3 Structural test apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.5.4 Implementation of MAP control . . . . . . . . . . . . . . . . . . . . . 83

3.5.5 Actuator electric loss characterization . . . . . . . . . . . . . . . . . . 85

3.5.6 MAP control validation . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4 MAP Control for Multiple Primary and Multiple Secondary 95

4.1 System Model For Theoretical Simulations . . . . . . . . . . . . . . . . . . . 96

4.1.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.1.2 Total input power minimization for multiple primary and secondary

patch excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.3 Net electrical power in secondary actuator at optimal control . . . . . 102

5
4.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.2.2 MAP Performance as a Function of Location of Secondary Actuators 104

4.2.3 MAP Performance as a Function of Number of Secondary Actuators . 106

4.2.4 Secondary Input Power at Optimal Control . . . . . . . . . . . . . . . 108

4.2.5 Effect of High Damping on MAP Control . . . . . . . . . . . . . . . . 109

4.3 Experimental Demonstration of MAP Control . . . . . . . . . . . . . . . . . 112

4.3.1 MAP verification with Impedance Head Measurements . . . . . . . . 112

4.3.2 MAP Testbed and Real-Time Implementation . . . . . . . . . . . . . 115

4.3.3 Experimental Validation of MAP Control with Multiple Excitations . 120

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5 MAP Feasibility for Rotorcraft Vibration Control 133

5.1 MAP Solution for Arbitrary Primary Phasing . . . . . . . . . . . . . . . . . 134

5.1.1 MAP Solution for Multiple Primary and Single Secondary . . . . . . 134

5.1.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6
5.1.3 MAP Solution for Multiple Primary and Multiple Secondary Sources 142

5.1.4 FEM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

5.2 Power Model for a Generic Actuator . . . . . . . . . . . . . . . . . . . . . . 150

5.3 Test Rig and Plate Characterization . . . . . . . . . . . . . . . . . . . . . . . 151

5.4 Experimental Validation of MAP Control . . . . . . . . . . . . . . . . . . . . 154

5.4.1 Change in Mode Shapes with MAP Optimal Control . . . . . . . . . 156

5.4.2 Validation of New MAP Algorithm . . . . . . . . . . . . . . . . . . . 157

5.5 Rotorcraft Airframe Mockup . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.5.1 Test Rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.5.2 Comparison between experimental mode shapes and FEM Analysis . 162

5.6 Effectiveness of MAP Control . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.6.1 MAP Control with Multiple Secondary Controllers and Multiple Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6 Effect of MAP Vibration Control on Radiated Noise For a Simple Interior

Structure 178

7
6.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

7 Conclusions and Next Steps 183

Bibliography 187

A Total input power due to primary and secondary patches 192

B Total Input Power and Global Energy 195

8
List of Figures

2.1 Helicopter interior noise profile. . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Key helicopter interior noise sources, components, and structures. . . . . . . 31

2.3 Measured radiation index for aircraft skin panels. . . . . . . . . . . . . . . . 32

2.4 Schematic of the AVC architecture developed by Sikorsky. . . . . . . . . . . 33

2.5 Number of acoustic modes for a 5 ft x 6 ft x 9 ft enclosure. . . . . . . . . . . 34

2.6 Schematic of the Sikosrsky ANC architecture for helicopter interior noise reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 Comparison between (a) Conventional feedback control and (b) proposed

MAP control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.8 Schematic of the potential architecture for gear-mesh structural AVC for interior noise reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

9
3.1 (a) Schematic of a simply supported rectangular plate excited by two PZT

actuators (b) Distributed moment generated by a patch. . . . . . . . . . . . 49

3.2 Schematic of the plate system for simulation. . . . . . . . . . . . . . . . . . . 57

3.3 Vibrational energy of the plate under unit primary excitation. . . . . . . . . 59

3.4 Resonance modes for ωn=1395 Hz and ωn=1893 Hz. The primary actuator

location is shown by the black box. . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Resonance modes for ωn=1413 Hz and ωn=1861 Hz. The secondary actuator

location is shown by the black box. The primary actuator location is shown

by the dashed box. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line) for the frequency range 550-750 Hz. Two resonant

frequencies (620 Hz and 724 Hz) and one off-resonant frequency (680 Hz) are

marked by the blue triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.8 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 620 Hz. . . . . 63

3.9 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 724 Hz. . . . . 64

3.10 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 680 Hz. . . . . 65

10
3.11 Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line) for the frequency range 1500-1800 Hz. One resonant

frequency (1626 Hz) and two off-resonant frequencies (1525 Hz and 1750 Hz)

are marked by the blue triangles. . . . . . . . . . . . . . . . . . . . . . . . . 66

3.12 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1626 Hz. . . . 67

3.13 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1525 Hz. . . . 68

3.14 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1750 Hz. . . . 69

3.15 Contour plot of the optimization function Φ. Dark blue areas indicate where

the function is a minimum. The dashed black box shows the original secondary

patch location (xs, ys) = (0.075, 0.075) while the solid black box shows the

optimized location (xs, ys) = (0.125, 0.135). The primary actuator is shown

by the solid brown box (top right). . . . . . . . . . . . . . . . . . . . . . . . 70

3.16 Vibrational energy of the plate. No Control (solid); Secondary actuator at

(xs, ys) = (0.075, 0.075) (red dashed line); Secondary actuator at (xs, ys) =

(0.125, 0.135) (blue dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.17 Vibrational energy reduction as function of the distance between the primary

and secondary patch for ω = 800 Hz. . . . . . . . . . . . . . . . . . . . . . . 71

3.18 Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) for multi-frequency excitation

with ω1 = 1000 Hz and ω2 = 1500 Hz. The bottom plot shows the components

of the real power in the secondary actuator for ω1 = 1000 Hz (solid) and

ω2 = 1500 (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

11
3.19 Real electric power in the primary actuator with and without MAP control. 78

3.20 (top) Vibrational energy as a function of the secondary voltage at ωn = 620

Hz. (bottom) Real electric power in the primary and secondary actuators and

the secondary loss as a function of the secondary voltage. . . . . . . . . . . . 79

3.21 Instrumentation of MAP experimental verification. . . . . . . . . . . . . . . 81

3.22 Test apparatus of simply supported plate for map validation. . . . . . . . . . 82

3.23 The aluminum plate test apparatus with a primary actuator and a secondary

actuator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.24 Real electric power of the primary (P ̄rp) and secondary (P ̄rs) actuators, their

dielectric losses at 10 V excitation. . . . . . . . . . . . . . . . . . . . . . . . 84

3.25 Material dielectric loss of a null actuator mounted on a solid aluminum block. 87

3.26 The power loss of null actuator mounted on aluminum block as MAP control

baseline metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.27 The power loss of the on-line actuator mounted on aluminum plate as MAP

control baseline metric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.28 Actuator power at four on-resonance frequencies of the plate. . . . . . . . . . 90

3.29 Actuator power at three off-resonance frequencies of the plate. . . . . . . . . 91

3.30 MAP prediction error analysis at on-resonance and off-resonance frequencies.

Vs-opt: predicted by MAP (interception), Vs-min dB: actual corresponding to

the minimum vibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

12
3.31 The power and loss of primary and secondary actuators under two resonance

frequency excitation, 539 Hz and 812 Hz, simultaneously. . . . . . . . . . . . 92

4.1 (a) Schematic of a simply supported rectangular plate excited by PZT actuators (b) Distributed moment generated by a patch. . . . . . . . . . . . . . . 96

4.2 Schematic of the plate system for simulation. . . . . . . . . . . . . . . . . . . 103

4.3 Plate excited by two primary (red) actuators and controlled by a single secondary (blue) actuator. Total power input with and without MAP control

shown on right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4 Effect of primary actuator locations on MAP control. The secondary actuator location is optimal for suppressing resonance peaks in the 500-2000 Hz

frequency range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Effect of primary actuator locations on MAP control. As the primary actuator locations are varied, the given secondary location is not optimal for

suppressing all the resonance peaks in the 500-2000 Hz range. . . . . . . . . 106

4.6 MAP control with two secondary patch actuator. . . . . . . . . . . . . . . . 107

4.7 MAP control with three secondary patch actuators. Adding one additional

secondary actuator improved MAP performance at a given frequency while

reducing vibration at other frequencies too. . . . . . . . . . . . . . . . . . . . 107

4.8 MAP control with three secondary patch actuators with a different secondary

actuator configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.9 Total, Primary and Secondary Input Powers for the configuration shown in

Fig. 4.5. (a) ω = 826 Hz (b) ω=724 Hz. . . . . . . . . . . . . . . . . . . . . 110

13
4.10 Total, Primary and Secondary Input Powers for the configuration shown in

Fig. 4.6. (a) ω = 800 Hz (b) ω=724 Hz. . . . . . . . . . . . . . . . . . . . . 110

4.11 Intersection of secondary electrical power and loss with damping equal to 1%.

The primary voltage is Vp = 10 volts and the excitation frequency ω = 620 Hz. 111

4.12 Intersection of secondary electrical power and loss with damping equal to 8%.

(a) Vp = 10 volts and (b) Increasing values of Vp. . . . . . . . . . . . . . . . . 112

4.13 Plate Instrumented with Impedance Head (shakers not shown). . . . . . . . 114

4.14 Impedance Head Measurements of Real Mechanical Powers. . . . . . . . . . 115

4.15 Actuator electric loss characterization. . . . . . . . . . . . . . . . . . . . . . 117

4.16 Frequency and mobility phase detection for mMAP control. . . . . . . . . . . 118

4.17 Frequency and mobility phase detection for MAP control. . . . . . . . . . . . 119

4.18 The aluminum plate test apparatus with two primary and two secondary

actuators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.19 Optimum search of mMAP at 1100 Hz mode. The purple vertical dashed lines

show MAP control voltage V ∗

s (left) and the actual vibration VsdB (right). . . 123

4.20 (a) 0 degree primary phasing and (b)30 degree primary phasing. Two dashed

vertical lines indicate the minimal vibration and MAP control, respective.

Vp1 = 0.5 V, Vp2 = 0.5 V, f = 1136 Hz, S1 phase = 180o. . . . . . . . . . . . 125

4.21 (a) 90 degree primary phasing and S1 phase = 180o and (b)90 degree primary

phasing with S1 phase = 225o. Vp1 = 0.5 V, Vp2 = 0.5 V, f = 1136 Hz. . . . 126

14
4.22 Convergence of MAP control vs primary phasing. . . . . . . . . . . . . . . . 127

4.23 Frequency response function for the composite panel. . . . . . . . . . . . . . 127

4.24 Composite skin panel with two primary and two secondary piezoelectric patch

actuator/sensors mounted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.25 Electric power measurement of four actuator-sensor on the composite panel

at 10 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.26 Effect of structural damping on MAP control: (a) low damping 1% (b) composite high damping 10%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.27 MAP control on high damping panel (Vp = 5 V, center accel). . . . . . . . . 131

4.28 Effect of augmented primary excitation on MAP control Vp = 10 V, Center

Accel. Pp1 is out of range. Power scale is set to compare with Vp = 5 V case. 131

5.1 Primary source locations shown with corresponding amplitude and phase.

Location A corresponds to the optimal secondary location while location B is

not optimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Resonance case ω = 826 Hz. The Ps = 0 line is shown in dark. The contour

lines for the total input power are superimposed. (a) Secondary located at A

(b) Secondary located at B . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3 Off-resonance case ω = 850 Hz. The Ps = 0 line is shown in dark. The

contour lines for the total input power are superimposed. . . . . . . . . . . . 141

5.4 Off-resonance case ω = 800 Hz. The Ps = 0 line is shown in dark. The

contour lines for the total input power are superimposed. . . . . . . . . . . . 142

15
5.5 Simply supported plate excited by 8 primary sources with excitation frequency

equal to 800 Hz. The total power attenuation is shown as a function of the

secondary location. Location A shows the optimal location of secondary with

respect to the neighboring mode shapes. Location B is plate center and hot

spot for the mode at 826 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.6 Simply supported plate excited by 8 primary sources with excitation frequency

equal to 800 Hz. The real secondary power is shown as a function of the

secondary location. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.7 Finite element model of test rig. . . . . . . . . . . . . . . . . . . . . . . . . . 149

5.8 Rig flexural kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.9 Flexural Kinetic Energy with and without MAP control. . . . . . . . . . . . 150

5.10 Radiated sound power with and without MAP control. . . . . . . . . . . . . 151

5.11 Test rig with a shaker and a PZT sensor/actuator pair. . . . . . . . . . . . . 152

5.12 Polytec Laser Scanning Doppler Vibrometer . . . . . . . . . . . . . . . . . . 153

5.13 FRF of the plate measured by LSVD with shaker excitation. Red circle indicates the frequency selected for MAP testing . . . . . . . . . . . . . . . . . . 153

5.14 Velocity modal shapes for (a) 624 Hz (b) 999 Hz and (c) 1052 Hz . . . . . . 154

5.15 MAP control with a pair of PZT patch sensor/actuators. Red circle indicates

MAP predicted optimal control at Vs=5.94V. Purple circle indicates optimal

control at Vs=5.70V as predicted by the accelerometer. . . . . . . . . . . . . 155

16
5.16 Normalized total kinetic power of the plate around optimal MAP control.

Blue circle indicates global kinetic power of the plate at Vs = 5.60 V, and the

normalized total energy is 28.1 at no control (Vs = 0). . . . . . . . . . . . . . 156

5.17 Velocity modal shapes in mm/s for no-control condition and with optimal

MAP control (Vs = 5.5V). The low left square is the primary (excitation) and

upper right the secondary (control). . . . . . . . . . . . . . . . . . . . . . . 157

5.18 Experimental mode shape corresponding to resonant frequency 994 Hz . . . 158

5.19 Plot showing implementation of optimal phase search. The optimal secondary

phase is the phase at which the secondary phase is minimum or maximum. . 158

5.20 MAP control with secondary phase = −31o . . . . . . . . . . . . . . . . . . . 159

5.21 MAP control with secondary phase = −26o . . . . . . . . . . . . . . . . . . . 160

5.22 Test Plate Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.23 Picture of Test Rig showing (a) Front Face and (b) Back Face . . . . . . . . 162

5.24 (a) FRF of the plate measured by LSVD and shaker excitation (b) Total plate

KE per N of force, from FEA . . . . . . . . . . . . . . . . . . . . . . . . . . 163

5.25 Mode shape comparison (a) Velocity modal shape at 947 Hz measured by

LSVD and shaker excitation (b) Calculated vibration mode shape at 940 Hz 163

5.26 Mode shape comparison (a) Velocity modal shape at 850 Hz measured by

LSVD and shaker excitation (b) Calculated vibration mode shape at 850 Hz 164

5.27 Mode shape comparison (a) Velocity modal shape at 1073 Hz measured by

LSVD and shaker excitation (b) Calculated vibration mode shape at 1073 Hz 164

17
5.28 Voltage sweep showing the optimal secondary voltage at 850 Hz while control

for 1073 Hz is off. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.29 Voltage sweep showing the optimal secondary voltage at 1073 Hz while control

for 850 Hz is active. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.30 Voltage sweep showing the optimal secondary voltage at 850 Hz while control

for 1073 Hz is on. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.31 Plot showing global vibration levels with and without MAP control . . . . . 168

5.32 Global vibration reduction with MAP control. . . . . . . . . . . . . . . . . . 168

5.33 Illustration of PZT patch locations. . . . . . . . . . . . . . . . . . . . . . . . 169

5.34 Experimental Deflection Shapes for Three PZT Control . . . . . . . . . . . . 171

5.35 Average plate response with and without MAP control . . . . . . . . . . . . 172

5.36 Experimental Deflection Shapes for Multi-frequency control using a Single

PZT Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

5.37 Average plate response with and without MAP control . . . . . . . . . . . . 174

5.38 Plot showing global vibration with and without MAP control . . . . . . . . 174

5.39 Plot showing S3 baseline when 6.75 V is applied. . . . . . . . . . . . . . . . . 175

5.40 Experimental Deflection Shapes for Off-Resonance Control using a Single PZT

Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

18
6.1 Schematic of the box structure. . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.2 System frequency responses for the box structure with no control. . . . . . . 180

6.3 Response fields at 505 Hz (no control). (a) Structural displacement (b) Acoustic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.4 The total kinetic energy and SPL for S1 MAP Control . . . . . . . . . . . . 181

19
List of Tables

3.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 Geometry and material properties for plate and PZT patch. . . . . . . . . . 84

3.3 MAP control effectiveness at resonance, off-resonance and multi-frequency

excitations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.1 Simulation Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2 Geometry and material properties for plate and PZT patch. . . . . . . . . . 122

4.3 Damping versus frequency for the composite panel . . . . . . . . . . . . . . . 128

5.1 Primary Excitation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.2 Primary Excitation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.3 Estimated Control Parameters (S1, S2, and S3) . . . . . . . . . . . . . . . . 170

20
5.4 Estimated Control Parameters (S3 only) . . . . . . . . . . . . . . . . . . . . 172

5.5 Primary Excitation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 175

5.6 Estimated Control Parameters (S3 only) . . . . . . . . . . . . . . . . . . . . 175

21
Nomenclature

Variables

Lx length of the plate

Ly width of the plate

H thickness of the plate

m mass of the plate

ρ mass density of the plate

E elastic modulus of the plate

μ poison ratio for the plate

η structural damping of the plate

a width of the PZT patch

h thickness of the PZT patch

d31 piezoelectric coupling coefficient between x and z axes

d32 piezoelectric coupling coefficient between y and z axes

Y11 elastic modulus of PZT patch

ε33 complex dielectric constant of the PZT patch

ε ̄33 real part of ε33

δ dielectric loss factor

νp poison ratio for the PZT patch

Δe discretization of the patch edge

M ̄ p line moment per unit length (complex amplitude) generated by the primary patch

M ̄ s line moment per unit length (complex amplitude) generated by the secondary patch

Mp line moment (complex amplitude) generated by the primary patch element

Ms line moment (complex amplitude) generated by the secondary patch element

V ̄ p voltage (complex amplitude) applied to primary patch

V ̄ s voltage (complex amplitude) applied to secondary patch

P ̄ time average total input power

P ̄p total input power due to the primary patch

P ̄s total input power due to the secondary patch

P ̄se net electrical power in the secondary patch

22
Ppp equivalent point angular mobility for the primary patch

Pss equivalent point angular mobility for the secondary patch

Pps equivalent transfer angular mobility for the primary patch due to the secondary patch

Psp equivalent transfer angular mobility for the secondary patch due to the primary patch

Spp real part of Ppp

Sss real part of Pss

Sps real part of Pps

Ssp real part of Psp

ω excitation frequency

ωn nth resonant frequency

f point force

fk component of f with frequency ωk

v transverse velocity

vk component of v with frequency ωk

P ̄k average input power corresponding to ωk

F complex amplitude of f

R real part of linear mobility

E global energy of plate

Ekin time averaged global kinetic energy

Epot time averaged global potential energy

Eopt

kin time averaged global vibrational energy with MAP control

Ered

kin normalized reduction in vibrational energy

Ek

kin time averaged global vibrational energy corresponding to ωk

23
Subscripts

e electrical

p primary excitation

s secondary excitation

pp at primary location due to primary excitation

ss at secondary location due to secondary excitation

ps at primary location due to secondary excitation

sp at secondary location due to primary excitation

Acronyms

MAP Minimum Actuation Power

AVC Active Vibration Control

ANC Active Noise Control

ACSR Active Control of Structural Response

ASAC Active Structural Acoustic Control

PZT Piezoelectric

SOW Statement of Work

24
Chapter 1

Executive Summary

The objective of this work was to develop and demonstrate an approach for active vibration

reduction on a rotorcraft structure to reduce noise in the passenger cabin of the rotorcraft.

Vibration reduction at frequencies where gear-mesh tones and harmonics occur (500–2000

Hz) due to structure-borne excitation from the main gearbox is of interest to the rotorcraft

cabin noise community. The proposed work developed and demonstrated a control methodology that is simpler to implement than traditional control approaches and potentially more

robust and reliable for on-board implementation. Successful completion of this 3 year program will assist NASA in addressing the goals of the Quiet Cabin Technical Challenge within

the Subsonic Rotary Wing project, where methods and concepts are sought-after that can

be applied to or integrated into the structure to reduce rotorcraft cabin noise.

Conventional active vibration control (AVC)/active noise control (ANC) typically incorporates a sensor array for monitoring the vibratory or acoustic response, a controller, and

control actuators. In many cases, applications of active control are hampered by the added

system complexity, reduced reliability, and potential weight penalty. A novel approach to

AVC/ANC, Minimum Actuation Power (MAP), is demonstrated that results in control of

a single scalar objective, simpler and faster real time implementation through monitoring

and adjusting the electrical input power of the control actuator and reduced hardware for

25
distributed vibratory response sensing, as compared with the conventional approach. MAP

control features the following principle characteristics; 1) formulating the AVC process as

real power control between vibration sources and actuators, 2) suppressing vibration by minimizing the total input power as the sole objective function, and 3) eliminating the need for

error sensing by using the reactive nature of actuator for self-sensing of vibratory response.

The theory of Minimum Actuation Power (MAP) was first developed for a structure excited by a single primary (disturbance) source and controlled by a single secondary (control)

source. It was shown that the real input(mechanical) power in the secondary source goes

exactly to zero when the total input power from both the primary and secondary source is

minimized. This leads to a novel way to implement MAP control without explicit knowledge about the magnitude and location of the primary excitation. In order to validate the

theoretical findings, a model for a simply supported plate excited by a primary piezo-electric

(PZT) patch actuator and controlled by a secondary piezo-electric patch actuator was developed. An electromechanical coupling model of piezoelectric patch actuator with a structure

is developed and integrated with plate MAP model. A simple relation between the real

electrical power and the real input power of the piezo-electric actuator was developed. This

enabled implementing MAP control using the real electric power in the secondary actuator.

The effectiveness of MAP control was demonstrated for resonant and off-resonant excitation

frequencies. It was shown through simulations that optimizing the location of the secondary

actuator can help control otherwise uncontrolled modes very effectively. The MAP theory

was extended to multi-frequency excitation and simulations were presented to validate the

theory. A test setup that mimics simply supported boundary conditions was built to validate the theoretical findings. To implement MAP using electrical power, tests were done to

characterize the power consumption of the plate and the dielectric loss of the piezo-electric

patch actuators. The MAP control algorithm was tested for on-resonance and off-resonance

frequencies and effectiveness of MAP for vibration reduction. We demonstrated up to 47 dB

vibration reduction at resonance and 7 dB vibration reduction at off-resonance, as measured

by a single accelerometer at the plate center. There was good qualitative agreement between

the experimental results and theoretical findings.

The MAP control approach was extended to multiple primary excitations with zero relative

phasing and multiple secondary excitations. It was shown that when the primary excitations

have the same phasing, then the real secondary power is zero when the total input power is

26
minimized. However, when the primary excitations have arbitrary phasing, the secondary

real power is not always zero when the total input power is minimized. One way the MAP

condition is satisfied is when the secondary sources are located symmetrically with respect to

the primary sources. In general, such a symmetry constraint is difficult to satisfy for arbitrary

locations of the primary sources. An electro-mechanical model for a simply supported plate

excited by multiple primary and multiple secondary PZT patch actuators was developed.

Performance of MAP control as a function of the location and number of the secondary

sources was assessed. It was seen that increasing the number of secondary sources improves

vibration reduction. Also, multiple secondary sources can be useful when vibration reduction

over a frequency band is sought. Experimental validation of MAP control was demonstrated

in the lab with two primary and two secondary excitations using an aluminum plate as well

as a composite panel. For example, we demonstrated 20 dB vibration reduction for the

aluminum plate and 12 dB vibration reduction for the composite plate, as measured by a

single accelerometer at the plate center.

The feasibility of MAP control approach for reducing airframe vibrations due to the rotorcraft main rotor transmission and the reduction of corresponding sound levels in the passenger cabin was assessed using a more meaningful metric for vibration reduction, the global

kinetic energy of the plate structure. MAP feasibility was assessed for multiple primary

excitation sources with arbitrary relative phasing and with multiple frequencies controlled

by multiple secondary sources. In order to circumvent the problem of placing the secondary

actuators symmetrically with respect to the primary sources to satisfy the MAP condition,

it was shown that placing the the secondary sources at the intersection of the node lines

of the neighboring modes can guarantee that the secondary real power is zero at optimal

conditions even when the primary sources have arbitrary phasing. An algorithm to compute

the optimal secondary phase and magnitude was developed for resonant and off-resonant

excitation frequencies. A stable iterative solution for finding the MAP solution for multiple

secondary sources was developed. The stable iterative solution, however, is not feasible for a

large number of secondary sources. An alternative feasible approach was developed to find

the MAP solution when the secondary sources are large in number. A test rig representative of a rotorcraft interior structure was designed, modeled and built. MAP control was

successfully demonstrated with primary excitation using shaker, in addition to PZT, with

multiple frequencies and multiple secondary actuators for control. It was shown that MAP

control achieved significant vibration reduction with average global vibration reduction of

27
the order of 15-20 dB for single frequency with one secondary actuator. Also, a single secondary actuator was optimized to control multiple frequencies, in some cases up to 3 with

8-10 dB average global vibration reduction for each frequency.

