Lagrange equations applied to flexural mode transducers

This paper deals with the flexural vibrations of composite piezoelectric transducers excited by electrode pairs configured to produce electric fields in phase opposition. Lagrange equations are used to analyze the electrodynamic characteristics of the transducers. These equations are essentially equations of dynamic equilibrium constructed in terms of derivatives of the kinetic and potential functions of the electromechanical system, themselves expressed in terms of generalized coordinates. For this class of transducers, the strains and electric field are chosen as the independent variables and the electric enthalpy function is the proper thermodynamic potential for use in the Lagrange equations. The present analysis takes account of the strain-induced electric fields in the composite transducer which arises when the strain and electric field have a common direction. The generalized electromechanical equations thus derived are suitably represented by an electric network based on the classical system of analogies. A set of generalized mode impedances is defined for each vibratory mode and is shown to play a significant role in the analog network. By applying known formulas to such a network, the electrodynamic characteristics of the transducer are readily obtained. Subject Classification: 85.52; 40.22.