Optimal Piezoelectric Potential Distribution for Controlling Multimode Vibrations

: Vibration damping is prominent in engineering; in fact, vibrations are related to many phenomena (e.g., the fatigue of structural elements). The advent of smart materials has signiﬁcantly increased the number of available solutions in this ﬁeld. Among smart materials, piezoelectric materials are most promising. However, their efﬁciency depends on their placement. There are many studies on their optimal placement for damping a particular mode, but few account for multimodal vibrations damping. In a previous work, an analytical method was proposed to ﬁnd the optimal placement of piezoelectric plates to control the multimode vibrations of a cantilever beam. In this study, the efﬁciency of the above method has been improved, considering all plates active simultaneously, regardless of the eigenmodes that are excited, and changing, instead of the plates, the potential distribution. The method results indicate the optimal potential distribution for different excited eigenmodes. The results have been compared with those obtained by experimental tests and numerical simulations, exhibiting very good agreement.


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 studies seem 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 Barboni 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.

Governing Equations for a Piezoelectric Coupled Beam
In Figure 1, a scheme of the piezoelectric coupled beam is reported. Where L e , L in , L a , and L p indicate the work of external, inertial, applied, and piezoelectric forces, respectively, the virtual work principle can be written as The calculus of δL p has been confirmed by many studies (one of the first studies in this regard was conducted by Crawley et al. ([7]) in 1987), and experimental evidence shows that, in the case of perfectly bonded material interfaces, the stresses that a plate applies to the beam are concentrated at the ends of the plate. Therefore, by applying two harmonic voltages opposite in the phase between the upper plate and the lower plate, such action can be modeled by a couple of flexural moments M concentrated at their ends ( Figure 2).
the flexural moment can be written as ( [7]) and δL p will be where x i represents the points where the potential changes its sign, so that δ(V i ) will be +1 or −1 depending on the potential distribution on the plate i ( Figure 2). The optimization problem can be traced back to the research of the optimal distribution of the δ(V i ) or, in other words, to the research of the optimal distribution of the points x i . By introducing the inertial, internal, and external forces in Equation (1) and applying the modal analysis technique, the following equilibrium equation is obtained M ..
where M and K are, respectively, the mass and the stiffness matrix, X is the vector of the modal amplitudes, and the vector B is T a . If the viscous damping is added, Equation (5) becomes: Assuming that φ i are the orthonormal modes, it follows that M is the identity matrix I and that K is the natural frequencies matrix ω . If C is related to the Rayleigh damping (C = αM + βK) where β = 0, Equation (7) can be written as ..
Since the flexural eigenmode of the cantilever beam achieves its maximum displacement amplitude at the free end, this point has been chosen as a reference point for the vibrations of the whole cantilever beam. Indicating with w F (t) and w p (t) the free end displacement due to, respectively, the load F and the piezoelectric load, the total displacement can be written as w T (t) = w F (t) + w p (t) as a consequence of the linearity of the system. Choosing a suitable spectrum load, it is possible to activate the piezoelectric plate to obtain w p (t) opposite in phase with respect to w F (t), so that, indicating with |w T |, |w F |, and |w p | the amplitudes of the relative displacements, |w T | = |w F | − |w p |. In this paper, the optimal potential distribution has been considered as the one that minimizes the amplitude of w T so that the optimal distribution will be the one that, for the same V, maximizes |w p |. It can be demonstrated that ( [22]), without taking into account the transient part, where N s are the excited modes. Choosing a load that excites two modes, indicating with i 1 and i 2 the excited modes and considering a parameter r that takes into account the distribution of the load between the two eigenfrequencies, the following can be written: To achieve the maximum efficiency (or, as mentioned above, in order to obtain w p (t) opposite in phase to w F (t)), the spectrum of the potential distribution has been chosen with the same characteristics of the load ( [22]) and parameterized by the same parameter: Substituting Equation (11) in Equation (9), which can be written in the form where x N p = L b has been chosen to cover the entire beam with the piezoelectric plates. In order to find the other points {x 1 , x 2 , ..., x N p −1 }, it is helpful to introduce the following function: This is obtained by a linear combination of two oscillating functions and presents an alternating maximum and minimum. It can be observed that the optimal potential distribution is obtained when the points x j coincide with the extrema of f i 1 ,i 2 (r, x). In particular, considering that the function it is convenient to choose δ(V N p ) = +1 in Equation (13). Moreover, comparing f i 1 ,i 2 (r, x) with Equation (13), it can be noticed that, to obtain the maximum value of | w p |, x N p −1 must coincide with the first minimum of f i 1 ,i 2 (r, x) closer to L b , choosing δ(V N p −1 ) = −1, the point x N p −2 must coincide with the first maximum of f i 1 ,i 2 (r, x) closer to x N p −1 , choosing δ(V N p −2 ) = +1, etc. In this manner, . (16) In Figure 3, the functions f i 1 ,i 2 for different combinations of the first four eigenmodes, and various values of r, are reported. The black lines represent the "paths" of an extrema point when r changes, i.e., at the intersections between the black lines, and f i 1 ,i 2 identifies the point where the potential must change its sign. In what follows, for the sake of example, the coupling between the first two modes is considered, and the black curves in the {r,x} plane are represented.
Choosing two values of r (e.g., r = 0.2 and r = 0.8) (Figure 4) it can be observed that, in the first case, there is no change of sign, and, in the second case, there is a switch atx 0.17. In Figure 5, the beam configurations with the sign distribution are reported.

