Research Article The Behaviour of Mistuned Piezoelectric Shunt Systems and Its Estimation

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1 Shock and Vibration Volume 216, Article ID , 18 ages htt://dx.doi.org/1.1155/216/ Research Article The Behaviour of Mistuned Piezoelectric Shunt Systems and Its Estimation M. Berardengo, 1 S. Manzoni, 2 and M. Vanali 1 1 Deartment of Industrial Engineering, Università degli Studi di Parma, Parco Area delle Scienze 181/A, Parma, Italy 2 DeartmentofMechanicalEngineering,PolitecnicodiMilano,ViaLaMasa34,2156Milan,Italy Corresondence should be addressed to S. Manzoni; stefano.manzoni@olimi.it Received 2 May 216; Revised 18 July 216; Acceted 26 July 216 Academic Editor: Nicola Caterino Coyright 216 M. Berardengo et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. This aer addresses monoharmonic vibration attenuation using iezoelectric transducers shunted with electric imedances consisting of a resistance and an inductance in series. This tye of vibration attenuation has several advantages but suffers from roblems related to ossible mistuning. In fact, when either the mechanical system to be controlled or the shunt electric imedance undergoes a change in their dynamical features, the attenuation erformance decreases significantly. This aer describes the influence of biases in the electric imedance arameters on the attenuation rovided by the shunt and rooses an aroximated model for a raid rediction of the vibration daming erformance in mistuned situations. The analytical and numerical results achieved within the aer are validated using exerimental tests on two different test structures. 1. Introduction Vibration attenuation in light structures is a widely studied toic and often takes advantage of the use of smart materials, which are characterised by useful roerties. Indeed, these materials are inexensive when comared to other control systems, and they are characterised by low weight. This last feature is a fundamental asect because it avoids introducing high load effects on the controlled structure. Among smart materials, iezoelectric elements (articularly iezoelectric laminates, which are used in this aer) are among the best materials to attenuate vibrations in bidimensional (e.g., lates) and monodimensional (e.g., beams) structures [1 3]. There are several control techniques for light structures that rely on this tye of actuator, and one of the most attractive is the shunt of the iezoelectric element. In this case, a roerly designed electrical network is shunted to the iezoelectric bender bonded to the structure. The ability of the iezoelectric element to convert mechanical energy into electrical energy and vice versa [4, 5] is used, which allows a assive attenuation of the structure s vibration. This method was initially roosed by Hagood and von Flotow [4]. This technique is extremely attractive because it is chea, it does notintroduceenergyintothesystem,thatis,itcannotleadto instability, and it does not require any feedback signal. When a monoharmonic control is required, the most effective shunt electric imedance consists of a resistance R andaninductancel in series [2, 4, 6 8] (resonant shunt or RL shunt). These two elements, along with the caacitance C of the iezoelectric actuator (i.e., the iezoelectric actuator is modelled here as a caacitance 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 damer (TMD) [2]. Therefore, this circuit is able to dam the structural vibration corresonding 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 exlain how to select the values of R and L to otimise the vibration attenuation. Hagood and von Flotow [4] roosed two different tuning strategies based on considerations on the shae of the system transfer function and on the ole lacement techniques for an undamed structure. Both these tuning methods are based on the classical TMD theory. Høgsberg and Krenk [9, 1] develoed another calibration method based on the ole lacement for RL circuits in series and

2 2 Shock and Vibration arallel. The values of R and L are selected to guarantee equal modal daming of the two modes of the electromechanical structure and good searation of the comlex oles. Thomas et al. [11] roosed two different methods, even for damed structures, that relied on the transfer function criteria and ole lacement and rovided closed formulas to estimate the attenuation erformance. Although all of the mentioned tuning strategies work extremely well, one significant issue of shunt daming using RL imedances is that this tye of electrical circuit is not adative. This in turn means that it is not ossible to follow ossible changes in the dynamic behaviour of either the vibrating structure (e.g., a temerature shift can change the eigenfrequency of the mode to be controlled) or the imedance itself (e.g., a temerature shift can cause a significant change in the R value [12]). Hence, this control technique often works in mistuned conditions, even when starting from a erfect tuning condition. This mistuning leads to severe worsening of the attenuation erformance. A few techniques based on adative circuits were roosedtoovercomethelimitationsduetouncertaintiesin the mechanical and electrical quantities. Based on the singlemode control, Hollkam and Starchville develoed a selftuning RL circuit that was able to follow any change in the frequency of the mode to be controlled [13]. This technique isbasedonasyntheticcircuit(whichrovidesboththe resistance and the inductance) consisting of two oerational amlifiers and a motorised otentiometer. Desite its effectiveness, this method only considers a mistuning due to a change in the eigenfrequency to be controlled and does not consider other tyes of changes or uncertainties, such as ones related to electrical arameters. Furthermore, this method is active, thus losing the advantage rovided by the assive shunt technique. Other recent studies by Zhou et al. [14, 15] attemted to determine methods to limit the roblem of mistuning by using nonlinear elements when the disturbance was harmonic and using more than one iezoelectric actuator bonded to the vibrating structure. Although these techniques can be effectively emloyed, their use imlies the loss of the two rimary features of the resonant iezoelectric shunt: linearity (and thus ease of use) and assivity. Therefore, the analysis of the erformances of traditional RL shunts in mistuned conditions still has significant relevance. Although the roblems related to mistuning are evidenced in literature [16 18], there have been few analyses on shunt robustness. These analyses are of significant interest for numerous engineering alications where electrical ower is often limited or even avoided, thus reventing the use of adatation systems for the shunt imedance (e.g., sace alications). In situations where assivity is requested, it is imortant to analyse the behaviour of the shunted system in theresenceofmistuningbecauseitworsenstheattenuation erformance. Recently, Berardengo et al. [19] studied the robustness of different otimisation methods for RL circuits and determined the most robust method. Based on the outcomes of [19], this aer aims to further investigate the robustness of RL shunt daming. The word robustness is intended here as the caability of the shunt imedance to attenuate the vibrations even when in mistuned conditions. Therefore, this aer analyses the behaviour of mistuned electromechanical systems, thus deicting the relationshi between the attenuation and the system arameters (e.g., couling coefficient, mechanical nondimensional daming ratio, and eigenfrequency) in tuned and mistuned conditions. Furthermore, this aer demonstrates that the loss of attenuation rimarily deends on only one bias (i.e., either the bias on the daming or the eigenfrequency of the electric resonant circuit) if the electrical daming is overestimated, whereas the effects of the two bias tyes (on the electrical eigenfrequency and daming) combine with each other when theelectricaldamingisunderestimated.basedonthese results, an aroximated analytical model is roosed to estimate the attenuation erformance with different amounts of mistuning using a small number of numerical simulations. To summarise, the goals of this aer are to investigate how mistuned systems (which are often encountered in real alications) behave and consequently roose an aroximated model that is able to redict the behaviour of the mistuned system with the least amount of numerical simulations. To reach the above goal, the authors highlight the relationshi between the attenuation and all of the roblem arameters and demonstrate that some of these relations can be aroximated linearly in a logarithmic scale. Moreover, the authors bring to evidence the cases where the loss of erformance deends on just one mistuning tye (i.e., either the bias on the daming or the eigenfrequency of the electric resonant circuit), even though mistuning occurs on both, as well as the cases where both the mistuning tyes have an influence. All of these observations allow for the develoment of the mentioned aroximated model for mistuned systems, which enhances the knowledge of their behaviour. Moreover, using this new simlified model, the authors demonstrate that an initially overestimated value of R is able to decrease the loss in erformance due to mistuning and exlain why this henomenon occurs. Additionally, this allows for guidelines to be rovided on how to tune the shunt arameters when a mistuning is exected. This aer is structured as follows. Section 2 discusses the model of the electromechanical system used in this aer. Section 3 highlights the linear relationshi between theattenuationandthesystemarameters,whichwillbe emloyed in Section 4 to analyse the effects of mistuning and roose an aroximated model to describe the attenuation erformance in the resence of mistuning. Lastly, Section 5 validates the revious results using exeriments. 2. Model of the Electromechanical System As mentioned in the revious section, the goal of this aer is to study the vibration attenuation of the controlled system in mistuned conditions. Thus, the most intuitive and used index to evaluate the attenuation erformance is the ratio between the maximum of the dynamic amlification modulus in uncontrolled and controlled conditions [11, 19]. Therefore, for the erformance analysis, it is necessary to derive the exression of the frequency resonse function of the electromechanical system and thus to introduce the model used to describe its electrodynamic behaviour.

