Research Article | Open Access
M. Berardengo, A. Cigada, S. Manzoni, M. Vanali, "Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis", Shock and Vibration, vol. 2015, Article ID 704265, 30 pages, 2015. https://doi.org/10.1155/2015/704265
Vibration Control by Means of Piezoelectric Actuators Shunted with LR Impedances: Performance and Robustness Analysis
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.
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  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  has suggested using a resistance and an inductance in parallel, and Park and Inman  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  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 . 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  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) . Thomas et al. also proposed closed formulas to evaluate the performance of the controller as function of the MEMCF .
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 . 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 . 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  and by employing nonlinear elements when the disturbance is harmonic .
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  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  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  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.
Two piezoelectric patches are usually needed to control a light structure by means of piezoelectric materials: one acting as sensor and the other as actuator (Figure 2). The sensor output (i.e., the voltage across the piezoelectric patch electrodes) is equal to (Figure 1) and it depends on the strain of the structure to which it is bonded. In turn, this voltage is the input to the controller which uses this signal as the reference for its control law, generating an output voltage , applied to the piezo-patch acting as actuator. This piezo-actuator will thus change its shape, thus applying a control force to the structure.
The control mechanism described here can also be made by using a single piezo-patch, which behaves at the same time as sensor and actuator. In this case a single piezoelectric element is shunted to an impedance  (Figure 3); the structure vibration will induce a voltage to the terminals of the actuator (equal to when the circuit is open: see Figure 1). Since the impedance is connected to the piezoelectric element, a current circulates and the voltage between the terminals of the impedance will no longer be , but it will be modified by the presence of the impedance. This voltage becomes the input to the piezo-element and thus induces a change of the strain of the patch (i.e., a control force is imposed to the structure). Hence, the voltage and the control action to the structure depend on how the impedance used to shunt the actuator is built. Thus, its layout and the values of its parameters must be carefully chosen, according to the specific application, in order to maximize the control effect.
This kind of control scheme (Figure 3) is chosen to develop the single mode control strategies presented in this paper.
2.2. Feedback Representation of the Control by Means of a Shunted Piezoelectric Actuator
Taken here into consideration is a structure controlled by a piezoelectric actuator shunted by an impedance and subject to the disturbance (Figure 4(a)) and its electrical equivalent, as shown in Figure 4(b). The disturbance induces a flexural motion of the elastic structure described by the variable , which represents the transversal displacement. As explained in , if the impedance is removed (open circuit), the voltage at the terminals of the actuator results equals (Figure 1(b)) and is entirely induced by the strain generated by the disturbance . In this case the voltage can be related to the disturbance by the transfer function :Otherwise, in the case no disturbances act on the structure and the impedance is replaced by a voltage source ; then depends only on the voltage . Thus, the relation between and can be expressed by the transfer function :When the disturbance acts on the structure and the voltage is applied to the terminals of the piezo-actuator, the voltage can be expressed as the sum of the two contributions, because of the linearity of the system. By applying the superposition principle, it is then possible to writeAs for Figure 4(b), the following expressions are achieved by applying the Kirchhoff and the Ohm laws:where in the Laplace domain.
Equations (3) and (4) allow representing the electromechanical structure, constituted by the elastic structure, the piezo-patch, and the impedance, as the feedback controlled system in Figure 5, where the controller in the Laplace domain is represented by the inner loop:Therefore, the impedance can be used to produce a given voltage capable of reducing the effect induced by the disturbance . The impedance used in this paper to develop a single mode control strategy is constituted by a resistance in series with an inductance .
As mentioned above, different tuning methodologies will be defined in Section 3, in order to choose the optimal values of the impedance parameters (i.e., and ), relying on transfer function considerations. The selected target transfer function shall allow describing the response of the elastic structure in terms of deflection to an external disturbance. Therefore, the expression of the closed loop transfer function between the disturbance and the displacement of the structure is needed.
As for Figure 5, the closed loop transfer function relating the voltage with the disturbance can be defined asOnce the transfer function between the voltage and the displacement (i.e., ) is defined, can be expressed asThus, the analytic expressions of , , and must be known, in order to derive the expression of and thus the dynamics of the coupled system (i.e., piezo-actuator + elastic structure) must be taken into account.
2.3. Dynamic Model of the Electromechanical System
The mathematical procedure, developed in this section, to derive the analytic relations between the electric and mechanical quantities involved in this control problem (i.e., , , , and ) is referred to a plate structure (Figure 6). This same approach can be used to derive these relationships for one-dimensional cases (i.e., beam structures) as well [24, 25]. The authors have decided to refer to the two-dimensional case but to report only the final results for the one-dimensional case, for sake of conciseness and because the experimental tests (Section 6) were performed on a plate.
