The problem solution of natural vibrations of electroelastic bodies with external electric circuits based on their electrical analogue

The optimization of dissipative properties of electroelastic bodies with external electric circuits suggests the selection of parameters for these circuits which can provide the most effective damping of vibrations at the prescribed frequency. Since the external circuits can be represented as a set of elements with concentrated parameters connected to the system with distributed parameters, solving the electroelastic problems for such systems using their full formulation requires high computational and time costs. In mechanics, there exist several approaches, which allow the representation of the mechanical systems with distributed parameters as discrete systems with concentrated parameters of the spring-mass-damper type. In this paper, an approach to the analysis of the dynamic behavior of electromechanical systems with external electric circuits is developed based on the equivalent electric circuits representing discrete electric systems with concentrated parameters. The above discrete systems are the analogues of the initial electromechanical systems in relation to frequency characteristics and electrical processes occurring in them. The solution to the problem on natural vibrations of electromechanical systems are the complex natural vibration frequencies, the real part of which defines the frequency of vibrations and imaginary part - the damping index. The present paper considers the problems of determining the magnitude of elements parameters of the equivalent circuit. The proposed approach made it possible to obtain the mathematical relations binding the limit values defining the dynamic characteristics of systems under consideration (boundaries of the frequency range for natural vibration frequencies change, maximal reachable damping index, the valued of resonant frequency of a system when the resistive external circuit is attached) with the values of natural vibration frequencies of the original structure with piezoelectric element operating in the open circuit and short circuit modes.