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

Abstract

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.

Effectiveness Analysis of the Beam Modes Active Vibration Protection with Different Number of Actuators

“…It was proved there, that the most eective actuators distribution is on the structure sub-domains with the largest curvatures; such distribution is called quasi-optimal one (QO). In the quoted papers, the QO-distribution is deduced based on the heuristic reasons and it was conrmed theoretically in [13]. So hereafter such distribution may be regarded as optimal one (O-distribution) and it is proved that the maximum eectiveness of the preduction of the beam separate modes may be achieved for O-distribution.…”

“…The p-reduction is a particular case of active vibration reduction (a-reduction) [13]. Both a-and p-reduction are realized with actuators.…”

Comparison of Formulas Obtained for Analytical and LQ Idea Approaches to Determine the Optimal Actuator Location in Active Multimodal Beam Vibration Reduction

“…There are several kinds of reduction: a-reduction ([a]ctive) -a special case of a-reduction is p-reduction ([p]rotection) -0-reduction (lack of reduction) and ireduction ([i]ntensify vibration) (Brański, Lipiński, 2011). There are also many structures of which the vibrations are reduced via actuators, from the simple beam (Augustyn et al, 2014;Brański, 2011), through the plate (Wiciak, 2007) to the very complicated structure (Kozień, Wiciak, 2008).…”

“…The a-reduction, via actuators, is described for example in (de Silva, 2000; Hansen, Snyder, 1997; Fuller et al, 1997;Wiciak, 2008). It depends on many factors enumerated in (Bruant, 2010;Brański, Lipiński, 2011;Brański, 2011). It seems that the most important factor is the distribution of the actuator on the structure, because once bonded actuator cannot change the place.…”

“…In the recent years, a great number of papers has been published on this subject, see for example (Guney It has been proved, in own papers, that for separate modes the most effective distribution of actuators is in the sub-areas with the largest curvatures. Moreover, the p-reduction of the vibration is possible (Brański, Lipiński, 2011;Brański et al, 2010). From mathematical point of view, this distribution is called an optimal one (O-distribution).…”