Problem on natural vibrations of electroviscoelastic bodies with external electric circuits and finite element relations for its implementation

In this paper the mathematical statement for solving of the problem of natural vibrations of piecewise homogeneous electro-viscoelastic bodies with a passive external circuit of different configurations is proposed. The results of the problem solution are complex natural vibration frequencies, which characterize the natural vibration frequencies of bodies and their damping rates. These characteristics can be efficiently used to analyze dynamic processes in smart systems, i.e. materials exhibiting a piezoelectric effect. Recently these systems have found many applications in different branches of science and industry. Their capability to control and govern the dynamic behavior of structures can be greatly expanded by connecting external circuits of different combinations of resistive, capacitive and inductive elements to piezoelectric elements. The search for the optimal parameters of smart-systems is based on the appropriate choice of piezoelectric element arrangements, electric circuit schemes and circuit elements. When considering problems with a large number of parameters, the modeling is one of the efficient approaches for finding the optimal solution. Application software packages currently available for solving natural vibration problems do not provide direct solution to the problem formulated here. Therefore, this study involves finite-element relations, including those for the original finite element representing an external circuit, which, taking into account the well-known types of finite elements for continuous media description, form the basis for constructing numerical algorithms. As an example, calculations of the complex natural vibration frequencies of a half cylinder with an attached piezoelectric element and an external series RL-circuit are performed.