The Behaviour of Mistuned Piezoelectric Shunt Systems and Its Estimation

This paper addresses monoharmonic vibration attenuation using piezoelectric transducers shunted with electric impedances consisting of a resistance and an inductance in series. This type of vibration attenuation has several advantages but suffers from problems related to possible mistuning. In fact, when either the mechanical system to be controlled or the shunt electric impedance undergoes a change in their dynamical features, the attenuation performance decreases significantly. This paper describes the influence of biases in the electric impedance parameters on the attenuation provided by the shunt and proposes an approximated model for a rapid prediction of the vibration damping performance in mistuned situations. The analytical and numerical results achieved within the paper are validated using experimental tests on two different test structures.


Introduction
Vibration attenuation in light structures is a widely studied topic and often takes advantage of the use of smart materials, which are characterised by useful properties.Indeed, these materials are inexpensive when compared to other control systems, and they are characterised by low weight.This last feature is a fundamental aspect because it avoids introducing high load effects on the controlled structure.Among smart materials, piezoelectric elements (particularly piezoelectric laminates, which are used in this paper) are among the best materials to attenuate vibrations in bidimensional (e.g., plates) and monodimensional (e.g., beams) structures [1][2][3].There are several control techniques for light structures that rely on this type of actuator, and one of the most attractive is the shunt of the piezoelectric element.In this case, a properly designed electrical network is shunted to the piezoelectric bender bonded to the structure.The ability of the piezoelectric element to convert mechanical energy into electrical energy and vice versa [4,5] is used, which allows a passive attenuation of the structure's vibration.This method was initially proposed by Hagood and von Flotow [4].This technique is extremely attractive because it is cheap, it does not introduce energy into the system, that is, it cannot lead to instability, and it does not require any feedback signal.
When a monoharmonic control is required, the most effective shunt electric impedance consists of a resistance  and an inductance  in series [2,4,[6][7][8] (resonant shunt or RL shunt).These two elements, along with the capacitance   of the piezoelectric actuator (i.e., the piezoelectric actuator is modelled here as a capacitance and a strain-induced voltage generator in series; see Section 2), constitute a resonant circuit, which is the electric equivalent of the mechanical tuned mass damper (TMD) [2].Therefore, this circuit is able to damp the structural vibration corresponding to a given eigenmode as soon as its dynamic features are tuned to those of the vibrating structure.
There are several methods in the literature that explain how to select the values of  and  to optimise the vibration attenuation.Hagood and von Flotow [4] proposed two different tuning strategies based on considerations on the shape of the system transfer function and on the pole placement techniques for an undamped structure.Both these tuning methods are based on the classical TMD theory.Høgsberg and Krenk [9,10] developed another calibration method based on the pole placement for RL circuits in series and parallel.The values of  and  are selected to guarantee equal modal damping of the two modes of the electromechanical structure and good separation of the complex poles.Thomas et al. [11] proposed two different methods, even for damped structures, that relied on the transfer function criteria and pole placement and provided closed formulas to estimate the attenuation performance.
Although all of the mentioned tuning strategies work extremely well, one significant issue of shunt damping using RL impedances is that this type of electrical circuit is not adaptive.This in turn means that it is not possible to follow possible changes in the dynamic behaviour of either the vibrating structure (e.g., a temperature shift can change the eigenfrequency of the mode to be controlled) or the impedance itself (e.g., a temperature shift can cause a significant change in the  value [12]).Hence, this control technique often works in mistuned conditions, even when starting from a perfect tuning condition.This mistuning leads to severe worsening of the attenuation performance.
A few techniques based on adaptive circuits were proposed to overcome the limitations due to uncertainties in the mechanical and electrical quantities.Based on the singlemode control, Hollkamp and Starchville developed a selftuning RL circuit that was able to follow any change in the frequency of the mode to be controlled [13].This technique is based on a synthetic circuit (which provides both the resistance and the inductance) consisting of two operational amplifiers and a motorised potentiometer.Despite its effectiveness, this method only considers a mistuning due to a change in the eigenfrequency to be controlled and does not consider other types of changes or uncertainties, such as ones related to electrical parameters.Furthermore, this method is active, thus losing the advantage provided by the passive shunt technique.Other recent studies by Zhou et al. [14,15] attempted to determine methods to limit the problem of mistuning by using nonlinear elements when the disturbance was harmonic and using more than one piezoelectric actuator bonded to the vibrating structure.Although these techniques can be effectively employed, their use implies the loss of the two primary features of the resonant piezoelectric shunt: linearity (and thus ease of use) and passivity.Therefore, the analysis of the performances of traditional RL shunts in mistuned conditions still has significant relevance.
Although the problems related to mistuning are evidenced in literature [16][17][18], there have been few analyses on shunt robustness.These analyses are of significant interest for numerous engineering applications where electrical power is often limited or even avoided, thus preventing the use of adaptation systems for the shunt impedance (e.g., space applications).In situations where passivity is requested, it is important to analyse the behaviour of the shunted system in the presence of mistuning because it worsens the attenuation performance.Recently, Berardengo et al. [19] studied the robustness of different optimisation methods for RL circuits and determined the most robust method.Based on the outcomes of [19], this paper aims to further investigate the robustness of RL shunt damping.The word robustness is intended here as the capability of the shunt impedance to attenuate the vibrations even when in mistuned conditions.Therefore, this paper analyses the behaviour of mistuned electromechanical systems, thus depicting the relationship between the attenuation and the system parameters (e.g., coupling coefficient, mechanical nondimensional damping ratio, and eigenfrequency) in tuned and mistuned conditions.Furthermore, this paper demonstrates that the loss of attenuation primarily depends on only one bias (i.e., either the bias on the damping or the eigenfrequency of the electric resonant circuit) if the electrical damping is overestimated, whereas the effects of the two bias types (on the electrical eigenfrequency and damping) combine with each other when the electrical damping is underestimated.Based on these results, an approximated analytical model is proposed to estimate the attenuation performance with different amounts of mistuning using a small number of numerical simulations.
To summarise, the goals of this paper are to investigate how mistuned systems (which are often encountered in real applications) behave and consequently propose an approximated model that is able to predict the behaviour of the mistuned system with the least amount of numerical simulations.To reach the above goal, the authors highlight the relationship between the attenuation and all of the problem parameters and demonstrate that some of these relations can be approximated linearly in a logarithmic scale.Moreover, the authors bring to evidence the cases where the loss of performance depends on just one mistuning type (i.e., either the bias on the damping or the eigenfrequency of the electric resonant circuit), even though mistuning occurs on both, as well as the cases where both the mistuning types have an influence.All of these observations allow for the development of the mentioned approximated model for mistuned systems, which enhances the knowledge of their behaviour.Moreover, using this new simplified model, the authors demonstrate that an initially overestimated value of  is able to decrease the loss in performance due to mistuning and explain why this phenomenon occurs.Additionally, this allows for guidelines to be provided on how to tune the shunt parameters when a mistuning is expected.
This paper is structured as follows.Section 2 discusses the model of the electromechanical system used in this paper.Section 3 highlights the linear relationship between the attenuation and the system parameters, which will be employed in Section 4 to analyse the effects of mistuning and propose an approximated model to describe the attenuation performance in the presence of mistuning.Lastly, Section 5 validates the previous results using experiments.