A fully coupled vibro-acoustic finite element model of a larger, more complex and modally

denser box type structure was used as a quick evaluation of the potential of MAP to reduce

noise in the passenger cabin due to the main rotor transmission excitation. It was shown

that MAP control resulted in 6.1 dB reduction in the total kinetic energy of the structure

while the averaged sound pressure levels (SPL) in the acoustic cavity dropped by 13.8 dB

at resonant excitation. This preliminary analysis clearly shows the potential of MAP in

reducing interior noise in a helicopter cabin.

28
Chapter 2

Introduction

The objective of the proposed work is to demonstrate active vibration reduction for interior

noise reduction through the development of a control methodology that is simpler to implement, more robust and reliable for on-board embedment, and requires less and/or low cost

hardware. The focus is on demonstrating significant vibration reduction for mid-frequency

(500 - 2000 Hz) cabin noise resulting from the gear-box transmission of structure-borne

excitation into helicopter interiors. Active vibration/noise control for multiple tone structural noise reduction conventionally employs a sensor array for monitoring the vibratory or

acoustic response, a controller with a computationally expensive algorithm, and multiple excitation actuators. In many cases, applications of active control are hampered by the added

system complexity, reduced reliability and weight penalty, especially for rotorcraft applications where weight restrictions and reliability are stringent. The proposed active vibration

and noise technology using minimum actuation power can significantly reduce the control

hardware requirement and greatly simplifies the control algorithm by using a simple and

unified objective control function and actuation self-sensing.

29
2.1 Motivation

Interior noise is an increasingly important discriminator in the commercial helicopter market,

with acceptable noise levels traditionally being achieved passively, albeit with substantial

weight penalties. Increasing performance demands (i.e., longer range, higher payloads) have

driven the pursuit of lighter-weight and higher-performance solutions, often compromising

the acoustic comfort inside the cabin. The main sources of high interior noise levels in a

helicopter are the main rotor, tail rotor, and main gearbox. The main and tail rotors generate

low frequency airborne noise, while the main gearbox generates mostly structure-borne noise

from 500 - 2000 Hz, see Fig. 2.1. This mid-frequency range is the focus of the proposed

research.

The main gearbox dynamic excitations, generated by meshing gear pairs, are a significant

source of vibration and cabin noise in a helicopter. The gearbox excitations are transmitted

through to the fuselage via rigid connections that do not appreciably attenuate vibratory

energy. These mid to high frequency vibrations typically include discrete gear-meshing frequencies that excite the frames and skins of the helicopter. Subsequently, the frame and skins

of the fuselage transmit the vibrations produced by the main gearbox and radiate sound into

the cabin interior. Figure 2.2 illustrates the key components of helicopter interior noise

sources, propagation paths, and radiation to the interior. Over the past several years, stiffer

and lighter weight composite structures have been replacing helicopter structures previously

constructed from metal. Specifically, the aircraft skin which was traditionally aluminum

has been replaced by composite sandwich panels which have a high Young’s Modulus and

high flexural wave speeds that cause the panels to be efficient sound radiators in the frequency range of the main gearbox as shown in Fig. 2.3. For such composite panels, typical

constrained layer damping passive noise treatment are not effective, necessitating either a

different panel structure or an active control system to reduce the radiated noise.

30
Figure 2.1: Helicopter interior noise profile.

Figure 2.2: Key helicopter interior noise sources, components, and structures.

2.2 Background

Controlling vibration propagation in the airframe is assumed to lead to improvements in

passenger comfort and a better environment for on-board electronics. Active Control of

31
Figure 2.3: Measured radiation index for aircraft skin panels.

Structural Response (ACSR) is a technology developed by Westland Helicopters Ltd., and

places its actuation between two hard-points in the fuselage to produce equal and opposite

forces between these two hard points. The ACSR approach utilizes these dual-point actuators to minimize the response of the fuselage to the non-propulsion vibratory loads. With a

strut-mounted gearbox, and by adding magnetostrictive actuators, Sutton et al. reduced the

transmitted kinetic energy to the fuselage of up to 40 dB at frequencies below 1250 [1]. Recently, this control of energy transmission through the strut was revisited with an active and

periodic strut design [2] or refined control algorithms [3]. Unlike the Westland helicopters

EH101, Sikorsky helicopters, such as the S-92A and UH-60, have the main gearbox bolted

directly to the airframe instead of using struts. As a consequence, Sikorsky has developed

an Active Vibration Control (AVC) architecture that involves the use of multiple airframemounted force actuators to provide canceling vibration, an approach that was successfully

demonstrated in-flight on a Sikorsky UH-60L aircraft to cancel 4P vibration [4]. Accelerometers mounted in the cockpit and cabin provided feedback to a controller to determine the

actuator commands required to minimize vibration. A schematic of the configuration is presented in Fig. 2.4. A similar AVC system is included as a baseline on all S-92A aircraft [5]

and was selected for incorporation into the Army UH-60M, the Navy MH-60R and MH-60S

32
aircrafts [6, 7]. Finally, AVC is currently being incorporated into the S-76D helicopter and

is also one of the enabling technologies for the Sikorsky X2 Technology Demonstrator.

Figure 2.4: Schematic of the AVC architecture developed by Sikorsky.

Active Noise Control (ANC) is an adequate technology to reduce noise in the low- to midfrequency range [8, 9, 10]. Using this approach in aircraft, many studies reported significant

noise reduction, with most of the difference in performance coming from the differences in

implementation and control algorithms [11, 12, 13, 14, 15]. Sikorsky Aircraft and the United

Technologies Research Center have also been actively engaged in the development and testing

of active noise control and active vibration control systems for a number of decades. In the

early 1990s, Sikorsky embarked on a multi-year IR & D program to develop a flight-worthy

ANC system to actively cancel gear-mesh noise inside the cabin interior. The classical ANC

application with speakers as actuators proved inadequate for a typical helicopter cabin due

to the high frequencies (> 700Hz) and corresponding large number of participating acoustic

modes that would require that the number of speakers be at least equal to the number of

relevant acoustic modes. The number of acoustic modes as a function of frequency for a

cabin the size of S-76 helicopter is shown in Fig. 2.5

Sikorsky instead pursued the more practical approach of using a pseudo choke-point methodology to prevent the structure-borne gear-mesh vibratory energy from entering the cabin [16].

A schematic of this approach is presented in Fig. 2.6 and involved placing small resonant

proof-mass actuators where the main transmission mounts to the airframe, and using micro33
Figure 2.5: Number of acoustic modes for a 5 ft x 6 ft x 9 ft enclosure.

phones inside the cabin as error sensors. Two ANC flight tests were conducted on a Sikorsky

S-76B commercial helicopter [17], the first of which is the first known successful flight test

of a high frequency gear-mesh ANC system on a helicopter. The development flight tests

demonstrated the capability of the gear-mesh ANC system to achieve tonal noise reductions

of 18 dB on average and peak reduction by about 23 dB. Based on the success of these

flight tests, a follow-on project was conducted to install a production-ready ANC system on

a production S-76B helicopter with a VIP Cocoon Interior. The production demonstration

flight test was successful in providing significant reductions in the primary gear-mesh tones.

Finally, if the vibration propagation cannot be controlled efficiently or sufficiently, an alternative strategy is to prevent the panels and skin from radiating noise inside the helicopter

cabin. Active Structural Acoustic Control (ASAC) [18, 19] alters the noise radiation of the

structure through the modification of the vibration pattern. Fuller et. al. [20] mounted

piezoceramic actuators directly to an aircraft frame and with only two error microphones

inside achieved a noise reduction of over 10dB inside the aircraft fuselage. However, it

was shown that the radiated sound could be controlled without having error sensors in the

acoustic field, having instead error sensors directly on the structure and a radiation model

[21, 22]. As an implementation, the velocity feedback architecture was shown to introduce

34
Figure 2.6: Schematic of the Sikosrsky ANC architecture for helicopter interior noise reduction.

active damping by feeding the output of the sensor directly back to the piezoceramic patch

actuator via a fixed gain [23]. Furthermore, when the sensor and actuator are collocated, the

system is stable. This velocity feedback architecture allows for a decentralized configuration,

and actuators of various shapes were successfully tested [23, 24].

With passive noise treatments being too bulky or ineffective to control noise radiation from

composite panels, active control of vibration and noise is a viable and attractive alternative.

Until now, commercial deployment of active control in rotorcraft has been limited due to high

costs, stability issues and overall system complexity. The proposed active control approach,

to be described in sections to follow, addresses and avoids these limitations.

35
2.3 Program goals

The objective of this work is to develop and demonstrate an approach for active vibration

reduction on a rotorcraft structure that will ultimately reduce noise in the passenger cabin

of the rotorcraft. The active control approach to be developed and demonstrated in this

work is based on the concept of actuation power minimization. This is a decentralized

control concept, meaning all sensing, control input calculation, and actuation is done local

to a specific actuator attached to the structure. This local control approach reduces the

need for extensive wiring to connect distributed actuators and sensors and eliminates the

need for a centralized control processing unit that coordinates the efforts of many actuators.

Although the simplicity of a decentralized control approach is desirable, the approach is not

without challenges. These include ensuring that locally optimized control inputs produce a

net reduction in a relevant global performance metric, such as sound power in the passenger

cabin. A second challenge is to keep the actuator force requirements within realizable limits

at the frequencies of interest.

The three-year program is aimed at achieving the following specific overall objectives:

1. Establish a fundamental understanding and expertise of the minimal power control

methodology and demonstrate feasibility on a proof-of-concept test in Year 1.

2. Develop modeling capability and a real time control algorithm and demonstrate the

technology’s effectiveness and ease of implementation on a rotorcraft interior panel

structure in Year 2.

3. Demonstrate a technical path toward applying the methodology to the gear meshing

noise of rotorcraft main rotor gear box and analytically prove the effectiveness in Year

3.

Specific year-wise goals as stated in the contract Statement of Work (SOW) are:

36
Phase 1 Theoretical Development and Proof-of-Concept Demonstration

1.1 Develop the theory of minimal actuation power (MAP) vibration control of a single

mechanical excitation (the primary excitation) of a structure using a single control

actuator (the secondary actuator).

1.2 Develop electromechanical models of the coupling between reactive secondary actuators and a host structure.

1.3 Develop a real-time control algorithm to implement MAP for a single secondary

actuator within the frequency range of 500-2000 Hz.

1.4 Experimentally demonstrate MAP for a single primary excitation and single secondary actuator on a simple structure at multiple frequencies.

1.5 Present theoretical and experimental studies of single excitation/single control MAP

to quantify control performance using metrics that include, but not be limited to,

the reduction of a response variable, the secondary actuator force requirements, and

the total input power from the primary and secondary excitations.

Phase 2 Minimal Power Control for Multiple Actuators and Multiple Excitations

2.1 Develop a model of MAP control for multiple primary excitations and multiple

secondary actuators. This shall include consideration of system performance as a

function of the phasing between primary excitations at a given frequency.

2.2 Develop models of the electromechanical coupling of multiple reactive actuator/sensors to a host structure.

2.3 Develop system simulation models of the MAP control approach with multiple reactive actuators to enable design and optimization of a multiple-actuator MAP control

system.

2.4 Experimentally demonstrate MAP vibration control on a rotorcraft interior panel

structure with multiple primary excitations and multiple secondary actuators.

37
Phase 3 Feasibility Assessment of Applying MAP for Vibration Reduction of

Airframe Structure

3.1 Assess the feasibility of the MAP control approach for reducing airframe vibrations

due to the rotorcraft main rotor transmission and the reduction of corresponding

sound levels in the passenger cabin. This assessment shall include, but not be limited

to, estimates of system performance versus secondary actuator force and location

within the frequency range of 500-2000 Hz.

3.2 Develop a power model of a generic reactive actuator for rigid body excitation.

3.3 Experimentally demonstrate MAP control for vibration reduction on a mock-up of

a rotorcraft airframe subject to simulated main rotor transmission excitation.

3.4 Document the technical path to apply MAP control to reduce noise in the passenger

cabin due to the main rotor transmission excitation.

2.4 Proposed Active Vibration Control Approach

Our active vibration control approach aims at reducing the interior noise through active

vibration control of the helicopter’s fuselage and interior structure over the vibration frequencies that are efficient sound radiators. A unique and unconventional concept to perform

the structural active vibration and noise control features the following principal characteristics:

• Formulating AVC process as an optimal control strategy that minimizes the total

input power into the structure from all the sources of excitation that includes both the

primary and secondary sources.

• Minimizing the total input power using the input power in the secondary sources only

without any knowledge of the primary sources.

• Eliminating the need for error sensor by using the reactive nature of actuator for selfsensing of vibratory response.

38
MAP active vibration control has the following advantages:

• Simple implementation. A single scalar quantity, the real power in the secondary

source, is used to minimize the total input power.

• Global vibration reduction. The total input power is an indicator of the global vibration

level.

• Reduced hardware. Since sensing and actuation is done at the secondary sources, the

approach does not need additional sensors.

• Unconditionally stable.

A comparison with a conventional configuration is shown in Fig. 2.7. It is envisioned that

implementation into a helicopter interior would potentially consistent of active vibration

sensing and control on skins and or frames resulting in lowered measured interior noise

levels, see illustration in Fig. 2.8. Note that the architecture in Fig. 2.8 tries to reduce

interior noise levels by directly attenuating the panel vibration whereas the architecture in

Fig. 2.6 reduces interior noise by sensing the interior noise using an array of micro-phones

and using that information to suppress the vibration at the source locations.

2.5 Report Outline

In §3, we develop the theory of MAP for a structure excited by a single primary excitation

and controlled by a single secondary excitation. We develop an electro-mechanical model

of a simply supported plate excited by a single primary PZT actuator and controlled by

a single secondary PZT actuator, then extend MAP theory for multi-frequency excitation.

Simulation and experimental results are presented to validate the theoretical findings at

resonance and off-resonance conditions.

39
Figure 2.7: Comparison between (a) Conventional feedback control and (b) proposed MAP

control.

In §4, we extend the theory of MAP for a structure excited by a multiple primary excitation

and controlled by multiple secondary excitations. We develop an electro-mechanical model

of a simply supported plate excited by multiple primary and secondary PZT actuators. The

effect of relative phasing between the primary sources on MAP condition is analyzed and

the performance of MAP control as a function of the number and location of the secondary

actuators is studied. Simulation and experimental results are presented to validate the

theoretical findings at resonance and off-resonance conditions.

In §5, we assess feasibility of MAP control for rotorcraft vibration reduction with realistic

excitation. Specifically, we look at feasibility of MAP for multiple primary sources with

arbitrary relative phasing controlled by multiple secondary sources. A test-rig that mimics

40
Figure 2.8: Schematic of the potential architecture for gear-mesh structural AVC for interior

noise reduction.

rotorcraft construction was designed, modeled and built. Experimental results are presented

that validate MAP control with multi-frequency shaker excitation and controlled by multiple

secondary actuators for resonance and off-resonance excitations.

In §6, we present a FEM analysis on a simple box type interior structure and demonstrate

the potential of MAP in reducing the interior noise by reducing the structural vibration.

In §7, we summarize the key accomplishments and provide next steps.

41
Chapter 3

MAP Control for Single Primary and

Single Secondary

In this chapter, we develop the theory of MAP for a structure excited by a single primary

source and controlled by a single secondary source. In §3.1, we review the relevant literature

in the areas of structural vibration control using power minimization. In §3.2, we discuss

the general theory of input power minimization and present the key idea behind MAP. A

model of simply supported plate excited by a primary PZT actuator and controlled by a

secondary PZT actuator is developed. The MAP theory is developed for multi-frequency

excitation and a special case of commensurate frequencies is considered. Simulations results

are presented in §3.3 to validate the theoretical findings. The performance of MAP control is

evaluated at resonant and off-resonant frequencies. In §3.4, a simple relation between the net

electrical power in the secondary actuator and the net input (mechanical) power is derived

that enables implementing MAP by monitoring the net electrical power. Experimental results

are presented in §3.5

42
3.1 Literature Review

This section summarizes the published works specific to our proposed goals of implementing

MAP control strategy and using piezoelectric actuators for vibration control of a panel

structure.

The concept of power minimization for the sound radiation of multiple acoustic monopoles

was presented by Nelson et al. [25] and it was found that in many instances, there exists

a unique set of secondary sources of complex source strengths that minimizes the overall

acoustic power in the system. Elliott et al. [26] further developed the methodology by

introducing two control strategies, i.e. the minimization of total input power by primary

and secondary actuators and the maximization of secondary actuator power absorption. For

the case of single primary and single secondary acoustic sources, Johnson and Elliott [27]

demonstrated that the real power of a secondary source was zero when optimally controlling

a primary source in an enclosed field. In addition, these authors noted when the two sources

were close to each other, greater global sound reduction could be achieved. As will be shown,

the proposed MAP approach follows the same reasoning and expends on it and applies it to

vibration control.

Bardou et al. [28] applied a similar strategy of considering the total power that enters a

structure to control the flexural vibration of plates. Bardou et al. showed that minimizing

the total power in the plate yields greater overall power attenuation than what results from

maximizing the secondary power absorption, for both an infinite and finite plate. Furthermore, the greatest attenuation was achieved if the secondary sources were relatively close

to the primary source, i.e. when the two excitation sources are close with respect to the

wavelength of the flexural wave, and in the case of infinite plate, the primary and secondary

sources cannot be too distant if global control is to be achieved. Benassi and Elliott [29] also

noted the effect of distance on global control, and used idealized passive treatments with

specified impedances as secondary reactive sources to attenuate the vibratory energy in a

plate structure.

Another major task in MAP development, the methodology of actuator/sensor interaction

with the vibrating host structure and its power consumption has been broadly reviewed

43
and analyzed. Although an electromagnetic actuator, which is of reactive nature, is capable

of performing the collocated sensing and actuation in principle, its low sensitivity, large

weight and implementation issues disqualify it as the first choice for the MAP demonstration.

The piezoelectric actuator, on the other hand, is considered the most promising device for

demonstrating the principle of the MAP AVC because of its intimately close coupling to the

structure and its distributed and non-invasive actuation.

A lot of work on piezoelectric power consumption has been reported since the first publication

of the piezoelectric collocated sensor actuator by Dosch [30]. Early research of piezoelectric

power consumption as actuator consists of two distinct approaches described in Liang et

al[31, 32], Zhou et al [33] and Hagood et al [34]. These approaches examined the power

consumption of actuators when used to excite the host structure and did not consider the

case of active control. Liang developed an equation of the one-dimensional piezoelectric

actuator. The response of the actuator was determined by prescribing boundary conditions

to the actuator that coupled it with the host structure. The resulting explicit solution of

the coupled electromechanical impedance has been widely used for structural health monitoring and actuation efficiency optimization in modal control and modal analysis. The

other modeling approach developed by Hagood et al, established a relation of piezoelectric

actuator mechanical outputs with the electric voltage and current inputs to the actuator.

Both models are proven to be accurate. Another comprehensive study by Warkentin [35]

presented a development of piezoelectric power consumption for active vibration control using the analytical models developed by Hagood et al. The conclusions in these papers are

that the electrical power of piezoelectric actuators is dependent on the mechanical motion

of the structure and the electrical characteristics of the piezoelectric material and geometry.

More recent work on the actuator power modeling includes [36, 37, 38]. Brennan et al. [36]

developed an ANC model that shows that electric power of a piezoelectric actuator in active

vibration control is independent of the mechanical motion of the host structure when the

structure is under optimal control. While the work was intended to define the electric power

requirement for the amplifier, the conclusion has significant physical implications relative

to MAP for AVC. Namely, when complete control is reached, the structure is motionless,

therefore the power consumption of the piezoelectric actuator is no longer a function of the

mechanical motion but a function of the geometry, material properties of actuators, and the

voltage and frequency of the control signal (power baseline). This principle sets the stage

for the electric MAP baseline control which simply seeks a blocked boundary condition to

44
the actuator without the need for knowledge of the primary excitation. The current development will build upon that work and bridge the gap between the global vibration reduction

and minimal electric power (baseline). The spatial optimization of the actuators for modal

control is another important aspect to consider[39, 40].

3.2 Theory of Minimum Actuation Power (MAP)

In this section, we develop the theory of minimum actuation power. In §3.2.1, we develop

the theory of MAP by considering a general structure that is excited by a set of primary

sources and controlled by a set of secondary sources. We consider point forces for sake

of illustration. However, the sources can be either forces or moments or a combination of

both. We derive the optimal secondary sources that minimize the total power input into the

structure. For the case with one primary source and one secondary source, we show that

at optimal condition the real power in the secondary source is exactly zero. In §3.2.2, we

extend the theory of MAP for a simply supported plate excited by a primary PZT patch

and controlled by a secondary PZT patch.

3.2.1 Input power minimization

Conside a two-dimensional structure (e.g., simply supported plate) that is excited by N

normal point forces

f T (t)=[f1 f2 ... fN ] (3.1)

and let

vT (t)=[v1 v2 ... vN ] (3.2)

45
be the the velocity vector consisting of the set of normal velocities at the point of application

of the forces. The time averaged total input power due to all the forces is given as1

P ̄ = 1

2

Re{F HV } (3.3)

where F and V represent the complex amplitudes of f and v. The total input power can

be expressed as

P ̄ = 1

2

Re{F HMF} (3.4)

where M is the mobility matrix. Using reciprocity, M is symmetric. In this case

P ̄ = 1

2

F HRe{M}F = 1

2

F HRF (3.5)

where R is the real part of M. If the force vector consists of primary and secondary sources

FT = {FT

p FTs } (3.6)

then R can be decomposed as

R =

Rpp Rps

Rsp Rss 

(3.7)

where Rsp = RT

ps Using (4.7) and (3.7), we can express the total input power as

P ̄ = 1

2

[F H

p RppFp + F Hp RpsFs + F Hs RspFp + F Hs RssFs] (3.8)

Note that P ̄ is a quadratic function of Fs. Minimizing P ̄ with respect to Fs, we get [41]

Fopt

s = −R−1ss RspFp (3.9)

1The averaged product of two sinusoidally varying quantities represented in complex form reduces to the

real part of the product of their complex amplitudes

u1(t)u2(t) = 1

2

Re{U1U∗

2 }

46
Key idea for MAP control

Let us consider the case where there is one primary and one secondary source. In this case,

the optimal secondary source is given by

Fopt

s = −Rsp

Rss

Fp (3.10)

The real power in the secondary source is given by

P ̄s = 1

2

Re{F∗

s MspFp + F∗s MssFs} (3.11)

Using (3.10), the real power in the secondary source at optimal value is

P ̄opt

s = 1

2

Re{−Rsp

Rss

F∗

p MspFp +

R2

sp

R2

ss

F∗

p MssFp}

= 1

2

(−Rsp

Rss

F∗

p RspFp +

R2

sp

R2

ss

F∗

p RssFp}

= 0 (3.12)

We make the following important observation:

For one primary force controlled by one secondary force, the real power P ̄s in the secondary

source is zero when Fs = Fopt

s where Fopts is the optimal value that minimizes the total input

power P ̄

This suggests a way to control the structure by only tuning the power in the secondary

actuator without explicit knowledge about the primary source. The control strategy is to

drive the real power in the secondary actuator to zero. This is the underlying principle of

MAP control.

47
3.2.2 MAP theory for a simply supported plate actuated by two

piezoelectric patches

We investigate active control of the transverse vibrations of a simply supported plate using

the Minimum Actuation Power (MAP) technique. The plate is excited by a primary PZT

patch actuator and controlled by a secondary PZT patch actuator. Analytical expressions

for the total input power due to the primary and secondary patches in terms of the equivalent

patch angular mobilities are presented. The optimal value of the secondary moment that

minimizes the total input power is computed. We show that at optimal value of the secondary

moment, the total input power due the secondary actuator is zero. This key observation is

the basis of MAP control.

System description

Consider a simply supported rectangular plate that is actuated by two identical PZT patch

actuators as shown in Fig 4.1(a). Each actuator is driven by a voltage source. The primary

patch acts as the disturbance while the secondary patch is the control actuator. Let Lx, Ly

and H be the length, width and thickness of the plate respectively. The plate is assumed

to be isotropic. The PZT patches are bonded to the plate surface and are assumed to be

square with width a and thickness h. When an electric field is applied to the patch in the

z direction, the patch expands and contracts in both the x and y directions. The resulting

strain field generates a distributed moment along the edges of the patch, labeled 1 to 4, as

shown in Fig 4.1(b). We assume that the patch is isotropic. Also, the patch is assumed to

be thin and the patch inertia and stiffness are ignored in the analysis.

Total input power from primary and secondary actuators

Let M ̄ p and M ̄ s be the moments per unit length (complex amplitude) generated by the

primary and secondary patch respectively. Let the primary and secondary patch edges be

48
(a)

(b)

plate

primary 

PZT patch

secondary 

PZT patch

plate

line moment

Figure 3.1: (a) Schematic of a simply supported rectangular plate excited by two PZT

actuators (b) Distributed moment generated by a patch.

discretized into an equal number of elements and let Δe be the length of the edge element.

Then, the primary and secondary moments generated by each element are

Mp = Δe M ̄ p

Ms = Δe M ̄ s (3.13)

The total power input from the primary and secondary patches can be written as

P ̄ = P ̄p + P ̄s (3.14)

where P ̄p and P ̄s are the total power inputs from the primary and secondary patches respectively.