3 Shock and Vibration 3 i s V=V C C Z V V V Figure 1: Electric equivalent of a iezoelectric actuator in oen circuit and shunted with imedance Z. The iezoelectric actuator is modelled here as a caacitance C and a strain-induced voltage generator in series (Figure 1). The induced voltage is V, whereas the voltage between the electrodes of the iezoelectric bender is V. V is equal to V when the iezoelectric actuator is oencircuited and null when the actuator is short-circuited. V takes different values when an imedance Z is shunted to the electrodes of the actuator (Figure 1) because a current i s flows in the circuit. Moheimani et al. [2, 21] roved that systems controlled by iezoelectric actuators shunted with electric imedances can be modelled as a double feedback loo (Figure 2). The inner loo of Figure 2 can be observed as a controller K, which can be exressed in the Lalace domain as follows: K (s) = V V = sc Z (s) 1+sC Z (s), (1) W K(s) G vw (s) + + i s V + sc Z(s) Gvv (s) + z W Z Actuator V where s is the Lalace variable. Because the shunt imedance Z considered in this study is a resistance R and an inductance L (see Section 1) in series, Z can be exressed in the Lalace domain as follows: Z (s) =Ls+R. (2) The two terms G VV and G Vw in Figure 2 are frequency resonse functions (FRFs). The former is the FRF between V and V, whereas the latter is between a disturbance W and V.ThesetwoFRFscanbeexressedbytheformulationsin the Lalace domain [2] as follows: G VV (s) = V V =γ ψ n ψ n s 2 +2ξ n ω n s+ωn 2 n=1 G Vw (s) = V W = γ Φ n (x F )ψ n K s 2 +2ξ n ω n s+ωn 2, n=1 where ω n is the nth eigenfrequency of the structure with the iezoelectric bender short-circuited; ξ n is the associated nondimensional daming ratio; Φ n is the nth eigenmode (3) Figure 2: Feedback reresentation of the shunt control and a structure subject to disturbance W and damed using a iezoactuator shunted to an electric imedance Z. of the structure (scaled to the unit modal mass); Φ n (x F ) reresents the value of the nth mode at the forcing oint x F ; ψ n is a term deending on the curvature of the nth mode in the area of the iezoelectric atch [2, 21], which assumes different formulations for mono- and bidimensional structures; and γ and K aretwoarametersbasedonthe geometric, mechanical, and electrical features of the structure and the iezoelectric actuator. The method for calculating ψ n, γ, and K for different ossible configurations (e.g., monodimensional and bidimensional structures, symmetric and antisymmetric configuration of the iezoelectric actuator)canbefoundin[19].

4 4 Shock and Vibration The closed-loo FRF between disturbance W (Figure 2) and V can be exressed as follows: T Vw (s) = V W = G Vw (s) 1+K(s) G VV (s). (4) Then, the closed-loo FRF between W and the transverse dislacement z ofthestructure (Figure2) ina givenoint x M,whichdescribesthebehaviourofthesystemdamedby the shunt, can be exressed as follows: T zw (s) x=xm = z(x M) W =G(s) x=x M T Vw (s) G VV (s) (5) G =G(s) Vw (s) x=xm 1+K(s) G VV (s) 1 G VV (s), where G(s) is the FRF between V and z [19], which can be given as follows: G(s,x M )= z(x M) V = K Φ n (x M )ψ n n=1s 2 +2ξ n ω n s+ωn 2. (6) Basedontheaforementionedtheoreticalaroach(see(5)), the formulation of T zw can be rearranged to achieve a comact exression. Thus, the eigenfrequency ω and the nondimensional daming ratio d i of the electric network (comosed by the series of C, L, and R) [19] can be conveniently defined as follows: 1 ω = (7) LC d i = R 2 C L. (8) By substituting (2), (7), and (8) into (1), the controller K can be exressed as a function of these two quantities as follows: K= s(s+2d iω ) s 2 +2d i ω s+ω 2. (9) For single degree of freedom systems, the FRF T zw as a function of the electrical eigenfrequency and daming can be derived by substituting (9), (6), and (3) into (5) as follows: T zw (s, x M )=Φ n (x M )Φ n (x F ) s 2 +2d i ω s+ω 2 (s 2 +2d i ω s+ω 2)(s2 +2ξ n ω n s+ωn 2)+γψ2 n s(s+2d iω ). (1) This formulation is valid for both beams and lates as well as for any layout of the iezoelectric actuator (e.g., single actuator, two colocated actuators) [19]. It should be noted that if the oles of this FRF are calculated considering the iezoelectric actuator in oen-circuit condition (i s =, Z= + ), then the exression of the oen-circuit eigenfrequency ω oc n canbewrittenasfollows: ω oc n = ω2 n +γψ2 n. (11) Hence, it is ossible to calculate the nth effective couling factor k n (defined as ((ω oc n )2 ωn 2)/ω2 n, e.g., [4, 11, 22]) using (11) as follows: k n = (ωoc n )2 ω 2 n ω 2 n = γψ 2 n ω n. (12) It should be noted that k n does not deend on the tye of shunt used but is a roerty of the system comosed ofthevibratingstructureandtheiezoelectricactuator;k n indicates the caability of the iezoelectric actuator, couled to a given structure, to transform mechanical energy into electrical energy. The erformance of the controlled system in otimal conditions will deend on the tuning strategy selected to fix the values of R and L. The one considered here is found as the most robust to ossible mistuning in [19]. It is based on considerations on the shae of the FRF of (1). Nevertheless, it will be shown that the results and the rocedure resented in this aer are valid for all tuning strategies that lead to a nearly flat shae of the FRF around the resonance frequency (see Section 3). The tuning criterion considered here fixes the values of R and L basedontherocedurebrieflysummarised here below: (i) The trend of T zw is indeendent of the daming factor of the electrical circuit d i at two frequency values ω A and ω B (see the corresonding oints A and B in Figure 3) (ω is the circular frequency) forundamedsystems[23].theotimalvalueofω ) can be found by imosing the same dynamic amlification modulus at these two frequencies. Thus, the exression for the electrical eigenfrequency can be achieved [19] as follows: (ω ot ω ot = ω 2 n +γψ2 n =ωoc n. (13) Then, the value of L can be found by combining (7) and (13). (ii) The otimal value of the daming d i (and thus of R) is found by imosing an equal dynamic amlification T zw at two different frequencies: ω A and a second frequency given by the square root of the arithmetic mean of ω 2 A and ω2 B. This frequency is found to be equal to the electrical frequency ω [19].Thus,the condition used to fix the value of d i can be given as follows: T zw ω A = T zw ω. (14) This condition is convenient to tune the shunt imedance because it allows a flat trend of T zw to occurinthefrequencybandaroundtheresonance (Figure 3). The value of the otimal electrical daming (d ot i ), which results from (14) (considering ξ n =), can be given as follows: i,ξ n = = γψ2 n 2(ω ot ) 2. (15) d ot Then, the value of R can be found using (8).