Model of the Electromechanical System
As mentioned in the previous section, the goal of this paper is to study the vibration attenuation of the controlled system in mistuned conditions.Thus, the most intuitive and used index to evaluate the attenuation performance is the ratio between the maximum of the dynamic amplification modulus in uncontrolled and controlled conditions [11,19].Therefore, for the performance analysis, it is necessary to derive the expression of the frequency response function of the electromechanical system and thus to introduce the model used to describe its electrodynamic behaviour.The piezoelectric actuator is modelled here as a capacitance   and a strain-induced voltage generator in series (Figure 1(a)).The induced voltage is   , whereas the voltage between the electrodes of the piezoelectric bender is . is equal to   when the piezoelectric actuator is opencircuited and null when the actuator is short-circuited. takes different values when an impedance  is shunted to the electrodes of the actuator (Figure 1(b)) because a current   flows in the circuit.Moheimani et al. [20,21] proved that systems controlled by piezoelectric actuators shunted with electric impedances can be modelled as a double feedback loop (Figure 2(a)).The inner loop of Figure 2(a) can be observed as a controller , which can be expressed in the Laplace domain as follows: where  is the Laplace variable.
Because the shunt impedance  considered in this study is a resistance  and an inductance L (see Section 1) in series, Z can be expressed in the Laplace domain as follows: The two terms  VV and  V in Figure 2(a) are frequency response functions (FRFs).The former is the FRF between  and   , whereas the latter is between a disturbance  and   .These two FRFs can be expressed by the formulations in the Laplace domain [20] as follows: where   is the th eigenfrequency of the structure with the piezoelectric bender short-circuited;   is the associated nondimensional damping ratio; Φ  is the th eigenmode of the structure (scaled to the unit modal mass); Φ  (  ) represents the value of the th mode at the forcing point   ;   is a term depending on the curvature of the th mode in the area of the piezoelectric patch [20,21], which assumes different formulations for mono-and bidimensional structures; and  and  are two parameters based on the geometric, mechanical, and electrical features of the structure and the piezoelectric actuator.The method for calculating   , , and  for different possible configurations (e.g., monodimensional and bidimensional structures, symmetric and antisymmetric configuration of the piezoelectric actuator) can be found in [19].
The closed-loop FRF between disturbance W (Figure 2(a)) and   can be expressed as follows: Then, the closed-loop FRF between  and the transverse displacement  of the structure (Figure 2(b)) in a given point   , which describes the behaviour of the system damped by the shunt, can be expressed as follows: where () is the FRF between  and  [19], which can be given as follows: Based on the aforementioned theoretical approach (see ( 5)), the formulation of   can be rearranged to achieve a compact expression.Thus, the eigenfrequency   and the nondimensional damping ratio   of the electric network (composed by the series of   , L, and R) [19] can be conveniently defined as follows: By substituting (2), (7), and (8) into (1), the controller  can be expressed as a function of these two quantities as follows: For single degree of freedom systems, the FRF   as a function of the electrical eigenfrequency and damping can be derived by substituting ( 9), (6), and (3) into (5) as follows: This formulation is valid for both beams and plates as well as for any layout of the piezoelectric actuator (e.g., single actuator, two colocated actuators) [19].It should be noted that if the poles of this FRF are calculated considering the piezoelectric actuator in open-circuit condition (  = 0,  = +∞), then the expression of the open-circuit eigenfrequency  oc  can be written as follows: Hence, it is possible to calculate the th effective coupling factor   (defined as √ (( oc  ) 2 −  2  )/ 2  , e.g., [4,11,22]) using (11) as follows: It should be noted that   does not depend on the type of shunt used but is a property of the system composed of the vibrating structure and the piezoelectric actuator;   indicates the capability of the piezoelectric actuator, coupled to a given structure, to transform mechanical energy into electrical energy.
The performance of the controlled system in optimal conditions will depend on the tuning strategy selected to fix the values of  and .The one considered here is found as the most robust to possible mistuning in [19].It is based on considerations on the shape of the FRF of (10).Nevertheless, it will be shown that the results and the procedure presented in this paper are valid for all tuning strategies that lead to a nearly flat shape of the FRF around the resonance frequency (see Section 3).The tuning criterion considered here fixes the values of  and  based on the procedure briefly summarised here below: (i) The trend of |  | is independent of the damping factor of the electrical circuit   at two frequency values   and   (see the corresponding points A and B in Figure 3(a)) ( is the circular frequency) for undamped systems [23].The optimal value of   ( opt  ) can be found by imposing the same dynamic amplification modulus at these two frequencies.Thus, the expression for the electrical eigenfrequency can be achieved [19] as follows: Then, the value of  can be found by combining (7) and ( 13).(ii) The optimal value of the damping   (and thus of R) is found by imposing an equal dynamic amplification |  | at two different frequencies:   and a second frequency given by the square root of the arithmetic mean of  2  and  2  .This frequency is found to be equal to the electrical frequency   [19].Thus, the condition used to fix the value of   can be given as follows: This condition is convenient to tune the shunt impedance because it allows a flat trend of |  | to occur in the frequency band around the resonance (Figure 3(b)).The value of the optimal electrical damping ( opt  ), which results from ( 14) (considering   = 0), can be given as follows: Then, the value of  can be found using (8).It should be noted that the use of ( 13) and ( 15) (which are yielded considering   = 0) in the case of damped systems introduces certain approximations.Nevertheless, these approximations can be assumed as negligible.In fact, according to [19], the maximum difference between the attenuation provided by ( 13) and ( 15) and the actual attenuation is less than 0.5 dB for most practical applications.Therefore, the use of ( 13) and ( 15) can be considered reliable even with damped systems.
The behaviour of mistuned systems will be studied in the following sections.Because there are no closed formulas to describe the vibration attenuation in mistuned conditions, the maximum of |  | must be found numerically using (10).The number of variables in this equation is high: five variables, that is,   ,   ,   ,   , and  2  .Hence, several simulations must be performed if a detailed description of the behaviour of different possible mistuned systems is desired (several values of   ,   , and  2  for each mode considered, defined by   and   ).Therefore, it is essential to decrease the number of variables to be considered in the simulations to reduce the effort of this numerical study.Therefore, Section 2.1 presents a normalisation of the system model to reduce the number of variables involved in the problem.