Let us assume that the primary and secondary patch edges are discretized into an equal

number of elements, say q. Let pi and si represent the coordinates of the center of the i

th

49
primary and secondary patch element respectively where i = 1,...,q. Let Mp and Ms be the

primary and secondary moments acting on each element respectively. Consider the point pi

on the primary patch. The angular velocity at this point is a sum of the angular velocity

induced by the primary and secondary patch moments and can be expressed as

Wpi = 

j

Wpipj +

j

Wpisj (3.15)

where Wpipj is the angular velocity induced at pi due to the moment Mp at location pj and

Wpisj is the angular velocity induced at pi due to the moment Ms at location sj . The angular

velocities Wpipj and Wpipj can be expressed as

Wpipj = PpipjMp

Wpisj = PpisjMs (3.16)

where Ppipj and Ppisj are the cross mobility terms given by [28]

Ppipj = ∞

n=1



ηωω2

n + jω(ω2n − ω2

)

 4 ψδpi n (pi) ψ

δpj

n (pj )

m [(ω2

n − ω2)2 + ω4nη2]

Ppisj = ∞

n=1



ηωω2

n + jω(ω2n − ω2

)

 4 ψδpi n (pi) ψ

δsj

n (sj )

m [(ω2

n − ω2)2 + ω4nη2] (3.17)

where η is the structural damping, m is the plate mass, ω is the excitation frequency (assumed

same for primary and secondary excitation) and ωn is the nth resonant frequency given by

ωn =



EH2

12(1 − μ2)ρ

	n1π

Lx


2

+

	n2π

Ly


2



(3.18)

where n refers to the mode index (n1, n2), E is the modulus of elasticity, μ is the Poison

ratio and ρ be the density of the plate material. The function ψδp

n (p) is given by

ψδp

n (p) = n1π

Lx

cos (δp) cos 	n1πxp

Lx




sin 	n2πyp

Ly




+

n2π

Ly

sin (δp) sin 	n1πxp

Lx




cos 	n2πyp

Ly




(3.19)

50
where (x, y) are the coordinates of the point p and δp is the angle made by the moment axis

with the x axis. Note that for the primary (or secondary) patch, δpi (or δsi )=0, π/2, π and

3π/2 for all elements i along edges 1, 2, 3 and 4 respectively. The total power input due to

the primary patch is given by

P ̄p = 1

2

Re M∗

p



i

Wpi



(3.20)

Using (A.1) and (A.2) in (A.6), the total input power due to the primary patch can be

expressed as

P ̄p = 1

2

Re M∗

pPppMp + M∗pPpsMs

 (3.21)

where Ppp represents the equivalent angular mobility due to the primary patch given by

Ppp = 

i



j

Ppipj (3.22)

and Pps represent the cross angular mobility given by

Pps = 

i



j

Ppisj (3.23)

Similarly, the total power input due to the secondary patch can be shown to be

P ̄s = 1

2

Re (M∗

s PspMp + M∗s PssMs) (3.24)

where Psp = Pps represents the cross angular mobility and

Pss = 

i



j

Psisj (3.25)

represents the equivalent angular mobility due to the secondary patch. It can be shown that

P ̄p = 1

2

Re M∗

pPppMp + M∗pPpsMs

 (3.26)

P ̄s = 1

2

Re (M∗

s PspMp + M∗s PssMs) (3.27)

51
where Ppp and Pss are the equivalent angular mobility terms due to the primary and secondary

patch respectively while Pps and Psp are the cross angular mobility terms. By reciprocity, we

have Pps = Psp. In other words, the total input power of a simply supported plate excited by

two PZT patches is equivalent to the total input power of a simply supported plate excited

by two point moments Mp and Ms with equivalent point and transfer mobilities Ppp, Pss, Pps

and Psp. The reader is referred to §B for a detailed derivation of the relations in (3.26)

and (3.27). From (4.5), (3.26) and (3.27), the total input power from both the primary and

secondary patch is given by

P ̄ = 1

2

Re M∗

pPppMp + M∗pPpsMs + M∗s PspMp + M∗s PssMs

 (3.28)

Using reciprocity of the cross mobility terms, (4.6) can be further simplified as

P ̄ = 1

2



M∗

pSppMp + M∗pSpsMs + M∗s SspMp + M∗s SssMs

 (3.29)

where Spp, Sps, Ssp and Sss are the real parts of Ppp, Pps, Psp and Pss respectively. Note that

Sps = Ssp.

Total input power minimization (MAP) control

We seek to find the optimal secondary excitation that will minimize the net input power P ̄.

Equation (3.29) is a quadratic equation of the secondary moment Ms and therefore a unique

minimum exists. Minimizing the total input power P ̄ with respect to Ms, we get the optimal

secondary moment as a function of the primary moment and the angular mobility terms

Mopt

s = −Ssp

Sss

Mp (3.30)

Since Ssp and Sss are real, (3.30) shows that Mopt

s is either in phase or out of phase with Mp

depending on the sign of −Ssp/Sss. An important implication of the above observation is that

one only needs to vary the amplitude of the secondary moment to search for the optimal value.

Equations (3.30) show that in order to compute the optimal secondary moment, we need

information about the primary excitation. From an experimental point of view, frequency

52
of the primary excitation can be easily extracted. However, extracting the magnitude of the

primary source is difficult. In that case, we cannot calculate the optimal values Mopt

s . In

what follows, we show how the net power in the secondary actuator can be used as a control

variable to implement MAP control without explicit knowledge about the strength of the

primary excitation. Using (3.30) in (3.27), the real power in the secondary source at optimal

value is given by

P ̄opt

s = 1

2

Re 	−Ssp

Sss

M∗

pPspMp +

S2

sp

S2

ss

M∗

pPssMp




= 1

2

	

−Ssp

Sss

M∗

pSspMp +

S2

sp

S2

ss

M∗

pSssMp




= 0 (3.31)

Thus, for a plate excited by a primary PZT actuator and controlled by a secondary PZT

actuator, the real power P ̄s in the secondary actuator is zero when Ms = Mopt

s where Mopts is

the optimal value of the secondary actuator moment that minimizes the total input power P ̄

from both the primary and secondary actuators. This suggests a way to control the structure

by only tuning the power in the secondary actuator without explicit knowledge about the

magnitude of the primary source. The control strategy is to drive the real power in the

secondary actuator to zero.

3.2.3 MAP control for multifrequency excitation

We show that the theory of MAP derived for single frequency excitation also holds when

you have multi-frequency excitation provided the frequencies are commensurate, i.e., integer

multiples of a fundamental frequency. In section 3.2.3, we develop the theory of MAP

for multi-frequency excitation by considering point forces acting on the structure. Though

we consider point forces for sake of illustration, the results hold for distributed moment

excitation as generated by a PZT patch. In section 3.3.7, we present simulation results for

a simply supported plate excited by a primary PZT actuator and controlled by a secondary

PZT actuator where the primary and secondary excitations are multi-frequency.

53
Let

f(t) = n

k=1

fk(t) (3.32)

be a point force excitation acting on the structure where fk is the kth component with

frequency ωk. Let us assume that ωk’s are commensurate. Using superposition, the response

at the excitation location can be expressed as

v(t) = n

k=1

vk(t) (3.33)

where vk is the response corresponding to the component fk of the excitation. The time

averaged input power due to the excitation is given by

P ̄ = f(t)v(t) = 1

T

 T

0

n

k=1

fk(t)

 n

k=1

vk(t)



dt (3.34)

where T is the period. Since the frequencies are commensurate, Parseval’s lemma [42] gives

P ̄ = n

k=1

1

T

 T

0

fk(t)vk(t) dt = n

k=1

P ̄k (3.35)

where

P ̄k = 1

T

 T

0

fk(t)vk(t) dt (3.36)

is the time-averaged input power corresponding to ωk. Equation (3.35) will hold even when

the system is excited by a primary and secondary force given by2

fp(t) = n

k=1

fpk(t)

fs(t) = n

k=1

fsk(t) (3.37)

2In general, (3.35) holds if the structure is excited by a set of primary and secondary forces with similar

harmonic components

54
where fpk and fsk are harmonic with frequency ωk. Note that in this case, P ̄k denotes the

total input power due to fpk and fsk. It follows from (3.35) that

min P ̄ ≡ {min P ̄1, min P ̄2,..., min P ̄n} (3.38)

That is, the total input power can be minimized by minimizing the input powers corresponding to each frequency. The optimal value of the secondary force that minimizes the total

input power P ̄ is given by

f opt

s (t) = n

k=1

f opt

sk (t). (3.39)

where the optimal complex amplitude Fopt

sk of f optsk (t) is given by

Fopt

sk = −R−1

ss (ωk)Rsp(ωk) Fopt

pk ∀ k (3.40)

where Fopt

pk is the complex amplitude of fpk(t). Equation (3.40) follows from the MAP results

for single frequency excitation. Thus, the MAP theory can be extended to multi-frequency

excitation.

Next, we show that minimization of total input power is equivalent to minimization of the

net vibrational energy in the case of multi-frequency excitation. For example, consider a

rectangular plate that is excited by primary and secondary forces as given by (3.37). By

superposition, the velocity at any point (x, y) on the plate can be expressed as

v(x, y, t) = n

k=1

vk(x, y, t) (3.41)

where vk(x, y, t) has frequency ωk and is a contribution of both fpk(t) and fsk(t). The global

vibrational energy of the plate is given by

Ekin(t) = 1

2

m ̄

 Lx

0

 Ly

0

v(x, y, t)

2 dx dy (3.42)

where  ̄m is the mass per unit area. The average vibrational energy is given by

Ekin = 1

2T m ̄

 Lx

0

 Ly

0

 T

0

n

k=1

vk(x, y, t)

 n

k=1

vk(x, y, t)



dt dx dy (3.43)

55
Using Parseval’s lemma gives

Ekin = n

k=1

1

2T m ̄

 Lx

0

 Ly

0

 T

0

vk(x, y, t)

2 dt dx dy = n

k=1

Ek

kin (3.44)

where

Ek

kin = 1

2T m ̄

 Lx

0

 Ly

0

 T

0

vk(x, y, t)

2 dt dx dy (3.45)

is the average vibrational energy corresponding to ωk. Using the theory of MAP for single

frequency, we have

min P ̄k ≡ min Ek

kin (3.46)

Using (3.38), (3.44) and (3.46), we get

min P ̄ ≡ min Ekin (3.47)

Equation (3.47) shows that minimizing the average input power is equivalent to minimizing

the average vibrational energy even when the excitation is multi-frequency provided that the

frequencies are commensurate.

3.3 Simulations

We present simulation results to validate the analytical findings. The effectiveness of the

control depends not only on the position of the secondary patch with respect to the mode

shapes but also whether the excitation frequency is close to a resonant frequency or away

from resonance. We present simulation results that help quantify the control effectiveness

as a function of the secondary position and the excitation frequency.

3.3.1 Simulation parameters

A schematic of plate system for simulation studies is shown in Fig. 4.2. Note that (xp, yp)

and (xs, ys) represent the coordinates of the centers of the primary and secondary patch

56
primary 

secondary 

Figure 3.2: Schematic of the plate system for simulation.

respectively. The location of the primary patch is fixed. We assume that the plate is made

of aluminum. Table 4.1 lists geometrical and material properties for the plate and PZT

patches.

3.3.2 Vibration response under primary excitation

We first study the response of the plate under primary excitation only. The secondary

actuator does not apply any control. The coordinates of the center of the primary actuator

are (xp, yp) = (0.275, 0.225). The frequency range for the excitation frequency is chosen to

be 100-2000 Hz. We assume that the voltage applied to the primary actuator is such that

it generates a unit moment at each element, that is, Mp = 1 Nm. There are 13 resonant

frequencies in this frequency range: ωn(Hz) = 620, 724, 826, 976, 1039, 1222, 1395, 1413,

1474, 1626, 1862, 1893 and 1982. Figure 3.3 shows the average vibrational energy of the

57
Table 3.1: Simulation Parameters

Variable Value

Lx 0.38 m

Ly 0.35 m

H 4.2e-3 m

E 70e9 Pa

η 0.01

ρ 2700 kg/m3

μ 0.33

a 0.05 m

h 0.13e-3 m

Y11 6.6e10 Pa

d31 -190e-12 m/V

ε ̄33 1.53e-8 F/m

νp 0.35

δ 0.02

plate when the primary excitation is applied. Interestingly, it can be seen from Fig. 3.3

that the resonances at 1395 Hz and 1893 Hz are not excited. This is due to the fact that

the location of the primary patch is close to the nodal lines of the mode shapes at 1395 Hz

and 1893 Hz. Figure 3.4 shows the modes shapes at 1395 Hz and 1893 Hz. The position of

the primary patch is also shown and it can be clearly seen that in both cases, the primary

patch is located at the intersection of two nodal lines. Therefore, the primary excitation is

not effective in exciting the modes at 1395 Hz and 1893 Hz.

3.3.3 Vibration response with MAP control

The center of the secondary actuator is located at (xs, ys) = (0.075, 0.075). Note that this

location is not optimized for any frequency range. Figure 3.5 shows the average vibrational

energy of the plate with and without MAP control. As seen, the control is most effective

close to each resonance (the greatest difference in vibrational energy between control off and

58
control on). Note that the modes at ωn = 1413 Hz and ωn = 1862 Hz are not controlled.

This is because the location of the secondary actuator is not optimal for controlling these

resonant modes. This is evident from Fig. 3.6 which shows the mode shapes for 1413 Hz

and 1862 Hz, and at both resonances the secondary actuator is on the node lines and away

from the mode shape peaks.

3.3.4 MAP control at resonant and off-resonant frequencies

The simulations show that the real part of the secondary power is zero when the total input

power is minimum at both resonant and off-resonant frequencies. The key difference is that

at resonant frequencies, the vibrational energy is also minimum when the total input power is

minimum. However, at off-resonant frequencies, the vibrational energy is not truly minimum

when the total input power is minimum (see Eq. B.11). A more detailed analysis of the

global vibrational energy and its relation to the total power input is presented in §B.

500 1000 1500 2000

10−3

10−2

10−1

100

101

Frequency (Hz)

Vibrational Energy (J)

Figure 3.3: Vibrational energy of the plate under unit primary excitation.

59
ωn

 = 1395 Hz Mode = 3,3

y

x

0 0.05 0.1 0.15 0.2 0.25 0.3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ωn = 1893 Hz Mode = 4,3

y

x

0 0.05 0.1 0.15 0.2 0.25 0.3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 3.4: Resonance modes for ωn=1395 Hz and ωn=1893 Hz. The primary actuator

location is shown by the black box.

500 1000 1500 2000 10−3

10−2

10−1

100

101

Frequency (Hz)

Vibrational Energy

1413 Hz 1862 Hz

(J)

Figure 3.5: Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line).

60
ωn

 = 1413 Hz Mode = 1,4

y

x

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

ωn = 1862 Hz Mode = 5,1

y

x

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 3.6: Resonance modes for ωn=1413 Hz and ωn=1861 Hz. The secondary actuator

location is shown by the black box. The primary actuator location is shown by the dashed

box.

For the sake of illustration, we focus on two frequency ranges: 550-750 Hz and 1500-1800

Hz. Fig 3.7 shows a zoomed-in view of the vibration response with and without control

for the frequency range 550-750 Hz. Two resonant frequencies (620 Hz and 724 Hz) and

one off-resonant frequency (680 Hz) were chosen to study the relation between vibrational

energy, total input power and real power in secondary actuator. At each frequency, we vary

the secondary moment Ms around the optimal value Mopt

s by defining

Ms = αMopt

s (3.48)

where α is a real scalar. Note that α = 0 implies zero control and α = 1 implies optimal

control.

Figures 3.8 and 3.9 show the variation of vibrational energy, total input power and real power

in secondary actuator for the resonant frequencies 620 Hz and 724 Hz respectively. As α

is varied from zero (no control), the vibrational energy and the total input power (primary

plus secondary) decrease till α = 1 and then increase. The real secondary power decreases

61
550 600 650 700 750 10−3

10−2

10−1

100

101

Frequency (Hz)

Vibrational Energy

620 Hz

680 Hz

724 Hz

(J)

Figure 3.7: Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line) for the frequency range 550-750 Hz. Two resonant frequencies (620

Hz and 724 Hz) and one off-resonant frequency (680 Hz) are marked by the blue triangles.

and crosses zero at α = 1. In other words, at α = 1, both the vibrational energy and the

net input power are minimum while the real power in the secondary is zero. On the other

hand, Fig. 3.10 shows that when the excitation frequency is away from resonance, the total

input power is minimum and the real power in the secondary actuator is zero when α = 1

but the vibrational energy is not necessarily minimum at α = 1 though it is close to the true

minimum.

Similar conclusions can be drawn when we consider the frequency range of 1500-1800 Hz.

Figure 3.11 show the vibration response with and without control for the above frequency

range. We choose one resonant frequency (1626 Hz) and two off-resonant frequencies (1525

Hz and 1750 Hz) for examining the variation of vibrational energy, total input power and

the real secondary power as α is varied.

Figures 3.12-3.14 show that at resonant frequencies, the real power in the secondary is zero

when the total input power is minimum. Also, the vibrational energy is a minimum at α = 1.

However, at off-resonant frequencies, the real power in the secondary is zero when the total

input power is minimum but the vibrational energy is not necessarily minimum at α = 1.

62
0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−4

10−2

100

102

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0

100

200

300

400

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 −100

0

100

200

300

α

Secondary Real Power (J) (W) (W)

Figure 3.8: Variation of the vibrational energy (top), the total input power (middle) and the

real power in secondary actuator (bottom) with α for ω = 620 Hz.

At off-resonance frequencies, setting the real power in the secondary actuator to zero will

not necessarily produce the greatest attenuation in vibrational energy.

63
0 0.2 0.4 0.6 0.8 1 1.2 1.4 10−3

10−2

10−1

100

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0

50

100

150

200

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 −50

0

50

100

α

Secondary Real Power (J) (W) (W)

Figure 3.9: Variation of the vibrational energy (top), the total input power (middle) and the

real power in secondary actuator (bottom) with α for ω = 724 Hz.

3.3.5 Optimal secondary location for a frequency band

Figure 3.5 shows that the secondary actuator position (xs, ys) = (0.075, 0.075) is not optimal

for the resonant modes at ωn = 1413 Hz and ωn = 1862 Hz. If the secondary actuator location

is chosen carefully, we can expect to control the above modes. We consider the frequency

band 1300-1900 Hz to demonstrate optimization of the secondary actuator position. First,

64
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

10−2.7

10−2.2

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1.5

2

2.5

3

3.5

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1

0

1

2

3

α

Secondary Real Power

(J) (W) (W)

Figure 3.10: Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 680 Hz.

we note that the peaks and valleys of the mode shape for a given resonance mode are optimal

locations for controlling that mode. Let φi(x, y) denoted the mode shape for the resonant

frequency ωi in the frequency band 1300-1900 Hz. We define the following optimization

function that is a weighted sum of the mode shapes:

Φ = −



i

βi abs(φi(x, y)) (3.49)

65
1500 1550 1600 1650 1700 1750 1800 10−2

10−1

100

101

Frequency (Hz)

Vibrational Energy

1626 Hz

1525 Hz 1750 Hz

(J)

Figure 3.11: Vibrational energy of the plate with MAP control (dashed line) and without

MAP control (solid line) for the frequency range 1500-1800 Hz. One resonant frequency

(1626 Hz) and two off-resonant frequencies (1525 Hz and 1750 Hz) are marked by the blue

triangles.

where βi is some weight. The optimal locations for controlling the resonant modes in the

frequency band are given by the minima of the function Φ. For the frequency band of interest,

that is 1300-1900 Hz, there are resonances at ωi = 1413 Hz, 1474 Hz, 1626 Hz and 1862 Hz.

Figure 3.15 shows the plot for Φ where the minima are given by the dark blue areas. For this

example, we weight the mode shapes equally, that is, βi = 1 for i = 1,..., 4. Two locations

of the secondary actuator are shown. One of the locations, shown by the dashed box, is the

non-optimal location given by (xs, ys) = (0.075, 0.075). The other location, shown by the

solid box, is chosen to be one of the four minima and is given by (xs, ys) = (0.125, 0.135).

Note that this optimal location is farthest from the primary location (xp, yp) = (0.25, 0.20).

Figure 3.16 compares the vibration response when the secondary actuator is located at

(xs, ys) = (0.075, 0.075) and (xs, ys) = (0.125, 0.135). Figure 3.16 clearly shows that by

optimizing the location of the secondary actuator, we are able to control all the modes in

the given frequency band.

66
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10−2

10−1

100

101

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

500

1000

1500

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1000

0

1000

2000

3000

α

Secondary Real Power (J) (W) (W)

Figure 3.12: Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1626 Hz.

3.3.6 MAP control as a function of separation between primary

and secondary patch

The effectiveness of MAP control not only depends upon the location of the secondary

actuator with respect to mode shapes, as we saw in the last section, but also on the distance

67
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

10−1.83

10−1.74

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 13

14

15

16

17

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −2

0

2

4

6

α

Secondary Real Power

(J)

(W) (W)

Figure 3.13: Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1525 Hz.

from the primary actuator. Let the reduction in vibrational energy be defined as:

Ered

kin = Ekin − Eopt

kin

Ekin

(3.50)

where Ered

kin is the normalized reduction in the vibrational energy. Eoptkin and Ekin are the

vibrational energies of the plate with and without MAP control respectively. Note that

Ered

kin is a measure of the effectiveness of MAP control. Let the excitation frequency be 800

68
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

10−1.79

10−1.76

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 16.4

16.6

16.8

17

17.2

17.4

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.5

0

0.5

1

1.5

2

α

Secondary Real Power (J) (W) (W)

Figure 3.14: Variation of the vibrational energy (top), the total input power (middle) and

the real power in secondary actuator (bottom) with α for ω = 1750 Hz.

Hz (off-resonant). Figure 3.17 shows the general trend of Ered

kin with distance between the

primary and secondary actuator. In general, the effectiveness of MAP control decreases as

the distance between the primary and secondary actuator increases. The control is most

effective when the actuators are collocated. However, this is not feasible in practice and the

best one can get is sub-optimal performance. Note that at nodal lines, the effectiveness of

any control is zero as seen by the dips in Fig. 3.17.

69
Figure 3.15: Contour plot of the optimization function Φ. Dark blue areas indicate where

the function is a minimum. The dashed black box shows the original secondary patch

location (xs, ys) = (0.075, 0.075) while the solid black box shows the optimized location

(xs, ys) = (0.125, 0.135). The primary actuator is shown by the solid brown box (top right).

3.3.7 Multifrequency excitation

We present results for MAP control when the primary excitation is multi-frequency. The

primary patch location is given by (xp, yp) = (0.275, 0.225) while the secondary patch is

located at (xs, ys) = (0.075, 0.075). Without loss of generality, assume that the primary

excitation has two frequencies component ω1 = 1000 Hz and ω2 = 1500 Hz. Note that ω1

and ω2 are commensurate (integer multiples of some base frequency). Let Mp1 = Mp2 = 1

where Mp1 and Mp2 are the complex amplitudes of the primary moment corresponding to ω1

and ω2 respectively. Let the components of the secondary moments be defined as a fraction

of the corresponding optimal moments as:

Ms1 = αMopt

s1

Ms2 = αMopt

s2 (3.51)

where Mopt

s1 and Mopts2 are given by (3.30) with ω replaced by ω1 and ω2 respectively. Recall

that α is a real scalar where α = 0 implies no control while α = 1 implies optimal control.

70
1300 1400 1500 1600 1700 1800 1900 10−3

10−2

10−1

100

101

Frequency (Hz)

Vibrational Energy (J)

Figure 3.16: Vibrational energy of the plate. No Control (solid); Secondary actuator at

(xs, ys) = (0.075, 0.075) (red dashed line); Secondary actuator at (xs, ys) = (0.125, 0.135)

(blue dashed line).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance between patch centers (m)

Vibrational Energy Reduction

Figure 3.17: Vibrational energy reduction as function of the distance between the primary

and secondary patch for ω = 800 Hz.

The top plot in Figure 3.18 shows the variation of the total vibrational energy E with α.

71
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 10−1.6

10−1.5

α

Vibrational Energy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 18

20

22

24

26

28

α

Total Real Power

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

2

4

6

8

10

α

Secondary Real Power

1000 Hz

1500 Hz

(J) (W) (W)

Figure 3.18: Variation of the vibrational energy (top), the total input power (middle) and the

real power in secondary actuator (bottom) for multi-frequency excitation with ω1 = 1000

Hz and ω2 = 1500 Hz. The bottom plot shows the components of the real power in the

secondary actuator for ω1 = 1000 Hz (solid) and ω2 = 1500 (dashed).

Note that E = E1 + E2. The middle plot shows the variation of the total input power

P ̄ = P ̄1 + P ̄2 while the bottom plot shows the variation of P ̄s1 and P ̄s2, components of the

real power in the secondary actuator corresponding to ω1 and ω2 respectively. The plots in

Fig. 3.18 clearly show that when the total input power is minimum, the real power in the

secondary actuator corresponding to each frequency is zero. Also, it can be seen that the

total vibrational energy is minimum when the total input power is minimum.

72
3.4 MAP Implementation with Electric Actuator

In an actual experiment, the net input power from the secondary actuator is hard to measure.

