Keywords: vibration control, composite beam, thermal loading, piezoelectric actuation.

Numerous studies have demonstrated the feasibility of suppressing structural deformation through application of an appropriate electric field to piezoelectric elements incorporated within the structure. The present authors [1,2], for example, studied control of forced vibrations of layered beams and plates subject to rapidly applied thermal and/or mechanical disturbances. It is shown in [2] that the induced deflections in a structure can be reduced significantly if constant amplitude electric pulses are applied to piezoelectric layers when the uncontrolled deflections reach peak values and discontinued after one-half the fundamental period of vibration. All layers of the structure were assumed to have identical thermal properties, and the amplitudes of the electric pulses used to control the response were determined by trial-and-error. The present paper extends the authors' earlier works by considering a simply-supported composite beam in which the thermal properties of a structural layer differ from those of the adjacent piezoelectric layers. An optimization procedure is employed to determine the characteristics of applied electric pulses for optimum vibration control.

The beam under consideration consists of a central thermoelastic structural layer onto which two outer piezothermoelastic layers are perfectly bonded. The bottom free surface of the beam is subject to a sudden temperature rise, while the top free surface is thermally insulated. The resulting transient temperature field produces a thermal moment that induces flexural vibration of the beam. Control of the vibration is provided by the electric moment resulting from electric pulse loading applied to the piezoelectric layers. In evaluating the electric moment, it is assumed that the electric field resulting from variations in stress or temperature is insignificant compared with that produced by the electric loading applied to control the vibration. In this investigation the electric loading consists of pulse voltages applied to the piezoelectric layers at time and terminated at time . For optimum vibration control there are three parameters to determine, namely , and pulse duration .

The Laplace transform technique is employed, first to solve the transient heat conduction problem for the temperature field throughout the beam, and secondly to solve the equation of motion for the transverse beam deflection. The resulting deflection is expressed as the superposition of the displacement due to the thermal moment and the displacement due to electric moment , i.e., . The thermally-induced displacement at the center of the beam is further decomposed into a quasi-static thermoelastic component and a thermoelastic vibration component .

The conditions imposed in order to suppress the thermoelastic vibration displacement include: (i) , where and represent, respectively, the values of and associated with the fundamental mode of vibration ; (ii) , where is the value of for the next higher mode of vibration ; and (iii) determination of parameter such that the quantity is a minimum, where is the ratio of the sum of the displacements of the electroelastic vibration for at the center of the beam to the maximum displacement of electroelastic vibration for . Conditions (i) - (iii) are used to obtain a set of relationships that can be used to determine the amplitude, time of initiation, and duration of electric pulses for optimum vibration suppression.

The proposed control strategy is applied to a beam consisting of an aluminum host layer and PZT piezoceramic layers. Numerical results show that the vibrations induced by a sudden temperature rise on the surface of the beam are virtually eliminated when the pulse parameters established in this paper are utilized.

1

T.R. Tauchert, S. Adali and V.E. Verijenko, "Piezo-Control of Forced Vibrations of a Thermoelastic Beam", Proceedings of the 4th International Congress on Thermal Stresses, Osaka, Japan, 477-480, 2001.

2

T.R. Tauchert and F. Ashida, "Control of Thermally-Induced Structural Vibrations via Piezoelectric Pulses", Proceedings of the IUTAM Symposium on Dynamics of Advanced Materials and Smart Structures, Yonezawa, Japan, 397-408, Kluwer Academic Publishers, 2003.

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