Normalisation of the Model.
First, all of the possible values of   and   can be defined as a function of the optimal ones  opt  and  opt  as follows: where  and V are the amount of mistuning on the electrical damping and eigenfrequency, respectively ( = 1 and V = 1 in the case of no mistuning).Then, ( 10) is considered: both the numerator and the denominator are divided by  4   , and  is expressed as  (j is the imaginary unit).After a few mathematical rearrangements (see Appendix A), a new expression of   in the frequency domain can be obtained.This new expression uses (13), (15), and ( 16) to express the electrical parameters as a function of their optimal values as follows: where  = /  is the nondimensional frequency.
The advantages provided by the use of ( 17) will be underlined in Section 3.

Attenuation Performance of the Optimally Tuned Shunt
As previously mentioned, the performance of the shunt in terms of vibration attenuation can be expressed as the ratio between the maximum amplitude of the uncontrolled system FRF and the maximum amplitude of the controlled system FRF (i.e., max(|  |); see (17)).
The FRF of the uncontrolled structure (i.e., with the piezoelectric patch in short-circuit) can be defined as follows [24]: Therefore, the attenuation performance, denoted here as att, can be expressed as follows: where, according to [11], max( ).The analytical expression of max(|  |) is rather complex; thus, it is convenient to define the index attk instead of att for the case of perfect tuning (  =  opt  ) as follows: The difference between att and attk is that, in the former case, the maximum amplitude of the controlled system FRF is considered (max(|  |)), whereas, in the latter case, the value of the system response As previously mentioned, the use of attk simplifies the notation and can be used to accurately approximate the value of att.In fact, in the case of perfect tuning, max is in the frequency range where the controlled FRF has a flat shape; see Figure 3(b)) [19].Hence, att ≃ att.
Based on (17), attk can be expressed as follows (see Appendix B): Thus, the attenuation in decibels ( dB ) can be expressed as follows: dB = 20 log 10 att Equation ( 22) only depends on two system parameters,   and   .Therefore, the properties of tuned systems can be studied considering only these two parameters.A similar approach is used for mistuned shunt systems (see Section 4).Hence, the normalisation proposed in Section 2.1 allows the model to be simplified, thus avoiding one of the variables (i.e., now only   and   are considered, whereas it would have been necessary to consider the three parameters   ,   , and  2  without the normalisation).
It is easy to see that ( 22) links the achievable attenuation to the problem parameters (i.e.,   and   ).Since   is fixed, (22) allows the attenuation to be predicted as a function of the value of   , thus suggesting which value should be used to obtain the desired attenuation performance.In fact, it can be recalled that   is a function of  2  (see ( 12)), and it can thus be modified by changing the geometrical, mechanical, and electrical characteristics of the actuator as well as its position [19].Furthermore,   can be also modified by connecting several piezoelectric actuators in series/parallel [25,26] and by using a negative capacitance [22,27].Therefore, the model used here is of general validity.
Equation ( 22) can be rearranged as follows: Now, three different situations in terms of the   value can be considered: (1)   of the same order of magnitude of   (the maximum value of   considered here is 1%): this is the case of extremely stiff and damped structures and/or badly positioned actuators.In this case, ( 23) can be approximated as follows: In fact,   is so low that 10 log 10 (1 +  2  ) can be approximated as 10 log 10 (1) = 0.
(3)   close to 1: this is the case where extremely flexible structures and/or the addition of a negative capacitance are considered.Equation ( 23) can be approximated as follows: Equation (26) demonstrates that, in this case, the relationship between  dB and log 10 (  ) is no longer linear.Nevertheless, in most practical applications, a linear relation can still be used.In fact, the term 10 log 10 (1 +  2  ) has a negligible contribution up to approximately   = 0.6.Its contribution becomes more evident, albeit small, only for higher   values (at   = 0.8, its contribution is approximately 2 dB).Hence, the term 10 log 10 (1 +  2  ) can be neglected, and the attenuation  dB as a function of log 10 (  ) can be approximated by the linear relation as follows: where  0 = 20 and  0 = −10 log 10 (8 2  (1 −  2  )).
Figure 4 shows the linear relationship between  dB and log 10 (  ) for different systems selected as an example.Additionally, the figure indicates that the area in which linearity is lost (corresponding to the case in which   is of the same order of magnitude of   , point 1 of the previous numbered list; see the left part of the green dashed line in the figure) corresponds to cases where  dB is extremely low (approximately 5 dB or lower).
The linear relationship demonstrated thus far (see ( 25) and ( 27)) can lead to the following notations: ( The central expression of (28) indicates that if   is incremented from value 1 to value 2 where 2/1 = , then the value of att increases by a factor , which signifies that consistent increases in att can be achieved with moderate increases (in terms of absolute value) of   when   is low.
Conversely, high increases in   (in terms of absolute value) produce a low increment of att when   is high.Hence, an asymptotic behaviour of the attenuation performance is demonstrated.
The next section considers mistuned shunt systems.

Robustness of the Shunt Damping: Performance in Mistuned Conditions
Section 1 explained that, in most cases, the shunted system operates in mistuned conditions because of the uncertainty in the estimated values of the system parameters (especially electrical quantities) or changes in either the mechanical system or the electrical network.This often leads to a control performance considerably lower than that expected; thus, a robustness analysis would be useful for understanding the behaviour of the controlled system and determining a method to limit this performance reduction.Therefore, the analysis of robustness attempts to investigate the worsening of performance due to mistuning and provides formulations for its prediction.
Based on (10), (13), and (15), the mistuning can be due to errors in the estimated values of   ,   , and  2  as well as the values of  and .It is easy to see that all of the different reasons for mistuning can be expressed as errors in the optimal values of   and   .Therefore, in this study, the actual values of   and   are expressed as changes from their optimal values.Therefore, the mistuning can be easily considered in (17) by fixing  and V at values other than 1 (values lower than 1 indicate underestimation, whereas those higher than 1 indicate overestimation; see (16)).The FRF of the mistuned shunt system can thus be described by (17), and the related vibration attenuation performance is measured by the index att of (19).The attenuation in these mistuned conditions can be expressed in decibels as  * dB (see Appendix C for certain clarifications for the symbols used) as follows: The value of  * dB can be found numerically to study the attenuation in different mistuned conditions.Thus, the values of |  | as a function of frequency must be calculated for a given situation (i.e., fixing the values of   ,   , , and V in ( 17));  and then the maximum of |  | can be found numerically.Lastly, att can be calculated using (19).
Nevertheless, the number of simulations needed is often high.In fact, several different values of  and V need to be tested to consider various different possible mistuning situations.Furthermore, numerous values of   must be considered; in fact, it is useful to understand if an increased   value allows the desired attenuation performance to be achieved, even in mistuned conditions.Furthermore, according to [19] and as it will be shown in Sections 4.1 and 4.3, it is often good practice to increase the initial value of the resistance with respect to its optimal value to improve the attenuation in mistuned conditions; hence,  values significantly higher than 1 need to be tested, thus leading to a large number of  values to be taken into account.Therefore, the number of required simulations can increase substantially.For example, when   values of   ,  V biased values of   , and   biased values of   have to be considered to fully study the given problem, the entire number of simulations   that must be performed to evaluate the attenuation in all of the possible cases results in    V   (e.g., if   is equal to 10 6 , the amount of computational time to perform all the simulations becomes longer than 10 hours on a normal laptop).
Therefore, the goal of the next sections is to propose a model to describe the attenuation in mistuned conditions  * dB .Sections 4.1 and 4.2 analyse the effects of errors on   and   , respectively.Then, Section 4.3 addresses situations where both errors (i.e., on   and   ) occur together.