Rather, one can easily measure the electrical power by monitoring the voltage and current

in the patch. We develop the model for electromechanical coupling between the secondary

actuator and the host structure. We relate the real electrical power in the secondary actuator

to the net input power through the dielectric loss. We show how the real electrical power in

the secondary actuator can be used to implement a real-time MAP control algorithm.

3.4.1 Electromechanical coupling between the PZT actuator and

the structure

The two dimensional strain field developed in the PZT patch actuator is commonly expressed by the following constitutive equation governing the electromechanical interaction

of piezoelectric material:

εx

εy



= 1

Y11 1 −νp

νp 1

 σx

σy



+

d31

d32 

E3 (3.52)

where εx and εx are the strains in the x and y directions respectively, σx and σx are the

stresses in the x and y directions respectively, E3 is the electric field applied to the patch in

the z direction, d31 and d32 are piezoelectric coupling coefficients, Y11 is the elastic modulus

and νp is the Poisson ratio of the piezoelectric material. Equation (3.53) can be expressed

in terms of the stress field as

σx

σy



= Y11

1 − ν2

p

	1 νp

νp 1

 εx

εy



−

d31

d32 

E3




(3.53)

The electric displacement in the z direction is expressed as [43]

D3 = ε33E3 + d31σx + d32σy (3.54)

73
where ε33 is the complex dielectric constant of the piezoelectric material. Assuming the PZT

is isotropic, that is d31 = d32 , we get

D3 =

	

ε33 − 2d2

31Y11

1 − νp




E3 +

d31Y11

1 − νp

(εx + εy) (3.55)

If V is the voltage applied to the PZT patch, we have

E3 = V

h (3.56)

where h is the thickness of the PZT patch. Using (3.56) in (3.55), we get

D3 =

	

ε33 − 2d2

31Y11

1 − νp


 V

h +

d31Y11

1 − νp

(εx + εy) (3.57)

The electric current in the PZT is given by

I = jω  x2

x1

 y2

y1

D3 dx dy (3.58)

where x1, x2, y1 and y2 are the patch coordinates. Using (3.57) in (3.58), we get

I = jω 	ε33 − 2d2

31Y11

1 − νp


 a2V

h +

d31Y11

1 − νp

 x2

x1

 y2

y1

jω (εx + εy) dx dy (3.59)

where a is the width of the patch. The strain rates can be written as

jωεx = H + h

2

∂2v

∂x2

jωεy = H + h

2

∂2v

∂y2 (3.60)

where v is the transverse velocity of the plate. Using (3.60) in (3.59), we get

I = jω 	ε33 − 2d2

31Y11

1 − νp


 a2V

h +

d31Y11

1 − νp

H + h

2

 x2

x1

 y2

y1

	∂2v

∂x2 +

∂2v

∂y2




dx dy (3.61)

74
Consider the first term in the integral in (3.61):

 x2

x1

 y2

y1

∂2v

∂x2 dx dy =

 y2

y1



∂v

∂x













(x2,y)

− ∂v

∂x













(x1,y)



dy

=

 y2

y1



w(x2,y) − w(x1,y)



dy (3.62)

where w is represents the angular velocity. If the patch is discretized along the edge, then

we get the following approximation:

 y2

y1



w(x2,y) − w(x1,y)



dy = Δe



1+3

wi (3.63)

where Δe is the length of the patch element and the summation is along edges 1 and 3 of

the patch. Thus, from (3.62) and (3.63), we get

 x2

x1

 y2

y1

∂2v

∂x2 dx dy = Δe



1+3

wi (3.64)

Similarly, we can show

 x2

x1

 y2

y1

∂2v

∂y2 dx dy = Δe



2+4

wi (3.65)

Using (3.64) and (3.65) in (3.61) gives

I = jω 	ε33 − 2d2

31Y11

1 − νp


 a2V

h +

d31Y11

1 − νp

(H + h)Δe

2



1+3

wi +

2+4

wi



= jω 	ε33 − 2d2

31Y11

1 − νp


 a2V

h +

d31Y11

1 − νp

(H + h)Δe

2



pzt

wi (3.66)

where 

pzt denotes summation along all the edges of the patch. Equation (3.66) can be

expressed in terms of complex amplitudes as

 ̄I = jω 	

ε33 − 2d2

31Y11

1 − νp


 a2V ̄

h +

d31Y11

1 − νp

(H + h)Δe

2



pzt

Wi (3.67)

75
where  ̄I and V denote the complex amplitudes of the PZT current and voltage respectively  ̄

while Wi denotes the complex amplitude of the angular velocity at the i

th element along the

patch edge. The real part of the electric power is given by

P ̄e = 1

2

Re(V ̄ ∗ ̄I) (3.68)

where ∗ denotes the complex conjugate. Using (3.67) and (3.69), we get

P ̄e = 1

2

Re jω 	ε33 − 2d2

31Y11

1 − νp


 a2V ̄ ∗

V ̄

h +

d31Y11

1 − νp

(H + h)Δe

2

V ̄ ∗

pzt

Wi



(3.69)

Note that for a thin patch, the moment per unit length (complex amplitude) can be expressed

as a function of the applied voltage (complex amplitude) as

M ̄ = d31Y11

1 − νp

(H + h)

2 V (3.70)  ̄

The net moment applied to the patch element is, therefore,

M = ΔeM ̄ = Δe

d31Y11

1 − νp

(H + h)

2 V (3.71)  ̄

Using (3.71) in (3.69), we get

P ̄e = 1

2

Re jω 	ε33 − 2d2

31Y11

1 − νp


 a2V ̄ ∗

V ̄

h + M∗

pzt

Wi



. (3.72)

Let

ε33 =  ̄ε33 (1 − jδ) (3.73)

where  ̄ε33 is the real part of the complex dielectric constant and δ is the dielectric loss factor.

Using (3.73) in (3.72), we get which simplifies to

P ̄e = 1

2

ω a2

h |V ̄ |

2 ε ̄33δ +

1

2

Re M∗

pzt

Wi



(3.74)

76
Recall that the net input power (mechanical) due to the patch is given by

P ̄ = 1

2

Re M∗

pzt

Wi



(3.75)

Using (3.75) in (3.74), we get

P ̄e = 1

2

ω a2

h |V ̄ |

2 ε ̄33δ + P ̄

P ̄e = P ̄ + P ̄ (3.76)

where P ̄ is the dielectric loss. Equation (5.51) gives the relation between the net input

electrical power and net input mechanical power.

3.4.2 Real-time control algorithm to implement MAP for a single

secondary actuator

From Eq. (5.51), the real power in the secondary actuator can be expressed as

P ̄se = P ̄s + P ̄s (3.77)

where

P ̄s = 1

2

ω a2

h |V ̄ s|

2 ε ̄33δ (3.78)

is the dielectric loss, P ̄se is the net electrical power in the secondary actuator, P ̄s is the net

mechanical input power from the secondary actuator and V ̄ s is the complex amplitude of the

voltage applied to the secondary actuator. Since P ̄s = 0 at optimal control, equation (3.77)

shows that at optimal value of control, the net electrical power in the secondary actuator is

given by the following baseline value:

P ̄se = P ̄s (3.79)

Equation (4.31) shows that setting P ̄se − P ̄s = 0 (with V ̄ s non-zero) provides a novel way to

implement MAP control.

77
Next, we present simulations to illustrate the implementation of MAP control. The primary

and secondary actuators are symmetrical with respect to the center of the plate. Specifically,

(xp, yp) = (0.305, 0.275) and (xs, ys) = (0.075, 0.075). The voltage applied to the primary

actuator is Vp =10 volts. Figure 3.19 shows the real electrical power in the primary actuator,

P ̄pe, with and without MAP control as a function of frequency. In case of perfect control, that

is, P ̄p = 0, the real electric power in the primary actuator is simply the dielectric loss. The

top plot in Fig. 3.20 shows the variation of the total vibrational energy with the secondary

voltage at the resonant frequency ωn = 620 Hz while the bottom plot shows the variation of

real electric power in the primary and secondary actuators and the secondary loss. Because

of symmetry in actuator location, the optimal secondary voltage is Vs = 10 V. It can be

seen clearly that the secondary voltage is optimal at the intersection of the secondary real

electric power and the secondary loss as given by Eq. 4.31.

500 1000 1500 2000 1

1.5

2

2.5

3

3.5

4

4.5

Frequency (Hz)

Primary Electrical Power (mW) 

No Control

MAP Control

Figure 3.19: Real electric power in the primary actuator with and without MAP control.

78
0 5 10 15 −180

−170

−160

−150

−140

−130

−120

−110

−100

Vs (V)

Vibrational Energy (dB J)

0 5 10 15 −1

0

1

2

3

4

5

Vs (V)

Power (mW)

Primary Electric Power

Secondary Electric Power

Secondary Loss

Figure 3.20: (top) Vibrational energy as a function of the secondary voltage at ωn = 620 Hz.

(bottom) Real electric power in the primary and secondary actuators and the secondary loss

as a function of the secondary voltage.

3.5 Experimental Demonstration of MAP

The theoretical findings just described provide the groundwork and guide an experimental

validation. A test setup that closely mimics simply supported boundary conditions was built.

One primary and one secondary patch actuator was symmetrically bonded to the plate structure. The symmetrical location of the actuators helps validation as the optimal secondary

voltage is expected to be equal in magnitude to the primary voltage. The theoretical findings

show that at optimal control, the net input power from the secondary actuator is identically

zero. The net input power from the secondary actuator can be indirectly computed by

79
measuring the real electric power supplied to the secondary actuator and calculating the

dielectric loss. If the dielectric loss is accurately characterized, then MAP control can be

implemented by solely monitoring the real electric power supplied to the secondary actuator.

3.5.1 Overview of experimental verification

We have shown that for a structure excited by a single primary actuator and controlled by

a single secondary actuator, MAP control for vibration minimization of a single harmonic

excitation can be practically implemented by monitoring, regulating and matching the real

electric power of the secondary actuator to a predetermined value. Recall that at optimal

control, the net input mechanical power from the secondary actuator into the structure

diminishes. The net input power is related to the real electric power by the following simple

relation:

P ̄s = P ̄se − P ̄s

where P ̄s is the net input power, P ̄se is the real electric power and P ̄s is the electric loss.

Thus, at optimal control, the real electric power in the secondary actuator ideally reduces

to its electric resistive loss, or the dielectric loss for a piezoelectric actuator. The process

of MAP control then simply reduces to the measurement and regulation of electric power ,

which is far easier than that of its mechanical counterparts, the vibratory angular velocity

and bending moment. It is important to note that the dielectric loss is a known function

of the piezoelectric material properties, excitation frequency, and the electric field applied

(voltage). The dissipative power of the actuator is a quadratic function of the driving

voltage. As the secondary actuator voltage increases, both the real power consumption and

the electrical loss increase, although at different rates. Therefore, the optimal control voltage

Vs could be determined by the interception of the real time power measurement with the

electrical loss, as illustrated in Fig. 3.20. For a given primary voltage Vp = 10 volts, it shows

variation of the power as the secondary voltage is increased from 0 to 15 volts. The optimal

secondary voltage for the minimum vibrational energy is determined by the intersection of

P ̄se and P ̄sl. The experimental implementation of the MAP control for each single frequency

consists of two steps: 1) determining the actuator electrical loss as the control baseline and

2) varying the secondary voltage with respect to the primary voltage until the measurement

of the real power matches a predetermined resistive power loss. The MAP control of multiple

80
frequencies is obtained by matching the total electric power of all the sinusoidal excitations

to the sum of the electric losses for all the frequencies.

3.5.2 Test instrumentation

Figure 3.21: Instrumentation of MAP experimental verification.

The experimental process for MAP verification consists of multiple channel frequency signal generating, voltage and frequency sweep, complex power measurement, and vibration

monitoring, as shown in Fig. 3.21. The two audio power amplifiers with line transformers

are used to drive the primary and secondary actuators having a common electric grounding.

The real electric power in the secondary actuator is calculated from the secondary voltage

and the current sampled by a precision resistor on the drive circuit. The real power of each

actuator is monitored and recorded as function of drive frequency, voltage, and phase inputs.

The vibration level of the plate structure is monitored by an accelerometer mounted on the

81
plate at a location away from most of the node lines of the modes (frequencies) being analyzed. The secondary power is then compared against the baseline electric loss prediction

while the vibration level is monitored. The measurement system is programmed in Labview. The power consumption of the piezoelectric actuators as function of frequency is also

independently characterized with an HP 4194 electric impedance analyzer.

3.5.3 Structural test apparatus

The experiment test apparatus is made of a simply supported thin aluminum plate. The

plate is sandwiched between two steel plate frames, as shown in Fig. 3.22. The aluminum

plate is V-notched around its four clamped edges to emulate flexible hinges with minimal

rotational stiffness. The two steel frames are bolted together to create highly rigid mass

blocks providing a nearly clamped boundary condition to the aluminum plate edges beyond

the notch, as shown in Fig. 3.23. Two thin piezoelectric patches are bonded to the plate as

the primary and secondary actuators.

Figure 3.22: Test apparatus of simply supported plate for map validation.

82
Figure 3.23: The aluminum plate test apparatus with a primary actuator and a secondary

actuator.

The piezoelectric patch actuators are made of PZT 5H material with nickel plating on both

side as electrodes. They are directly bonded to the aluminum plate with low viscosity

adhesive Loctite 495. A uniform pressure is applied to the actuators by vacuum bag during

the adhesive curing to minimize the thickness of the adhesive layer and minimize the electric

and dielectric losses. Since the adhesive is semi-conductive, the two actuators therefore use

the aluminum plate as a common electrode to complete their drive circuit. The dimensions

and material properties for the plate and PZT patch are provided in Table 4.2.

3.5.4 Implementation of MAP control

A critical step for MAP validation is to establish the control baseline, which is to be derived

from the electric power loss of the secondary actuator. We have surveyed the actuator power

consumption characteristics in the frequency range of interest by an electric impedance

analyzer. The HP 4194 electric impedance analyzer measures complex impedance as a

function of frequency for low voltages (2.8 Vp-p). Figure 3.24 shows a frequency sweep of

83
Table 3.2: Geometry and material properties for plate and PZT patch.

Variable Value

Lx 0.38 m

Ly 0.35 m

H 4.2e-3 m

E 70e9 Pa

η 0.01

ρ 2700 kg/m3

μ 0.33

a 0.05 m

h 0.13e-3 m

Y11 6.6e10 Pa

d31 -190e-12 m/V

ε ̄33 1.53e-8 F/m

νp 0.35

the real electric power deduced from the complex impedance measurement of both actuators

mounted on the plate.

Figure 3.24: Real electric power of the primary (P ̄rp) and secondary (P ̄rs) actuators, their

dielectric losses at 10 V excitation.

84
As a result of the electromechanical coupling of the piezoelectric actuators to the plate, the

real electric power plots of both actuators P ̄p and P ̄s reveal not only the electric power loss of

the actuators themselves but also the structural dynamics and power coupling variation with

frequency. A few observations can be made by this survey. The real electric power, linearly

proportional to the mechanical power of the actuator, maps the plate vibration amplitude

well. The actuator power increases substantially when the plate is resonance and collapses

to a small baseline level when off-resonance. The actuator power off-resonance shows a

general linear dependence with frequency. Although the primary and secondary actuators

are symmetrical relative to the center of the plate and possess an identical geometry, their

power consumption differs noticeably at resonance frequencies. This implies variation in

their control authority when one is used as a primary and the other is the secondary.

Figure 3.24 also compares the electric loss of the actuators as potential control baselines.

One of them (Loss C) is predicted by an analytical model (Equation 5.51) assuming a constant piezoelectric material loss tangent and the other (Loss M) predicted by curve fitting

the power measurement P ̄rs and P ̄rp at their off-resonance frequencies, respectively. The

significant difference between these two loss estimates as control baseline candidates leads us

to believe that a more comprehensive actuator characterization is necessary in order to validate the MAP control algorithm by the power interception approach mentioned early. The

red dots indicate resonant frequencies and purple dots indicate the off-resonant frequencies

selected for MAP validation later. The results are discussed in sections to follow.

3.5.5 Actuator electric loss characterization

We considered three methods to estimate the MAP control baseline from the actuator electric

loss.

1. Analytical linear relation in Equation 5.51 taking piezoelectric material loss tangent

as constant (orange dash line in Fig. 3.24. This model, although simple and easy to

follow, lacks sufficient fidelity when used in the power interception algorithm for MAP

control. Measurement of the piezoelectric material by the electric impedance analyzer

85
reveals a variation of 250% in the loss tangent of PZT 5H piezoelectric in frequencies

between 100 Hz and 2000 Hz, in constract to the nominal constants of 0.02 published by

the material manufactures and academia as shown in Fig. 3.25. The loss tangent also

largely depends on the electric field even at the low field intensity (0.2∼2V/mil) and

varies by more than 50% when the voltage changes by one order of magnitude (1∼10V).

2. Empirically curve fitting to the power measurement of the on-line actuator (mounted

on the plate) at its off-resonance frequencies (orange solid line in Fig. 3.24). The baseline thus obtained is a direct measurement and a system calibration of the actuator on

the structure represents the true power characteristics at all frequencies.

3. Empirically curve fitting to an off-line or null actuator. The off-line actuator has the

same size as the on-line but is mounted to a solid and thick aluminum block with very

high flexural stiffness. When actuated, it induces nearly zero strain in the block so the

patch actuator essentially behaves like a pure electric capacitor with only dielectric loss.

Therefore, curve fitting to its power measurement represents a good control baseline

of the optimal control condition, that is, when the vibration completely diminishes.

The observed strong dependence on the material property and mismatch of the power in the

initial MAP testing suggests that we may not be able to use the simple dielectric loss model

for the control power estimate and would need to map out experimentally the actuator’s

power loss in the entire frequency range of interest as a function of the applied electric

field. We have constructed a null actuator, characterized its loss as a function of frequency

and voltage, for comparison to the actuator on the plate. Figures 3.26 and 3.27 show the

electric power survey of the plate-mounted actuator and null actuator, respectively. The

power is measured between resonance frequencies of the plate, from 400 Hz to 2000 Hz, for

drive voltages of 1V-10V. A large difference can be seen between the loss estimates for the

two actuators. The null actuator consumes less power since its measurement was taken at

nearly zero residual vibration and will likely represent the lower boundary of MAP control

baseline. Comparatively, the plate-mounted actuator consumes more power for the same

frequency and drive voltage, and therefore constitutes a poor estimate for control baseline

as the vibration at the optimal control never truly becomes zero.

86
3.5.6 MAP control validation

The MAP control of plate vibrations is performed using the instrument setup described

previously in Fig. 3.21. The MAP validation is done by sweeping the secondary voltage

while monitoring the plate vibration at various frequencies, both resonant and non-resonant,

between 400 Hz and 2000 Hz. The secondary voltage at which the measured P ̄s equals the

control baseline P ̄s-opt is then considered as the optimal voltage. For this study, the power

loss measured with the actuator mounted on the aluminum plate was used. For the MAP

control testing to follow, the baseline loss values at each resonance are calculated by linear

interpolation between loss values computed at two adjacent non-resonance frequencies. The

MAP control criterion was tested for validation on both single sinusoidal excitation and

dual sinusoidal excitation. For most of the frequencies selected for testing, the MAP control

criterion holds whether they are on-resonance or off-resonance. The degree of control authority demonstrated very much depends on the placement of the control actuator (secondary)

relative to the source (or primary actuator).

Figure 3.25: Material dielectric loss of a null actuator mounted on a solid aluminum block.

87
Figure 3.26: The power loss of null actuator mounted on aluminum block as MAP control

baseline metric.

Figure 3.27: The power loss of the on-line actuator mounted on aluminum plate as MAP

control baseline metric.

Single frequency: on-resonance and off-resonance

Figure 3.28 presents the results of MAP control at four of the plate’s resonance frequencies;

539, 816, 1136 and 1514 Hz, respectively, as illustrated by the red dots in Fig. 3.24. Each

88
plot shows the primary, secondary power, the control baseline and vibration in dB as function of secondary voltage Vs while the primary voltage Vp is fixed to 10 V. The interception

of the P ̄s with the control baseline is defined as the optimal control voltage Vs-opt predicted

by MAP. The secondary voltage that minimizes the plate vibration is Vs-min dB. The difference between the two voltages represents a deviation of the MAP prediction from the

measurement. The case of 1136 Hz shows nearly perfect agreement between the prediction

and the measurement. The primary and secondary power also meet at the optimal secondary

voltage, reflecting the fact that the boundary conditions and actuator placement are nearly

symmetric. The results at 539 Hz illustrate a good match between optimal and minimum

plate vibration. However, the MAP result under-predicts the optimal control voltage at both

812 Hz and 1514 Hz. While the vibration minimum occurs at 10 V and 9 V respectively, the

secondary power intercepts the baseline at 8.6 V and 8.2 V, respectively.

Figure 3.29 presents the results of MAP control at three frequencies between the plate

resonance frequencies: 580, 1050 and 1450 Hz, as illustrated by the purple dots in Fig. 3.24.

Each plot shows the primary, secondary power, the control baseline and vibration in dB as

function of secondary voltage Vs while the primary voltage Vp is fixed to 10 V. As can be

seen, the MAP optimal control voltage is a fair match to the voltage at minimum vibration

for 589 and 1050 Hz. The prediction off-resonance overshoots the target, which is in contrast

to the case on-resonance at1450 Hz, the MAP criterion fails as the secondary power does

not intercept the baseline at all.

The performance of MAP control is summarized in Fig. 3.30 which compares the measured

control voltage (intersection of real electric power with the control baseline) to the measured

minimum vibration voltage. As observed, the control error varies with frequency up to 15% of

the targeted values. The MAP prediction at resonance frequencies appears to under-estimate

the minimum vibration voltage, while the opposite happens between resonances. The overall

deviation between the MAP optimal voltage and the voltage at minimum vibration averages

at 4%.

89
(a) (b)

(c) (d)

Figure 3.28: Actuator power at four on-resonance frequencies of the plate.

Multiple frequency excitation

To validate the effectiveness of MAP control on a multiple frequency excitation, we have

simultaneously applied two frequencies, 539 Hz and 812 Hz to the primary and secondary

actuators. The analytical model of the multiple frequency excitation case shows that the total

power and total loss (and therefore the control baseline) are the sums of that of each frequency

components when the two frequencies are commensurate. Figure 3.31 shows summation of

the primary power, secondary power, the control baselines by each frequency component.

90
(a) (b)

(c)

Figure 3.29: Actuator power at three off-resonance frequencies of the plate.

Two resonances of the plate are chosen for this verification. The vibration reduction in this

case is calculated by the change in total energy at both frequencies. The primary voltage is

fixed to 5V while the secondary is swept from 1 V to 5 V for both frequencies. The MAP

optimal control voltage is Vs-opt=4.2 V for both frequencies while the vibration minimum

occurs at 4.5 V, as shown by Fig. 3.31. A total vibration reduction of 15 dB is observed at

MAP optimal voltage. Consistent with the trend seen earlier at individual frequencies, the

MAP voltage is below the minimum vibration voltage by 4.5%.

91
Figure 3.30: MAP prediction error analysis at on-resonance and off-resonance frequencies.

Vs-opt: predicted by MAP (interception), Vs-min dB: actual corresponding to the minimum

vibration.

Figure 3.31: The power and loss of primary and secondary actuators under two resonance

frequency excitation, 539 Hz and 812 Hz, simultaneously.

Table 3.3 summarizes the control effectiveness of MAP control at resonance and off-resonance

frequencies.

92
Table 3.3: MAP control effectiveness at resonance, off-resonance and multi-frequency excitations.

Mode On-Resonance Off-Resonance Dual Frequency

Freq, Hz 539 816 1136 1514 580 1050 1450 539 + 816

MAP Prediction Error % -6 -15 -2 -8 6 7 n/a -4.5

dB in reduction -33.7 -47.2 -39.1 -42.2 -1.7 -7 n/a -20

The experimental investigation of the simply supported plate with a single primary and

single secondary actuator has demonstrated the effectiveness of the proposed MAP control

algorithm for active structural vibration control. It has also identified some of the potential

challenges with the implementation. Using an empirically derived actuator electric loss

model as the control baseline, the MAP control approach produces good vibration reduction

in a broad frequency range of the structure except at one tested frequency. The difference

between accuracy of the MAP optimal voltage and the voltage corresponding to minimum

vibration varies by frequency and modal status (resonance or non-resonance) of the structure

with observed average difference below 10%. The difference between the MAP voltage and

the minimum vibration voltage depended on whether the excitation frequency was between

resonances or at a resonance.

Vibration reduction produced by the secondary actuator is very effective at structural resonance mode for both single frequency and dual frequency vibration (> 20dB). However, it is

much less effective between resonances off-resonances (> 1.7 dB). This does not necessarily

suggest that MAP control is not effective at off-resonance frequencies. The effectiveness of

MAP control depends on the location of the secondary actuator with respect to the primary

actuator. Optimizing the location of the secondary actuator can substantially improve the

performance of MAP control between resonances (see §3.3.6).

The most critical step in experimental implementation of the MAP is identified as the estimate of the control baseline with the specific piezoelectric actuator on the structure. It is

essential that the electric loss metric be based on or calibrated against an actual measurement

of the actuator being used for control.