5 Shock and Vibration T zw (m/n) B T zw 1 3 A ω/ω n d i =.4% d i =.5% d i =.9% Frequency Short-circuit Otimal shunt Figure 3: T zw for an undamed elastic structure and trend of T zw for a generic system with the otimal value of d i. It should be noted that the use of (13) and (15) (which are yielded considering ξ n = )inthecaseofdamed systems introduces certain aroximations. Nevertheless, these aroximations can be assumed as negligible. In fact, according to [19], the maximum difference between the attenuation rovided by (13) and (15) and the actual attenuation is less than.5 db for most ractical alications. Therefore, the use of (13) and (15) can be considered reliable even with damed systems. Thebehaviourofmistunedsystemswillbestudiedinthe followingsections.becausetherearenoclosedformulasto describe the vibration attenuation in mistuned conditions, the maximum of T zw must be found numerically using (1).Thenumberofvariablesinthisequationishigh:five variables, that is, ω n, ω, ξ n, d i,andγψ 2 n.hence,several simulations must be erformed if a detailed descrition of the behaviour of different ossible mistuned systems is desired (several values of ω, d i,andγψ 2 n for each mode considered, defined by ω n and ξ n ). 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 resents a normalisation of the system model to reduce the number of variables involved in the roblem Normalisation of the Model. First, all of the ossible values of ω and d i canbedefinedasafunctionoftheotimal ones ω ot and dot i as follows: d i =ld ot i ω = Vω ot, (16) where l and V are the amount of mistuning on the electrical daming and eigenfrequency, resectively (l =1and V =1in thecaseofnomistuning). Then, (1) is considered: both the numerator and the denominator are divided by ω 4 n,ands is exressed as jω (j is the imaginary unit). After a few mathematical rearrangements (see Aendix A), a new exression of T zw in the frequency domain can be obtained. This new exression uses (13), (15), and (16) to exress the electrical arameters as a function of their otimal values as follows: T zw (x M )= Φ n (x M )Φ n (x F ) ω 2 n φ 2 + 2jφk n lv + V 2 (1 + k 2 n ) ( φ 2 + 2jφk n lv + V 2 (1 + kn 2)) ( φ2 +2jξ n φ + 1) + jφk 2 n (jφ + 2k n lv), (17) where φ=ω/ω n is the nondimensional frequency.

6 6 Shock and Vibration The advantages rovided by the use of (17) will be underlined in Section Attenuation Performance of the Otimally Tuned Shunt As reviously mentioned, the erformance of the shunt in terms of vibration attenuation can be exressed as the ratio between the maximum amlitude of the uncontrolled system FRF and the maximum amlitude of the controlled system FRF (i.e., max( T zw ); see (17)). The FRF of the uncontrolled structure (i.e., with the iezoelectric atch in short-circuit) can be defined as follows [24]: G zw = Φ n (x M )Φ n (x F ) (18) (s 2 +2ξ n ω n s+ωn 2 ). Therefore, the attenuation erformance, denoted here as att, canbeexressedasfollows: att = max ( G zw ) max ( T zw ) = Φ n (x M )Φ n (x F ) /(2ξ nω 2 n 1 ξ2 n ) max ( T zw ), (19) where, according to [11], max( G zw ) = Φ n (x M )Φ n (x F ) / (2ξ n ω 2 n 1 ξ2 n ). The analytical exression of max( T zw ) is rather comlex; thus, it is convenient to define the index attk instead of att for the case of erfect tuning (ω =ω ot ) as follows: attk = max ( G zw ) T = ( max ( G zw ) 2 zw ω T ). (2) zw ω The difference between att and attk is that, in the former case, the maximum amlitude of the controlled system FRF is considered (max( T zw )), whereas, in the latter case, the value of the system resonse T zw at ω is considered ( T zw ω ). As reviously mentioned, the use of attk simlifies the notation and can be used to accurately aroximate the value of att. In fact, in the case of erfect tuning, max ( T zw ) T zw ω because of the flat shae of T zw around ω n (ω =ω ot =ω oc n is in the frequency range where the controlled FRF has a flat shae; see Figure 3) [19]. Hence, attk att. Based on (17), attk can be exressed as follows (see Aendix B): attk =(k n +2 2ξ n ) (1 + k2 n ) 8ξ 2 n (1 ξ2 n ). (21) Thus, the attenuation in decibels (A db ) can be exressed as follows: A db =2log 1 attk [ =2log 1 (k n +2 2ξ n ) (1 + k2 n ) ] (22) 8ξn 2 (1 ξ2 n ). [ ] Equation (22) only deends on two system arameters, ξ n and k n. Therefore, the roerties of tuned systems can be studied considering only these two arameters. A similar aroach is usedformistunedshuntsystems(seesection4).hence,the normalisation roosed in Section 2.1 allows the model to be simlified, thus avoiding one of the variables (i.e., now only ξ n and k n are considered, whereas it would have been necessary to consider the three arameters ξ n, ω n,andγψ 2 n without the normalisation). It is easy to see that (22) links the achievable attenuation to the roblem arameters (i.e., k n and ξ n ). Since ξ n is fixed, (22) allows the attenuation to be redicted as a function of the value of k n,thussuggestingwhichvalueshouldbeusedto obtain the desired attenuation erformance. In fact, it can be recalled that k n is a function of γψ 2 n (see (12)), and it can thus be modified by changing the geometrical, mechanical, and electrical characteristics of the actuator as well as its osition [19]. Furthermore, k n can be also modified by connecting several iezoelectric actuators in series/arallel [25, 26] and by using a negative caacitance [22, 27]. Therefore, the model used here is of general validity. Equation(22)canberearrangedasfollows: A db =2log 1 (k n +2 2ξ n )+1log 1 (1 + k 2 n ) 1log 1 (8ξ 2 n (1 ξ2 n )). (23) Now, three different situations in terms of the k n value can be considered: (1) k n of the same order of magnitude of ξ n (the maximum value of ξ n considered here is 1%): this is the case of extremely stiff and damed structures and/or badly ositioned actuators. In this case, (23) can be aroximated as follows: A db =2log 1 (k n +2 2ξ n ) (24) 1log 1 (8ξ 2 n (1 ξ2 n )). In fact, k n is so low that 1 log 1 (1 + k 2 n ) can be aroximated as 1 log 1 (1) =. (2) 2 2ξ n k n 1:thisisthetyicalcase[19].Itis ossible to make the following simlifications: (k n + 2 2ξ n ) k n and 1+k 2 n 1.Thus,formost engineering alications, (23) can be aroximated as follows: A db =2log 1 (k n ) 1log 1 (8ξ 2 n (1 ξ2 n )) =m log 1 (k n )+q, (25)