Mistuning on the Electrical
Damping.This section only considers mistuning on   .Figure 5 depicts the curves relating log 10   and  * dB for different systems and different errors on   (i.e., different  values and V = 1).These curves were found numerically using ( 29), (19), and (17) (see Section 4).In fact, a general analytical solution is not possible because the equations are of a high order, and thus the solution cannot be expressed through closed analytical formulas and must be calculated numerically case by case.It should be noted that the use of the normalised model of ( 17) still allows a decrease in the number of variables to be considered: four variables in the normalised model (i.e.,   ,   , , and V; see ( 17)) versus five variables in the nonnormalised model (i.e.,   ,   ,  2  , , and V; see (10)).
The primary property of the plots in Figure 5 is that the main effect of mistuning is to shift the curves with respect to the case of  = 1; however, all of these curves can still be approximated as straight lines.In fact, the lines associated with   = 1% lose their linear trend in correspondence of low values of  * dB (i.e., for approximately  * dB < 4 dB); nevertheless, these curves can still be considered piecewise linear.In fact, if an interval for   equal to one order of magnitude is considered (it is hard to change   for one order of magnitude [25] or more, even using negative capacitances [27]), the curves can be well approximated as lines.
The lines in Figure 5 prove that the effects of the change in the intercept are significantly higher than the effects of the change in the angular coefficient (i.e., the lines primarily shift due to a nonunitary value of , parallel lines).In other words, the sensitivity of the attenuation performance on the value of the coupling coefficient tends to be independent of the level of mistuning.Hence, for a given system, the improvement in the attenuation achieved by increasing the value of the coupling coefficient is the same whether the shunt is tuned or not.
The relationship between log 10   and  * dB for a given system can be expressed as follows: where Ã * dB is the estimate of  * dB and  *  and  *  are the intercept and the angular coefficient of the line, respectively, which are both a function of , as evidenced in (30) (see also Appendix C for certain clarifications of the symbols used).It should be noted that  *  is indicated as dependent on , even if this dependency is slight (see above), for the sake of completeness.
If the trend of  * dB as a function of  is depicted for different values of   , a few further interesting facts can be noted (see Figure 6).All of the curves of this new figure demonstrate nearly the same features: the attenuation loss is limited for  > 1; furthermore, in this range of , the rate of the loss is nearly constant.Conversely, for  < 1, the rate becomes increasingly larger by decreasing the value .It is possible to approximatively state that, for  < 0.5, the rate of the attenuation loss increases.This result is a first sign of the benefits provided by the use of overestimated electrical damping   values (and thus overestimated  values).In fact, even if an overestimated   value causes a worsening in the attenuation performance if compared to the tuned situation, this worsening is not that high (see Figure 6); overall, if a mistuning occurs in situations where the starting   value is overestimated deliberately, the attenuation loss due to the mistuning is low.The use of initially overestimated   values will be considered again in Section 4.3.
A further interesting point is that the trend of  * dB as a function of  can be modelled as the combination of two fourth-order polynomials, one for  < 1 and another for  > 1, regardless of the system considered (see Figure 6).
The calculation for each of these fourth-order polynomials requires the knowledge of  * dB for five values of .Therefore, for the given values of   and   , it is sufficient to calculate ten points (,  * dB ) using ( 29), (19), and (17) (i.e., five for  < 1 and five for  > 1) to determine the trend of  * dB for an extended range of  values (e.g.,  = 0.01 ÷ 2, as indicated in Figure 6).
Based on (30) and Figure 6, the following procedure can be applied when the behaviour of a mistuned shunt system in a range of   values between   and   is studied.
(i) Calculate five pairs (,  * dB ) using ( 29), (19), and (17) for  < 1 and   =   and determine the interpolating polynomial.Then, repeat the same procedure for  > 1.It is now possible to know the value of  * dB for any value of  at   =   .
(ii) Calculate five pairs (,  * dB ) using ( 29), (19), and (17) for  < 1 and   =   and determine the interpolating polynomial.Then, repeat the same procedure for Therefore, it is possible to estimate the attenuation for any value of  and   (between   and   ) with only twenty simulations.
The accuracy of this procedure was verified using a Monte Carlo test with more than 10 5 simulations comparing the attenuation values Ã * dB achieved using this procedure and the  * dB values obtained using ( 29), (19), and (17).The difference Δ is defined as Ã * dB −  * dB .For each simulation, the values of   ,   ,   , and   were extracted from uniform distributions.The bounds of the distributions are presented in Table 1 and were chosen in order to take into account the most part of the practical applications.Table 2 lists the results, hence proving the reliability of the proposed procedure.It should be noted that   = √ 3  was used in the simulations.This corresponds to a change of  2   within an interval equal to three times the starting value, which is quite a broad interval.If a wider interval of   must be considered and the same accuracy is desired, it is possible to analyse the system in two different ranges.For example, if   = 3  , the entire range can be split as follows:   ÷  ℎ and  ℎ ÷   with  ℎ = √ 3  .This requires using thirty simulations instead of twenty to describe the behaviour of the mistuned shunt system.

