Andrew J' Kurdila, Xiaoyan Zhang, Richard J' Prazenica

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Averaging Analysis of State-Switched Piezostructur
al Systems
Andrew J. Kurdila, Xiaoyan Zhang, Richard J.
Prazenica Department of Mechanical and Aerospace
Engineering University of Florida George
Lesieutre Department of Aerospace
Engineering Pennsylvania State University Chris
Niezrecki Department of Mechanical
Engineering University of Massachussets
Lowell Presented at the 2005 SPIE Smart
Structures and Materials Conference 7 - 10 March 2
005San Diego,  CA 
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Overview

• 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

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Motivation
Piezoceramic Inertial Actuator (PIA) Davis
Lesieutre, JSV, 2000
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Motivation
Vibration 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

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Comments and Open Questions i)
Frequency domain analysis Critical for
evaluation of filtering properties. ii)
Laplace domain initial conditions often
neglected Steady 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 the electromechanical system and
switching strategy? v) What design / analysis
methods for pulse width modulated (PWM) systems?
Clark, Kurdila 2002 Kurdila, Lesieutre 2002
Todays Presentation
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Stability of State Switched Systems Multiple
Lyapunov Function Methods
Two State Stability Stiff out, Soft in Clark,
Kurdila et al. 2002
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Stability of State Switched Systems
Three State Stability Maximum Voltage Kurdila,
Lesieutre et al. 2002
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Model
Piezoceramic Vibration Absorber
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Governing Equations
Idealization
Piezo Constitutive Law
Dielectric constant at constant stress
piezoelectric constant
Mechanical compliance at constant electric field
Electromechanical Equations
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Governing Equations
Represented as Discrete Capacitance Value Ck
Piecewise Affine Control System
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Averaging Analysis

• Two types of averaging theorems

1. Slow systems state variables vary slowly
with time
2. Mixed systems include slow variables and
fast variables

• Our Problem Averaged state space model for
  slow systems

• Theorem For slowly-varying systems, over
  time scale

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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, Ngtgt1

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State-Switching Control Strategy

• Duty Cycle Fraction of T when switch is
  closed - determines an
• equivalent capacitance or stiffness
  (resulting in a notch frequency)

C.O.V.

• Averaging of the capacitance terms

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Averaging Analysis
Averaged Equations of Motion
Averaged Terms
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Averaging Results Time Domain

• Simulation example ma/ms 1/1000
• Comparison of averaged response and true
  simulated response
• Varying duty cycle (D0 open circuit, D1
  short circuit)

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Averaging Results Frequency Domain

• D1 short circuit (lowest frequency)
• D0 open circuit (highest frequency)
• Effective Filter Bandwidth 745.5 Hz

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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