7 Shock and Vibration 7 A db (db) The linear relationshi demonstrated thus far (see (25) and(27))canleadtothefollowingnotations: log 1 attk log 1 (k n )+q /2 attk k n (1) q /2 k n attk. 2 (1) q (28) log 1 (k n ) ξ n =.5% ξ n =.1% ξ n =1% Figure 4: Relationshi between A db and log 1 k n for different systems (i.e., different ξ n values). where m = 2 and q = 1log 1 (8ξ 2 n (1 ξ2 n )). Equation (25) demonstrates a linear relationshi between A db and log 1 (k n ). (3) k n close to 1: this is the case where extremely flexible structures and/or the addition of a negative caacitance are considered. Equation (23) can be aroximated as follows: A db =2log 1 (k n )+1log 1 (1 + k 2 n ) (26) 1log 1 (8ξ 2 n (1 ξ2 n )). Equation (26) demonstrates that, in this case, the relationshi between A db and log 1 (k n ) is no longer linear. Nevertheless, in most ractical alications, a linear relation can still be used. In fact, the term 1 log 1 (1 + k 2 n ) has a negligible contribution u to aroximately k n =.6. Its contribution becomes more evident, albeit small, only for higher k n values (at k n =.8, its contribution is aroximately 2 db). Hence, the term 1 log 1 (1+k 2 n ) canbeneglected,and the attenuation A db as a function of log 1 (k n ) can be aroximated by the linear relation as follows: A db m log 1 (k n ) +q, (27) where m =2and q = 1log 1 (8ξ 2 n (1 ξ2 n )). Figure 4 shows the linear relationshi between A db and log 1 (k n ) for different systems selected as an examle. Additionally, the figure indicates that the area in which linearity is lost (corresonding to the case in which k n is of the same order of magnitude of ξ n,oint1oftherevious numbered list; see the left art of the green dashed line in the figure) corresonds to cases where A db is extremely low (aroximately 5 db or lower). The central exression of (28) indicates that if k n is incremented from value f1 to value f2 where f2/f1 = g, then the value of attk increases by a factor g, whichsignifiesthat consistent increases in attk can be achieved with moderate increases (in terms of absolute value) of k n when k n is low. Conversely, high increases in k n (in terms of absolute value) roduce a low increment of attk when k n is high. Hence, an asymtotic behaviour of the attenuation erformance is demonstrated. The next section considers mistuned shunt systems. 4. Robustness of the Shunt Daming: Performance in Mistuned Conditions Section 1 exlained that, in most cases, the shunted system oerates in mistuned conditions because of the uncertainty in the estimated values of the system arameters (esecially electrical quantities) or changes in either the mechanical system or the electrical network. This often leads to a control erformance considerably lower than that exected; thus, a robustness analysis would be useful for understanding the behaviour of the controlled system and determining a method to limit this erformance reduction. Therefore, the analysis of robustness attemts to investigate the worsening of erformance due to mistuning and rovides formulations for its rediction. Based on (1), (13), and (15), the mistuning can be due to errors in the estimated values of ξ n, ω n,andγψ 2 n as well as the values of R and L. Itiseasytoseethatallofthe different reasons for mistuning can be exressed as errors in the otimal values of ω and d i. Therefore, in this study, the actual values of ω and d i are exressed as changes from their otimal values. Therefore, the mistuning can be easily considered in (17) by fixing l 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 erformance is measured by the index att of (19). The attenuation in these mistuned conditions can be exressed in decibels as A db (see Aendix C for certain clarifications for the symbols used) as follows: A db =2log 1 att l=1,v =1. (29) The value of A db can be found numerically to study the attenuation in different mistuned conditions. Thus, the values of T zw as a function of frequency must be calculated for a given situation (i.e., fixing the values of k n, ξ n, l,andvin (17));

8 8 Shock and Vibration A db, A db (db) A db, A db (db) log 1 (k n ) log 1 (k n ) l =.2 l =.7 l=1 l = 1.3 l = 1.8 l =.2 l =.7 l=1 l = 1.3 l = 1.8 Figure 5: Relationshi between A db or A db and log 1 k n for different ξ n values:.1% and 1%. andthenthemaximumof T zw 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 l and V need to be tested to consider various different ossible mistuning situations. Furthermore, numerous values of k n must be considered; in fact, it is useful to understand if an increased k n value allows the desired attenuation erformance 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 ractice to increase the initial value of the resistance with resect to its otimal value to imrove the attenuation in mistuned conditions; hence, l values significantly higher than 1 need to be tested, thus leading to a large number of l valuestobetakenintoaccount.therefore,thenumberof required simulations can increase substantially. For examle, when N k values of k n, N V biased values of ω,andn l biased values of d i have to be considered to fully study the given roblem, the entire number of simulations N s that must be erformed to evaluate the attenuation in all of the ossible cases results in N N V N l (e.g., if N s is equal to 1 6,theamount of comutational time to erform all the simulations becomes longer than 1 hours on a normal lato). Therefore, the goal of the next sections is to roose a model to describe the attenuation in mistuned conditions A db. Sections 4.1 and 4.2 analyse the effects of errors on d i and ω, resectively. Then, Section 4.3 addresses situations where both errors (i.e., on d i and ω )occurtogether Mistuning on the Electrical Daming. This section only considers mistuning on d i. Figure 5 deicts the curves relating log 1 k n and A db for different systems and different errors on d i (i.e., different l 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 ossible because the equations are of a high order, and thus the solution cannot be exressed 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., ξ n, k n, l,andv;see(17))versusfive variables in the nonnormalised model (i.e., ξ n, ω n, γψ 2 n, l,and V; see (1)). The rimary roerty of the lots in Figure 5 is that the main effect of mistuning is to shift the curves with resect to the case of l=1;however,allofthesecurvescanstillbe aroximated as straight lines. In fact, the lines associated with ξ n = 1% lose their linear trend in corresondence of low values of A db (i.e., for aroximately A db < 4dB); nevertheless, these curves can still be considered iecewise linear. In fact, if an interval for k n equal to one order of magnitude is considered (it is hard to change k n for one order of magnitude [25] or more, even using negative caacitances [27]), the curves can be well aroximated as lines. The lines in Figure 5 rove that the effects of the change in the intercet are significantly higher than the effects of the change in the angular coefficient (i.e., the lines rimarily shift