Mistuning on the Electrical Eigenfrequency.
Figure 7 illustrates the same information as Figure 5 but for mistuning on   (i.e., V ̸ = 1 and  = 1).It should be noted that the resulting curves tend to increase their curvature.Nevertheless, the linearity is lost only when  * dB becomes lower than approximately 4 or 5 dB.However, the curves can be still piecewise approximated as lines with enough accuracy for wide ranges of log 10 (  ).The effect of a nonunitary value of V is to highly increase the angular coefficient of the lines.Consequently, increasing the value of   is even more effective in enhancing the attenuation in the case of mistuning on   than in the case of tuned systems.
The curves of Figure 7 can be expressed as follows: where Ã * dB is again the estimate of  * dB and  * V and  * V are the intercept and the angular coefficient of the lines, respectively, The + are the points related to the case of  > 1 calculated using ( 29), (19), and (17), which are then interpolated by fourth-order polynomials (solid curves).The * are the points related to the case of  < 1 calculated using ( 29), (19), and (17), which are then interpolated by fourth-order polynomials (solid curves).The + are the points related to the case of V > 1 calculated using ( 29), (19), and (17), which are then interpolated by fourth-order polynomials (solid curves).The * are the points related to the case of V < 1 calculated again using ( 29), (19), and (17), which are then interpolated by fourth-order polynomials (solid curves).If the trend of  * dB is shown as a function of V for different values of   , a few further interesting facts can be noted (see Figure 8).The same percentage value of mistuning leads to a different decrease in the performance if it is related to an overestimation or underestimation of the optimal value of the electrical eigenfrequency.In fact, values of V lower than 1 cause higher losses in the attenuation than values greater than 1 (e.g., compare the curves at V = 0.75 and V = 1.25).
A further interesting point is that the trend of  * dB as a function of V can be modelled as a fourth-order polynomial for both V < 1 and V > 1, regardless of the system considered (see Figure 8).Therefore, if the study of the behaviour of a mistuned shunt system in a range of   values between   and   is considered, the same procedure discussed in Section 4.1 (see the list in Section 4.1) can be applied, and it is possible to estimate the attenuation for any value of V and   (between   and   ) with only twenty simulations.Indeed, it is possible to write Again, a Monte Carlo test was performed with more than 10 5 simulations comparing the attenuation values Ã * dB achieved using this procedure and the  * dB values obtained using ( 29), (19), and (17).For each simulation, the values of   ,   ,   , and V  were extracted from uniform distributions (see Table 3 for the bounds of the distributions, which were chosen in order to take into account the most part of the practical applications), and   was fixed to √ 3  , as performed in Section 4.1.Table 4 lists the results, thus proving the reliability of the proposed procedure.It should be noted that the range of V  is narrower than that used in Section 4.1 for   .The reason is that the optimal value of   depends on more variables than the optimal value of   , thus leading to more uncertainty (see (15) and ( 13)).

