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Andrew J. Kurdila, Xiaoyan Zhang, Richard J. PrazenicaDepartment of Mechanical and Aerospace Engineering
University of Florida
George LesieutreDepartment of Aerospace Engineering
Pennsylvania State University
Chris NiezreckiDepartment of Mechanical Engineering
University of Massachussets Lowell
Presented at the 2005 SPIESmart Structures and Materials Conference
7 - 10 March 2005San Diego, CA
Averaging Analysis of State-SwitchedPiezostructural Systems
• Motivation: Tunable Vibration Absorbers- Discrete Notch Filters- Continuous Filtering
• Critical Issues- Stability of state-switched systems- Characterizing the system response (time and frequency domains)
• Governing Equations
• Averaging Analysis
• Numerical Examples
• Conclusions
Overview
Piezoceramic Inertial Actuator (PIA)
Davis & Lesieutre, JSV, 2000
Motivation
tuning bandwidth0.01
0.1
1
10
100
1000
5004003002001000
Frequency [Hz]
-180
-90
0
90
180
baseline passive (short) passive ( = 1) passive (open)
(forcing voltage = 12.6 V, sweep rate = 312.5 Hz/s)
MotivationVibration Suppression: Discrete Notch Filtering
Davis & Lesieutre, JSV, 2000
• Series of discrete notch filters defined by equivalent capacitance• Current Study: equivalent capacitance achieved by varying the duty cycle of a single switch – continuous notch filtering• Filtering bandwidth defined by short-circuit and open-circuit cases
Comments and Open Questions
i) Frequency domain analysis: Critical for evaluation of filtering properties.
ii) Laplace domain: initial conditions often neglectedSteady state only desired.
iii) Effects of switching on system stability: Quasi-steady? Fast Switching? O(KHz, MHz)!
iv) How do we define closed loop stability of theelectromechanical system and switching strategy?
v) What design / analysis methods for pulse widthmodulated (PWM) systems?
Today’s Presentation
Clark, Kurdila 2002 Kurdila, Lesieutre 2002
1 2 21 2
2
1( )
2i i i
XV X V k X mX
X
TransitionSets
Stability of State Switched Systems
Multiple Lyapunov Function Methods
Two State Stability: “Stiff out, Soft in”
Clark, Kurdila et al. 2002
V t K x t m x t C x t
V AC
C
k sc a k
Jk
j
( ) ( ) ( ) ( )
( )
*
**
*
FHG
IKJ
1
2
1
2
1
212
22
32
2
System k( )
D
V t K x m x C x
V AC
C
j sc a j
jk
j
( )
( )
*
**
*
1
2
1
2
1
212
22
32
BC
V t K x m x C x
C
CV A
k s a k
k
jj
( )
( )
*
*
**
1
2
1
2
1
212
22
32
System k( )
System j( )
V t K x m x C x
Vj A
C C
sc a j
j k
( )
( )
*
*
1
2
1
2
1
212
22
32
System j( )
Three State Stability: “Maximum Voltage”Kurdila, Lesieutre et al. 2002
Stability of State Switched Systems
Actuator mass
Structure mass
PZT
Ks
sy
C
am
m s
yi
s
ya
PZT
ModelPiezoceramic Vibration Absorber
,a D a aF F m y , ,( ) ( )s s a D s D s s im y t F F F k y y
FD F
ms
,s DF ( )s s ik y y
sy
F
ma
,a DF
ay
Governing Equations
233
3 dKL
AC sc
T
p
Piezo Constitutive Law
Ksc A
L S33E
Electromechanical Equations
3T33333 EdD
3333E333 EdS
Mechanical complianceat constant electric field
Dielectric constant atconstant stress
piezoelectric constant
Idealization
33
33
33
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
a sc sca a s a s
a a a
a sc s s scs a s a s s i s i
s s s s s
sca s
p k
C K K dy y y y y V t
m m m
C K C K K dy t y y y y y y y y V t
m m m m m
K dV t y y
C C
Governing Equations
33
33
33 33
0 0 1 0 0
0 0 0 1 0 0 0
0 0
0 0( ) ( )
0 00 0 0
asc sc a a sc
sa a a a a
