Semi-analytical solutions for optimal distributions of sensors and actuators in smart structure vibration control

In this paper, the optimal design of vibration control system for smart structures has been investigated semi-analytically via the optimization of geometric parameters like the placements and sizes of piezoelectric sensors and actuators (S/As) bonded on the structures. The criterion based on the maximization of energy dissipation was adopted for the optimization of the control system. Based on the sensing and actuating equations, the total energy stored in the system which is used as the objective function was analytically derived with design variables explicitly presented. Two cases of single and combined vibration modes were addressed for a simply supported beam and a simply supported cylindrical shell. For single vibration mode, the optimal distributions of the piezoelectric S/As could be obtained analytically. However, the Sequential Quadratic Programming (SQP) method has to be employed to solve those which violated the prescribed constraints and to solve the case of combined vibration modes. The results of three examples, which include a simply supported beam, a simply supported cylindrical shell and a simply supported plate, showed good agreement with those obtained by the Genetic Algorithm (GA) method. Moreover, in comparison with the GA method, the proposed method is more effective in obtaining better optimization results and is much more efficient in terms of computation time.