Mistuning on Both the Electrical Eigenfrequency and the
Damping Ratio.Sections 4.1 and 4.2 have treated cases in which the mistuning is related to either   or   .Nevertheless, in actual applications, both of the mistuning effects are expected to appear together.In these general mistuned situations (i.e.,  ̸ = 1 and V ̸ = 1), the performance of the shunt system demonstrates a different behaviour for values of  higher or lower than 1.This result can be clearly evidenced using the double-logarithmic representation already utilised in Figures 5 and 7.In fact, Figure 9 depicts the different behaviour for certain systems selected as examples: (i) For systems where  > 1, the loss of attenuation is essentially due to the mistuning causing the highest loss (see the subplots on the right side).
(ii) For systems where  < 1, the loss of attenuation can be derived as the sum of the losses caused by both the mistuning types (see the subplots on the left side).Now, for a given system (i.e., fixed values of   and   ) and fixed values of  and V (named   and V  ), the following indexes can be defined: where   expresses the loss of attenuation when a mistuning occurs on the values of both   and   ;   expresses the loss of attenuation when a mistuning occurs only on the value of   ; and   expresses the loss of attenuation when a mistuning occurs only on the value of   .
Based on the abovementioned considerations related to Figure 9 (see the list above in this section), ỹ (the estimate of   ) can be calculated as follows: According to (35), Ã * dB (i.e., the estimate of  * dB ) can be defined as follows: * dB | =1,V=V  and  * dB | =  ,V=1 can be estimated using ( 32) and (30), respectively. dB is given in (22).Hence, (36) allows the attenuation to be estimated for any values of , V, and   (between   and   ) using only forty-two simulations based on ( 17), (19), and (29).In fact, (36) allows the behaviour of the mistuned shunt systems to be analysed with a bias on both   and   by considering the mistuning on   and   separately.Twenty simulations are needed to study the behaviour of the system with  ̸ = 1 and V = 1 (see Section 4.1), twenty for the case  = 1 and V ̸ = 1 (see Section 4.2), and two for the case  = 1 and V = 1 (i.e., one with   =   and the other with   =   ); furthermore, the  dB (i.e., the case with  = 1 and V = 1) values for   =   and   =   can also be calculated using (22).In the case of   (total number of cases to be considered) equal to 10 6 , the amount of time required to perform all the simulations decreases from more than 10 hours (see the end of Section 4) to a few minutes or less (approximately 30 s).Hence, the study of the behaviour of the system in mistuned conditions becomes very fast, thus allowing to quickly analyse the effects of different  values on the attenuation performance and to choose the best one for the given application.
Therefore, by rearranging (36) using ( 30) and ( 32), the final form of the approximated model able to describe the behaviour of the mistuned shunt systems can be achieved as follows: where  *  ,  *  ,  * V , and  * V are defined in (31) and (33).The accuracy of this model was tested again using a Monte Carlo simulation with more than 10 6 cases, thus comparing the attenuation values Ã * dB achieved using this procedure and the  * dB values obtained using ( 29), (19), and (17).For each simulation, the values of   ,   ,   , V  , and   were extracted  from uniform distributions (see Table 5;   = √ 3  ).Table 6 presents the results (which have a Gaussian distribution), thus proving the reliability of the proposed procedure.
Certain benefits provided by the use of initially overestimated   values have already been discussed for the cases of bias just on   in Section 4.1.Here, the discussion can be extended to the more general case of mistuning on both   and   (which is the typical situation).Also in this case the use of an initially overestimated   value allows the loss of attenuation to decrease.In fact, when   is overestimated, only one bias has significant effects, whereas the other does not have much influence (see above in this section and Figure 9).Conversely, when the   value is lower than its optimal value, the attenuation loss due to mistuning is more severe.Therefore, this property of the mistuned systems along with those already shown in Section 4.1 highlights that the use of initially overestimated   values (and thus initially overestimated  values) allows the robustness to increase, thus lowering the loss of attenuation due to mistuning, which is typically experienced starting from the optimal   value.Furthermore, this can allow the analysis of the mistuned system to become faster because the study of its behaviour can focus on values of  higher than 1 (because an initially overestimated   value is used on purpose) and possibly slightly lower than 1 (e.g., greater than 0.5).Clearly, these are just guidelines because each practical case could require a different solution.Nevertheless, the points demonstrated thus far clearly indicate how robustness can be typically increased and how the proposed model can help in the tuning process.
The model presented so far has been validated by experimental tests shown in the next section.

Experimental Tests