asc s s ssc a a s sc
ss ss s s a s
sc sc
p k p k
yK K C C K d
ym m m m m
x t x tyK K K CK C C C K d
ym mm m m m m
VK d K d
C C C C
i
i
y
y
Represented asDiscrete CapacitanceValue Ck
( ) ( ) ( ) kk kx t A x t B u t F Piecewise Affine Control System:
Averaging Analysis• Two types of averaging theorems
( , )x f t x
• Our Problem: Averaged state space model for slow systems
0 ( )avg avgx f x0
1( ) lim ( , )
t T
Tt
f x f t x dtT
( , , )
( , , )
x f t x y
y g t x y
1. Slow systems: state variables vary slowly with time
nx R2. Mixed systems: include slow variables and fast variables
• Theorem: For slowly-varying systems, over time scale : 1/
abs avgx t x t O
Assumptions:
Averaging Analysis
(1) The switching function has period T• Switching rate depends on hardware used to realize the switch in the shunt circuit• Period T may be measured in microseconds
(2) The base motion and its time derivative have characteristic time constants dictated by the structural response
• If the frequency of the base motion is O(10) – O(1000) Hz, the period may be measured in milliseconds• There is a three order-of-magnitude difference between the switching and structural periods• Structural period is given by NT, N>>1
iy t iy t
State-Switching Control Strategy
h(t)
0 DT T t0 D 1
( )h
1
0 1 11
1 1 1 1 1ˆ ˆ ˆ( ) ( ) ( )
NN n D n
n n Dnp p p
d d dN NC Ch C Ch C Ch
33330
1 1ˆ( )
Nsc
scp pp
K d D Dd K d
N C C CC Ch
• Averaging of the capacitance terms:
• Duty Cycle: Fraction of T when switch is closed - determines an equivalent capacitance or stiffness (resulting in a notch frequency)
C.O.V.
Tt
33
33
33 330 0
0 0 1 0 0
0 0 0 1 0
( )( )
1 10 0 0
sc sc a a sc
a a a a a
sc ssc a a s sc
s s s a s
sc scp p p p
K K C C K d
m m m m md x tx tK KK C C C K ddt
m m m m m
D D D DK d K d
C C C C C C
Averaging Analysis
0
0
0
( ) ( )
0
s si i
s s
C Kavg y t avg y t
m m
Averaged Terms
Averaged Equations of Motion:
Averaging Results: Time Domain
0 2 4 6 8 10 12
x 10-7
-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-8
Time(s)
Dis
plac
emen
t(m
)
Avergaged system: Actuator mass response for different duty cycle
Duty cycle = 0
Duty cycle = 1
Duty cycle = 0.6
Duty cycle = 0.2
• Simulation example: ma/ms = 1/1000• Comparison of averaged response and true simulated response• Varying duty cycle (D=0: open circuit, D=1: short circuit)
Averaging Results: Frequency Domain
2
2 2 2
ˆ ( )( )
ˆ ( )( )s s a a
a s s a a ai
Y k k mj
k k m k m kY
• D=1: short circuit (lowest frequency)• D=0: open circuit (highest frequency)• Effective Filter Bandwidth: 745.5 Hz
0 0.5 1 1.5 2 2.5 3 3.5
x 10-10
8.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8x 10
7
Ck
Ka
Effective stiffness Ka
Conclusions
• Objective: develop an analysis framework for studying the
vibration of switched piezostructural systems
• Approach: apply averaging analysis, a well-established tool for analyzing switched power supplies, to vibration absorbers
• Averaging method assumes 2 time scales:- Pulse width modulation (PWM) time scale - Structural system time scale (3 order-of-magnitudes larger)
• Results of averaging analysis:- Compact expression of vibration response as a function of the
duty cycle D- New concept for creating vibration absorbers based on PWM
(continuous filtering as opposed to discrete notch filters)- Need for experiments to validate this approach
• Future work: apply to energy harvesting topologies