Rayleigh-Ritz formulation for active control of the acoustics in cabin enclosures

This numerical study presents a detailed optimal control design based on the Rayleigh-Ritz approach for the smart plate-cavity system. Linear quadratic (LQ) theory with output feedback is considered on the basis of the state space model of the system. A vibroacoustic model, which includes a rectangular shaped cavity, enclosed with a five rigid walls and a flexible smart plate with discrete piezoelectric sensor/actuator pairs bonded to its surface. Classical laminated plate theory is used to model the composite plate and electroelastic theory is used model the discrete piezoelectric patches. Eigenfunctions of a clamped-clamped beam are used as the Ritz functions for the panel and the rigid walled cavity modes are used the model the acoustic cavity. The dynamic equations of motion for the coupled smart panel-cavity system are derived using Hamilton's principle. The forcing term due to the cavity acoustic pressure is determined by using virtual work considerations. For the present study, five collocated pairs of sensor/actuator pairs are attached to the plate at a predetermined placement scheme. The performance index considered for the design of the optimal controller includes both the displacement of the panel and the pressure inside the cavity. Numerical simulation is used to predict the reduction in the sound pressure level inside an enclosure radiated from this optimally controlled plate. The Rayleigh-Ritz approach is found to be faster and a more efficient method for designing control system for simple plate-cavity systems when compared to other numerical methods such as the finite element method.