This section describes the experimental tests performed to validate the results shown in the previous sections.Two test structures have been used to investigate different values of the   and   parameters and different values of vibration attenuation.The first structure is an aluminium plate (in freefree condition by suspension) with the shunted piezobender bonded at about its centre (see Figure 10(a)).A bidimensional structure was used because it is a more complex test case when compared to monodimensional structures often used in other studies.The plate length is 600 mm, the width is 400 mm, and the thickness is 8 mm (this set-up is the same  as that used in the experiments of [19]).The capacitance   is 0.02 F.Several modes were taken into account in the tests.The one (among others tested) considered here as an example has the following modal parameters (identified by experimental modal analysis [28]):   = 530.67⋅ 2 rad/s,   = 0.22%,   = 0.0081, and  2  = 725 rad 2 /s 2 .Actually, the value of   was estimated by testing the system in both shortand open-circuit conditions (see (12)).
These values of  2  and   were achieved using a negative capacitance [27], which allowed their initial low values to increase.Furthermore, another value of   (i.e., 0.0240) was tested by further boosting the negative capacitance performance.The disturbance to the structure was provided by a dynamometric impact hammer, and the response was measured using a piezoelectric accelerometer.
The second structure is an aluminium cantilever beam (159 mm length, 25 mm width, and 1 mm thickness) with a piezoelectric patch bonded corresponding to the clamped end (see Figure 10(b)).Its capacitance   is 31 nF.Again, several modes were considered during the tests; here, the results related to the first mode are presented for the sake of conciseness.It has the following modal parameters, again identified using an experimental modal analysis:   = 32.61⋅ 2 rad/s,   = 0.40%,   = 0.2002, and  2  = 944.6 rad 2 /s 2 .Furthermore, other tests were performed by increasing the values of   up to 0.5108 using a negative capacitance.
Because this second test structure was extremely light, noncontact methods were used to provide excitation and to measure the response.Indeed, an electromagnetic device was used to excite the structure [29], and the response was measured using a laser velocimeter focused on the beam tip.The tests were performed by exciting the beam with a random signal [30] up to 1.6 kHz.
The tests were performed using synthetic impedance based on operational amplifiers [11,31,32] to build the inductor.Actually, certain tests on the plate were performed using an additional method: the entire shunt impedance was simulated using a high-speed Field Programmable Gate Array (FPGA) device (in this second case, a colocated piezoelectric patch was used to provide the input voltage to the simulated shunt impedance).The use of the FPGA device allowed for the full control of the parameters of the electric shunt impedance.Nevertheless, the two techniques led to similar results; therefore, those achieved using the synthetic impedance are presented here, since this technique introduces the highest level of uncertainty between the two.Therefore, the authors believed it to be the most representative to demonstrate the model effectiveness.
First, the reliability of the model, represented by ( 10) and ( 17), was verified.Figure 11 depicts the FRFs for the mode at approximately 530 Hz of the plate, achieved with different configurations of the shunt (i.e., using different  and V values).The numerical FRFs match the experimental curves, thus confirming the accuracy of the numerical model.The curves are not plotted on the same graph for the sake of clarity in the figure.Nonetheless, Figure 12 depicts a few of the experimental and numerical FRFs of Figure 11 on the same plot for an easy comparison.Then, the reliability of the proposed approximated model (see (37)) for predicting the attenuation in mistuned conditions was tested.Tables 7  and 8 list the comparisons between experimental attenuations, numerical attenuations calculated using the theoretical model of ( 17), (19), and ( 29), and attenuations estimated using the proposed approximated model of (37) for the plate and the beam.To build the approximated model of (37), the values of   and   must be fixed.Three different situations were tested: one where   was close to   (  = 1.1  , named case 1), one where   was close to   (  = 1.63  , named case 2), and a further one where   was nearly halfway (  = 1.37  , named case 3).In all of the three cases,   was fixed to √ 3  .The experimental attenuations are defined in the tables as EA, whereas the numerical ones (see (17), (19), and ( 29)) are defined as NA for the sake of conciseness.Moreover, the attenuation provided by the model of (37) in cases 1, 2, and 3 is named MA1, MA2, and MA3, respectively.The match among all of the results is good.The results related to the MA1, MA2, and MA3 cases are always close to each other, and the maximum difference if compared to the NA results is on the order of 0.5 dB.Because the EA results differ from the NA results at a maximum of 1.2 dB, the proposed model of (37) is considered to be validated.

