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