An analytical investigation of the active control of the transmission of sound through plates

An analytical model of a vibrating plate is used to explore the mechanisms of control involved in the reduction of harmonic sound transmission through the active control of plate vibration. A general model of the plate vibrations (previously used by Berry et al.) is presented, with the boundary conditions of the plate being modelled as variable rotational and translational elastic restraints at the edges of the plate. A variational formulation of the problem is presented and the equation 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 edges are considered. For the rigid panel it is found that large attenuations in transmitted sound can be obtained if either three control actuators are used (one for each rigid body mode) or the motion of the panel is restricted to piston type translation along an axis perpendicular 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” where the relative amplitudes and phases of plate modes are adjusted to produce a vibration distribution of low radiation efficiency. The latter mechanism is shown to be both subtle (often involving little change in mode amplitudes and little change in the overall vibration amplitude of the plate) and very effective (accounting for a large proportion of the attenuations in radiated sound at most frequencies). For the frequency range of interest (0.92 < ka < 2.75) large attenuations in the transmitted sound could always be achieved if the number and positioning of the secondary forces is correctly chosen. The simulations also demonstrate the phenomenon of “spill-over”, where inefficiently radiating plate modes are strongly excited with the introduction of control often resulting in an overall increase in the plate vibration.