9 Shock and Vibration 9 due to a nonunitary value of l, arallel lines). In other words, the sensitivity of the attenuation erformance on the value of the couling coefficient tends to be indeendent of the level of mistuning. Hence, for a given system, the imrovement in the attenuation achieved by increasing the value of the couling coefficient is the same whether the shunt is tuned or not. The relationshi between log 1 k n and A db for a given system can be exressed as follows: A db q l (l) +m l (l) log 1 (k n )= A db, (3) where A db is the estimate of A db and q l and m l are the intercet and the angular coefficient of the line, resectively, which are both a function of l,asevidencedin(3)(seealso Aendix C for certain clarifications of the symbols used). It should be noted that m l is indicated as deendent on l, even if this deendency is slight (see above), for the sake of comleteness. If the trend of A db as a function of l is deicted for different values of k n, a few further interesting facts can be noted (see Figure 6). All of the curves of this new figure demonstratenearlythesamefeatures:theattenuationlossis limited for l>1;furthermore,inthisrangeofl, therateof thelossisnearlyconstant.conversely,forl < 1,therate becomes increasingly larger by decreasing the value l. Itis ossible to aroximatively state that, for l <.5, therate of the attenuation loss increases. This result is a first sign of the benefits rovided by the use of overestimated electrical daming d i values (and thus overestimated R values). In fact, even if an overestimated d i value causes a worsening in the attenuation erformance if comared to the tuned situation, this worsening is not that high (see Figure 6); overall, if a mistuning occurs in situations where the starting d i value is overestimated deliberately, the attenuation loss due to the mistuning is low. The use of initially overestimated d i values will be considered again in Section 4.3. A further interesting oint is that the trend of A db as afunctionofl can be modelled as the combination of two fourth-order olynomials, one for l<1and another for l>1, regardless of the system considered (see Figure 6). The calculation for each of these fourth-order olynomialsrequirestheknowledgeofa db for five values of l. Therefore, for the given values of ξ n and k n,itissufficientto calculate ten oints (l, A db ) using (29), (19), and (17) (i.e., five for l<1and five for l>1) to determine the trend of A db for an extended range of l values (e.g., l =.1 2, asindicated in Figure 6). Based on (3) and Figure 6, the following rocedure can bealiedwhenthebehaviourofamistunedshuntsystemin arangeofk n values between k A and k B is studied. (i) Calculate five airs (l, A db ) using (29), (19), and (17) for l<1and k n =k A and determine the interolating olynomial. Then, reeat the same rocedure for l> 1. It is now ossible to know the value of A db for any value of l at k n =k A. (ii) Calculate five airs (l, A db ) using (29), (19), and (17) for l<1and k n =k B and determine the interolating olynomial. Then, reeat the same rocedure for Table 1: Bounds for the Monte Carlo simulations for l =1and V =1. ξ n k A k n l x Min.5%.1 k A.5 Max 1%.35 k B 1.5 l>1.itisnowossibletoknowthevalueofa db for any value of l at k n =k B. (iii) The value of A db for a generic value k n between k A and k B and a generic value l=l x can be comuted using (3), where m l = A db l=l x,k n =k B A db l=l x,k n =k A log 1 k B log 1 k A q l =A db l=l x,k n =k B m l log 1 k B. (31) Therefore, it is ossible to estimate the attenuation for any value of l and k n (between k A and k B )withonlytwenty simulations. The accuracy of this rocedure was verified using a Monte Carlo test with more than 1 5 simulations comaring the attenuation values A db achieved using this rocedure and the A db values obtained using (29), (19), and (17). The difference Δ is defined as A db A db. For each simulation, the values of ξ n, k A, k n,andl x were extracted from uniform distributions. TheboundsofthedistributionsareresentedinTable1and were chosen in order to take into account the most art of the ractical alications. Table 2 lists the results, hence roving the reliability of the roosed rocedure. It should be noted that k B = 3k A was used in the simulations. This corresonds to a change of γψ 2 n within an interval equal to three times the starting value, which is quite a broad interval. If a wider interval of k n must be considered and the same accuracy is desired, it is ossible to analyse the system in two different ranges. For examle, if k B = 3k A,theentirerangecanbe slit as follows: k A k h and k h k B with k h = 3k A.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 ω (i.e., V = 1 and l = 1). It should be noted that the resulting curves tend to increase their curvature. Nevertheless, the linearity is lost only when A db becomes lower than aroximately 4 or 5 db. However, the curves can be still iecewise aroximated as lines with enough accuracy forwiderangesoflog 1 (k n ).Theeffectofanonunitaryvalue of V is to highly increase the angular coefficient of the lines. Consequently,increasingthevalueofk n is even more effective in enhancing the attenuation in the case of mistuning on ω than in the case of tuned systems. The curves of Figure 7 can be exressed as follows: A db q V (V) +m V (V) log 1 (k n)= A db, (32) where A db is again the estimate of A db and q V and m V are the intercet and the angular coefficient of the lines, resectively,

10 1 Shock and Vibration A db (db) A db (db) 6 k n =.5 4 k n =.1 2 k n =.1 k n = l 4 3 k n = k n =.1 k n =.1 k n = l (c) A db (db) A db (db) 5 k n = k n =.1 2 k n =.1 1 k n = l 3 k n =.5 2 k n =.1 1 k n =.1 k n = l (d) Figure 6: Relationshi between A db and l for different k n values and different ξ n values:.5%,.1%,.5% (c), and 1% (d). The + are theointsrelatedtothecaseofl>1calculated using (29), (19), and (17), which are then interolated by fourth-order olynomials (solid curves). The are the oints related to the case of l<1calculated using (29), (19), and (17), which are then interolated by fourth-order olynomials (solid curves) A db, A db (db) 3 A db, A db (db) log 1 (k n ) log 1 (k n ) =.75 =.95 =1 = 1.5 = 1.25 =.75 =.95 =1 = 1.5 = 1.25 Figure 7: Relationshi between A db or A db and log 1 k n for different ξ n values:.1% and 1%.

