AN ANALYTICAL INVESTIGATION OF THE ACTIVE CONTROL OF THE TRANSMISSIONOF SOUND THROUGH PLATES

An analytical model of a vibrating plate is used to explore the mechan isms of control involved in the reduction of harmonic sound transmissi on through the active control of plate vibration. A general model of t he plate vibrations (previously used by Berry et al.) is presented, wi th the boundary conditions of the plate being modelled as variable rot ational and translational elastic restraints at the edges of the plate . A variational formulation of the problem is presented and the equati on of motion of the plate is derived by using a Rayleigh-Ritz method. Both an elastic plate with clamped edges and a rigid panel with free e dges are considered. For the rigid panel it is found that large attenu ations in transmitted sound can be obtained if either three control ac tuators are used (one for each rigid body mode) or the motion of the p anel is restricted to piston type translation along an axis Perpendicu lar to the panel. For the elastic plate two mechanisms of control are observed: ''modal control'' where the amplitudes of those modes which dominate sound radiation are reduced, and ''modal rearrangement'' wher e the relative amplitudes and phases of plate modes are adjusted to pr oduce a vibration distribution of low radiation efficiency. The latter mechanism is shown to be both subtle (often involving little change i n mode amplitudes and little change in the overall vibration amplitude of the plate) and very effective (accounting for a large proportion o f the attenuations in radiated sound at most frequencies). For the fre quency range of interest (0.92 < ka < 2.75) large attenuations in the transmitted sound could always be achieved if the number and positioni ng of the secondary forces is correctly chosen. The simulations also d emonstrate the phenomenon of ''spill-over'', where inefficiently radia ting plate modes are strongly excited with the introduction of control often resulting in an overall increase in the plate vibration.