Method for determining the parameters of an electrical signal for controlling forced steady-state vibrations of electroviscoelastic bodies. Mathematical relations

This paper presents a method for determining the magnitude of the electric potential generated on the electrodated surface of a piezoelectric element, which is part of a piece-wise homogeneous electroviscoelastic structure, necessary for the formation of a control action when actively controlling its dynamic behavior in the mode of forced steady-state vibrations in order to minimize the amplitude of vibrations at the selected resonant frequency. By mathematical transformations of the equations describing the intrinsic and forced vibrations of such electroviscoelastic bodies, the relations expressing the relationship between the values of the displacement of the nodes and the electric potential on the electroded surface of the piezoelectric element are derived. These formulas allow us to determine the magnitude of the potential that must be applied to the piezoelectric element in order to best dampen a given vibration mode of the structure. As a result of numerical experiments obtained by using the ANSYS finite element analysis software package, and the usability of the results of solving the problem of natural vibrations to find the optimal value of the potential characterizing the control electrical action aimed at damping the specified modes in the mode of forced steady-state vibrations is confirmed. The effectiveness of the obtained analytical dependencies is demonstrated by the example of a cantilevered viscoelastic plate with a piezoelectric element located on its surface. The proposed approach makes it possible to significantly reduce time and resource costs of the mathematical modeling of active control of forced steady-state oscillations of electroviscoelastic bodies, to determine the requirements for the hardware implementation of actuators and controllers of the control unit of such smart-systems.