The analytical formulation of the transfer functions , , and can be achieved by employing the expressions which relate the voltage induced at the terminals of the piezo-patch in open circuit (i.e., ) to its deformation and the equations of motion of the electromechanical system [3, 24].
The first of these equations (i.e., the relation between the voltage induced at the terminals of the piezo-patch in open circuit and its deformation) can be derived considering that the voltage between the piezo-actuator terminals in open circuit can be expressed as function of the charge induced on the surfaces of the patch: and the charge can be obtained as the surface integral of the electric displacement in direction, named :where is the surface of the piezoelectric patch. From the equation describing the direct piezoelectric effect , the expression of the electric displacement can be derived as function of the stresses on the piezo-patch in and directions ( and ) and of the piezoelectric strain constants and : Assuming the piezoelectric material bears similar properties in and directions, the two piezoelectric constants can be considered equal: . Furthermore, the stresses (, ) and strains (, ) in the piezoelectric patch in open circuit can be described by the following expressions:where and are the thickness of the plate and of the piezo-actuator, respectively, and are Young’s and Poisson’s moduli of the piezo-patch, and is the transverse deflection of the coupled structure.
By substituting (9), (10), and (11) in (8), the relation between the deformation of the piezoelectric patch and the voltage induced at its terminals is derived:where , , , and are the coordinates of the actuator extremities (Figure 6).
Now that the first equation has been derived (i.e., the relation between the voltage induced at the terminals of the piezo-patch in open circuit and its deformation), the equations of motion of the plate are necessary to calculate the transfer functions , , and .
As explained in the previous section, , , and represent the transfer functions related to the cases in which either the disturbance or the applied voltage is acting on the system. This means that the response of the elastic structure to the forces induced by the applied voltage and the disturbance separately must be known, in order to calculate these transfer functions.
The dynamic equation of a plate subject to a forcing term is represented by the following Partial Differential Equation (PDE) :where is the density of the plate material andwhere and are Poisson’s coefficient and Young’s coefficient of the plate material, respectively. Equations (13) and (12) expressed in modal coordinates and represented in Laplace domain allow calculating the transfer function between the voltage and the disturbance :where is the th eigenfrequency of the plate and is the associated nondimensional damping ratio. is the th eigenmode (normalised to unit modal mass) of the plate, represents the value of the th mode at the forcing point , and for the two-dimensional case iswhere is a term dependent on the curvature of the th mode and assumes different formulations for 1D or 2D cases; see Table 1 (the other parameters in Table 1 are explained in this section further on).
As for and , the forcing term of (13) has to be replaced by the forcing action generated on the structure by the piezo-actuator. The equation governing the dynamic of the plate subject to the moments applied by the piezo-actuator can be described by the following PDE :where the moments per unit length and represent the forcing term due to the piezoelectric patch (Figure 6).
The forcing term must be defined in order to solve the dynamic equation (17), to make explicit the dependency of the moments on the voltage and to calculate and .
The flexural moment applied to the structure is due to the deformation of the piezo-patch caused by the voltage applied to its terminals. Thus, the expression linking the voltage applied to the terminals of the piezoelectric patch and the moments generated must be derived. For sake of thoroughness, this paper explains the procedure to obtain this term, because for some particular cases (illustrated further on in this section) the available literature does not elucidate the results but is limited to suggesting the procedure to obtain them.
The moments acting on the plate can be described by the following expression:where and are the stresses acting on the plate in and directions and and are the transverse cross-sections of the plate (i.e., and referring to Figure 6).
The system represented in Figure 7, with two colocated piezoelectric actuators and a phase of 180° between the two of them, shall be discussed now. Relying on the hypothesis of a homogeneous plate, the strain in and directions can be considered the same and equals (see Figure 7). The stresses in the plate and in the piezo-actuators can be expressed aswhere the symbol refers to the stresses in the actuator at the bottom. In (19) the piezoelectric actuator is assumed to have the same properties in and direction, and the unconstrained strain of the piezoelectric actuator due to an applied voltage is given byThe expression of the coefficient , describing the deformation in the plate (Figure 7), must be derived to calculate the analytic expressions of the stresses in the plate, which in turn are necessary to calculate the moments. Such an expression can be computed by applying the moment equilibrium in and directions [3, 24, 25]:Just one of the two equilibria can be taken into consideration to derive the expression of , since the plate is assumed to be homogeneous (i.e., and ).