11 Shock and Vibration k n =.5 4 k n =.5 A db (db) k n =.1 k n =.1 k n = A db (db) k n =.1.8 k n =.1 k n = A db (db) k n =.5 k n =.1.8 k n =.1 k n = (c) A db (db) 2 1 k n =.5.8 k n =.1 k n =.1 k n = (d) Figure 8: Relationshi between A db and V for different k n values and different ξ n values:.5%,.1%,.5% (c), and 1% (d). The + are theointsrelatedtothecaseofv >1calculated using (29), (19), and (17), which are then interolated by fourth-order olynomials (solid curves). The are the oints related to the case of V <1calculated again using (29), (19), and (17), which are then interolated by fourth-order olynomials (solid curves). Table 2: Results of the Monte Carlo simulations for the case l =1and V =1. Mean value of Δ [db] Standard deviation of Δ [db] Minimum value of Δ [db] Maximum value of Δ [db] Table 3: Bounds for the Monte Carlo simulations for V =1and l=1. ξ n k A k n V x Min.5%.1 k A.9 Max 1%.35 k B 1.1 which are both functions of V (see Aendix C for certain clarifications of the symbols used). If the trend of A db is shown as a function of V for different values of k n, a few further interesting facts can be noted (see Figure8).Thesameercentagevalueofmistuningleadsto adifferentdecreaseintheerformanceifitisrelatedtoan overestimation or underestimation of the otimal 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., comare the curves at V =.75 and V = 1.25). A further interesting oint is that the trend of A db as a function of V can be modelled as a fourth-order olynomial for both V <1and 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 k n values between k A and k B is considered, the same rocedure discussed in Section 4.1 (see the list in Section 4.1) can be alied, and it is ossible to estimate the attenuation for any value of V and k n (between k A and k B )withonlytwentysimulations.indeed,itisossibleto write m V = A db V=V x,k n =k B A db V=V x,k n =k A log 1 k B log 1 k A q V =A db V=V x,k n =k B m V log 1 k B. (33) Again, a Monte Carlo test was erformed with more than 1 5 simulations comaring the attenuation values A db achieved using this rocedure and the A db values obtained using (29), (19),and(17).Foreachsimulation,thevaluesofξ n, k A, k n,and V x 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 art of the ractical alications), and k B was fixed to 3k A, as erformed in Section 4.1. Table 4 lists the results, thus roving the reliability of the roosed

12 12 Shock and Vibration Table 4: Results of the Monte Carlo simulations for V =1and l=1. Mean value of Δ [db] Standard deviation of Δ [db] Minimum value of Δ [db] Maximum value of Δ [db] rocedure. It should be noted that the range of V x is narrower than that used in Section 4.1 for l x.thereasonisthatthe otimal value of d i deends on more variables than the otimal value of ω,thusleadingtomoreuncertainty(see (15) and (13)) Mistuning on Both the Electrical Eigenfrequency and the Daming Ratio. Sections 4.1 and 4.2 have treated cases in which the mistuning is related to either d i or ω. Nevertheless, in actual alications, both of the mistuning effects are exected to aear together. In these general mistuned situations (i.e., l = 1 and V =1), the erformance of the shunt system demonstrates a different behaviour for values of l higher or lower than 1. This resultcanbeclearlyevidencedusingthedouble-logarithmic reresentation already utilised in Figures 5 and 7. In fact, Figure 9 deicts the different behaviour for certain systems selected as examles: (i) For systems where l > 1, the loss of attenuation is essentially due to the mistuning causing the highest loss (see the sublots on the right side). (ii) For systems where l<1, the loss of attenuation can be derived as the sum of the losses caused by both the mistuning tyes (see the sublots on the left side). Now, for a given system (i.e., fixed values of ξ n and k n )and fixed values of l and V (named l x and V x ), the following indexes can be defined: y t =A db A db l=l x,v=v x y o =A db A db l=1,v=v x y d =A db A db l=l x,v=1, (34) where y t exresses the loss of attenuation when a mistuning occurs on the values of both ω and d i ; y o exresses the loss of attenuation when a mistuning occurs only on the value of ω ;andy d exresses the loss of attenuation when a mistuning occurs only on the value of d i. Based on the abovementioned considerations related to Figure9(seethelistaboveinthissection), y t (the estimate of y t ) can be calculated as follows: y t = { max (y o,y d ), l x >1 { (35) y { o +y d, l x <1. According to (35), A db (i.e., the estimate of A db )canbe defined as follows: A db l=l x,v=v x = { min (A db l=1,v=v x,a db l=l x,v=1 ), l x >1 { A { db l=1,v=v x +A db l=l x,v=1 A db, l x <1. (36) A db l=1,v=v x and A db l=l x,v=1 can be estimated using (32) and (3), resectively. A db is given in (22). Hence, (36) allows the attenuation to be estimated for any values of l, V, andk n (between k A and k B ) 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 ω and d i by considering the mistuning on ω and d i searately. Twenty simulations are needed to study the behaviour of the system with l = 1 and V = 1 (see Section 4.1), twenty for the case l = 1 and V = 1 (see Section 4.2), and two for the case l = 1 and V = 1 (i.e., one with k n = k A and the other with k n =k B ); furthermore, the A db (i.e., the case with l = 1 and V = 1)valuesfork n = k A and k n = k B can also be calculated using (22). In the case of N s (total number of cases to be considered) equal to 1 6,theamountoftime required to erform all the simulations decreases from more than 1 hours (see the end of Section 4) to a few minutes or less (aroximately 3 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 R values on the attenuation erformance and to choose the best one for the given alication. Therefore, by rearranging (36) using (3) and (32), the final form of the aroximated model able to describe the behaviour of the mistuned shunt systems can be achieved as follows: A = { min (q V (V x)+m V (V x) log 1 (k n ),q l (l x )+m l (l x ) log 1 (k n )), l x >1 db l=l x,v=v x,k n { q { V (V x)+m V (V x) log 1 (k n )+q l (l x )+m l (l x ) log 1 (k n ) A db kn, l x <1, (37) where m l, q l, m V,andq V are defined in (31) and (33). The accuracy of this model was tested again using a Monte Carlo simulation with more than 1 6 cases, thus comaring the attenuation values A db achieved using this rocedure and the A db values obtained using (29), (19), and (17). For each simulation, the values of ξ n, k A, k n, V x,andl x were extracted

13 Shock and Vibration A db, A db (db) 2 1 A db, A db (db) log 1 (k n ) log 1 (k n ) l =.5, = 1.1 l =.5, =1 l=1, = 1.1 l=1, =1 l = 1.5, = 1.1 l = 1.5, =1 l=1, = 1.1 l=1, = A db, A db (db) 3 2 A db, A db (db) log 1 (k n ) log 1 (k n ) l =.5, = 1.1 l =.5, =1 l=1, = 1.1 l=1, =1 l = 1.5, = 1.1 l = 1.5, =1 l=1, = 1.1 l=1, =1 (c) (d) Figure 9: Relationshi between A db or A db and log 1 k n for different ξ n values, l,andv: ξ n = 1%, l =.5, and V =1.1,ξ n = 1%, l =1.5,and V =1.1,ξ n =.1%, l =.5, and V =1.1(c),andξ n =.1%, l =1.5,andV = 1.1 (d). from uniform distributions (see Table 5; k B = 3k A ). Table 6 resents the results (which have a Gaussian distribution), thus roving the reliability of the roosed rocedure. Certain benefits rovided by the use of initially overestimated d i values have already been discussed for the cases of bias just on d i in Section 4.1. Here, the discussion can be extended to the more general case of mistuning on both d i and ω (which is the tyical situation). Also in this case the use of an initially overestimated d i value allows the loss of attenuation to decrease. In fact, when d i 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 d i value is lower than its otimal value, the attenuation loss due to mistuning is more severe. Therefore, this roerty of the mistuned systems along with those already shown in Section 4.1 highlights that the use of initially overestimated d i values (and thus initially overestimated R values) allows the robustness to increase, thus lowering the loss of attenuation due to mistuning, which is tyically exerienced starting from the otimal d i 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 l higher than 1 (because an initially overestimated d i valueisusedonurose)andossibly slightly lower than 1 (e.g., greater than.5). Clearly, these are just guidelines because each ractical case could require a different solution. Nevertheless, the oints demonstrated thus far clearly indicate how robustness can be tyically increased and how the roosed model can hel in the tuning rocess. The model resented so far has been validated by exerimental tests shown in the next section. 5. Exerimental Tests This section describes the exerimental tests erformed to validate the results shown in the revious sections. Two test structures have been used to investigate different values of the ξ n and k n arameters and different values of vibration attenuation. The first structure is an aluminium late (in freefree condition by susension) with the shunted iezobender bonded at about its centre (see Figure 1). A bidimensional structure was used because it is a more comlex test case when comared to monodimensional structures often used in other studies. The late length is 6 mm, the width is 4 mm, and the thickness is 8 mm (this set-u is the same