93
3.6 Summary

We summarize this chapter by stating the key accomplishments

• The theory of MAP was developed for a structure excited by a single primary excitation

and controlled by a single secondary excitation.

• An analytical model for a simply supported plate excited and controlled by piezoelectric

actuator was developed.

• An electromechanical coupling model of a piezoelectric patch actuator on a structure

is developed and integrated with the MAP model.

• MAP AVC was demonstrated on a simply supported plate experimentally at discrete

excitation frequencies, both resonant and off-resonant.

94
Chapter 4

MAP Control for Multiple Primary

and Multiple Secondary

In chapter §3, we laid the foundations of MAP control and developed the theory for single

primary and single secondary actuator. Simulations and experimental results were presented

to validate the theory. In this chapter, we focus on extending the theory of MAP control to

multiple primary excitations and multiple secondary actuators with an emphasis on developing modeling capability and a real time control algorithm to demonstrate the technology’s

effectiveness and ease of implementation on a rotorcraft interior panel structure. In §4.1, we

develop the system model for MAP control of a simply supported plate that is excited by

multiple primary piezo-electric (PZT) patch actuators and controlled by multiple secondary

PZT patch actuators. In §4.2, we present numerical simulations for a simply supported

plate excited by two primary patches and controlled by two secondary patches and evaluate

MAP control performance as a function of the number and location of secondary patch actuators. In §4.3, we report experimental results that demonstrate MAP control of multiple

primary excitations with multiple secondary actuators and quantify MAP performance and

effectiveness in terms of vibration reduction and secondary power requirements.

95
(a)

(b)

plate

primary 

PZT patch

secondary 

PZT patch

plate

line moment

Figure 4.1: (a) Schematic of a simply supported rectangular plate excited by PZT actuators

(b) Distributed moment generated by a patch.

4.1 System Model For Theoretical Simulations

4.1.1 System description

We consider a simply supported rectangular plate that is excited by Np primary PZT patch

actuators and controlled by Ns secondary PZT patch actuators. For sake of illustration, Fig

4.1(a) shows the plate excited by one primary PZT patch and one secondary PZT patch.

Each actuator is driven by a voltage source. The primary patch acts as the disturbance while

the secondary patch is the control actuator. Let Lx, Ly and H be the length, width and

thickness of the plate respectively. The PZT patches are bonded to the plate surface and are

96
assumed to be square with width a and thickness h. When an electric field is applied to the

patch in the z direction, the patch expands and contracts in both the x and y directions. The

resulting strain field generates a distributed moment along the edges of the patch, labeled 1

to 4, as shown in Fig 4.1(b).

We make the following assumptions:

• The plate is isotropic.

• The patch is thin and the patch inertia and stiffness are ignored.

• The primary and secondary patches are identical.

• No losses are added due to the patch bond to plate

• The boundary conditions are simply supported.

4.1.2 Total input power minimization for multiple primary and

secondary patch excitation

In the Multi-Actuator Data package submitted previously this year, we developed the theory

of MAP for multiple primary sources controlled by multiple secondary sources. For simplicity,

we assumed the primary and secondary sources to be point forces. However, the experimental

setup uses PZT patches as actuators that generate line moments along the patch edges

instead of a point force. For completeness, we present the theory of MAP for multiple

primary and multiple secondary patch actuators to match the experiments. It should be

noted that the analysis and some key observations are similar to the point forces previously

reported but it is important to close the loop on the theory and experiments. The point

forces are replaced by moments and the linear mobilities replaced by equivalent angular

mobilities.

The theoretical development of input power minimization for multiple primary and multiple

secondary actuators follows closely the analysis for a single primary actuator controlled by

97
single secondary actuator. Let Mˆ pi and Mˆ si be the moments per unit length (complex

amplitude) generated by the i

th primary and ith secondary patch respectively. Let the

primary and secondary patch edges be discretized into an equal number of elements and let

Δe be the length of the edge element. Then, the moments generated by each element of the

i

th primary and ith secondary patch, respectively, are

Mpi = Δe Mˆ pi

Msi = Δe Mˆ si (4.1)

Let

MT

p = [Mp1 Mp2 ... MpNp ] (4.2)

represent the moment vector due to Np primary patches and

MT

s = [Ms1 Ms2 ... MsNs ] (4.3)

represent the moment vector due to Ns secondary patches. We assume that the primary and

the secondary patch moments are harmonic with frequency ω. Let

MT = [MT

p MTs ] (4.4)

be the total moment vector. The total power input P ̄ is simply the sum of the power inputs

from all the primary and secondary patches. That is,

P ̄ = P ̄p + P ̄s (4.5)

where P ̄p and P ̄s are the total power inputs from all the primary and secondary patches

respectively. The total input power can be expressed as

P ̄ = 1

2

Re{MHPM} (4.6)

where P is the equivalent rotational mobility matrix. The reader is referred to §B for a

detailed derivation of (4.6) and the analytical expression for P . P is symmetric due to

reciprocity. In this case

P ̄ = 1

2

MHRe{P }M = 1

2

MHSM (4.7)

98
where S is the real part of P . Decomposing into primary and secondary patch components,

we get

MT = {MT

p MTs } (4.8)

where Mp and Ms are the complex amplitudes of the primary and secondary patches respectively. Similarly, S can be decomposed as

S =

Spp Sps

Ssp Sss 

(4.9)

where Ssp = ST

ps. Using (4.7), (4.8) and (4.9), we can express the total input power as

P ̄ = 1

2

[MH

p SppMp + MHp SpsMs + MHs SspMp + MHs SssMs] (4.10)

Note that P ̄ is a quadratic function of Ms. Minimizing P ̄ with respect to Ms, we get the

optimal secondary patch moments are [41]

M∗

s = −S−1ss SspMp (4.11)

The optimal secondary patch moments depend not only on the magnitude and phase of the

primary patch moments but also on location of the primary patches with respect to the

secondary patches. In general, the primary patch moments are unknown. In what follows,

we assume that we know the location of the primary patches. However, we do not know the

magnitude and phase of the primary patch moments.

The net input power from the secondary patches is

P ̄s = 1

2

Re{MH

s PspMp + MHs PssMs} (4.12)

Since Pss is symmetric, we get

Re(MH

s PssMs) = MHs Re(Pss)Ms = MHs SssMs (4.13)

Using (4.13) in (4.12), we get

P ̄s = 1

2

Re{MH

s PspMp} +

1

2

MH

s SssMs (4.14)

We consider the following two cases.

99
Primary patch moments in-phase or 180o out-of-phase

If the primary moments are in phase (or 180o out-of-phase), then

Mp = M ̄ peiφ ⇒ MH

p = M ̄ Tp e−iφ (4.15)

The real input power from the secondary patches at optimal condition is

P ̄∗

s = 1

2

Re{−MH

p SpsS−1ss PspMp} +

1

2

MH

p SpsS−1ss SssS−1ss SspMp

= −1

2

Re{M ̄ T

p SpsS−1ss PspM ̄ p} +

1

2

M ̄ T

p SpsS−1ss SspM ̄ p

= −1

2

M ̄ T

p SpsS−1ss SspM ̄ p +

1

2

M ̄ T

p SpsS−1ss SspM ̄ p

= 0 (4.16)

Thus, the real secondary power is zero when the the primary moments are either in-phase

or 180o out-of-phase.

Primary patch moments with arbitrary phases

In this case, the net input power from the secondary patches at optimal condition is

P ̄∗

s = −1

2

Re{MH

p SpsS−1ss PspMp} +

1

2

MH

p SpsS−1ss SspMp (4.17)

Note that P ̄∗

s = 0 when SpsS−1ss Psp is symmetric. If SpsS−1ss Psp is symmetric, then we have

Re{MH

p SpsS−1ss PspMp} = MHp Re{SpsS−1ss Psp}Mp

which gives

P ̄∗

s = −1

2

MH

p Re{SpsS−1ss Psp}Mp +

1

2

MH

p SpsS−1ss SspMp (4.18)

= −1

2

MH

p SpsS−1ss SspMp +

1

2

MH

p SpsS−1ss SspMp (4.19)

= 0 (4.20)

100
Now, for SpsS−1

ss Psp to be symmetric we need

SpsS−1

ss Psp = PpsS−1ss Ssp (4.21)

Using Psp = Ssp + jXsp, we get

SpsS−1

ss (Ssp + jXsp)=(Sps + jXps)S−1ss Ssp (4.22)

which gives

SpsS−1

ss Xsp = XpsS−1ss Ssp (4.23)

Equation (4.23) holds if

Ssp = βXsp (4.24)

In order to better understand the above result, consider the case where there are two primary

patches and one secondary patch. In this case,

Psp = [Ssp1 + jXsp1 Ssp2 + jXsp2 ] (4.25)

We get

SpsS−1

ss Psp = 1

Sss 

Ssp1

Ssp2



[Ssp1 + jXsp1 Ssp2 + jXsp2 ]

= 1

Sss S2

sp1 + jSsp1Xsp1 Ssp1 Ssp2 + jSsp1Xsp2

Ssp1 Ssp2 + jSsp2Xsp1 S2

sp2 + jSsp2Xsp2



(4.26)

From (4.26), for SpsS−1

ss Msp to be symmetric, we get

Ssp1Xsp2 = Ssp2Xsp1 ⇒

Ssp1

Xsp1

= Ssp2

Xsp2

= β(say) (4.27)

Equivalently,

Ssp = 

Ssp1 Ssp2



= 

βXsp1 βXsp2



= βXsp (4.28)

101
Equation (4.28) puts a constraint on the location of the secondary patch with respect to the

primary patches in order to have the real secondary power equal to zero. It implies that

ψ11 = ψ12 (4.29)

where ψ11 = arg(Psp1 ) and ψ12 = arg(Psp2 ). That is, the phase introduced due to the

equivalent rotational cross-mobility is the same for both the primary-secondary pairs, (p1−s)

and (p2 − s).

4.1.3 Net electrical power in secondary actuator at optimal control

In an actual experiment, the net input (mechanical) power from the secondary patch actuators is hard to measure. Rather, one can easily measure the electrical power by monitoring

the voltage and current in the patches. The relation between the net electrical power and

the net input power for the i

th secondary is given

P ̄e

si = 1

2

ω a2

h |Vsi|

2 ε ̄33δ + P ̄si (4.30)

where P ̄e

si is the net electrical power in the i

th secondary patch, V ̄ si is the complex amplitude

of the voltage applied to the secondary actuator,  ̄ε33 is the real part of the complex dielectric

constant and δ is the loss factor. See §3.4 for detailed derivation of the relation between

net input power and net electrical power for a generic PZT patch. Since P ̄si = 0 at optimal

control, equation (4.30) shows that at optimal control, the net electrical power is equal to

the dielectric loss in the patch:

P ̄e

si = 1

2

ω a2

h |Vsi|

2 ε ̄33δ (4.31)

102
4.2 Numerical Simulations

Here, we present simulation results for a simply supported plate that is excited by two

primary patch actuators and controlled by one or more secondary patch actuators. The

effectiveness of MAP control depends not only on the position of the secondary patches with

respect to the primary patch actuators but also on the number of secondary patch actuators.

4.2.1 Simulation parameters

primary 

secondary 

Figure 4.2: Schematic of the plate system for simulation.

A schematic of a simply supported plate system for the simulation studies is shown in Fig.

4.2. For sake of illustration here, one primary and one secondary patch actuators are shown.

The coordinates of the i

th primary and secondary patch centers are (xpi, ypi) and (xsi, ysi)

103
respectively. We assume that the plate is made of aluminum. Table 4.1 lists geometrical and

material properties for the plate and PZT patches.

Table 4.1: Simulation Parameters.

Variable Value

Lx 0.38 m

Ly 0.35 m

H 4.2e-3 m

E 70e9 Pa

η 0.01

ρ 2700 kg/m3

μ 0.33

a 0.05 m

h 0.13e-3 m

Y11 6.6e10 Pa

d31 -190e-12 m/V

ε ̄33 1.53e-8 F/m

νp 0.35

δ 0.022

4.2.2 MAP Performance as a Function of Location of Secondary

Actuators

The primary focus of this study is to evaluate MAP performance as a function of the number and location of secondary actuators for a given set of primary actuators. We assume

that the primary patch voltages are either in phase or 180 degrees out of phase. We first

consider the plate being excited by two primary patches and controlled by a single secondary patch. Figure 4.3 shows the total power input with and without MAP control for a

particular configuration of the primary and secondary actuators. At resonance frequencies,

the total power input is proportional to the global vibrational energy. Therefore, minimizing the total power input is equivalent to minimizing the global vibrational energy. The

104
locations of the primary and secondary actuators are arbitrarily chosen. In this example,

(xp1, yp1) = (0.5Lx/4, 0.5Ly/4), (xp2, yp2) = (3.5Lx/4, 3.5Ly/4) and (xs1, ys1)=(Lx/2, Ly/2).

The primary patch voltages are Vp1 = 10 V and Vp2 = 10 V. It can be seen that while some of

the resonance peaks are attenuated, some are not. This depends on the position of the secondary actuator with respect to the primary actuators. Consider two more cases as shown in

Figures 4.4 and 4.5. In Fig. 4.4, (xp1, yp1) = (2Lx/4, 0.5Ly/4), (xp2, yp2) = (2Lx/4, 3.5Ly/4)

and (xs1, ys1)=(Lx/2, Ly/2) while in Fig. 4.5, (xp1, yp1) = (0.5Lx/4, 0.5Ly/4), (xp2, yp2) =

(3.5Lx/4, 0.5Ly/4) and (xs1, ys1)=(Lx/2, Ly/2). In both cases, Vp1 = 10 V and Vp2 = 10 V.

From Fig. 4.4, it can be seen that given the location of the primary actuators, the secondary

actuator location is optimal for suppressing all the resonance peaks in the 500-2000 Hz frequency range. However, for the primary actuator locations shown in Fig. 4.5, the location

of the secondary actuator is not optimal. Some of the resonance frequencies are attenuated

while some are uncontrolled. In general, one may need more than one secondary actuator

for improved MAP performance over a broad range of frequencies.

Figure 4.3: Plate excited by two primary (red) actuators and controlled by a single secondary

(blue) actuator. Total power input with and without MAP control shown on right.

105
Figure 4.4: Effect of primary actuator locations on MAP control. The secondary actuator

location is optimal for suppressing resonance peaks in the 500-2000 Hz frequency range

Figure 4.5: Effect of primary actuator locations on MAP control. As the primary actuator

locations are varied, the given secondary location is not optimal for suppressing all the

resonance peaks in the 500-2000 Hz range.

4.2.3 MAP Performance as a Function of Number of Secondary

Actuators

It is expected that MAP performance will improve as the number of secondary actuators is

increased. We fix the location of the primary actuators as shown in Fig. 4.5. We start with

106
Figure 4.6: MAP control with two secondary patch actuator.

Figure 4.7: MAP control with three secondary patch actuators. Adding one additional secondary actuator improved MAP performance at a given frequency while reducing vibration

at other frequencies too.

one secondary actuator and add one extra secondary actuator each time. Figures 4.5-4.7

show the configuration with one, two and three secondary actuators respectively. It can

be clearly seen from figures 4.5-4.7 that MAP performance significantly improves as more

secondary actuators are added. The resonance frequencies that were not controlled by one

secondary actuator, as shown in Fig. 4.5, are effectively controlled by adding more secondary

actuators.

107
Figure 4.8: MAP control with three secondary patch actuators with a different secondary

actuator configuration.

Given a fixed number of secondary actuators, MAP performance will depend on the location

of the secondary actuators. Optimal location of the secondary actuators may also depend

upon the location of the primary actuators and the frequencies that need to be controlled.

This can be seen from Eq. 4.11 that shows the dependence of the optimal secondary moment

on the cross angular mobility Ssp which depends on the location of the secondary actuators

with respect to the primary actuators. This is illustrated by comparing the MAP performance for two different secondary actuator configurations, as shown in Figures 4.7 and 4.8.

One of the secondary actuators is moved to a different location. It can be seen that changing

the location of one of the secondary actuators in blue made some of the resonance peaks,

otherwise controlled, uncontrollable and deteriorated MAP performance. The location of

the secondary actuators can be optimized by maximizing the sum of the mode shapes of the

frequencies to be controlled.

4.2.4 Secondary Input Power at Optimal Control

When the primary actuators are in phase or 180 degrees out of phase, the total secondary

input power is zero at optimal control regardless of location of the secondary actuators. If

the primary actuators have arbitrary phase, then the secondary location has to satisfy a

108
constraint, as given by Eq. 4.29, for the secondary power to be zero at optimal control.

Moreover, if there are more than one secondary actuators, then the input power from each

secondary actuator is zero at optimal control. This important observation is key to developing a stable iterative algorithm to find the optimal secondary voltages. Note that the

stable iterative algorithm can be developed even when the primary actuators have arbitrary

phasing. Figure 4.9 shows the input powers from the primary and secondary sources for the

actuator configuration shown in Fig. 4.5 at two different excitation frequencies ω = 826 Hz

and ω =724 Hz. It can be seen that the total secondary input power (in this case, there

is only one secondary actuator) is identically zero when the total input power is minimum.

Note that α is a scaling factor defined as α = Vsi/V ∗

si for all i where Vsi is the voltage applied

to the i

th secondary actuator and V ∗

si is the optimal value of the i

th actuator. Thus, α = 0

implies no control whereas α = 1 implies optimal control. In this case, Vp1 = Vp2 = 10 V.

V ∗

s1 = 7.058 V and V ∗s1 = 7.06 for ω = 826 Hz and ω = 724 Hz respectively.

Figure 4.10 shows the input powers from the primary and secondary sources for the actuator

configuration shown in Fig. 4.6 at two different frequencies: ω = 800 Hz (off-resonance)

and ω =724 Hz (resonance). Since there are two secondary actuators, we plot the input

powers from either actuator along with the total input power and the primary input power.

It can be seen that the secondary input power from either actuators is identically zero when

the total input power is minimum. In this case, Vp1 = 10 V and Vp2 = 5 V. (V ∗

s1, V ∗s2) =

(5.4395, 0.7616) V and (V ∗

s1, V ∗s2) = (4.0181, 1.8153) V for ω = 800 Hz and ω = 724 Hz

respectively.

4.2.5 Effect of High Damping on MAP Control

For simplicity, we consider one primary excitation controlled by one secondary patch actuator. We first consider low structural damping with η = 0.01. Figure 4.11 shows the

secondary electrical power, the secondary loss and the total input power as a function of

the secondary voltage. The primary voltage is Vp = 10 volts and the excitation frequency is

ω = 620 Hz. It can be seen that for this case of low damping, the intersection between the

secondary electrical power and loss curves is clear and easy to detect.

109
(a) (b)

Figure 4.9: Total, Primary and Secondary Input Powers for the configuration shown in Fig.

4.5. (a) ω = 826 Hz (b) ω=724 Hz.

(a) (b)

Figure 4.10: Total, Primary and Secondary Input Powers for the configuration shown in Fig.

4.6. (a) ω = 800 Hz (b) ω=724 Hz.

110
Figure 4.11: Intersection of secondary electrical power and loss with damping equal to 1%.

The primary voltage is Vp = 10 volts and the excitation frequency ω = 620 Hz.

As damping is increased without increasing the primary excitation level, the secondary electrical power approaches the loss curve. This makes determination of the intersection point

difficult. This is illustrated in Fig. 4.12(a) where the damping is artificially increased to

8 %. The primary excitation level was kept at Vp = 10 volts. In this case, the primary

source is not strong enough to induce the excitation level sensed by the secondary actuator

for implementing MAP control. However, if the primary excitation level is increased, enough

excitation will be sensed by the secondary actuator. The separation between the secondary

electrical and loss curves will increase resulting in a clean intersection. This is illustrated in

Fig. 4.12(b) where the primary excitation voltage is increased from 10 volts to 30 volts for

the same high damping value of 8%.

111
(a) (b)

Figure 4.12: Intersection of secondary electrical power and loss with damping equal to 8%.

(a) Vp = 10 volts and (b) Increasing values of Vp.

4.3 Experimental Demonstration of MAP Control

This section describes experimental validation of MAP control for multiple primary and

secondary actuators. In §4.3.1, we first describe MAP validation using impedance head

measurements to demonstrate how the secondary input power is zero when the total input

power is minimized. Impedance head measurements allow us to calculate input powers

directly. In §4.3.2, the testbed for real-time MAP implementation is described. Finally

in §4.3.3, experimental results for MAP control with two primary and two secondary PZT

actuators are presented.

4.3.1 MAP verification with Impedance Head Measurements

Minimum actuation power is verified by measuring the actual mechanical powers that are

injected into the vibrating structure by primary and secondary sources. This verification

112
step demonstrates the foundations of MAP without considering piezoelectric actuators or

how the mechanical power in the secondary sources is sensed. The mechanical power injected

into a structure is easily computed using an impedance head that measures simultaneously

acceleration (or velocity) and force according to

Π = 1

2

Re (fx ̇

∗

) = 1

2

Re 	fx∗

jω 


(4.32)

The impedance head at both primary and secondary sources allows for the estimation of the

total mechanical power injected into the structure by simple summation of the individual

mechanical powers at each source. This assumes most if not all the mechanical power is

contained in flexural waves as measured by the impedance heads.

Impedance Head Experimental Setup

Verification of MAP using impedance head was performed on a freely suspended aluminum

plate. The plates dimensions were 15.1 inches x 14.9 inches x 0.125 inches. Impedance

heads (PCB model 288D01) were mounted onto the structure at both primary and secondary sources. A picture of the test setup is presented in Fig. 4.13. Free-floating shakers

injected mechanical power through the impedance heads, and normal to the plate surface.

A LabVIEW code measured acceleration and force at all impedance heads and calculated

the respective mechanical power. Single frequency excitations were generated and fed to the

sources. With a set amplitude and phase of the primary source(s), the phase and amplitude

of the secondary source were varied until MAP was achieved. Total mechanical power in the

structure was calculated by simple summation of the individual powers of all sources.

113
Figure 4.13: Plate Instrumented with Impedance Head (shakers not shown).

Results

The measured mechanical power for a single primary and single secondary sources when the

secondary phase and amplitude are varied are presented in Fig. 4.14. The primary excitation

was at 843Hz, a plate resonant frequency. Locations of the primary and secondary sources

ensured both sources coupled well with this structural mode. The global structural power is

also presented. Due to the aluminum plates very low damping, the phase of the secondary

source is approximately either 0 or 180 degrees out of phase compared to the primary source.

In the figure, phase information is contained in the sign of the amplitude voltage, with a

negative value representing a 180 phase angle. For this particular source configuration, at

MAP, the secondary has is 180 degrees out of phase compared to the primary. As expected,

as gain is increased (becomes more negative in this case), the secondary’s real power becomes

negative before increasing again, becomes 0 where the global power is minimized, and then

becomes positive where the total power in the structure increases. This verifies analytical

calculations as presented in §4.1.2.

114
Figure 4.14: Impedance Head Measurements of Real Mechanical Powers.

While impedance heads measured point force and acceleration,the theory also holds for

moment excitations. This test setup provides a platform for the simultaneous development of

MAP using piezoelectric actuators, a methodology that uses electrical quantities to estimate

mechanical power without any impedance heads, and the development of control strategies

with multiple primary and secondary sources and using MAP as a control objective.

4.3.2 MAP Testbed and Real-Time Implementation

We have developed control software and an instrument system that enables semiautomatic

MAP implementation and allows for testing of various primary and secondary excitation and

sensing. In order to implement MAP control to real structures without knowledge of the

primary excitation and to facilitate more efficient tests with different configurations, it was

considered important to establish a flexible hardware setup and to develop comprehensive

software modules that automate the entire procedure. The control software and test system

was upgraded to control four actuators simultaneously. Sections 4.3.2 and 4.3.2 contain de115
tailed descriptiosn of two important software modules, a control objective function estimator

and a frequency detector. In §4.3.2, integration of the newly developed software modules

with the main MAP control software is described in detail.

Determination of Actuator Aggregated Loss

A software module was created to estimate aggregated loss of the secondary actuator including dielectric loss of the PZT and loss due to the residual vibration of the structure left

from the control. The software module is shown in Fig. 4.15. This module was developed in

LabVIEW and allows users to select voltage and frequency ranges before actuator excitation.

The software has two loops, one for voltage sweep and the other for frequency sweep. First,

the excitation voltage is fixed to a starting voltage, and the frequency is swept within the

specified range. As frequency increases, the real power at each frequency is calculated directly from voltage at a power amplifier and current at a resister connected to the actuator.

After this step, voltage is increased and the frequency sweep is repeated. Finally, a raw

power loss map is generated and stored as an array variable and sent to a two dimensional

regression module for the aggregated loss estimation.

The two dimensional regression module provides curve fitting to the power measurement of

the actuator. The embedded algorithm searches for local minima in the measurement which

are induced by off-resonance frequencies. Then, interpolation between the fitting points is

performed to create a complete control objective function with respect to frequency and

voltage. A heavy dependence of material properties and power mismatch were observed

in the initial MAP testing. This stimulated the need for mapping out actuator power loss

experimentally over the entire range of frequencies and electric field instead of using a simple

dielectric loss model. The 2D map is stored as a spread sheet and read when the MAP control

is performed.

