Shock and Vibration

Volume 2015 (2015), Article ID 704265, 30 pages

http://dx.doi.org/10.1155/2015/704265

## Vibration Control by Means of Piezoelectric Actuators Shunted with *LR* Impedances: Performance and Robustness Analysis

^{1}Department of Mechanical Engineering, Politecnico di Milano, Via La Masa 34, 20156 Milan, Italy^{2}Department of Industrial Engineering, Università degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy

Received 29 October 2014; Revised 8 April 2015; Accepted 14 April 2015

Academic Editor: Lei Zuo

Copyright © 2015 M. Berardengo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with passive monomodal vibration control by shunting piezoelectric actuators to electric impedances constituting the series of a resistance and an inductance. Although this kind of vibration attenuation strategy has long been employed, there are still unsolved problems; particularly, this kind of control does suffer from issues relative to robustness because the features of the electric impedance cannot be adapted to changes of the system. This work investigates different algorithms that can be employed to optimise the values of the electric components of the shunt impedance. Some of these algorithms derive from the theory of the tuned mass dampers. First a performance analysis is provided, comparing the attenuation achievable with these algorithms. Then, an analysis and comparison of the same algorithms in terms of robustness are carried out. The approach adopted herein allows identifying the algorithm capable of providing the highest degree of robustness and explains the solutions that can be employed to resolve some of the issues concerning the practical implementation of this control technique. The analytical and numerical results presented in the paper have been validated experimentally by means of a proper test setup.

#### 1. Introduction

Mitigation of vibrations in structures is a crucial issue in several fields, such as electronics, automotive, space, and manufacturing. It can lead to higher quality products, it improves durability by protecting components from fatigue and failure, it achieves reduced maintenance costs, and it improves comfort to people in terms of noise and vibration.

In this scenario, the control of light structures is of particular importance. Many industrial and engineering applications indeed rely on lightweight structures subject to a harsh dynamic environment. Usually these structures have low damping values: thus the vibration level induced can be very high. Furthermore, they are lightweight so that the actuators used for control purposes often introduce high load effects.

Due to these issues, many active and passive strategies have been developed to increase the damping of these structures. Among the passive strategies, up to 20 years ago, the most common were the introduction of high loss factor viscoelastic materials within the structure or connection to a mechanical vibration absorber. In 1991 Hagood and von Flotow [1] proposed a new method to passively increase the structural damping, by relying on piezoelectric materials shunted to a proper electrical network. Since this advance, a variety of passive and semipassive vibration reduction techniques based on piezoelectric materials have been proposed.

The capability of piezoelectric materials to convert mechanical energy into electrical and* vice versa* can indeed be employed in different ways: they can be used as sensors or actuators for active and passive control strategies. As regards the passive or semipassive control techniques, these materials allow either dissipating or reusing the electrical energy induced by the mechanical deflection, by means of a suitable electrical device. In the latter case, an electrical network has to be properly designed and shunted to a piezo-actuator, in order to generate an action opposite to the motion of the hosting structure.

The simplest passive shunt-circuits for single mode control are the resistive () and the resistive-inductive ( or resonant shunt, in which a resistance and an inductance are connected in series), which are the electrical equivalents of a Lanchester damper and a Tuned Mass Damper (TMD), respectively [2, 3]. Such network layouts are widely used for their effectiveness. In addition to these circuits, other types of shunt impedance have been developed; among others, Wu [4] has suggested using a resistance and an inductance in parallel, and Park and Inman [5] studied together both series and parallel circuits.

These passive techniques based on shunted piezoelectric actuators are particularly appealing to the scope of suppressing vibration: they require no power to be effective, which even allows coupling them to energy harvesting systems. Moreover, they cause little additional weight: a strategic issue for most light structures (e.g., space structures). Furthermore, these methods do not require either digital or analog expensive control systems and feedback sensors; they are always stable and easy to implement. On the other hand, these approaches are less flexible than active control strategies: therefore they must be carefully designed and optimised for each specific application. It is also noticed that and impedances can be coupled to a negative capacitance [6–8] in order to increase their attenuation performances. Such an approach allows increasing the vibration attenuation provided by the shunt impedance but poses some issues related to the stability of the electromechanical system because a circuit based on operational amplifiers is used to build the negative impedance. Therefore, the shunt impedance becomes semiactive.