14 14 Shock and Vibration Figure 1: Exerimental set-us: late and beam. Table 5: Bounds for the Monte Carlo simulations for V =1and l =1. ξ n k A k n l x V x Min.5%.1 k A.5.9 Max 1%.35 k B as that used in the exeriments of [19]). The caacitance C is.2 μf.severalmodesweretakenintoaccountinthe tests. The one (among others tested) considered here as an examle has the following modal arameters (identified by exerimental modal analysis [28]): ω n = π rad/s, ξ n =.22%, k n =.81,andγψ 2 n = 725 rad2 /s 2.Actually,the value of k n was estimated by testing the system in both shortand oen-circuit conditions (see (12)). These values of γψ 2 n and k n were achieved using a negative caacitance [27], which allowed their initial low values to increase. Furthermore, another value of k n (i.e.,.24) was tested by further boosting the negative caacitance erformance. The disturbance to the structure was rovided by a dynamometric imact hammer, and the resonse was measured using a iezoelectric accelerometer. The second structure is an aluminium cantilever beam (159 mm length, 25 mm width, and 1 mm thickness) with a iezoelectric atch bonded corresonding to the clamed end (see Figure 1). Its caacitance C is 31 nf. Again, several modes were considered during the tests; here, the results related to the first mode are resented for the sake of conciseness. It has the following modal arameters, again identified using an exerimental modal analysis: ω n = π rad/s, ξ n =.4%, k n =.22, andγψ 2 n = rad2 /s 2. Furthermore, other tests were erformed by increasing the values of k n u to.518 using a negative caacitance. Because this second test structure was extremely light, noncontact methods were used to rovide excitation and to measure the resonse. Indeed, an electromagnetic device was used to excite the structure [29], and the resonse was measured using a laser velocimeter focused on the beam ti. The tests were erformed by exciting the beam with a random signal [3] u to 1.6 khz. The tests were erformed using synthetic imedance based on oerational amlifiers [11, 31, 32] to build the inductor. Actually, certain tests on the late were erformed using an additional method: the entire shunt imedance was simulated using a high-seed Field Programmable Gate Array (FPGA) device (in this second case, a colocated iezoelectric atch was used to rovide the inut voltage to the simulated shunt imedance). The use of the FPGA device allowed for the full control of the arameters of the electric shunt imedance. Nevertheless, the two techniques ledtosimilarresults;therefore,thoseachievedusingthe synthetic imedance are resented here, since this technique introduces the highest level of uncertainty between the two. Therefore, the authors believed it to be the most reresentative to demonstrate the model effectiveness. First, the reliability of the model, reresented by (1) and (17), was verified. Figure 11 deicts the FRFs for the mode at aroximately 53 Hz of the late, achieved with differentconfigurationsoftheshunt(i.e.,usingdifferentl and V values). The numerical FRFs match the exerimental curves, thus confirming the accuracy of the numerical model. Thecurvesarenotlottedonthesamegrahforthesake of clarity in the figure. Nonetheless, Figure 12 deicts a few of the exerimental and numerical FRFs of Figure 11 on the same lot for an easy comarison. Then, the reliability of the roosed aroximated model (see (37)) for redicting the attenuation in mistuned conditions was tested. Tables 7 and 8 list the comarisons between exerimental attenuations, numerical attenuations calculated using the theoretical model of (17), (19), and (29), and attenuations estimated using the roosed aroximated model of (37) for the late and the beam. To build the aroximated model of (37), the values of k A and k B must be fixed. Three different situations were tested: one where k A was close to k n (k n = 1.1k A,namedcase 1), one where k B was close to k n (k n = 1.63k A,namedcase2), and a further one where k n was nearly halfway (k n = 1.37k A, namedcase3).inallofthethreecases,k B was fixed to 3k A. The exerimental 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 rovided by the model of (37) in cases 1, 2, and 3 is named MA1, MA2, and MA3, resectively. The match amongalloftheresultsisgood.theresultsrelatedtothema1, MA2, and MA3 cases are always close to each other, and the maximum difference if comared to the NA results is on the order of.5 db. Because the EA results differ from the NA resultsatamaximumof1.2db,theroosedmodelof(37)is considered to be validated. 6. Conclusion This aer addresses monomodal vibration attenuation using iezoelectric transducers shunted to imedances 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 roerly realised, this control technique is not adative, and its erformances thus decrease as soon as a mistuning occurs. Theaeranalysesthebehaviourofmistunedelectromechanical systems, demonstrating that a linear relationshi between the attenuation and the logarithm of the effective

15 Shock and Vibration T zw (db) 15 T zw (db) Frequency (Hz) Frequency (Hz) Short-circuit = 1.15, l =.26 = 1.25, l =.26 =.998, l =.42 =1, l =.42 =1, l =.23 Short-circuit = 1.15, l =.26 = 1.25, l =.26 =.998, l =.42 =1, l =.42 =1, l =.23 Figure 11: Exerimental and numerical FRFs for the late (k n =.81). Table 6: Results of the Monte Carlo simulations for V =1and l =1. Mean value of Δ [db] Standard deviation of Δ [db] Minimum value of Δ [db] Maximum value of Δ [db] T zw (db) Short-circuit, ex Short-circuit, num = 1.15, l =.26, ex Frequency (Hz) = 1.15, l =.26, num =.998, l =.42, ex =.998, l =.42, num Figure12:ExerimentalandnumericalFRFsforthelate(k n =.81). couling coefficient exists when a erfect tuning is reached. The same linear behaviour exists when there is mistuning on either the electrical eigenfrequency or daming. Moreover, the aer indicates how the loss of attenuation essentially deends on only one bias if the electrical daming is overestimated and describes how the effects of the two bias tyes (on the electrical eigenfrequency and daming) combine with each other when the daming is underestimated. This allows an aroximated model to be achieved for describing the behaviour of mistuned shunt systems, which was initially validated numerically using Monte Carlo simulations and then exerimentally 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. Aendix A. T zw Normalised Analytical Exression This aendix rovides the mathematical rocess that allows (17) to be derived from (1). Based on (1) and assing from the Lalace domain to the frequency domain (i.e., s=jω), the exression of T zw can be written as follows:

