Electromechanical transducer and stochastic damages. P. Will University of Applied Sciences Mittweida, Germany

Transcription

1 Electromechanical transducer and stochastic damages P. Will University of Applied Sciences Mittweida, Germany Keywords: Microdamaging, piezoelectric damper Abstract The paper focuses on the numerical simulation of a piezoelectric damper under stochastic microdamaging. Its influence on a tuned mechatronic transducer is studied using a local cumulative damage measure. 1. Introduction In reality, there is some degree of randomness associated with microscopic failure initiation and damage propagation in composite structures. Accordingly, a local probability density for failure initiation was published by the author at the last MicroMat Conference. This cumulative, pointwise failure distribution [1,2] [ ] ( ) ( Q ) = - ( ) 0 D xyzq,,, Ú Vo 1 PqV, o l xyzqdq,,, 0 D 1 (1) simulates the local, stochastic damage. The symbol l defines the local failure rate in an infinitesimal subdomain at point (x,y,z). The term in the square brackets is the survival probability of the whole volume V o up to a current load q. Note that the survival probability as well as the local failure rate themselfes vary with the damage level D, Therefore the definition (1) becomes the structure of an integral equation. The local measure D accounts for both the accumulation of microdamaging and the load redistribution to undestroyed subregions. Taking into account the sense of D it has been proved [2] among other things that brittle fracture is no more governed by a singularity. Any singularity dominated stress field is preceded by a crack tip damage zone and so it disappears. The following reflections test the effect of local damaging upon mechatronic transducers. A monomorph, piezoelectric transformer which operates in bending vibration as damper within a mechanical structure is simulated numerically. The transformer is provided for damping the second resonant mode of selected structures. 2. Simulation Figure (1) schematically illustrates the characteristics of the piezoelectric transformer. The analysis focuses on the theoretical simulation of controlling mechanical measurements (Q,v) by electric couple reactions (U,I). Local microdamaging and load redistribution are taken into account. The numerical simulation bases on an experimental strength distribution.

2 ÊQˆ Ë v 1 Ê1 Z t Ë0 1 mech 2 ˆ t Ê ˆ Ê ˆ Ë 0 1 ËiwC 1 ÊUˆ R L Ë I Figure 1: piezoelectric damper The complex, mechanical impedance characterizes inertial forces, the compliance n, and velocity-proportional damping (degree D) of the piezoelectric component. Z = 1 Ê mech D i n + Ê W Ë - w ˆ ˆ 2 w Ë w W (2) It is expressed in terms of the excitation angular frequency W normalized by the natural angular frequency. The parameter C denotes the capacity of the piezoelectric element. Applying the input part with load Q and deflection rate v generates AC voltage U by the coefficient of transformation t. Note that the piezo ceramic itself is damped slightly. Just feedback coupling due to the circuit elements R, L improves the damping behaviour essentially. Tuning parameters in optimizing this feedback with regard to the resonant mode w are two electronic angular frequencies: w CR = 1 R w RL = RC L (3) In order to avoid the heterogeneous electro-mechanical structure of the damper model an lumped-constant equivalent circuit may be used. U 1 = Q/t I 1 = vt L m R d C n C U I All mechanical components (mass, internal damping, compliance) should be substituted by electric equivalents (inductivity, resistance, capacity) in this case. The following section informs about results of the simulation. It analyses changes in damping behaviour under local failure. Random damaging in the outer fibres of the transformer under pure bending is indicated by a change in the second moment of the area [3]:

3 ( ) = - ( ) IQ A o 3. Results [ ] ÚÚ 1 2 D zq, zda (4) Figure (2) illustrates the effect of local damaged cross sectional areas which offer reduced resistance to bending. pure bending load Figure 2: second moment of (damaged) area determined by numerical simulation Initially, it decreases at small rates with the damage level. When the local damage penetration approaches the centre of the section the second moment of the area decreases rapidly for very small increases in load. The structure fails by collapse at this limit load. pure bending second moment of (damaged) area natural frequencies (normalized) load Figure 3: natural frequencies correlated with second moment of the area

4 The resonant angular frequency w of the undamped system varies with the local damage level D too. Assuming a bending vibration mode it correlates with the load [4] like I 1/2 vs. Q (s. Figure 3). Finally, the coefficient of transformation t changes as the net cross sectional area of the piezo ceramic. Applying these intermediate results yields figure (4) which illustrates the mechanical admittance of the whole damper as a function of excitation frequency W. The doted curve denotes the influence of microdamaging on damper s tuning. v / Q 1.5 W / w structure: without damper with piezoelectric damper (in tune) with damaged damper (out of tune) Figure 4: frequency response The diagram takes into account back coupling reactions due to the load impedance (R,L). Small changes in the second moment of the area (10%) and in the coefficient of transformation due to local damaging cause already considerable unbalances of the damper. Assuming experimental confirmation theoretical simulation is able to predict the modified transformation behaviour of heterogeneous, electro-mechanical transformers under random damage. References 1. Will P., Helbig S., Lokales kumuliertes Schadensmaß - Rissspitzen ohne Singularität, Mat.-wiss. u. Werkstofftech. 28(1997)10, Will P., Helbig S., Ein lokales Schadensmaß zur stochastischen Modellierung des Versagensverhaltens von spröden Werkstoffen, Proceedings: Workshop der AG Stochastische Modelle für Qualität, Zuverlässigkeit und Sicherheit (e.v.), puplished in: Wissenschaftliche Berichte, Wissenschaftliche Zeitschrift der Hochschule Mittweida, 5 (1999), 19-22

5 3. Will P., Helbig S., Wachstum stochastischer Schäden im inhomogenen Spannungsfeld, Material Mechanics Fracture Mechanics Micro Mechanics (Anniversery Volume in Honour of B. Michels 50th Birthday), (Eds. Winkler T., Schubert A.), Fraunhofer IZM, Berlin, (1999) Will P., Lämmel B., Kleine Formelsammlung Technische Mechanik, 2. verbesserte und erweiterte Auflage, Fachbuchverlag Leipzig im Carl Hanser Verlag (1998), Author s address: Prof. Dr. P. Will University of Applied Sciences Media and Electrical Engineering D09648 Mittweida, Germany PO box 91