Several strategies have been suggested to optimize the impedance parameters so to achieve the best performances from single mode controllers. Hagood and von Flotow [1] proposed tuning strategies based on transfer function criteria and on pole placement, to tune the numerical values of the shunt impedance in undamped structures. Both these tuning methods are based on the classic TMD theory, given that the circuit is the electrical equivalent of the TMD. In the first method abovementioned, that is, transfer function criterion, the inductance value is found by imposing some constraints on the transfer function shape of the coupled system. Instead, in the latter pole placement technique, the resistance and inductance values are set so that the complex poles of the shunted piezoelectric system reach the leftmost excursion in the* s*-plane. Høgsberg and Krenk developed a balanced calibration method for series and parallel circuits, based on pole placement [9]. In this case, the and values are chosen in order to guarantee equal modal damping of the two modes of the electromechanical structure and thus good separation of complex poles. This option guarantees a good compromise between high damping and performance in terms of response reduction with limited damping. Successful empirical methods to tune the impedance [10] have also been proposed in order to optimize the performance of the system, only with knowing the geometric, mechanical, and electrical characteristics of the hosting structure and actuator.

In addition to the impedance optimization, several studies have investigated the influence of the geometry and placement of the piezoelectric actuator on the performances of the controller. A number of studies to optimize geometry and position of the piezo-actuator have been carried out on the basis of finite element model analysis [11–14]. Instead Ducarne et al. proposed a method to optimise the geometry and the placement of the piezo-actuator, in order to increase the damping efficiency by maximizing the Modal Electro-Mechanical Coupling Factor (MEMCF) [15]. Thomas et al. also proposed closed formulas to evaluate the performance of the controller as function of the MEMCF [16].

For several reasons the performances of these passive control strategies are lower than the ones employing active control, even though they are optimized so to achieve the best vibration attenuation: firstly the power involved in the control is lower than in the case of active control, secondly the absence of a feedback control linked to an error signal does not allow improving control performance during actuation. Consequently, these strategies are much more conditioned by the uncertainties and sensitive to the changes of the parameters involved, thus leading to poor results when the strategies are not well tuned to the specific application. Therefore, in most cases it is not possible to successfully apply the methods available in literature in order to choose the right values of the impedance parameters. In most of the cases the practical applications require empirical tuning or adjustment of the theoretical optimal values. This is in agreement with the results obtained by Thomas et al. in their experimental tests [16]. In fact, they had to adjust the theoretical values calculated with their optimization method in order to achieve the highest attenuation values. The uncertainty which affects the mechanical and electrical parameters of both the structure and the piezoelectric actuators is indeed extremely high in the practical application. Furthermore, mistuning can occur even when starting from a perfectly tuned condition: for instance, if the environmental temperature changes, the eigenfrequency of the system to control will shift and a mistuning will thus occur. Therefore, the chances of having to work in mistuned conditions are very high in practical cases and this causes worse vibration attenuation performance.

Some techniques based on adaptive circuits have been proposed to overcome the limitations due to uncertainty on mechanical and electrical quantities leading to mistuning. As regarding the single mode control, Hollkamp and Starchville developed a self-tuning circuit able to follow any change in frequency of the mode to control [17]. This technique is based on a synthetic circuit (which provides both the inductance and the resistance of the circuit) constituted by two operational amplifiers and a motorized potentiometer. A change of the input voltage to the motorized potentiometer results in a change of the electrical resonance, so that the control system can follow the mechanical resonance change, allowing the correct tuning of the impedance. Nevertheless, this circuit contains active components needing a power supply: thus this strategy cannot be considered passive. Furthermore, the only uncertainty taken into consideration in the abovementioned referenced work is the one relative to the frequency of the controlled mode, while the uncertainty relative to the shunt parameters and to the electrical quantities of the piezo-actuator is not taken into account. Other recent works by Zhou et al. investigated methods to limit the problem of mistuning, by binding more than one piezo-actuator to the vibrating structure [18] and by employing nonlinear elements when the disturbance is harmonic [19].