Equation (21) shows that the value of the coefficient changes according to the stresses in the plate (i.e., ). In turn, the stress distribution depends on the layout of the electromechanical structure. The most common configurations for this kind of problem are antisymmetric or asymmetric. The antisymmetric configuration is constituted by two colocated actuators or by colocated sensor-actuator pair. In the antisymmetric configuration instead a single piezo-electric element acting as an actuator is bonded to the structure. The expression of the coefficient in the asymmetric configuration for a plate has not yet been discussed in its explicit formulation in existing literature; moreover this is the configuration for the plate control by means of shunted piezo-actuators: thus the authors have derived its formulation (see Table 2) and employed it in the herein paper. The strain condition in the plate is asymmetric with this layout and can be decomposed in two parts: flexural and longitudinal (Figure 8). The equilibria of moments and axial forces must be imposed to calculate and ; see Figure 8.
The expressions of for all the abovementioned configurations, for both beams and plates, are listed in Table 2.
As for the expressions of in Table 2, it is opportune to express as , in the following formulations.
The moments per unit length acting on the plate, in directions and , in (17) are equal (i.e., ). Thus, the moment per unit length acting on the plate can be related to the voltage applied to the terminals of the actuator using (20):where .
Having derived the expression of the moment as function of the applied voltage (23), the expressions of and can be achieved using (17) and (23) together with a modal coordinate representation in the Laplace domain:where represents the value of the th mode at the measuring point :Moreover, in (15) can be expressed as function of the parameters and shown in Table 1, leading toAll of the aforementioned expressions can be used for both beams and plates, only differing in the values attributed to the coefficients , , and . These values, for the 1- and 2-dimensional cases, are shown in Table 1.
2.4. Formulation of the Frequency Response Function
Having defined the transfer functions , , and , the frequency response function between the disturbance and the response can be calculated by means of (7).
Equation (7) is usually expressed in a slightly different form in literature, indeed using the following formulation:whereIt is recalled that the terms and can be calculated as explained in Table 1 for mono- and bidimensional structures.
The expressions of (7) and (27) are fully coincident. The use of in place of in (28) is compensated by changing the sign of the term in (27). Equation (27) will herein be used from now on in place of (7), for the sake of adhering to the commonplace convention employed in literature.
Relying on the theory described above, an opportune formulation of shall be achieved and employed for all the subsequent calculations and considerations, by virtue of its clarity. The function is the target function for the control strategies that are proposed in the following section. These strategies are based on transfer function considerations and in particular on considerations over the shape of and this allows making full use of the strategies developed for tuning TMDs. In fact, the piezo-actuator shunted to an circuit can be considered as the electric equivalent of the TMD . Therefore, tuning strategies similar to those developed for the TMD natural frequency and damping (e.g., ) can be used to tune the resistance and inductance of the shunting impedance. Hence, a formulation similar to the one used for TMDs must be derived in order to employ this approach.
The expression of the controller must be known in order to calculate the transfer function (see (27)). is represented by the inner loop of the feedback representation (Figure 5) of the controlled plate (5).
represents the impedance used to shunt the piezoelectric actuator and in this case it is constituted by the series of a resistance and an inductance . The differential equation linking the current and the voltage at the piezo-actuator terminals can be expressed as . This equation can be rearranged in the Laplace domain as .
Equation (5) thus can be formulated asThe expression of should be defined in terms of the electrical frequency (i.e., the eigenfrequency of the electrical circuit) and of the electrical damping (i.e., the nondimensional damping ratio of the electrical circuit), so to achieve an expression of similar to those used for TMD systems . Therefore, it is opportune to define the electrical damping and the electrical frequency asThe controller can be expressed as a function of these two quantities, substituting (31) and (32) into (30):Concerning single degree-of-freedom systems, the transfer function between the disturbance and the displacement can be derived by substituting (24), (28), (26), and (33) into (27): This formulation is valid for both beams and plates and can be used for any of the configurations listed in Table 2, simply by choosing the appropriate expressions for and . Therefore, the optimization criteria proposed in Section 3 take on general validity and can be used regardless of the configuration and the type of structure. Moreover, the formulation of (34) presents two advantages over the formulation function of and . The first one is that this formulation allows making use of the tuning formulations developed for TMDs. The second is that it simplifies significantly the mathematical treatment developed in Section 3, used to derive the optimal values of and .
It is worth remarking that this kind of approach (summarized in (24), (25), (26), and (34)) can be easily extended to the case in which more than one actuator is used. The presence of several actuators, each shunted to an impedance , can be accounted for adopting a vector formulation for :If the actuators involved in the control of the structure are not identical (i.e., they have different geometrical and electrical characteristics), the parameters and must be modified according to each actuator.