116
Figure 4.15: Actuator electric loss characterization.

Detection of Frequency and Structural Mobility Phase

A software module was created to use the secondary actuator (control actuator) as a passive

sensor to determine frequencies to suppress. In the experimental setup, a primary function

generator creates a continuous sinusoidal waveform function with respect to fp, Vp, and φp

specified by a user. Then, the current through a resistor connected to the PZT is read by

a digitizer. A built-in LabVIEW function then computes the Fast Fourier Transform of the

current signal to determine dominant frequency components in the signal. The FFT and

peak frequency detection thus allow the primary frequency fp to be determined without

using primary actuator information. The frequency of the secondary actuator, fs, is then

set equal to fp.

The cross mobility phase between the primary and secondary actuators is determined by

comparing the phase between Vp and Is. This is done in the off-line plate characterization

prior to the MAP control. It can also be extracted from plate’s analytical models. The

117
Figure 4.16: Frequency and mobility phase detection for mMAP control.

mobility phase detection technique is empirical and relies on experiments. Note that the

cross-correlation function in Fig. 4.16 is also a built-in LabVIEW function that provides the

phase at a select frequency between two time domain signals. When piezoelectric transducers

are used, the phase of Is can be changed by π/2 due to capacitance of the transducers. In

this case, an actuator phase correction is needed. fs and mobility between the primary and

the secondary sources are the output of this module to be used for the final MAP control.

MAP control independent of the primary excitation

The experimental process for MAP verification consists of multiple channel signal generation, acquisition and vibration monitoring as shown in Fig. 4.17. The updates to the test

setup include: 1) hardware update to enable multi-actuator excitation and sensing, and 2)

the development of a software module that incorporates objective function acquisition and

frequency detection to real-time MAP control. The overall measurement system has been

developed based on LabVIEW 8.6 software that controls an NI PXI chassis. The chassis contains analog to digital converters and waveform generators connected to four audio power

amplifiers with line transformers. The real power of each actuator is monitored and recorded

118
Figure 4.17: Frequency and mobility phase detection for MAP control.

as a function of drive frequency, voltage, and phase with respect to phase reference defined

by the primary actuator. The real electric power in the secondary actuator is calculated from

the secondary voltage and the current sampled by a precision resistor on the drive circuit.

The vibration level of the plate structure is monitored simultaneously by an accelerometer

mounted on the plate at a location away from most of the node lines of the modes being

analyzed (see Fig. 4.18).

Figure 4.17 illustrates how a secondary actuator is controlled to track the optimal control

frequency, phase, and voltage. The objective function for the secondary actuator is the sum

of electrical loss and mechanical loss (aggregated loss) as a function of frequency and voltage

and is predetermined. A frequency detector analyzes signals measured by the secondary

actuator and determines frequency contents of the disturbance in the structure fs. Then, the

software determines optimal phase (φs) for the secondary actuator. The optimal secondary

119
voltage is determined at two different secondary phases, ψs and ψs +90 degrees. The ratio of

the two voltage values is used to estimate φs. Once φs is determined, the secondary voltage

is swept again to find the optimal control voltage. The developed software is designed to

conduct frequency, phase and optimal control voltage estimation with a few clicks by users

in pursuit of fully automated real-time MAP control.

4.3.3 Experimental Validation of MAP Control with Multiple Excitations

The principle of MAP for single primary (excitation) and single secondary (control) has been

experimentally validated and demonstrated via piezoelectric collocated sensor and actuator

on a simple metal plate. This year, we have extended MAP control theory to multiple

primary excitations and multiple secondary actuators (mMAP) and validated it through

the enhanced control software and hardware platform with emphasis on the phase control

and optimization. We have also initiated MAP demonstration on a more realistic helicopter

structure and with high damping.

Theoretical investigations conducted early in Year 2 identified that: 1) principle of MAP is

valid for mMAP and vibration suppression is equally effective if the primary and secondary

actuation configuration can meet certain symmetry conditions, 2) MAP control implementation algorithm through gain control (magnitude optimization) on single excitation is also

applicable to the multiple excitation, and 3) for a more realistic excitation where arbitrary

phasing of the primaries is dominant, the secondary phase optimization becomes an essential

and necessary part of mMAP control.

Description of mMAP validation

The experimental setup described in $ 3.5 was used for the mMAP investigation and develop fundamental understanding. To accommodate the mMAP test, two piezoelectric patch

actuators are added and the four actuators are now symmetrically located on four corners

120
Figure 4.18: The aluminum plate test apparatus with two primary and two secondary actuators.

respectively to meet the nominal symmetry requirement of mMAP control. This placement

also facilities flexibility in pairing actuators to form various symmetric and asymmetric configurations, see Fig. 4.18. Also depicted in this figure is the accelerometer used to sense

local vibration. The two new actuators (P2 and S2) are bonded to the aluminum plate with

direct electrodes in back as the common grounding to minimize the impact of the adhesive

between the actuators and the plate on the actuator aggregated power loss. This is critical

to determining optimal control of the secondary actuators. The dimensions and material

properties of the plate and PZT patch are provided in Table 4.2.

The process of mMAP control (with gain only) is similar to the single MAP except that only

one variable is swept at a time while the others are kept constant in magnitude and phase.

The control objective is the actuator aggregated loss for the two control actuators. These

losses are predetermined in the form of a 2D matrix of power loss as a function of frequency

and voltage.

In searching for the optimal control voltage of the secondary actuators, the two primary

actuators excite the plate with a constant voltage Vp at select frequencies. The voltage of

the first secondary actuator, Vs1, is swept at a constant frequency while comparing its power

121
Table 4.2: Geometry and material properties for plate and PZT patch.

Variable Value

Lx 0.38 m

Ly 0.35 m

H 4.2e-3 m

E 70e9 Pa

η 0.01

ρ 2700 kg/m3

μ 0.33

a 0.05 m

h 0.13e-3 m

Y11 6.6e10 Pa

d31 -190e-12 m/V

ε ̄33 1.53e-8 F/m

νp 0.35

to its actuator aggregated loss until Ps1, the power in secondary actuator, equals the loss

power, at which point vibration should be a minimum. The corresponding voltage V ∗

s1 is

the estimated optimal MAP control voltage for the first control actuator. The next step

is to excite this secondary actuator at a constant V ∗

s1 while sweeping the voltage input to

the second control actuator voltage Vs2 until Ps2 equals its aggregated loss at Vs2=V ∗

s2. The

vibration of the plate is monitored by an accelerometer in the center of the plate. Applying

V ∗

s1 and V ∗s2 to the control actuators S1 and S2, respectively, further reduces the vibration

to a new minimum. A few iterations between the two control actuators may be required to

converge the control inputs to their global optimal values and further reduce the vibration.

Multiple Primary Excitation with Simple Phases

As the first example, we demonstrate performance of MAP with two primary excitations

and two secondary controls on the lightly damped aluminum plate. Figure 4.19 shows the

optimum search of the second control actuator with both primary actuators P1 and P2 on

122
and the first control actuator S1 excited at a fixed V ∗

s2. With both control actuators at the

optimal control, the vibration is reduced by 20 dB.

Figure 4.19: Optimum search of mMAP at 1100 Hz mode. The purple vertical dashed lines

show MAP control voltage V ∗

s (left) and the actual vibration VsdB (right).

In this test, MAP control criterion meets the actual vibration measurement with reasonable

agreement, that is, the minimal vibration occurs near the secondary voltage value where

secondary power equals the actuator aggregated loss (intersection) and the net power input

to the plate is minimum. The phase relation of the excitations is kept very simple. Both

secondary actuators are set to 0o and the phase of the primaries are 180o. As predicted by

the model for low damping and single excitation, mMAP control has demonstrated small

sensitivity to the phase of secondary actuation

Multi Excitation with Primary Arbitrary Phases

Unique and distinctive characteristics of mMAP control are fully revealed when arbitrary

phasing is imposed on the primary sources. The determination of the optional control now

relies on optimizing both the magnitude and phase. The phenomenon of phase dependency

123
is fully observed in the experimental results shown in Figures 4.20–4.21.

The aluminum plate is excited by both primaries and only one secondary performs the control

to illustrate the complex phasing challenge. The MAP optimal control is again indicated

by the intersection of the secondary power to its aggregated power loss (zero net power)

and the total input power is indicated by the sum of all actuator power. The input to the

secondary actuator is swept while the inputs to the two primaries are kept constant. The

vibration level is monitored by the accelerometer at the plate center. Several observations

can be made from these test results.

Figure 4.20(a) shows that the mMAP computes the optimal control voltage less accurately

with two secondary actuators than with a single secondary actuator. In Figure 4.20(a), V ∗

s

nearly equals VsdB while in Figure 4.19, these two voltages are slightly separated, although

in both cases the primary phasings are simply set to zero. Figures 4.20(a)–(b) and 4.21(a)

show the radical change of the MAP optimal control with respect to the minimal vibration

measurement and the total power input as the primary phasing varies from 0o to 90o and the

secondary phase is fixed to 180o. In addition, the vibration level dip near the optimal control

is very sharp for zero primary phasing. It gradually flattens out as the primary phasing shifts

to 90o. Similarly, the alignment of minimal input power to the minimal vibration also shifts

but at a milder rate.

It is interesting to note that the optimal control V ∗

s and the minimal vibration Vsdb flips back

to a near perfect alignment if 45o additional phase is imposed to the secondary actuator, as

seen in Fig. 4.21(b). Meanwhile, the vibration measurement dip also returns to this original

sharpness as the secondary phase is optimal. Note the optimal phase 45o is half the primary

phasing of 90o.

The phase dependency of mMAP control is quantitatively summarized in Fig. 4.22. We use

the ratio of the actual minimal vibration voltage Vsdb over the MAP control voltage V ∗

s as

an indicator of the MAP convergence and plot it as a function of the primary phasing, with

unity indicating perfect agreement between the two voltages. The indicator climbs to 2.5

when the primaries are out of phase by 90o. Setting the phase to 45o brings the convergence

indicator back to 1. These results illustrate that for mMAP control with arbitrary phasing,

the secondary phase must be controlled and optimized while for near zero primary phasing,

124
(a) (b)

Figure 4.20: (a) 0 degree primary phasing and (b)30 degree primary phasing. Two dashed

vertical lines indicate the minimal vibration and MAP control, respective. Vp1 = 0.5 V,

Vp2 = 0.5 V, f = 1136 Hz, S1 phase = 180o.

MAP control can be performed with the secondary phase neglected without a minimal control

error.

125
(a) (b)

Figure 4.21: (a) 90 degree primary phasing and S1 phase = 180o and (b)90 degree primary

phasing with S1 phase = 225o. Vp1 = 0.5 V, Vp2 = 0.5 V, f = 1136 Hz.

MAP Demonstration on Composite Structure

A composite test rig was instrumented for multiple excitation control and preliminary tests

preparation for the real time mMAP demonstration on a more realistic helicopter structure.

The composite panel, a mock up of a helicopter fuselage skin, consists of two bays and single

plate base panel. The composite panel consists of a panel skin with 6 plies of graphite for

a total thickness of 0.045 inches and a Kevlar Core 0.75 inches thick and the weight of the

panel is 14.48 lbs. The frequency response function, determined using a tap test, is used to

126
Figure 4.22: Convergence of MAP control vs primary phasing.

Figure 4.23: Frequency response function for the composite panel.

extract the damping values as shown in Fig. 4.23.

The damping value for different frequencies for the composite panel is given in Table 4.3.

Structural damping was extracted using a multi-degree of freedom modal curve fitting technique within the IDEAS software.

Four piezoelectric patch actuator/sensors are mounted on symmetrically about the center of

the panel as primary and secondary actuators as shown in Fig. 4.24. The panel is hung from

127
Table 4.3: Damping versus frequency for the composite panel

F req(Hz) 293 539 874 1068 1473

η% 3.0 2.6 4.0 8.9 4.8

Figure 4.24: Composite skin panel with two primary and two secondary piezoelectric patch

actuator/sensors mounted.

two bungee cords to simulate a free-free boundary conditions. Actuators are mounted on

the panel using an improved electroding method in which the copper foil is directly soldered

to the PZT electrode. The actuators are driven by audio amplifiers.

Compared to the aluminum panel tested previously, two distinctive power consumption

features are observed. First, the composite panel shows heavier structural damping as high

as 10% so that the dynamic range of the electric power (on resonance vs off resonance) of

each actuator around a resonance is significantly reduced (much smaller resonance power

peaks). Second, the off-resonance power changes by a factor of four from 400 Hz to 1000

Hz, as seen in Fig. 4.25. While both panel damping and the piezoelectric dielectric loss

are responsible for this frequency dependence, the former is believed to be dominant as the

piezoelectric loss varies by less than 10% (as determined experimentally) in this frequency

range. The power spectrum also shows there is very little identifiable modal activity beyond

1000 Hz due to heavy damping.

128
Figure 4.25: Electric power measurement of four actuator-sensor on the composite panel at

10 V.

For MAP control, the high structural damping constitutes a major challenge. Effective sensing and control of vibration by the secondary actuators relies on the structure’s transfer

mobility which is a function of material damping and structural configuration. The high

damping weakens the communication between the primary and secondary. Figure 4.26 analytically shows the effect of the high structural damping on the MAP control. As the damping

increases, the secondary actuator power approaches its aggregated loss in shape and value

and the intersection of them becomes indistinct. This could result in a large uncertainty

in experimental determination of the MAP control point when both the control circuit and

aggregated loss have a limited signal to noise ratio.

Nonetheless, to validate the controllability on a high damping structure, we have performed

a preliminary MAP test on the composite panel using the single primary and single secondary configuration. Figure 4.27 shows that the vibration reduction is comparable to the

low damping case (-12 dB), however, the result agrees with the simulation results. The

aggregated loss and secondary power curves run nearly parallel to each other in the figure

and the experimental noise creates multiple intersections between these curves. Therefore,

detecting the optimal control voltage becomes a challenge.

129
(a) (b)

Figure 4.26: Effect of structural damping on MAP control: (a) low damping 1% (b) composite high damping 10%.

To enhance MAP implementation, two techniques are being sought. One is to increase the

primary excitation strength and to force more communication between the primary and

secondary. The second technique is to aggressively filter the aggregated loss in order to

obtain clean intersections. Figure 4.28 shows an effort to enhance the MAP determination

with heavy damping loss. When the primary excitation is doubled, the secondary power Ps2

is now more negative and the two power curves are more separated compared to the Vp=5

case. However, because of the instrument voltage range limitation, the new intersection

between the secondary power and secondary loss cannot be observed. We anticipate that

the two curves intersect near 15 V.

130
Figure 4.27: MAP control on high damping panel (Vp = 5 V, center accel).

Figure 4.28: Effect of augmented primary excitation on MAP control Vp = 10 V, Center

Accel. Pp1 is out of range. Power scale is set to compare with Vp = 5 V case.

4.4 Summary

We summarize the key accomplishments presented in this chapter:

• The theory of MAP was extended for multiple primary excitations controlled by multiple secondary excitations.

131
• The effect of primary phasing on MAP control was investigated. It was shown that

when the primary sources have zero relative phasing, the real secondary power is zero

at optimal control. However, when the primary sources have arbitrary relative phasing,

the secondary real power need not be zero at optimal conditions. Symmetry conditions

under which the secondary real power is guaranteed to be zero at optimal control were

derived. However, satisfying the symmetry constraints on the secondary locations with

respect to the primary sources may be difficult in practice.

• An electro-mechanical model for a simply supported plate excited by multiple primary

and multiple secondary PZT patch actuators was developed.

• MAP control with multiple primary and multiple secondary PZT actuators was demonstrated experimentally on a composite panel

132
Chapter 5

MAP Feasibility for Rotorcraft

Vibration Control

In chapter §4, we developed the theory of MAP for multiple primary sources controlled by

multiple secondary sources. It was shown that when the phasing between the primary sources

is zero, then the secondary real power is zero at optimal control. However, when the primary

sources have arbitrary relative phasing, then the secondary real power is not zero. This poses

a challenge as MAP control is hard to implement if the secondary real power is not zero

at optimal control. Also, when the primary sources have arbitrary phasing, computing the

optimal secondary phase is not straight forward. Finally, extending MAP control to multiple

secondary sources requires a feasible approach to find the optimal solution especially when

the secondary sources are numerous. The main focus of this chapter is to assess the feasibility

of MAP control for realistic structures having a distributed primary excitation with arbitrary

relative phasing and possibly multiple frequencies controlled by a large number of secondary

sources.

133
5.1 MAP Solution for Arbitrary Primary Phasing

In this section, we present new theoretical results for MAP control implementation for multiple primary sources with completely arbitrary phasing. In §5.1.1, we present the theory

for multiple primary sources and one secondary source. We show that placing the secondary

actuator at the intersection of the node lines of the neighboring modes, that is, the modes

before and after the mode of interest, guarantees that the secondary real power is zero at

optimal condition. We also show that the optimal secondary phase and amplitude can be exactly calculated without any knowledge about the primary sources and phasing at resonance

and off-resonance frequencies. In §5.1.2, we present numerical simulations to validate the

theoretical findings for resonance and off-resonance cases. In §5.1.3, we describe an iterative

algorithm to find the optimal secondary sources and prove asymptotic convergence of the

algorithm to the optimal solution. However, this approach is not feasible for large number

of secondary sources. We present an alternative approach to find the optimal solution for

large number of secondary sources.

5.1.1 MAP Solution for Multiple Primary and Single Secondary

Consider a two-dimensional structure that is excited by N primary sources (point forces)

p1, p2,...,pN . The primary sources are assumed to have arbitrary phasing and given by

FT

p = [Fp1 Fp2 ... FpN ]=[fp1ejφp1 fp2ejφp2 ... fpN ejφpN ]

where fpi and φpi are, respectively, the amplitude and phase of the i

th primary source.

Assume that the structure is controlled by one secondary source given by

Fs = fsejφs

where fs and φs are the amplitude and phase of the secondary excitation. Note that the

optimal secondary source is

F∗

s = f ∗s ejφ∗

s = −R−1

ss RspFp (5.1)

134
where f ∗

s and φ∗s are the optimal secondary amplitude and phase, Rss is the real part of the

point mobility and Rsp is the real part of the transfer mobility Msp given by

Rsp = 

Rsp1 Rsp2 ··· RspN



. (5.2)

The MAP condition requires that the optimal secondary solution (f ∗

s , φ∗s) lies on the Ps = 0

curve where Ps is the real secondary input power. In general, for arbitrary primary phasing,

the optimal solution does not lie on the Ps = 0 curve. However, we show that optimizing the

location of the secondary actuator guarantees that the optimal secondary solution lies on the

Ps = 0 curve. This is a significant step towards implementation of MAP control technique

for realistic excitations.

Consider the transfer mobility between the secondary source and the i

th primary source

Mspi = ∞

n=1



ηωω2

n + jω(ω2n − ω2

)

 4φn(xs, ys)φn(xpi , ypi )

m [(ω2

n − ω2)2 + ω4nη2] (5.3)

where n represents the mode index (n1, n2), ω is the excitation frequency, ωn is the nth

resonant frequency, η is the damping, φn is the nth mode shape, (xpi , ypi ) is the location of

the i

th primary and (xs, ys) is the secondary location. When ω is close to ωk, then most of

the modal contribution can be assumed to come from the resonant frequencies ωk, ωk−1 and

ωk+1. Equation (5.3) can be written as

Mspi ≈ 

k+1

n=k−1



ηωω2

n + jω(ω2n − ω2

)

 4φn(xs, ys)φn(xpi , ypi )

m [(ω2

n − ω2)2 + ω4nη2] (5.4)

Now, if we choose the location of the secondary source such that it lies on the intersection

of the nodal lines of the mode shapes corresponding to ωk−1 and ωk+1, then we have

φk−1(xs, ys)=0 & φk+1(xs, ys) = 0 (5.5)

Using (5.5) in (5.4), we get

Mspi = 

ηωω2

k + jω(ω2k − ω2

)

 4φk(xs, ys)φk(xpi , ypi )

m [(ω2

k − ω2)2 + ω4kη2] (5.6)

= (a + jb)δspi (5.7)

135
where

a = ηωω2

k, b = ω(ω2k − ω2

), δspi = 4φk(xs, ys)φk(xpi , ypi )

m [(ω2

k − ω2)2 + ω4kη2] (5.8)

Next, we show that given (5.7), the optimal secondary solution lies on the Ps = 0 curve.

The real secondary power can be expressed as

Ps = 1

2

Re F H

s MspFp



+

1

2

F H

s RssFs (5.9)

where using (5.7), Msp can be expressed as

Msp = (a + jb)



δsp1 δsp2 ··· δspN



= (a + jb)δsp (5.10)

which gives

Rsp = a



δsp1 δsp2 ··· δspN



= aδsp (5.11)

Using (5.1) in (5.9), we get the secondary power at optimal condition as

P∗

s = − 1

2Rss

Re F H

p RpsMspFp



+

1

2Rss

F H

p RpsRspFp (5.12)

Using (5.10) and (5.11), it can be seen that

RpsMsp = a(a + jb)δpsδsp (5.13)

is symmetric. Since RpsMsp is symmetric, (5.12) reduces to

P∗

s = − 1

2Rss



F H

p RpsRspFp



+

1

2Rss

F H

p RpsRspFp (5.14)

= 0 (5.15)

Thus, the optimal secondary solution lies on the Ps = 0 curve. Now, if we know the optimal

secondary phase, then the optimal secondary amplitude can be easily calculated by sweeping

the secondary amplitude. Next, we show how the secondary optimal phase can be computed.

Note that Msp, as given in (5.10), can be expressed as

Msp = 

β1 β2 ··· βN



ejψ (5.16)

136
where βi = mag((a + jb)δspi )) and

ψ = tan−1 b

a

= tan−1 (ω2

k − ω2)

ηω2

k

(5.17)

which gives

Rsp = 

β1 β2 ··· βN



cos ψ (5.18)

Using (5.1) and (5.18), the optimal secondary source can be written as

f ∗

s ejφ∗

s = cos ψ



N

i=1

μiejφi (5.19)

where

μi = −βifpi

Rss

(5.20)

Rearranging, we get

f ∗

s = cos ψ



N

i=1

μiej(φi−φ∗

s) (5.21)

Since f ∗

s is real, setting the imaginary part to zero provides a condition that the optimal

phase has to satisfy



N

i=1

μi sin(φi − φ∗

s) = 0 (5.22)

The real secondary power can be expressed as

Ps = 1

2

Rssf 2

s +

1

2



N

i=1

fsβifpi cos(φi + ψ − φs) (5.23)

Setting Ps = 0 and expressing fs in terms of φs, we get the equation for the Ps = 0 curve

fs(φs) = 

N

i=1

−βifpi

Rss

cos(φi + ψ − φs)

= 

N

i=1

μi cos(φi + ψ − φs) (5.24)

137
Differentiating (5.24) with respect to φs gives

∂fs

∂φs

= 

N

i=1

μi sin(φi + ψ − φs) (5.25)

If φ ̄s is a solution to the equation

∂fs

∂φs

= 0 (5.26)

then,(5.25) gives



N

i=1

μi sin(φi + ψ − φ ̄s) = 0 (5.27)

Comparing (5.22) and (5.27), the optimal secondary phase is given by

φ∗

s = φ ̄s − ψ or φ∗s = π + φ ̄s − ψ (5.28)

A special case is ω = ωk (resonance). In this case,

ψ = tan−1 (ω2

k − ω2)

ηω2

k

= 0 (5.29)

which gives the optimal secondary phase as

φ∗

s = φ ̄s or φ∗s = π + φ ̄s (5.30)

The new MAP algorithm for multiple primary sources and one secondary source is summarized:

• Suppose that ω ≈ ωk. Place the secondary source at the nodal intersection of the mode

shapes for ωk−1 and ωk+1.

• Find the phase φ ̄s where the Ps = 0 curve is a minimum or a maximum.

• Calculate ψ according to (5.17).

• Calculate the optimal secondary phase φ∗

s according to (5.28).

• Set φs = φ∗

s and sweep fs till Ps = 0 to give the optimal f ∗s .

138
Figure 5.1: Primary source locations shown with corresponding amplitude and phase. Location A corresponds to the optimal secondary location while location B is not optimal

5.1.2 Simulation Results

We consider three primary point excitations with arbitrary phasing and one secondary point

excitation acting on the simply-supported aluminum plate previously considered in other

simulation studies. The primary sources with the relative phasing are shown in Fig 5.1.

We assume that the excitation frequency is close to the resonant frequency of ωk = 826

Hz. The corresponding mode number is (1,3). Note that the neighboring frequencies are

ωk−1 = 724 Hz with mode number (3,1) and ωk+1 = 976 Hz with mode number (3,2).

Therefore, according to the theory, the optimal secondary location is at the intersection of the

nodal lines for modes (3,1) and (3,2) as shown by location A in Figure 1. Location B shows

a non-optimal secondary location. We consider three cases: (1) ω = 826 Hz (resonance), (2)

ω = 800 Hz (off-resonance and left of 826 Hz) and ω = 850 Hz (off-resonance and right of

826 Hz).

Figure 5.2 shows the simulation for resonance case where Ps = 0 line is plotted in the fs −φs

space. The contour lines for the total input power are superimposed. In Figure 5.2(a), it can

be seen that the turning point of Ps = 0 curve (equivalent to minimum or maximum points

139
Figure 5.2: Resonance case ω = 826 Hz. The Ps = 0 line is shown in dark. The contour

lines for the total input power are superimposed. (a) Secondary located at A (b) Secondary

located at B

when phase is plotted on the x-axis) corresponds to the optimal secondary shown in red, as

suggested by the theoretical results. As expected, the total input power is minimized at the

optimal condition. Figure 5.2(b) shows the simulation for the non-optimal location. It can

be seen that the optimal solution (shown in red) does not lie on the Ps = 0 curve.

