Natural vibrations and hydroelastic stability of a plate with a piezoelectric element connected to an external Rl-circuit

The possibility of passive damping of harmonic vibrations and controlling the stability boundary of the plate interacting with the flowing fluid is investigated. The key idea behind the applied vibration control method is to connect the piezoelectric element located on the surface of the structure to an external shunt circuit. The selection of parameters for such circuit, providing the highest rate of vibration damping or the maximum change in the critical velocity of the fluid flow, is performed by solving a series of eigenvalue problems. Two mathematical formulations are considered. The first formulation is based on the three-dimensional equations of the linear theory of piezoelasticity, and the second one is a simplification of these equations with the aim of using them in conjunction with the theory of thin plates. The dynamics of an ideal fluid in both cases is described by a wave equation formulated for the perturbation velocity potential. Together with the impermeability condition and the boundary conditions it is transformed to the weak form. The hydrodynamic pressure is calculated by the linearized Bernoulli formula. The developed finite-element algorithms are verified and their computational efficiency is compared. A change in the complex eigenvalues of the electromechanical system is analyzed depending on the resistance and inductance of the electric circuit connected to the piezoelectric element. The values of these parameters providing the best damping of resonant vibrations of a rectangular plate interacting with the flowing liquid have been selected based on the solution of the optimization problem. The numerical studies have shown that the selected values lead to a smaller change in the frequency spectrum of the original system and provide a higher rate of vibration damping as compared to the values found by the known analytical expressions. The numerical investigation has been performed for two variants of boundary conditions set at the edges of the structure. It is demonstrated that the use of a passive electrical circuit cannot affect the loss of stability by divergence, but is able to change the critical flutter velocity by a few percent.