Analytical Determination of the PZTs Distribution in Active Beam Vibration Protection Problem

The paper concerns an active vibration protection (p-reduction) of the structure via piezoelectric transducers; p-reduction corresponds to an active vibration reduction (a-reduction). The quantity and effectiveness of the (a- or p-) reduction, among other parameters, depend on the piezoelectric transducers distribution on the structure. The best results are obtained bonding piezoelectric transducers to the structure in the sub-domains with the largest curvatures; it is so-called quasi-optimal distribution of the piezoelectric transducers. Up to now, the quasi-optimal distribution was determined based on heuristic reasons only. The aim of the paper is to confirm quasi-optimal distribution in analytical way. The beam clamped at one end, vibrating with first three modes separately, is chosen as the research object. It is assumed that the piezoelectric transducers are exactly the same. Demanding the vibration amplitude to be equal to zero (i.e. p-reduction condition), the general formula for interacting forces piezoelectric transducers-beam is derived. Next, such an appropriate distribution of piezoelectric transducers is searched analytically, that the minimal forces are achieved; it leads to the best reduction effectiveness. It turned out that the analytical method pointed out quasi-optimal distribution of the piezoelectric transducers. The validation of theoretical considerations is confirmed numerically.