Figures (5.3) and (5.4) show the simulation for the off-resonance cases. The calculated

optimal phase is indicated by the blue line. The intersection of the blue line with the Ps = 0

curve gives the estimated optimal secondary source which is close to the true optimal given

by the red dot.

Next, we try to understand the trade-offs between total power attenuation and MAP implementation based on optimal secondary location with respect to the mode shapes. We

consider a simply supported plate that is excited by 8 primary sources (point forces) along

the edges as shown in Fig 5.5. The primary locations are indicated by red dots. The relative

strength and phasing between the primary sources is arbitrary. For the simulations, we chose

the primary sources to have the same amplitude but different phasing. Specifically, the phas140
Figure 5.3: Off-resonance case ω = 850 Hz. The Ps = 0 line is shown in dark. The contour

lines for the total input power are superimposed.

ing was p = [0, 45, 30, 20, -30, -30, -60, 120] degrees with the ordering starting from bottom

left and moving counter-clockwise. We consider off-resonance excitation frequency of 800 Hz

which is close to the resonant frequency of 826 Hz. The secondary source location is varied in

the interior of the plate and the optimal control is applied at each location assuming that the

primary sources are known. The total power attenuation Pred = (Pnc − Pc)/Pnc, where Pnc

is the total input power without any control and Pc is the input power with optimal control,

is plotted. Note that Pred=1 for no control and Pred=0 for complete control. We compare

the power attenuation at two different locations. Location A is optimal with respect to the

mode shapes adjacent to 826 Hz. Location B is the plate center and hot spot for 826 Hz. It

can be seen that the power attenuation at A is close to the maximum attenuation you can

get. If one uses location B for the secondary, then we will get good power attenuation in the

ideal case where the primary sources are known. However, for MAP control we rely only on

the secondary power for control. Fig 5.6 shows the variation of the real secondary power at

optimal control.

141
Figure 5.4: Off-resonance case ω = 800 Hz. The Ps = 0 line is shown in dark. The contour

lines for the total input power are superimposed.

It can be seen that the secondary real power at A is zero, as predicted by the theory, and

hence MAP control can be implemented. However, at location B, the secondary real power

is not close to zero and therefore MAP control is not suitable even though the attenuation

is very good at B.

5.1.3 MAP Solution for Multiple Primary and Multiple Secondary

Sources

In §5.1.1, we presented an algorithm to find the optimal secondary source when multiple

primary sources with arbitrary phasing are present. Given that we know the optimal solution

for one secondary source, we present two approaches to find the MAP solution for multiple

142
Figure 5.5: Simply supported plate excited by 8 primary sources with excitation frequency

equal to 800 Hz. The total power attenuation is shown as a function of the secondary location.

Location A shows the optimal location of secondary with respect to the neighboring mode

shapes. Location B is plate center and hot spot for the mode at 826 Hz.

secondary sources. We first present an iterative algorithm to find the optimal secondary

sources when multiple secondary sources are used for MAP control. We show that the

iterative algorithm is asymptotically stable and converges to the optimal value. The iterative

solution, however, is not feasible for large number of secondary sources. We present an

alternative approach that is feasible for large number of secondary sources and more useful

in practice.

143
Figure 5.6: Simply supported plate excited by 8 primary sources with excitation frequency

equal to 800 Hz. The real secondary power is shown as a function of the secondary location.

Iterative Algorithm

Without loss of generality, consider N primary sources (point forces) p1, p2,...,pN with

arbitrary phasing

FT

p = [Fp1 Fp2 ... FpN ]=[fp1ejφp1 fp2ejφp2 ... fpN ejφpN ] (5.31)

For purposes of illustration, we consider two secondary sources s1 and s2 given by

FT

s = [Fs1 Fs2]=[fs1ejφs1 fs2ejφs2 ] (5.32)

Recall that the optimal secondary sources are given by

F∗

s = −R−1ss RspFp (5.33)

144
where the point and cross-mobility Rss and Rsp are given by

Rss =



Rs1s1 Rs1s2

Rs2s1 Rs2s2



(5.34)

and

Rsp =



Rs1p1 Rs1p2 ··· Rs1pN

Rs2p1 Rs2p2 ··· Rs2pN



=



Rs1p

Rs2p



(5.35)

The iterative algorithm is as follows:

1. Assuming only s1 to be active, find the optimal Fo

s1 .

2. Fix s1 to be F0

s1 and find the optimal s2, F0s2 , by treating s1 as a primary source.

3. Fix s2 to be F0

s2 and find the optimal s1, F1s1 , by treating s2 as a primary source.

4. Fix s1 to be F1

s1 and find the optimal s2, F1s2 , by treating s1 as a primary source.

5. Repeat the iteration procedure till |Fk+1

si − Fksi

| ≤  where i = 1, 2 and  is some

convergence metric.

The iterative procedure can be represented as a map:

Fk

si → Fk+1si (5.36)

for i = 1, 2. Next, we show that the iterative map given in Eq. 5.36 is asymptotically stable

and converges to the true optimal value given in Eq 5.33.

Stability of the Iterative Algorithm

Assuming s1 only, the optimal value of s1 is given by

F0

s1 = −Rs1pFp

Rs1s1

= α1 (say) (5.37)

145
Treating s1 as a primary source equal to F0

s1 , we get the optimal s2 as

F0

s2 = − Rs2p

Rs2s2

Fp − Rs1s2

Rs2s2

F0

s1 = α2 + β2F0s1 (5.38)

Treating s2 as a primary source equal to F0

s2 , we get the optimal s1 as

F1

s1 = α1 − Rs1s2

Rs1s1

F0

s2

= α1 + β1F0

s2 (5.39)

Using (5.38) in (5.39), we get

F1

s1 = α1 + β1α2 + β1β2F0s1 (5.40)

Similarly, F1

s2 can be expressed as

F1

s2 = α2 + β2α1 + β1β2F0s2 (5.41)

From (5.40) and (5.41), in general, we have the following iterative map:

Fk+1

s1 = α1 + β1α2 + β1β2Fks1

Fk+1

s2 = α2 + β2α1 + β1β2Fks2 (5.42)

Next, we show that the steady state solution of the iterative map given in (5.42) corresponds

to the true optimal solution. From (5.42), the steady state solution satisfies

F∗

s1 = α1 + β1α2 + β1β2F∗s1

F∗

s2 = α2 + β2α1 + β1β2F∗

s2 (5.43)

which gives

F∗

s1 = 1

1 − β1β2

(α1 + β1α2)

F∗

s2 = 1

1 − β1β2

(α2 + β2α1) (5.44)

146
Using expressions for α1, α2, β1 and β2, we get

F∗

s =



F∗

s1

F∗

s2



= − 1

Rs1s1Rs2s2 − R2

s1s2

Rs2s2 −Rs1s2

−Rs1s2 Rs1s1



RspFp

= −R−1

ss RspFp (5.45)

which shows that the equilibrium point of the iterative map is the optimal secondary source

distribution. Next, we show that the iterative scheme is asymptotically stable. Consider the

map for Fs1

Fk+1

s1 = α1 + β1α2 + β1β2Fks1

The map is linear and the eigenvalue is λ = β1β2. Asymptotic stability is guaranteed if

λ < 1. Therefore, we need

β1β2 = R2

s1s2

Rs1s1Rs2s2

< 1

Since Rss is positive definite, we have

Rs1s1Rs2s2 − R2

s1s2 > 0 ⇒

R2

s1s2

Rs1s1Rs2s2

< 1

which guarantees asymptotic stability.

Alternative Solution for Multiple Secondary Sources

The iterative approach presented in the previous section is not feasible for large number of

secondary sources. For large number of secondary sources, we need a more practical solution.

In what follows, we present an alternative approach that is feasible when the secondary

sources are large in number. Since the location of the secondary sources in known, we

assume we can compute Rss experimentally before running MAP control. Recall that the

optimal secondary sources are

F∗

s = −R−1ss RspFp (5.46)

Note that we assume Rss is known. However, RspFp is not known. First, compute the

optimal secondary sources one at a time. That is,

147
Fˆsj = − 1

Rsj sj

RsjpFp ∀ j = 1, 2,...,m (5.47)

Writing (5.47) compactly, we get

Fˆs = −diag(Rss)

−1

RspFp (5.48)

where

FˆT

s = [Fˆs1 Fˆs2 ... Fˆsm]

Equation (5.48) can be re-written as

RspFp = −diag(Rss)Fˆs (5.49)

Using (5.49) in (5.46), we get the optimal secondary sources as

F∗

s = R−1ss diag(Rss)Fˆs (5.50)

Thus, for m secondary sources, one needs to run MAP test m times to calculate the optimal

secondary sources.

5.1.4 FEM Model

A FEM model was built to verify MAP control authority and global vibration reduction

and also guide the experimental test rig design and validation. The design consists of a

thin aluminum plate riveted to rigid aluminum angles and is representative of a realistic

airframe structure. Point force or line moment excitations were applied to the angles as

primary excitation. The force amplitude and phase of a secondary point force excitation

148
Figure 5.7: Finite element model of test rig.

were swept until the non trivial MAP point was found. The location of the secondary source

was not optimized. A picture of the finite element model is presented in Fig. 5.7. Numerical

boundary conditions were free-free.

Flexural kinetic energies for a single primary point force excitation for both the system and

the plate subcomponent are presented in Fig. 5.8. Frequency responses exhibit many modes

in the 500 to 2500 Hz interval. The system’s flexural kinetic energy with and without MAP

control around three of the structure’s resonant frequencies (937, 1012, and 1215 Hz) are

presented in Fig.5.9. MAP produced over 8 dB of attenuation at 937 and 1215 Hz, and

no control at 1012 Hz. This lack of control for that particular resonance was due to the

secondary excitation being located on a nodal line of the underlying structural mode.

The effect of the MAP control on the radiated sound field is presented in Fig. 5.10. When

modes could be controlled, the radiated sound power was reduced by 9.5 and 8.5 dB at 937

Hz and 1215 Hz, respectively. Figures 5.8 and 5.9 show that there is a direct relationship

between structural vibration and interior noise.

149
Figure 5.8: Rig flexural kinetic energy

Figure 5.9: Flexural Kinetic Energy with and without MAP control.

5.2 Power Model for a Generic Actuator

The power model of a generic PZT actuator was developed in §3.4. Specifically, we relate

the net electrical power in the actuator, the quantity that is measured, to the net mechanical

150
Figure 5.10: Radiated sound power with and without MAP control.

power, the quantity that is controlled, through the dielectric loss using the following equation:

P ̄e = 1

2

ω a2

h |V ̄ |

2 ε ̄33δ + P ̄ (5.51)

Equation (5.51) shows that the net input electrical power P ̄e is equal to the sum of the

dielectric loss and the net input mechanical power P ̄. Note that even though Eq. (5.51)

was derived for a PZT actuator, a similar relation can be derived for other actuators, for

example, proof-mass actuators.

5.3 Test Rig and Plate Characterization

The main experimental efforts this year include: 1) verification of global power minimization

using MAP, 2) implementation of MAP control on a realistic and sophisticated aircraft

structure, and 3) experimental validation of the new MAP theory developed in §5.1.

151
Figure 5.11: Test rig with a shaker and a PZT sensor/actuator pair.

A test apparatus was constructed to mimic a representative rotorcraft skin structure with

more realistic vibration load paths from bulkhead to skin. An aluminum plate, 12 x 12 x 1/8

inches, was riveted by 28 aircraft grade rivets to a welded 1 x 1 inch aluminum angle frame.

This design introduces contact surface and Coulomb damping to the test rig. The plate

assembly is suspended by 4 bungee cords to mimic a free-free boundary condition, as shown

in Figure 5.11. Two PZT patch sensor/actuators (2.5 x 2.5 x 0.005 inches) are bonded to the

upper left and low right corners, respectively, one acting as a primary actuator (disturbance)

and the other acting as a secondary actuator (control). A mini-shaker is attached to the

plate via a piano wire stinger at the aluminum side wall as an alternative primary actuator

to mimic distributed vibration load along the edges of the side walls to the aluminum plate.

The structural dynamic characterization of the newly constructed plate was conducted with a

Polytec PSV-400/OFV5000 Laser Scanning Doppler Vibrometer (LSDV) as shown in Figure

5.12. The LSDV scans the plate surface in a grid of 320 x 320 nodes, and a laser pointer dwells

on each node for a certain period to measure average vibration velocities orthogonal to the

plane. The plate is excited by either the shaker or the primary PZT with a broad frequency

band noise signal. The frequency response function (FRF) of the vibration velocities with

respect to the excitation at each node is then calculated, and mode shapes of any selected

frequencies are extracted from the FRFs.

152
Figure 5.12: Polytec Laser Scanning Doppler Vibrometer

Figure 5.13: FRF of the plate measured by LSVD with shaker excitation. Red circle indicates

the frequency selected for MAP testing

Figure 5.13 shows the measured mobility at the center of the plate for a shaker excitation at

the edge of the side. There are over 20 strong vibration modes in the 0-3000 Hz frequency

range. The modal cluster around 1000 Hz is of particular interest and is selected for MAP

153
control and analysis. The velocity mode shapes of 3 frequencies, 624 Hz, 999 Hz and 1052

Hz were extracted from the FRFs at all the nodes, as shown in Figure 5.14. The (2,2) mode

of the plate at 999 Hz is selected for MAP control due to symmetry of the mode shape with

respect to PZT actuator locations.

Figure 5.14: Velocity modal shapes for (a) 624 Hz (b) 999 Hz and (c) 1052 Hz

5.4 Experimental Validation of MAP Control

The validation of the global energy minimum at optimal control was conducted using a pair

of PZT patches rather than using the shaker (as primary) and a PZT patch (secondary) due

to some uncertainty and slight asymmetry of the mode shapes imposed by the attachment

of the shaker. A frequency of 999 Hz (2,2 mode) was selected for MAP control. As can be

seen from Figure 5.13, this frequency is closely coupled to other two modes. Minimization

of global kinetic energy of the plate at optimal MAP control is verified by:

• Measuring velocity mode shapes of the plate at various excitation levels

• Calculating global kinetic energy of the plate

Pk = 

l

i=1

n

j=1

1

2

mi,jv2

i,j (5.52)

154
Figure 5.15: MAP control with a pair of PZT patch sensor/actuators. Red circle indicates MAP predicted optimal control at Vs=5.94V. Purple circle indicates optimal control

at Vs=5.70V as predicted by the accelerometer.

where vi,j and mi,j are the vibration velocity and the mass of plate element at location

i and j, respectively and l × n is the LSDV scan grid size.

• Identifying the optimal MAP control voltage V ∗

s by sweeping Vs applied to the secondary actuator.

• Checking the correlation between V ∗

s , minimum Pk and minimum acceleration measurement.

The harmonic primary excitation is kept constant at Vp = 5 V with primary phase equal to 0o.

The optimal secondary phase was estimated to be 183.8o with respect to the primary. This

is close to the true optimal of 180o (or 0o). The secondary voltage Vs is swept from 0 to 10

volts while its phase is kept fixed at 183.8o. The secondary power is acquired and compared

to the predetermined aggregated power loss. Meanwhile, the acceleration of the plate center

is monitored by the accelerometer. Figure 5.15 shows that the MAP criterion (intersection

of the baseline with the secondary real electric power) predicts the optimal control to be

Vs = 5.94 V while the acceleration reading indicates maximum vibration reduction(from -26

155
Figure 5.16: Normalized total kinetic power of the plate around optimal MAP control. Blue

circle indicates global kinetic power of the plate at Vs = 5.60 V, and the normalized total

energy is 28.1 at no control (Vs = 0).

dB to -42 dB) at Vs = 5.70 V. Figure 5.16, from the LSDV measurements, shows the total

kinetic energy of the plate calculated from the mode shapes at different secondary voltages

and normalized with respect to the total power at optimal control, that is, at Vs = 5.94 V

and shows that the global power minimum occurs when Vs = 5.60 V. It can be seen that

the acceleration measurement predicts the global energy minimum very well even though it

is slightly off the hot spot of the mode shape.

5.4.1 Change in Mode Shapes with MAP Optimal Control

The effect of MAP control on mode shapes of the plate at 999 Hz is shown in Figure 5.17. The

MAP control significantly reduces the vibration magnitude in the entire plate. It was also

observed that the PZT actuators placed on the hot spots of the mode shape do not change

the mode shape of a resonance frequency while significantly reducing the amplitude of the

vibration. This test has demonstrated that optimal MAP control corresponds to minimum

156
Figure 5.17: Velocity modal shapes in mm/s for no-control condition and with optimal

MAP control (Vs = 5.5V). The low left square is the primary (excitation) and upper right

the secondary (control).

global kinematic energy of the plate. The test also demonstrates that MAP control is capable

of reducing the vibration of a plate with complexity in structure and damping mechanism

effectively.

5.4.2 Validation of New MAP Algorithm

The main idea behind the new algorithm is to optimally locate the secondary actuator

such that the cross mobility terms can be approximated as a single mode. This guarantees

that MAP solution holds, that is, the optimal solution lies on the Ps = 0 curve for any

arbitrary phasing of the primary actuators. The optimal secondary amplitude and phase

can be determined for resonance and off-resonance cases. We used the shaker as the primary

excitation and one of the PZT patches as the secondary actuator as shown in Fig. 5.11, §5.3.

The excitation frequency was kept at 994 Hz. Figure 5.18 shows the experimental mode

shape corresponding to 994 Hz. The secondary PZT actuator s1 is shown in bottom left and

is located close to the hot spot and is optimal for 994 Hz.

The first step in the new MAP algorithm is to find the optimal secondary phase. For resonant

excitation, the optimal phase is given by the secondary phase at which the secondary power

157
Figure 5.18: Experimental mode shape corresponding to resonant frequency 994 Hz

Figure 5.19: Plot showing implementation of optimal phase search. The optimal secondary

phase is the phase at which the secondary phase is minimum or maximum.

is either a minimum or maximum for a given voltage. The primary excitation was 5 V and

the phase was set to zero. With the secondary voltage equal to 3 V, the secondary phase was

varied to find the phase for which the secondary power is a minimum or maximum. Figure

158
5.19 shows the variation of the secondary power Ps with respect to the secondary phase. It

can be seen that the estimated optimal phase is −31o where Ps is minimum, indicated by

the red circle in Figure 5.19. Figure 5.19 also shows that the accelerometer reading, located

close to the plate center, indicates that the vibration minimum occurs at −26o (purple circle)

which is close to the estimated value of −31o.

Figure 5.20: MAP control with secondary phase = −31o

The next step is to fix the secondary phase at the optimal phase and sweep the secondary

voltage to get the optimal voltage, indicated by intersection of the secondary power and the

baseline loss curve. Figure 5.20 shows MAP performance when the secondary phase was set

to −31o. It can be seen from the accelerometer reading that a vibration reduction of 20 dB

is achieved. When the secondary phase was set to −26o, the MAP performance was greatly

enhanced leading to a vibration reduction of 44 dB, see Figure 5.21.

In summary, the new MAP algorithm provides a novel way to find the optimal secondary

phase and amplitude without any knowledge about the primary sources.

159
Figure 5.21: MAP control with secondary phase = −26o

5.5 Rotorcraft Airframe Mockup

Modal analysis of the riveted structure revealed asymmetrical mode shapes and split resonance peaks that likely were resulting from asymmetrical boundary conditions due to the

rivets and small variations in this fabricated structure. These experimental modes shapes

could not be easily predicted using finite element models. To position secondary actuators

for MAP, it is best to know the structural mode shapes to guarantee optimal control. Hence,

to minimize the uncertainty in the mode shapes, we constructed a test structure without

rivets but of similar geometry than the original panel. This new structure could be easily

modeled and had predictable mode shapes and good separation of resonance frequencies.

160
Figure 5.22: Test Plate Dimensions

5.5.1 Test Rig

The new test plate was fabricated through a milling process to enable seamless integration

of angle frames to the plate. Detailed dimensions of the test plate are shown in Fig. 5.22.

The assembly is suspended in air by 4 bungee cords to mimic free-free boundary conditions,

as shown in Fig. 5.23. A mini-shaker is attached to the plate via a piano wire stinger at the

aluminum side wall to mimic distributed vibration load along the edge of the plate.

161
Figure 5.23: Picture of Test Rig showing (a) Front Face and (b) Back Face

5.5.2 Comparison between experimental mode shapes and FEM

Analysis

The velocity frequency response function (FRF) of the plate is acquired by Laser Scanning

Doppler Vibrometer (LSDV) and the plate is excited with broad band noise by the minishaker. Figure 5.24(a) shows FRF of the plate measured by LSDV and shaker excitation. A

finite element analysis (FEA) of the plate was done to compare the frequency response and

mode shapes. The frequency response from FEA is shown in Fig. 5.24(b). The experimental

modal frequencies were within 1% of FEA results for frequencies within 500-2000 Hz range.

We also compared the mode shapes for different resonant frequencies. For example, Fig.

5.25(a) shows the experimental mode shape, a (2,2) mode, for 947 Hz. Figure 5.25(b) shows

that mode shape obtained from the FEA model at 940 Hz which is in excellent agreement

with the experimental mode shape. Such good agreement is also evident at other resonant

frequencies. For example, figures 5.26 and 5.27 compare the experimental mode shape with

the mode shape calculated using FEA for 850 Hz and 1073 Hz respectively.

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Figure 5.24: (a) FRF of the plate measured by LSVD and shaker excitation (b) Total plate

KE per N of force, from FEA

Figure 5.25: Mode shape comparison (a) Velocity modal shape at 947 Hz measured by LSVD

and shaker excitation (b) Calculated vibration mode shape at 940 Hz

5.6 Effectiveness of MAP Control

In this section, we present experimental results that demonstrate the effectiveness of MAP

control for single and multi-frequency excitation. We use the shaker as the primary excitation

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Figure 5.26: Mode shape comparison (a) Velocity modal shape at 850 Hz measured by LSVD

and shaker excitation (b) Calculated vibration mode shape at 850 Hz

Figure 5.27: Mode shape comparison (a) Velocity modal shape at 1073 Hz measured by

LSVD and shaker excitation (b) Calculated vibration mode shape at 1073 Hz

source and a PZT patch, located at the plate center, as the secondary excitation. The

primary excitation has two resonant frequencies, 850 Hz and 1073 Hz. Figure 5.26 shows

that the PZT location is optimal with respect with respect to the mode at 850 Hz whereas

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Table 5.1: Primary Excitation Parameters

ωp11 850 Hz ωp12 1073 Hz

Vp11 2.5 V Vp12 4 V

φp11 30o φp12 0o

Figure 5.28: Voltage sweep showing the optimal secondary voltage at 850 Hz while control

for 1073 Hz is off.

figure 5.27 shows that the PZT location, though good, is not the best location for the mode

at 1073 Hz. As we discuss later, this is a good test of the effectiveness of MAP for suboptimal locations of the PZT actuator. The primary excitation parameters are shown in

Table 5.1. An accelerometer located at the plate center measured the local vibration levels

whereas the global vibration was measured using the laser vibrometer.

850 Hz control with 1073 Hz not controlled

We use a frequency-by-frequency approach for controlling multi-frequency excitation. First,

the optimal secondary voltage and phase are determined for the mode at 850 Hz while the

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mode at 1073 Hz is not controlled. The optimal secondary phase corresponding to 850 Hz was

found to φs11 = 25.09o. With the secondary phase equal to the optimal phase, the voltage is

varied till the secondary real power intersects the baseline curve to give the optimal voltage.

See Fig. 5.28. It can be seen that at the optimal voltage, the local vibration is a minimum

resulting in a 30 dB reduction.

1073 Hz control with 850 Hz controlled

With the control for 850 Hz on, the mode at 1073 Hz is controlled next. In this case, the

optimal phase was determined to be φs12 = −6.198o. With the secondary phase equal to the

optimal phase, the voltage was varied to find the optimal secondary voltage at 1073 Hz as

seen in Fig. 5.29. The local vibration level is minimum at optimal secondary voltage and

the local vibration reduction is close to 20 dB.

Figure 5.29: Voltage sweep showing the optimal secondary voltage at 1073 Hz while control

for 850 Hz is active.

166
850 Hz control with 1073 Hz controlled

Finally, we control the 850 Hz excitation again while 1,073 Hz is under control. The optimal

phase was estimated to be φs11 = 26.010o with local vibration reduction of 38.31 dB, which

is 8.31 dB more than the single frequency case. Note that the local vibration reduction

improved as a result of better optimal phase estimation. See Fig. 5.30. The vibration

reduction for 1073 Hz was unaffected.

Figure 5.30: Voltage sweep showing the optimal secondary voltage at 850 Hz while control

for 1073 Hz is on.

Global Vibration Reduction

Figures 5.29-5.30 show that MAP control results in significant local vibration reduction

at both 850 Hz and 1073 Hz. However, it is important to verify whether we get similar

performance globally. The global vibration levels were measured using the laser vibrometer.

