1. 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 2005 San Diego, CA Averaging Analysis of State-Switched Piezostructural Systems

2. 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

3. Piezoceramic Inertial Actuator (PIA) Davis & Lesieutre, JSV, 2000 Motivation

4. 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

5. 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? Today’s Presentation Clark, Kurdila 2002 Kurdila, Lesieutre 2002

6. Stability of State Switched Systems Multiple Lyapunov Function Methods Two State Stability: “Stiff out, Soft in” Clark, Kurdila et al. 2002

7. Three State Stability: “Maximum Voltage” Kurdila, Lesieutre et al. 2002 Stability of State Switched Systems

8. Model Piezoceramic Vibration Absorber F D F m s F m a

9. Governing Equations Piezo Constitutive Law Electromechanical Equations Mechanical compliance at constant electric field Dielectric constant at constant stress piezoelectric constant Idealization

10. Governing Equations Represented as Discrete Capacitance Value C k Piecewise Affine Control System:

11. Averaging Analysis Two types of averaging theorems Our Problem: Averaged state space model for slow systems 1. Slow systems: state variables vary slowly with time 2. Mixed systems: include slow variables and fast variables Theorem: For slowly-varying systems, over time scale :

12. 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

13. State-Switching Control Strategy h (t) 0 DT T t 0 D 1 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.

14. Averaging Analysis Averaged Terms Averaged Equations of Motion:

15. Averaging Results: Time Domain 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)

16. Averaging Results: Frequency Domain D=1: short circuit (lowest frequency) D=0: open circuit (highest frequency) Effective Filter Bandwidth: 745.5 Hz

17. 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