In this note, the results are presented of some analytical and numerical studies on the active control of non-linear panel vibrations and sound radiation due to wall pressure fluctuation. In our previous work [1], we treated the problem of passive control of non-linear panel vibration by boundary damping without the wall pressure excitation. It was shown there that a slight boundary damping can result in an exponentially fast decay in vibrational energy. However, with a persistent excitation, the passive control is ineffective. It seems necessary to apply an active control force as an effective counter-measure. The main purpose of this note is to study, based on a non-linear panel model, the effectiveness of such control to suppress the panel vibration and sound radiation induced by the unsteady pressure forcing.
Without control, non-linear motion of elastic panels has been studied by E. H. Dowell [2, 3], Nayfeh and Mook [4] and others. Experimental and numerical studies of such non-linear interaction problems with and without control were carried out by Maestrello and his collaborators [5][6][7]. Analytically, optimal control of beams or panels with concentrated forces as actuators was treated by Su and Tadjbakhsh [8], by neglecting the non-linear tension term and the wall pressure excitation. However, so far, little analytical work on optimal control of non-linear panel vibrations and sound radiation has been undertaken.
In this note we deal with the forced vibration due to wall pressure alone. The control consists of a distributed force applied normally to one side of the wall. For simplicity, the flexible panel is assumed to be hinged to rigid plates at both ends (see the schematic diagram in Figure 1). In section 2, the coupled equations governing the non-linear panel vibration and acoustic radiation problem are given. The optimal control problem is formulated in section 3. For the optimality criterion, a time-average cost or objective functional is introduced to measure the performance in controlling the vibration and sound radiation. In section 4, by applying the variational method, we derive the optimality equation for the control force distribution which is coupled with the controlled equations of motion. By using an eigenfunction expansion, the modal control problem is discussed in section 5. Then we solve a truncated modal control problem numerically by the shooting method for a two-point boundary value problem in time domain. The numerical results are described in section 6 and shown in Figures 2-13. In the last section, the main results of the note are summarized and discussed to reach the conclusions of this study.