Figure 5.31 show the global vibration in dB with and without control. Figure 5.32 show

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Figure 5.31: Plot showing global vibration levels with and without MAP control

the global vibration reduction. It can be seen that vibration reduction is around 22-23 dB

for 850 Hz excitation whereas the 1073 Hz excitation was reduced by 11-12 dB. Though the

global vibration reduction is not as high as the local reduction, as expected, it is still very

significant.

Figure 5.32: Global vibration reduction with MAP control.

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Figure 5.33: Illustration of PZT patch locations.

The above results demonstrate the effectiveness of MAP control for single and multi-frequency

excitation. Also, note that dB reduction for the mode at 1073 Hz is less than the reduction at

850 Hz. This is due to the fact that the PZT location for 1073 Hz is sub-optimal. However,

it is important to note that a global reduction of 11-12 dB is significant and demonstrates

effectiveness of MAP control even when the PZT location is not optimal.

5.6.1 MAP Control with Multiple Secondary Controllers and Multiple Frequency

The LabVIEW code developed for testing was modified to excite and control three frequency

signals with arbitrary phases. The shaker was used as the primary excitation source, and

three PZT patches shown in Fig. 5.33 were used as secondary excitation sources. S1 is placed

at the center of the plate where optimal vibration reduction for 851.1 Hz (3 by 1 mode) is

expected. Similarly, S2 is located at the modal peak of 1,320 Hz. P1 and S3 are positioned at

the intersection of 866, 940, and 1310 Hz based on FE analysis. The primary excitation has

three resonant frequencies, 851.1, 945.2, and 1319.8 Hz. The primary excitation parameters

are shown in Table 5.2. Table 5.3 shows estimated MAP parameters after phase and voltage

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search for the MAP control.

Table 5.2: Primary Excitation Parameters

FSh1 851.1 Hz FSh2 945.2 Hz FSh3 1319.8 Hz

VSh1 2.5 V VSh2 4.0 V VSh3 3.0 V

φSh1 30◦ φSh2 0◦ φSh3 60◦

Table 5.3: Estimated Control Parameters (S1, S2, and S3)

FS1 851.1 Hz FS3 945.2 Hz FS2 1319.8 Hz

VS1 -0.525 V VS3 -0.59 V VS2 -1.78 V

φS1 26.84◦ φS3 135.83◦ φS2 58.86◦

With the control strategy established, the plate was excited, and the vibration response

measured using the vibrometer at 99 (9 x 11 grid) locations on the test plate, phase-referenced

to an accelerometer located at the back side of the S3 center. The resulting mode shapes

are captured in Fig. 5.34. The columns (left to right) summarize the different phases of

multi-frequency MAP control: no control and three PZT control. The spatially averaged

vibration response is depicted in Fig. 5.35. The control case reduces the global plate response

at all three frequencies of 851.1, 945.2 and 1319.8 Hz with significant reductions. For the

baseline case, the average vibratory responses were around 60 to 65 dB (0 dB at 1 m/s) for

the no control case as depicted in Fig. 5.35. When control was applied to the three modes,

the vibratory response exhibited significant reductions for each mode (19, 4.4, and 9.9 dB

reduction, respectively).

MAP Control with Strategically Positioned Single Secondary

Experimental results presented here demonstrate the vibration reduction capability of a single secondary for multiple-frequency disturbance. The input to the shaker is kept unchanged

(see Table 5.2), while only S3 is used to control three frequencies. As previously described,

S3 is placed at the intersection of three mode shapes (851.1, 945.2 and 1319.8 Hz). Phase

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(a) 851 Hz (Baseline) (b) 851 Hz (Control)

(c) 945 Hz (Baseline) (d) 945 Hz (Control)

(e) 1320 Hz (Baseline) (f) 1320 Hz (Control)

Figure 5.34: Experimental Deflection Shapes for Three PZT Control

and voltage search for control parameter estimation was done before global vibration measurement as shown in Table 5.4.

171
Figure 5.35: Average plate response with and without MAP control

Table 5.4: Estimated Control Parameters (S3 only)

FS31 851.1 Hz FS32 945.2 Hz FS33 1319.8 Hz

VS31 0.755 V VS32 -0.59 V VS33 0.975 V

φS31 29.84◦ φS32 135.83◦ φS33 43.67◦

Figure 5.36 shows the mode shapes before and after MAP control is applied for the respective

frequencies. The spatially averaged vibration response is depicted in Fig. 5.37. The control

cases reduce the global plate response at all three frequencies of 851.2, 943.7 and 1204.2 Hz

with significant reductions as summarized in Fig. 5.38. It is noticeable that even a single PZT

(S3) alone can control multiple frequencies equivalently to three PZTs to which individual

frequencies are assigned. In such a condition, a single control placed at the intersection of

multiple (i.e., three and more) frequency modes can be used effectively to reduce overall

vibration levels.

MAP for Off-Resonance Control

The excitation frequency was set to be harmonics of 300 Hz (600, 900, and 1,200 Hz),

simulating gearbox vibrations not associated with natural frequencies of the test rig. Only

S3 was used for control. The primary excitation parameters are shown in Table 5.5 and

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(a) 851 Hz (Baseline) (b) 851 Hz (Control)

(c) 945 Hz (Baseline) (d) 945 Hz (Control)

(e) 1320 Hz (Baseline) (f) 1320 Hz (Control)

Figure 5.36: Experimental Deflection Shapes for Multi-frequency control using a Single PZT

Control

173
Figure 5.37: Average plate response with and without MAP control

Figure 5.38: Plot showing global vibration with and without MAP control

Table 5.6 shows estimated MAP parameters after phase and voltage search for the MAP

control. For the off resonance control, the challenge is that the amount of possible vibration

reduction is dependent on the controller characteristic. Figure 5.39 is the baseline curve of S3

PZT patch showing power directly measured from the PZT with respect to frequency. The

red solid line is the mechanical response of the test rig which can be suppressed to baseline

level (dotted line). At off-resonance frequencies, the amount of structural vibration is very

close to the baseline power level, which complicates the MAP control algorithm search for

optimal control parameters.

Figure 5.40 shows vibration plots before and after the MAP control is turned on. A 7.5 dB

reduction was achieved at 1200 Hz, which is close to a resonance frequency of 1204 Hz. In

174
Table 5.5: Primary Excitation Parameters

FSh1 600 Hz FSh2 900 Hz FSh3 1200 Hz

VSh1 1.0 V VSh2 6.0 V VSh3 3.0 V

φSh1 30◦ φSh2 0◦ φSh3 60◦

Table 5.6: Estimated Control Parameters (S3 only)

FS31 600 Hz FS32 900 Hz FS33 1200 Hz

VS31 -5.3 V VS32 0.37 V VS33 0.8 V

φS31 −30◦ φS32 −80◦ φS33 45◦

Figure 5.39: Plot showing S3 baseline when 6.75 V is applied.

600 Hz and 900 Hz cases, while local mode shape changes were evident, no significant global

reduction was made. For application of the MAP control algorithm to real structures, it

is suggested to closely analyze the external vibration sources, and structural responses of

the target structure. Then controller position can be determined based on a trade study to

reduce global vibration response due to resonance and off-resonance frequencies.

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(a) 600 Hz (Baseline) (b) 600 Hz (Control)

(c) 900 Hz (Baseline) (d) 900 Hz (Control)

(e) 1200 Hz (Baseline) (f) 1200 Hz (Control)

Figure 5.40: Experimental Deflection Shapes for Off-Resonance Control using a Single PZT

Control

176
5.7 Summary

The key accomplishments of the MAP feasibility analysis and rotorcraft vibration reduction

are:

• The feasibility of the MAP control approach was assessed for multiple primary sources

with arbitrary phasing. It was shown that optimizing the location of the secondary

actuator with respect to the mode shapes guarantees that the secondary real power is

zero at optimal control for resonance and off-resonance frequencies. An algorithm to

find the optimal phase and magnitude of the secondary source was developed.

• A stable iterative scheme was developed to find the MAP solution for multiple secondary sources. The iterative algorithm is attractive for small number of secondary

sources. An alternative approach was developed that is feasible for large number of

secondary sources.

• Designed, modeled and built a test rig to mimic rotorcraft construction. Demonstrated

MAP control with primary excitation using shaker with multiple frequencies and multiple secondary actuators for control.

• This research has led to an improvement in the technology readiness level (TRL) from

1 to 3-4.

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Chapter 6

Effect of MAP Vibration Control on

Radiated Noise For a Simple Interior

Structure

We present some preliminary analysis showing the potential benefit of MAP control to reduce noise in the passenger cabin due to the main rotor transmission excitation, the NASA

program goal. Helicopter gearbox excitations are transmitted to the fuselage via rigid connections at the transmission well. Subsequently, the frame and skins of the fuselage transmit

the vibrations produced by the main gearbox and radiate sound into the cabin interior.

While representative of helicopter interior skin panel, the plate structure used for the development of MAP control was relatively small and had few structural modes in the 500 to

2500 Hz range. A fully coupled vibro-acoustic finite element model of a larger, more realistic

and modally denser structure was used to evaluate MAP potential. Although recognized as

a rather simplistic representation of a real helicopter interior, this model is a step closer to

evaluating MAP performance on radiated interior noise levels. Up to this point, the MAP

development program focused on demonstrating vibration reductions that are expected to

lead to noise reduction.

178
6.1 Model description

An acoustic cavity of dimension 1 × 1.1 × 1.03m3 is enclosed in an aluminum structure

composed of 2-mm thick panels with 20 × 20mm2 beams in all corners. The panels are

modeled as shells, whereas the reinforcement beams are solid elements. A schematic of the

structure is presented in Fig. 6.1. The base of the structure was simply supported.

Figure 6.1: Schematic of the box structure.

A primary point force excitation P1 is applied to one panel of the structure. MAP control

was applied using a single secondary point force. The secondary was either located on the

same panel as the primary (S1), or on an adjacent panel (S2). The locations of (S1) or (S2)

were not optimized for a broad range control or single mode control, but assigned by visual

inspection on hotspots of the forced response at 505 Hz (P1 only).

179
6.2 Results

The performance of the controller was evaluated by using the structural kinetic energy in

all the panels, as well as the averaged sound pressure levels (SPL) in the acoustic cavity.

The frequency responses without control are presented in Fig. 6.2. The many peaks in the

structural responses suggest that the modal density is already relatively high, whereas there

is still a lot of separation between peaks in the acoustic response.

Figure 6.2: System frequency responses for the box structure with no control.

For this study, the structure was excited at 505 Hz, a frequency close to a resonance of the

acoustic cavity, and also in a range where the structure is modally dense. The structural

and acoustical fields at 505 Hz are presented in Fig. 6.3. Results when MAP control is

applied at S1 are shown in Fig. 6.4. The MAP condition, where the structural kinetic

energy is minimized, is noted with the vertical dashed line. The total kinetic energy level

was reduced by 6.1 dB, and the averaged SPL dropped by 13.8 dB. While the kinetic energy

180
was minimized at MAP, a greater SPL reduction (15.5 dB) could be achieved for slightly

larger force amplitude since the acoustic resonance is not exactly at 505 Hz and MAP controls

the underlying dominant structural mode.

Figure 6.3: Response fields at 505 Hz (no control). (a) Structural displacement (b) Acoustic

pressure

Figure 6.4: The total kinetic energy and SPL for S1 MAP Control

With MAP control applied at S2, the total kinetic energy level was reduced by 7.0 dB, and

the averaged SPL dropped by 13.6 dB. Thus, MAP control can be very effective at reducing

181
the injected power even when the secondary actuation is located away from the primary

sources and in a system where energy has many paths to propagate.

Results of the MAP control as applied on the complex structure demonstrate that MAP

has good potential at reducing sound inside a cavity, even if the secondary forces are not

in close proximity to the primary excitations. The study was limited to a single point of

control and should be expanded to multiple actuators and multiple frequencies to evaluate

the performance of the control. In addition, the control was applied at a frequency close to

both a structural and acoustic resonance. Other cases of interest include frequencies where

at most one of the domains is very close to a resonance.

182
Chapter 7

Conclusions and Next Steps

A novel active vibration control strategy called Minimum Actuation Power (MAP) is presented for a structure excited by multiple primary (excitation) sources and controlled by

multiple secondary (control) sources. MAP is an optimal control strategy that minimizes

the total input power into the structure by monitoring the input power from the secondary

source only and therefore does not need any knowledge about the primary sources.

The theory of MAP was first developed for a structure excited by a single primary source

and controlled by a single secondary source. It was shown that the real input(mechanical)

power in the secondary source goes exactly to zero when the total input power from both

the primary and secondary source is minimized. This leads to a novel way to implement

MAP control without explicit knowledge about the magnitude and location of the primary

excitation. In order to validate the theoretical findings, a model for a simply supported plate

excited by a primary piezo-electric (PZT) patch actuator and controlled by a secondary piezoelectric patch actuator was developed. An electromechanical coupling model of piezoelectric

patch actuator with a plate structure is developed. A simple relation between the real

electrical power and the real input power of the piezo-electric actuator was developed. This

enabled implementing MAP control using the real electric power in the secondary actuator.

The effectiveness of MAP control was demonstrated for resonant and off-resonant excitation

183
frequencies. It was first shown through simulations that optimizing the location of the

secondary actuator can help control otherwise uncontrolled modes very effectively. The MAP

theory was extended to multi-frequency excitation and simulations were presented to validate

the theory. A test setup that mimics simply supported boundary conditions was built to

validate the theoretical findings. To implement MAP using electrical power, tests were done

to characterize the power consumption of the plate and the dielectric loss of the piezo-electric

patch actuators. The MAP control algorithm was tested for on-resonance and off-resonance

frequencies and effectiveness of MAP for vibration reduction. We demonstrated up to 47 dB

vibration reduction at resonance and 7 dB vibration reduction at off-resonance, as measured

by a single accelerometer at the plate center. There was good qualitative agreement between

the experimental results and theoretical findings.

The MAP control approach was extended to multiple primary excitations with zero relative

phasing and multiple secondary excitations. It was shown that when the primary excitations

have the same phasing, then the real secondary power is zero when the total input power is

minimized. However, when the primary excitations have arbitrary phasing, the secondary

real power is not always zero when the total input power is minimized. It was shown that

the MAP condition is exactly satisfied when the secondary sources are located symmetrically

with respect to the primary sources. However, such a constraint on the secondary source

locations is hard to satisfy in practice. Extending the single actuator model, we developed

an electro-mechanical model for a simply supported plate excited by multiple primary and

multiple secondary PZT patch actuators. Performance of MAP control as a function of the

location and number of the secondary sources was assessed. It was seen that increasing the

number of secondary sources improves vibration reduction. Also, multiple secondary sources

can be useful when vibration reduction over a frequency band is sought. Experimental validation of MAP control was demonstrated in the lab with two primary and two secondary

excitations using an aluminum plate as well as a composite panel. For example, we demonstrated 20 dB vibration reduction for the aluminum plate and 12 dB vibration reduction for

the composite plate, as measured by a single accelerometer at the plate center. A test for

real-time implementation was developed to automate MAP control strategy.

The feasibility of MAP control approach for reducing airframe vibrations due to the rotorcraft main rotor transmission and the reduction of corresponding sound levels in the passenger cabin was assessed using a more meaningful metric for vibration reduction, the global

184
kinetic energy of the plate structure. MAP feasibility was assessed for multiple primary

excitation sources with arbitrary relative phasing and with multiple frequencies controlled

by multiple secondary sources. In order to circumvent the problem of placing the secondary

actuators symmetrically with respect to the primary sources to satisfy the MAP condition,

it was shown that placing the the secondary sources at the intersection of the node lines

of the neighboring modes can guarantee that the secondary real power is zero at optimal

conditions even when the primary sources have arbitrary phasing. An algorithm to compute

the optimal secondary phase and magnitude was developed for resonant and off-resonant

excitation frequencies. A stable iterative solution for finding the MAP solution for multiple

secondary sources was developed. The stable iterative solution, however, is not feasible for

large number of secondary sources. An alternative feasible approach was developed to find

the MAP solution when the secondary sources are large in number. A test rig to mimic

rotorcraft construction was designed, modeled and built. MAP control with primary excitation using shaker, in addition to PZT, with multiple frequencies and multiple secondary

actuators for control was demonstrated. It was shown that MAP control achieved significant

vibration reduction with average global vibration reduction of the order of 15-20 dB for

single frequency with one secondary actuator. A single secondary actuator was optimized to

control multiple frequencies, in some cases up to 3 frequencies with 8-10 dB average global

vibration reduction for each frequency.

A fully coupled vibro-acoustic finite element model of a larger, more complex and modally

denser box type structure was used as a quick evaluation of the potential of MAP to reduce

noise in the passenger cabin due to the main rotor transmission excitation. It was shown

that MAP control resulted in 6.1 dB reduction in the total kinetic energy of the structure

while the averaged sound pressure levels (SPL) in the acoustic cavity dropped by 13.8 dB

at resonant excitation. This preliminary analysis clearly shows the potential of MAP in

reducing interior noise in a helicopter cabin.

Next Steps

Possible future directions include establishing physical system design rules for actuator density that leverages the highly distributed actuation opportunities offered by MAP and op185
timal actuator positions on a generic panel. It is important to quantify the cost/benefit

of MAP on an aircraft-level system for both tonal and broadband sources using analysis

and tests. Finally, there is a need for better understanding of the vibration path and the

structural contributions to interior noise using a detailed diagnostic survey of vibro-acoustic

problem and the impact of composite structures and improved gear designs or gear noise

reduction technologies should be studied to assess the interior noise challenges of the future.

186
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1999.

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332, 1996.

191
Appendix A

Total input power due to primary and

secondary patches

Let us assume that the primary and secondary patch edges are discretized into an equal

number of elements, say q. Let pi and si represent the coordinates of the center of the i

th

primary and secondary patch element respectively where i = 1,...,q. Let Mp and Ms be the

primary and secondary moments acting on each element respectively. Consider the point pi

on the primary patch. The angular velocity at this point is a sum of the angular velocity

induced by the the primary and secondary patch moments and can be expressed as

Wpi = 

j

Wpipj +

j

Wpisj (A.1)

where Wpipj is the angular velocity induced at pi due to the moment Mp at location pj and

Wpisj is the angular velocity induced at pi due to the moment Ms at location sj . The angular

192
velocities Wpipj and Wpipj can be expressed as

Wpipj = PpipjMp

Wpisj = PpisjMs (A.2)

where Ppipj and Ppisj are the cross mobility terms given by [28]

Ppipj = ∞

n=1



ηωω2

n + jω(ω2n − ω2

)

 4 ψδpi n (pi) ψ

δpj

n (pj )

m [(ω2

n − ω2)2 + ω4nη2]

Ppisj = ∞

n=1



ηωω2

n + jω(ω2n − ω2

)

 4 ψδpi n (pi) ψ

δsj

n (sj )

m [(ω2

n − ω2)2 + ω4nη2] (A.3)

where η is the structural damping, m is the plate mass, ω is the excitation frequency (assumed

same for primary and secondary excitation) and ωn is the nth resonant frequency given by

ωn =



EH2

12(1 − μ2)ρ

	n1π

Lx


2

+

	n2π

Ly


2



(A.4)

where n refers to the mode index (n1, n2), E is the modulus of elasticity, μ is the Poison

ratio and ρ be the density of the plate material. The function ψδp

n (p) is given by

ψδp

n (p) = n1π

Lx

cos (δp) cos 	n1πxp

Lx




sin 	n2πyp

Ly




+

n2π

Ly

sin (δp) sin 	n1πxp

Lx




cos 	n2πyp

Ly




(A.5)

where (x, y) are the coordinates of the point p and δp is the angle made by the moment axis

with the x axis. Note that for the primary (or secondary) patch, δpi (or δsi )=0, π/2, π and

3π/2 for all elements i along edges 1, 2, 3 and 4 respectively.

The total power input due to the primary patch is given by

P ̄p = 1

2

Re M∗

p



i

Wpi



(A.6)

Using (A.1) and (A.2) in (A.6), the total input power due to the primary patch can be

expressed as

P ̄p = 1

2

Re M∗

pPppMp + M∗pPpsMs

 (A.7)

193
where Ppp represents the equivalent angular mobility due to the primary patch given by

Ppp = 

i



j

Ppipj (A.8)

and Pps represent the cross angular mobility given by

Pps = 

i



j

Ppisj (A.9)

Similarly, the total power input due to the secondary patch can be shown to be

P ̄s = 1

2

Re (M∗

s PspMp + M∗s PssMs) (A.10)

where Psp = Pps represents the cross angular mobility and

Pss = 

i



j

Psisj (A.11)

represents the equivalent angular mobility due to the secondary patch.

194
Appendix B

Total Input Power and Global Energy

The modal response of the structure can be written as

w(ω, x, y) = 

i

qi(ω)φi(x, y) (B.1)

where φi(x, y) is the mode shape,

qi(ω) = 1

m

fi

Ωi

is the complex modal amplitude, m is the mass,

fi = 

k

Fk(ω)φi(xk, yk)

is the modal force and

Ωi(ω) = ω2

i (1 + jηi) − ω2

195
is the modal frequency factor where ωi is the natural frequency and ηi is the loss factor. The

overall time averaged kinetic energy and potential energy of the structure can be written as

[41]

Ekin = mω2

4



i

|qi|

2 (B.2)

Epot = m

4



i

w2

i |qi|

2 (B.3)

Similarly, the total input power can be expressed in terms of the modal amplitudes as

P ̄ = mω

2



i

ηiw2

i |qi|

2 (B.4)

Iin general there is no simple relation between P ̄ and Ekin and Epot. However, if the forcing

frequency is close to the natural frequency, then the total input power can be relation to the

total vibrational (kinetic) energy. If ω ≈ ωi, then a single mode dominates the response. In

such a case, we have

Ekin = Epot ≈

mω2

i

4 |qi|

2 (B.5)

P ̄ ≈

mωi

2 ηiw2

i |qi|

2 (B.6)

Thus, we have

P ̄ = 2ηiωiEkin ⇒ Ekin = P ̄

2ηiωi

(B.7)

Thus

min P ̄ ≡ min Ekin (B.8)

That is, minimizing the total input power is equivalent to minimizing the total vibrational

energy.

When the excitation frequency is not close to the resonant frequency, that is, ω 
≈ ωi, then

we can relate the total input power to the potential energy of the structure. Assume that

196
the loss factor is the same for all modes. That is, ηi = η for all i. In this case, comparing

(B.3) and (B.4), we get

P ̄ = 2ηωEpot (B.9)

The net input power is proportional to the potential energy at all frequencies. Thus, in this

case

min P ̄ ≡ min Epot (B.10)

That is, minimizing the net input power is equivalent to minimizing the total potential

energy

If there is high modal overlap, it can be argued that

Epot ≈ Ekin

for all frequencies except below the first resonance frequency. The argument is as follows

[41]

“Below the resonance potential energy of a particular mode is larger than kinetic, while the

opposite applies above the resonance. In the case of high modal overlap, at any particular

excitation frequency, the (lower) kinetic energy of modes having the eigenvalues above the

excitation frequency will become compensated by the (higher) kinetic energy of lower modes.

Inverse compensation effect will apply to potential energy. As a result, the global kinetic

and potential energies in modally overlapped systems can be shown to match closely at all

frequencies except below the first resonance where the compensation cannot take place”

That is,

P ̄ ≈ 2ηωEkin (B.11)

for all ω ≥ ω1. Thus,

min P ̄ ≡ min Ekin (B.12)

Thus, in general, if there is high modal overlap and if the loss factor is same for all modes,

then minimizing the total input power is equivalent to minimizing the total vibrational

energy for all frequencies except below the first resonance frequency.

197
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 4. TITLE AND SUBTITLE

Active Vibration Control for Helicopter Interior Noise Reduction Using 

Power Minimization 

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NNL11AA06C

 6. AUTHOR(S)

Mendoza, J.; Chevva, K.; Sun, F.; Blanc, A.; Kim, S. B.

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NASA Langley Research Center 

Hampton, Virginia 23681 

 

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This report describes work performed by United Technologies Research Center (UTRC) for NASA Langley Research Center (LaRC) under Contract 

NNL11AA06C. The objective of this program is to develop technology to reduce helicopter interior noise resulting from multiple gear meshing frequencies. A 

novel active vibration control approach called Minimum Actuation Power (MAP) is developed. MAP is an optimal control strategy that minimizes the total 

input power into a structure by monitoring and varying the input power of controlling sources. MAP control was implemented without explicit knowledge of 

the phasing and magnitude of the excitation sources by driving the real part of the input power from the controlling sources to zero. It is shown that this occurs 

when the total mechanical input power from the excitation and controlling sources is a minimum. MAP theory is developed for multiple excitation sources with 

arbitrary relative phasing for single or multiple discrete frequencies and controlled by a single or multiple controlling sources. Simulations and experimental 

results demonstrate the feasibility of MAP for structural vibration reduction of a realistic rotorcraft interior structure. MAP control resulted in significant 

average global vibration reduction of a single frequency and multiple frequency excitations with one controlling actuator. Simulations also demonstrate the 

potential effectiveness of the observed vibration reductions on interior radiated noise.

15. SUBJECT TERMS

control; excitation; noise; reduction; simulation; structural; vibration

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