Conclusion
This paper addresses monomodal vibration attenuation using piezoelectric transducers shunted to impedances consisting of an inductance and a resistance in series.Although this method works well when the tuning between the mechanical system and the electrical network is properly realised, this control technique is not adaptive, and its performances thus decrease as soon as a mistuning occurs.
The paper analyses the behaviour of mistuned electromechanical systems, demonstrating that a linear relationship between the attenuation and the logarithm of the effective   coupling coefficient exists when a perfect tuning is reached.The same linear behaviour exists when there is mistuning on either the electrical eigenfrequency or damping.Moreover, the paper indicates how the loss of attenuation essentially depends on only one bias if the electrical damping is overestimated and describes how the effects of the two bias types (on the electrical eigenfrequency and damping) combine with each other when the damping is underestimated.
This allows an approximated model to be achieved for describing the behaviour of mistuned shunt systems, which was initially validated numerically using Monte Carlo simulations and then experimentally through the use of two test structures.Furthermore, the use of overestimated resistance values is demonstrated to limit the loss of attenuation due to mistuning.

B. att𝑘 Analytical Expression
The mathematical process used to express att (see (21)) is explained here.The expression of   in ( 17) can be rearranged by separating the real and imaginary parts at the numerator and denominator as follows:

C. List of the Symbols
This appendix clarifies the meaning of the symbols used.
The symbol * represents a generic mistuned condition.The symbol ∼ represents an estimate of the considered quantity.
* dB expresses the attenuation in decibels achieved in case of mistuning (this is evidenced by the asterisk).
When  dB and  * dB are evaluated at specific points (i.e., given values of , V, or   ), the following expressions are used:  dB |   and  * dB | =1,V=V  , as examples.The former expression indicates that  dB is computed in correspondence with a given value   of the effective coupling factor, whereas the latter indicates that  * dB is computed for  = 1 and V = V  .Ã * dB and ÃdB represent the values of  * dB and  dB , respectively, which are estimated using the model proposed in the paper, that is, by (30), (32), and (37).
As for the angular coefficient  and the intercept  of the linear relations presented in the paper (e.g., (25), ( 27), (30), and (32)), when they have a subscript 0, they are calculated for a perfectly tuned shunt impedance.Conversely, when they

Figure 1 :
Figure 1: Electric equivalent of a piezoelectric actuator in open circuit (a) and shunted with impedance Z (b).

Figure 2 :
Figure 2: Feedback representation of the shunt control (a) and a structure subject to disturbance  and damped using a piezoactuator shunted to an electric impedance Z (b).

Figure 3 :
Figure 3: |  | for an undamped elastic structure (a) and trend of |  | for a generic system with the optimal value of   (b).

Figure 4 :
Figure 4: Relationship between  dB and log 10   for different systems (i.e., different   values).

Table 2 :
Results of the Monte Carlo simulations for the case  ̸ = 1 and V = 1.

Table 3 :
Bounds for the Monte Carlo simulations for V ̸ = 1 and  = 1.

Table 4 :
Results of the Monte Carlo simulations for V ̸ = 1 and  = 1.