Although the passive control strategies by means of piezoelectric actuators have been widely studied in the last twenty years, there is still much need for improvement, because of some criticalities. The most relevant ones are summarized below and will each be discussed in detail further in this section:(1)Most of the methods to tune the shunt impedance, available in literature, require the estimation of the natural frequency of the electromechanical system in open and short circuit conditions or the estimation of the Electro-Mechanical Coupling Factor (EMCF).(2)In most cases, the impedance optimization algorithms provide numerical values of the parameters, which are nonetheless unfeasible in practice. As for tuning, very large inductors are necessary for the more commonplace mechanical frequencies; moreover the resistance values are often so small that the sole resistance of the cables and of the piezo-actuator [20] together results to be higher than the value of the optimal shunt resistance itself. Therefore, it is necessary to implement synthetic circuits by means of operational amplifiers, in order to overcome the limitation due to the high value of the inductance . This solution nonetheless leads to the problem that this synthetic circuit requires power supply. Moreover, the problem of the low value of the resistance has seldom been studied in literature.(3)Although it is well known [21–23] that the performance of the control strategy varies significantly in case of uncertainty on the mechanical and electrical parameters of both structure and actuator, a robustness analysis has not yet been carried out in any of the works available in literature. Furthermore, the behaviour of the optimisation methods, in case of mistuning, has never been analysed in terms of attenuation performances.The aim of this paper is thus to resolve some of these shortcomings.

Relative to the issue in point 1 of the list presented above, the problem should be broken down into different considerations. Firstly, when a numerical estimate of the EMCF value is needed, approximated closed formulas may be used when available [15] or else it is possible to measure the natural frequency of the short and open circuit. In this latter case the piezo-actuator has to be chosen and bonded* a priori* and only subsequently the impedance can be tuned. Therefore, the optimization procedure is carried out in two different steps: the optimization of the actuator placement and the optimisation of the impedance parameters. Though some optimization methods are available for the placement and the geometry of the actuator [11–15], this procedure precludes the possibility to perform a more general analysis taking into account at the same time the shunt-impedance parameters, the geometric parameters, and the position of the actuator. But such an analysis can be of great importance for a number of reasons: a specific desired performance may be achieved by different configurations, not necessarily the optimum one. Moreover, sometimes a solution comparable to the optimal one in terms of vibration reduction can also be achieved with an electromechanical configuration different from the optimal one in terms of geometry and position of the actuator, by properly tuning the impedance parameters. This solution can be hardly achieved if the optimization is carried out in two separate steps. A comprehensive analysis in which every parameter is optimized at the same time would also allow to estimate the performance of the controller* a priori* and therefore to highlight whether such a kind of control strategy can be effective enough or not.

The second point in the list of criticalities abovementioned concerns the values of and deriving from the optimization methods: the problem is that their values often result to be too small and too large, respectively, in order to be obtained by physical passive components. This leads to the need of implementing the impedance through operational amplifiers, in turn requiring power supply, even though the power necessary is actually very low. The comprehensive approach proposed above could clarify if the values of and can be changed in order to become feasible, by changing other system parameters (e.g., geometric, mechanical, and electrical parameters, position of the actuator) maintaining the same performance of the controller.

By combining this general analysis with an analysis of robustness to mechanical and electrical uncertainties, proposed in point number 3 of the list of criticalities, a clearer and complete insight on the problem under analysis can be achieved.

This paper proposes an analytical treatment that enables the user to investigate all these aspects. A comprehensive approach, as discussed above, has been developed: it aims at obviating to the aforementioned criticalities by sustaining the tuning algorithm by means of a performance and robustness analysis.

The model employed to describe the dynamic behaviour of the coupled electromechanical system plays an important role in the development of this procedure. It must provide clear formulations which allow performing a global analysis highlighting simultaneously the influence of the position of the piezo-actuator, of its geometry, and of the shunt impedance parameters. Then, all the analyses underlined before can be carried out.