These characteristics make this a particularly efficient approach, with an extended generality and suitable for treating a wide range of problems: the elastic structures taken in consideration can be either a beam or a plate indifferently and the control can be implemented by one or more actuators.
3. Tuning Strategies for Damped Elastic Structures
Closed analytic formulas for tuning the resistance and the inductance of the shunting impedance are presented in this section. These tuning methodologies are based on transfer function considerations and exploit TMD theory. In Section 3.1 the optimal values of the electric eigenfrequency and the inductance (31) are derived. Subsequently, Section 3.2 presents three different tuning strategies for the electrical damping and the resistance .
3.1. Tuning Strategy for the Electrical Eigenfrequency
The tuning method implemented to derive the optimal electrical eigenfrequency is based on a well-known criterion for undamped structures. Indeed two specific frequencies for undamped structures can be reckoned: one higher and another lower than the mechanical resonance (resp., and ). The behaviour of is independent of the damping factor of the electrical circuit , at the two frequency values and (Figure 9(a)). The optimal value of is found by imposing the same dynamic amplification modulus at these two frequencies :This kind of method assumes to be working with an undamped elastic structure. In the case of a damped structure, there is not a single intersection point between curves with different damping values (Figure 9(b)). Nevertheless, the use of frequencies and guarantees low approximation: because the frequency values of the intersection points of the different curves fall within limited frequency bands and do not differ greatly from and , respectively (Figure 9(b)). Furthermore, this optimization strategy was compared against a numerical optimization strategy in which the optimal values for and were found numerically by minimizing the maximum value of . The results provided by the strategy herein proposed do not differ significantly from the optimization values found by numerical minimization (see Section 4): therefore the hypothesis of undamped elastic structure for tuning the electrical frequency does not introduce noteworthy approximations.
Therefore, the dynamic amplification modulus at and must be known, in order to find the optimum value of . It is noteworthy that is independent of the value of at these two frequencies, and therefore the response at these two frequencies can be determined as the limit . Relying on (34) where represents the circular frequency and is the imaginary unit. must be fixed to and in (37) and the + and − signs are referred to the value of and , respectively. Moreover (37) is yielded considering .
Hence, the substitution of (37) in (36) leads toAnd the following expression is achieved by rearranging (38):Furthermore (34) can be expressed (neglecting the structural damping ) aswhere , , , and areAs highlighted by Krenk in , the dynamic amplification modulus at and is independent of the damping, and this can be mathematically expressed asSubstituting , and in (42) givesThe use of the + sign leads to the trivial solution , where there is no motion and therefore no damping force. The minus sign instead gives the following relation:Then, the following expression is achieved by substituting (39) in (44):This formulation allows calculating the optimum value of .
Finally, if we consider that the electrical frequency can be expressed as a function of the inductance of the shunting circuit by using (31), the optimal value of is
3.2. Tuning Strategies for the Electrical Damping Ratio
Three strategies have been developed to tune the value of the nondimensional damping ratio (and thus the value of the resistance ) and are designed allowing for damped elastic structures, which are seldom accounted for in literature. The first two methodologies discussed here are based on standard tuning criteria ([29, 30]) for the TMD devices, while the third one is based on considerations on the shape of as function of the electrical damping . The tuning criteria listed below are explained in Figure 10 and described in detail in Sections 3.2.1, 3.2.2, and 3.2.3:(1)Optimization 1: ,(2)Optimization 2: ,(3)Optimization 3: .
3.2.1. Optimization 1
The first optimization criterion proposed for the electrical damping makes use of the procedure developed by Krenk for TMDs . The optimal value of the damping is found by imposing equal dynamic amplification at two different frequencies: and at a frequency given by the square root of the arithmetic mean between and . This frequency is found to be equal to the electrical frequency . Indeed, relying on (44) and (45), the following expression is yielded:
Unlike the procedure proposed by Krenk , the authors of this paper have decided to reckon with damped systems, in order to reach a formulation that can account for the whole dynamic behaviour of the system.
The dynamic amplification value at and must be calculated to find the value for . Its expression can be derived by substituting the expression of (45) in (43) with negative sign and solving it with respect to and :where is obtained using the negative sign, while the positive.
Rearranging (34), can be expressed aswhere , and areTherefore, the square of the dynamic amplification modulus can be expressed as a function of the parameters , and :The dynamic amplification modulus in and can be derived by substituting (47) and (48) in (51). Then, the equation needed to find the electrical damping ratio can be derived by imposingThe complete equation, function of the electrical and mechanical parameters, shall not be included in this paper, for sake of conciseness.