16 16 Shock and Vibration Table 7: Attenuations for the mode of the late. EA [db] NA [db] MA1 [db] MA2 [db] MA3 [db] l =.6, V = 1 l =.6, V =.9925 l =.8, V = 1.1 l = 1.2, V =.998 k n = k n = k n = k n = k n = k n = k n = k n = k n = k n = T zw (x M )=Φ n (x M )Φ n (x F ) ω 2 + 2jωd i ω +ω 2 ( ω 2 + 2jωd i ω +ω 2 )( ω2 + 2jωξ n ω n +ω 2 n )+γψ2 n jω (jω + 2d iω ). (A.1) Substituting (16) into (A.1) and using (13) and (15), the exression of T zw can be further modified as follows: T zw (x M )=Φ n (x M )Φ n (x F ) ( ω 2 + 2jωld ot i Vω ot ω 2 + 2jωld ot i Vω ot +(Vωot )2 +(Vω ot ) 2 )( ω 2 + 2jωξ n ω n +ωn 2)+γψ2 n Φ n (x M )Φ n (x F )[ ω 2 + 2jωlV γψn 2/2 + V2 (ω 2 n +γψ2 n )] =. ( ω 2 + 2jωlV γψn 2/2 + V2 (ωn 2 +γψ2 n )) ( ω2 + 2jωξ n ω n +ωn 2)+γψ2 n jω (jω + 2lV γψ2 n /2) jω (jω + 2ldot i Vω ot ) (A.2) By dividing both the numerator and the denominator of φ=ω/ω (A.2) by a factor ω 4 n, defining the nondimensional frequency n, and using (12), the final exression of T zw can be obtained as follows (see (17)): T zw (x M )= ω4 n ωn 4 Φ n (x M )Φ n (x F )[ ω 2 + 2jωlV γψ 2 n /2 + V2 (ω 2 n +γψ2 n )] ( ω 2 + 2jωlV γψ 2 n /2 + V2 (ω 2 n +γψ2 n )) ( ω2 + 2jωξ n ω n +ω 2 n )+γψ2 n jω (jω + 2lV γψ2 n /2) [Φ n (x M )Φ n (x F )/ω 2 n ][ ω2 /ω 2 n + 2j (ωlv γψn 2/ω2 n )+V2 (ω 2 n +γψ2 n )/ω2 n ] = (A.3) ( ω 2 /ωn 2 + 2j (ωlv γψn 2/ω2 n )+V2 (ωn 2 +γψ2 n )/ω2 n )( ω2 + 2jωξ n ω n +ωn 2)+(γψ2 n ω/ω3 n ) j (j (ω/ω n)+ 2(lV γψn 2/ω n)) = Φ n (x M )Φ n (x F ) ω 2 n φ 2 + 2jφk n lv + V 2 (1 + k 2 n ) ( φ 2 + 2jφk n lv + V 2 (1 + k 2 n )) ( φ2 +2jξ n φ + 1) + jφk 2 n (jφ + 2k n lv).

17 Shock and Vibration 17 Table 8: Attenuations for the mode of the beam. EA [db] NA [db] MA1 [db] MA2 [db] MA3 [db] l =.6, V = 1.1 l = 1.5, V = 1 l =.6, V =.8 l = 1.2, V =.8 k n = k n = k n = k n = k n = k n = k n = k n = k n = k n = B. attk Analytical Exression The mathematical rocess used to exress attk (see (21)) is exlained here. The exression of T zw in (17) can be rearranged by searating the real and imaginary arts at the numerator and denominator as follows: where T zw (x M ) = Φ n (x M )Φ n (x F ) ω 2 n A zw +jb zw C zw +jd zw, (B.1) The exression of (B.3) can be evaluated in ω = ω =ω ot (the case of erfect tuning is considered here), which in turn corresonds to φ=ω ot /ω n =ω oc n /ω n = 1+kn 2: T Φ zw (x M ) ω = n (x M )Φ n (x F ) ωn 2 2 (k n +2 2ξ n ) 1+kn 2. (B.4) According to [11], max( G zw ) = Φ n (x M )Φ n (x F ) / (2ξ n ω 2 n 1 ξ2 n ). Therefore A zw = φ 2 + V 2 (1 + k 2 n ), attk = max ( G zw ) 2 (1 + k T =(k n +2 2ξ n ) n ) zw ω 8ξn 2 (1 ξ2 n ). (B.5) B zw = 2φk n lv, C zw =( φ 2 + V 2 (1 + k 2 n )) ( φ2 +1) 2 2φ 2 k n lvξ n φ 2 k 2 n, D zw = 2φk n lv ( φ 2 +1) +2ξ n φ( φ 2 + V 2 (1 + k 2 n )) + 2φk 3 n lv. According to [23], T zw can be exressed as follows: T zw (x M ) 2 =( Φ 2 n (x M )Φ n (x F ) ωn 2 ) A2 zw C2 zw +B2 zw D2 zw +B2 zw C2 zw +A2 zw D2 zw (C 2 zw +D2 zw )2. (B.2) (B.3) C. List of the Symbols This aendix clarifies the meaning of the symbols used. The symbol reresents a generic mistuned condition. The symbol reresents an estimate of the considered quantity. A db exresses the attenuation in decibels achieved in case of erfect tuning (i.e., l=1,v =1). A db exresses the attenuation in decibels achieved in case of mistuning (this is evidenced by the asterisk). When A db and A db are evaluated at secific oints (i.e., given values of l, V,ork n ), the following exressions are used: A db kn and A db l=1,v=v x, as examles. The former exression indicates that A db is comuted in corresondence with a given value k n of the effective couling factor, whereas the latter indicates that A db is comuted for l=1and V = V x. A db and A db reresent the values of A db and A db, resectively, which are estimated using the model roosed in the aer, that is, by (3), (32), and (37). As for the angular coefficient m and the intercet q of the linear relations resented in the aer (e.g., (25), (27), (3), and (32)), when they have a subscrit, they are calculated for a erfectly tuned shunt imedance. Conversely, when they

18 18 Shock and Vibration are calculated for a mistuned system, they have an as a suerscrit and a subscrit equal to l (for a mistuning on d i and with ω erfectly tuned) or V (for a mistuning on ω and with d i erfectly tuned). Cometing Interests The authors declare that there are no cometing interests regarding the ublication of this aer. References [1] A. Preumont, Vibration Control of Active Structures: An Introduction, Kluwer Academic, Dordrecht, Netherlands, 2nd edition, 22. [2] A. Preumont, Mechatronics Dynamics of Electromechanical and Piezoelectric Systems, Sringer, Dordrecht, The Netherlands, 26. [3] C.Fuller,S.Elliott,andP.Nelson,Active Control of Vibration, Acadamic Press, London, UK, [4] N. W. 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