The Idea of the Selection of PZT-Beam Interaction Forces in Active Vibration Protection Problem

Abstract

The paper concerns an active vibration protection (p-reduction) of the structure. This problem corresponds to the active vibration reduction (a-reduction). The quantity and effectivness of the a-/p-reduction, measured with reduction and effectiveness coefficient respectively, depends on, inter alia, the PZT distribution on the structure subdomains with the largest curvatures (quasi optimal distribution) are considered. The aim of this paper is to determine such interacting forces PZT-structure, assuming QO distribution of PZTs, which maximize the effectiveness of p-reduction. The beam clamped at one end, vibrating separately with first three modes, is chosen as the research object. The interacting forces are searched requiring that the shear force and bending moment at the clamped side are equal to zero. The total p-reduction is achieved for separate modes. Assuming the QO distribution of the PZTs, the best p-effectiveness is achieved. The validation of theoretical considerations is confirmed numerically.

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

“…As it was pointed out in [1,13,15], reduction of A f ;ν leads to reduction of the curvature κ f (x), the shear force Q(x), and the bending moment M (x); hereafter all these quantities are described jointly as…”

“…However, these techniques do not provide explicit distribution of the actuators and consistently, do not assure maximum eectiveness of the reduction. However, this problem is solved in [1,1012]. 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).…”

“…It turned out that the values of forces were minimal, so this means that the minimum energy is added to the system and consequently, the maximum eectiveness of the p-reduction is assured. An eectiveness measure of the reduction is an eectiveness coecient dened in [1,2,4,7,915].…”

“…It is proved in [1] that even one actuator bonded anywhere on the beam provides preduction of separate modes but the eectiveness may be poor.…”

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

“…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).…”