Experimental and Numerical Tests
In order to test the model, numerical and experimental investigations were carried out. An experimental workbench was developed and two arrays of 13 PZT plates (PPs) were attached by means of an epoxy glue on both sides of an aluminium alloy cantilever beam. The dimensions of beam and PZT elements are summarized in Table 1, while a picture of the beam coupled with the PPs is shown in Figure 6. By means of an ad hoc developed power supply circuit, the V(t) voltage of Equation (11) is provided to each piezo plate. Every PP is wired to a mechanical relay that is actuated by a micro controller. The relays can switch independently the sign of each PP supply voltage V(t) in order to execute a specific PZT activation pattern (i.e., the potential distribution of Table 2. The voltage V(t) is provided by an arbitrary function generator (Yokogawa FG420) and properly amplified and inverted by two amplifier circuits specifically developed by the authors. The circuit diagram of the amplifiers is shown in Figure 7 and provides appropriate voltages to actuate the piezoelectric plates. It basically consists of two pairs of complementary transistors (MJE340, MJE350), 10 resistors, and three fast diodes; such a circuit is able to achieve an amplification factor of 10 up to 300 V pp . These features, along with their inexpensiveness and their ease of implementation, make these amplifiers particularly suitable for piezoelectric plate actuation. An ICP micro-accelerometer (PCB 352A56) was placed at the free end of the beam and a data acquisition board (NI USB 6251), and a Labview subroutine was used for collecting and processing the measured data. The sampling rate was set at 15 kHz and 4 s of acceleration was acquired for each PP activation pattern. The FFT analysis of the measured data was performed for each PP activation pattern, so that the amplitudes of the acceleration at ω i 1 and ω i 2 could be assessed. Finally, the total tip displacement was calculated for each PP activation pattern in order to identify the optimal experimental potential distributions of the PPs. The overall measurement chain scheme is illustrated in Figure 8.  The experimental trials were repeated several times for every PP activation pattern, and four combinations of ω i 1 and ω i 2 were tested. The r ratio (the parameter r takes into account the distribution of the load between the two eigenfrequencies (see Equation (10))) was varied by step of 0.1 from 0 to 1. The actual experimental setup is shown in Figure 9, and its uncertainty (i.e., the accelerometer, the acquisition board, the power supply system, and the data processing) is estimated to be 5% according to previous studies [23,24].    A COMSOL R FEM code was chosen to conduct the numerical simulations. The first four modes, and their pairings, were considered. For every combination (Table 2), and every value of r, the amplitude displacement at the free end of the beam was calculated by the frequency response function. Among all of these, the potential distribution, which implied the maximum amplitude, was considered optimal. The characteristics of the beam and of the piezoelectric elements are reported in Table 1.

Results and Discussion
The model, numerical, and experimental results are reported in Figures 10-13. It can be observed that there is a good agreement between the numerical and experimental results and those predicted by the model. However, the position evaluation error is due to two principal factors. The first is that the model proposed is a continuum model, which implies that it includes the possibility to change the sign of the potential at every point x. On the other hand, the numerical and experimental simulations inevitably provide a discrete approach, and the sign can be changed only in a discrete number of points. In the specimen piezoelectric coupled beam described above, there are 13 points where the sign can be changed, and these do not always coincide with those provided by the continuum model. In the worst cases, the continuum model points are in the center of one of the 13 couples of the piezoelectric plates. In this case, the error committed is 1 26 L b , which is the maximum error within the considered domain. In some cases, this error is larger, but this happens only at the boundary of the domain inx = 1 for f 1,2 and f 1,3 . The reason for this is due to the shape of f 1,2 and f 1,3 inx = 1 (Figure 3) and the uncertainty of the measuring equipment (estimated about 5% as described above). In fact, all functions f i 1 ,i 2 have the first derivative equal to zero inx = 1, but f 1,2 and f 1,3 are quite flat for a long interval nearx = 1. Since the difference in | w p | between two different potential combinations is connected with the difference that the f i 1 ,i 2 assume in these configurations, if this difference is below the 5%, the measuring equipment is not capable of capturing it and the position evaluation error can be considerable ( Figure 14). In Figures 10-13, it can be observed that, when the modal order number increases the number of the "line changing sign", the corresponding number of points N p also increases. In fact, increasing the modal order number increases the number of the extrema of the derivative of the flexural displacement or, in other words, the extrema of the section rotation of the beam (the Euler Bernoulli beam is considered here). Equation (12) shows that the maximum efficiency of the piezoelectric plates is obtained when the sign change occurs in these extrema points. In Figures  15-18, some configurations with the potential distribution (lines of the same color represent areas with the same sign) are represented. The increase of N p with r is due to the fact that, when r rises, the flexural deformation switches from the modal shape i 1 to i 2 > i 1 .

Conclusions
In this paper, research on the optimal placement of piezoelectric plates for damping flexural vibrations is conjoined with research on the optimal potential distribution. In this way, a more efficient system is obtained, because PZT plates can always work and make a contribution for every combination of loads. In this regard, a new mathematical model, an improvement on an existing method, is proposed. The model outcomes are compared with the results of numerical simulations and experimental data, and a good agreement among them was found. Future works will focus on the implementation of the model in cases of rotating cantilever beams. The interest in these studies could be, e.g., the damping of the blade vibrations in turbomachinery [25][26][27][28].
Author Contributions: Fabio Botta designed and developed the model. Andrea Rossi, Fabio Botta, and Andrea Scorza carried out the experimental tests. Fabio Botta carried out the numerical simulations.

Conflicts of Interest:
The authors declare no conflict of interest.