Solving (52) with respect to the electrical damping leads to a sixth-order equation which does not provide a closed analytic formulation for . Nonetheless, the exact values of the electrical damping can be yielded, if all the values of the other variables are known.
On the contrary, a closed analytic formulation can indeed be derived for , in the case of undamped elastic structures. The same procedure described above leads to an electrical damping equal to
3.2.2. Optimization 2
The second optimization criterion for the electrical damping is based on a standard optimization criterion for the damping element in TMD systems. The optimal value for is found by setting the maximum value of at . Therefore, the square root of (51) is derived with respect to the frequency , and then the resulting function is set equal to zero at (48):The optimal value for is derived by solving (54) with respect to the electrical damping . The complete expression of (54) as function of is not expounded in this paper because of its complexity and because it is a sixth-order equation in . Therefore, also in this case a closed formulation to derive the electrical damping cannot be achieved.
3.2.3. Optimization 3
Unlike the optimization strategies discussed above, both based on TMD tuning techniques, the third optimisation strategy here discussed originates from considerations on the trend of the dynamic amplification modulus in and as function of the electrical damping.
In fact, has a minimum for a specific value of (Figure 11). Hence, this optimization criterion is based the use of the electrical damping value which minimises the dynamic amplification modulus at . This condition is expressed by the following relation:where is found by substituting (45) and (48) in (51).
The three optimization methods for electrical damping discussed in this paper each lead to different values of and therefore to different shapes of the dynamic amplification modulus (Figure 10). Figure 10 illustrates that the third method generates lower damping values than the first method and that the second strategy instead leads to a behaviour midway between the two. The following sections present performance and robustness analyses of the three optimisation methods, displaying advantages and disadvantages of each strategy, in terms of vibration reduction and robustness against uncertainties on the electrical and mechanical parameters.
4. Performance Analysis
Three different tuning methodologies for the shunt impedance parameters are expounded in Section 3. Each of them leads to different values for the resistance and therefore to different levels of damping; consequently the performance, in terms of vibration attenuation, changes according to the tuning strategy employed. This section deals with the analysis of the performance of these tuning methodologies in the entire domain of application of this type of control method. The domain considered in this analysis is described in Section 4.1: it has been chosen so to take into account the majority of cases possible in actual applications of light and thin structures. Subsequently, in Section 4.2, a comparison is drawn between the optimal damping values required by each of the three tuning strategies and the optimum achieved through a numerical minimization. In addition, a performance analysis in terms of vibration reduction is discussed in Section 4.3, showing the effectiveness of each of the tuning strategies in the whole domain analysed. All of these analyses have also been carried out on two additional tuning strategies: optimisation 1 and optimisation 2 in the case of no structural damping (i.e., carrying out all the calculations to obtain the value of , considering a null mechanical damping, e.g., (53), and then applying the solution found to a mechanical system with a nonnull mechanical damping). Finally, the effect of the parameter on the performance of the control system is studied in detail and explained in Section 4.4.
4.1. Domain Description
A wide application domain is taken into consideration in the study of the behaviour of the tuning strategies analysed here in order to make this study as general as possible. Cases taken into consideration include extreme situations, their opposites, and the range of intermediate ones, so as to account for the majority of actual real applications involving shunted piezoelectric actuator controls: highly flexible and extremely rigid structures, very high and very low natural frequencies to be controlled, best and worst position of the piezo-actuator for controlling a given mode (i.e., ), and different geometries and materials of the elastic structure and of the actuator. Of course, this approach introduces even near-implausible situations into the analysis, but it does allow generalising the conclusions arising from the analysis and to exclude the eventuality of different behaviours for test cases not taken into consideration. The range of values considered for each parameter is shown in Table 3, where the values of were derived from the geometrical and material characteristics according to the formulas of Table 1 and for different kinds of constraint. Equations (45), (52), (53), (54), and (55) show that the shunting impedance parameters (i.e., and ) and the dynamic amplification (34) depend solely on the problem parameters in Table 3. Thus, all the cases included in the domain described by the quantities in Table 3 can be represented by modifying the problem parameters , and . Each of these parameter was altered by increments of 50 rad2/s2, rad/s (i.e., 100 Hz), and , respectively, and a simulation was performed for each combination.
All the results of the simulations (see Section 4.2) showed a monotonic trend with respect to the three problem parameters , and ; hence only a selection of representative cases is reported in this paper, also for sake of conciseness. Three values of , and have been selected: corresponding to a high, medium, and low level of the parameters, respectively (see Table 4). This approach leads to the analysis of 27 cases, numbered 1 to 27 (Table 5). The rationale behind the selection of the values in Table 5 shall be clarified further on in this paper.