This model chosen takes advantage of the one proposed by Moheimani and Fleming [3] and has several benefits. Firstly the control action performed by the shunted electrical network is seen as a feedback loop: this allows applying the classic control theory to the electromechanical system. Moreover, this model is able to describe at the same time the behaviour of both the elastic structure and the piezoelectric actuator, which in turn are coupled with the shunted impedance. This kind of modelling takes into account both the electromechanical structure (piezoactuator + structure) and the shunt impedance. Also, this model can describe, with a single mathematical description, both 1-dimensional (e.g., beams) and 2-dimensional (e.g., plates) structures.

By the use of this analytical model, this paper demonstrates that there is one specific parameter which affects the control performances and the effectiveness of the control strategy. Such a parameter depends only on the mechanical, geometrical, and electrical characteristics of the structure and of the actuator. This parameter, together with the shunt impedance parameters, can be then modified and properly tuned in order to achieve the target performances. As explained above, the simultaneous tuning of these parameters can be advantageous. Furthermore, this approach brings to light the ineffectiveness of the control techniques based on shunted piezoelectric actuators in the cases where the natural frequencies and the damping of the mode to be controlled exceed given values.

In this scenario, three different methodologies to tune the impedance parameters have been developed, all relying on transfer function considerations. Analytic closed formulas to derive the optimal values of the resistance and inductance of the shunt circuit for damped light structures were then derived. Although all of these strategies prove to be very effective when there are no uncertainties on the parameters, a robustness analysis shows that one of these three tuning algorithms is more robust than the others to uncertainties on electrical and mechanical parameters.

All the results have been experimentally validated. Since the approach developed in this paper results is valid for both beams and plates, the authors have decided to build a test setup, with an aluminium plate and a piezo patch bonded close to its centre, which provides a more complex case-study than those commonly treated in literature (i.e., often the 1-dimensional case is preferred).

This paper is structured as follows. The general electromechanical model for an elastic structure with piezoelectric elements coupled to an electric circuit is described in Section 2. The three different tuning methodologies based on transfer function considerations are presented in Section 3, and analytic formulas to tune and are derived. Section 4 illustrates the performance analysis of the mentioned optimisation methods and explains the effect of the electric and mechanical characteristics of the structure and of the piezo-actuator on the and values. The robustness analysis of the optimization methods is presented in Section 5. Finally, Section 6 illustrates and explains the experimental tests carried out on a plate, and a simplified formulation for the most robust tuning procedure is proposed in Section 7.

#### 2. Electromechanical Model

This section treats the analytical modelling of the whole electromechanical structure constituted by the elastic structure, the piezo-actuator, and the shunting impedance. Though some of the issues treated in this section are already known and discussed in literature [3, 24, 25], the authors have decided to provide a concise recapitulation of them, for sake of clarity; such an abridgement is moreover meant to make the paper more readily accessible and makes for a better understanding of the improvements contributing to such a model by this paper. This section is subdivided into four parts: the first part describes the electric model of the piezoelectric actuator is described; part two highlights the feedback nature of the controlled system; and the third part provides the dynamic model of the coupled system, and analytical formulations are derived for cases that have yet to be analysed in literature. Finally, this model is used to achieve a new formulation of the frequency response function of the controlled structure, in the fourth subsection.

##### 2.1. Electric Equivalent Scheme of a Piezoelectric Actuator

Piezoelectric materials are materials such that an applied stress is capable of generating a charge on the surfaces of the piezoelectric element, and an applied voltage generates a strain. Thanks to the latter working principle, the shape of the solid can be modified depending on the charge induced on the surfaces of the piezoelectric element. These two effects (called piezoelectric effects, direct and inverse, resp.) entail to employ these materials as both sensors and actuators, making them extremely interesting in applications for vibration control.

One of the models which can be used to describe the electrical behaviour of piezoelectric materials is a series of a capacitor and a strain-dependent voltage generator [26], as shown in Figure 1.