Topological design of piezoelectric actuator layer for linear quadratic regulator control of thin-shell structures under transient excitation

Abstract

The purpose of this work is to develop a topology optimization scheme for thin-shell piezoelectric smart structures under the linear quadratic regulator optimal control for suppressing transient vibrations. The transient responses of the dynamic system are found by the finite element method based on the classical plate theory and a time-integration algorithm. A mode superposition method is employed to improve the computational efficiency. The pseudo-densities indicating the layout of piezoelectric actuators are taken as the design variables. The sensitivity analysis for a general time-integral function of transient structure response under optimal control is derived with the adjoint-variable method. A Lyapunov equation is solved to determine the derivative of the feedback gain in the sensitivity analysis. In the numerical examples, the integral of the structural response over a specified time interval is chosen as the objective function. Then, the optimization problem is solved with a gradient-based programming algorithm. The optimization results illustrate the validity of the proposed method. It is shown that the effects of active control have been substantially improved through optimization. The influences of some key factors on the optimized design are also discussed.