# Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations
## 1 Introduction  
Vibrations in flexible and lightweight structures are often a matter of concern for mechanical and aeronautical engineers since they promote the formation of cracks that weaken the mechanical components and lower their fatigue resistance. In order to increase their efficiency, the aerospace and mechanical industries have made extensive use of lightweight, robust, and low-cost materials. In gas turbine engines, for example, the vibrations result from the interaction between the fluid and the blades, and vibration reduction systems are widely used to increase the fatigue life of mechanical components [1; 2; 3; 4]. Damping systems can be classified into passive and active: The former exhibit good effectiveness, low complexity, and low cost, but show a small bandwidth. The latter present a large bandwidth and can adapt their features to time-dependent loads, so these are more efficient than a passive system. Smart materials developed in the last several decades have shown a great potential for monitoring and control applications and, among these piezoelectric materials, are the most frequently used when a high frequency and a high transient response is required [5; 6]. These materials are unique because their mechanical behavior is related to their electric behavior and vice versa [7]. In particular, the strains induce an electric field (a direct effect) and vice versa (an inverse effect). This peculiarity enhanced its applicability in many engineering sectors. One of the materials most commonly used in vibration control applications is lead zirconate titanate (PZT) [8; 9]. However, the efficiency of the PZT plates, to damp single-mode or multimode vibrations, depends strictly on their placement. PZT element optimal placement usually concerns the minimization or maximization of an objective function [10; 11]. Early studies on single-mode attenuation were carried out by [7]. The evidence reveals that the element should be placed in the highest average deformation area, and further studiesseem to support this result [12; 13]. If a cantilever beam is considered, the maximum damping effect occurs when the element is placed close to the joint [14; 15; 16]. In the simply supported beam case, it has been demonstrated by [17] that the maximum attenuation of a single mode occurs if the PZT lies in the region between the nodal lines. The studies of [PERSON] et al. [18] show that, in order to excite a specific single flexural mode of an Euler-Bernoulli beam, the actuator must be placed within two consecutive zero curvature points. Further studies have been performed to identify the optimal size and positioning of two pairs of PZT plates, in the case of different constraint conditions [19]. Other researchers analyzed the interaction between damping efficacy and the material and bonding layer thicknesses [20], and the outcomes suggest that the actuators have to be placed in an extensive strain zone. Single-mode and multimodal vibrations were examined by [21]. Their work focused on two pairs of piezoelectric actuators placed on a beam, and a new index of control has been proposed.  
In this paper, the concept of the optimal placement is traced back to the concept of the optimal potential distribution, which is the potential distribution on all the piezoelectric plates that cover all the beam, where maximum damping is achieved. In this way, a more efficient system is obtained. In fact, in this manner, the piezoelectric plates will always work to damp the vibrations (in every condition and combination of modes) and have the potential that changes its distribution so that they can always work in optimal conditions. In a previous paper [22], one of the authors proposed a new method to damp the multimodal vibrations. In this work, a further new model is developed and verified by experimental tests and numerical simulations. The results show that there is good agreement between those of the model and those of the experimental tests and numerical simulations.