A nonlinear tunable piezoelectric resonant shunt using a bilinear component: theory and experiment

In this article, we propose a new concept for tuning a resonant piezoelectric shunt absorber thanks to the use of a nonsmooth electronic component. It consists in adding a voltage source in the resonant shunt circuit, which is a bilinear function of the voltage across the piezoelectric patch. The main advantage is the ability to change the electrical resonance frequency with the bilinear component gain, enabling a tuning as well as a possible reduction in the required inductance value. Furthermore, because of the intrinsic nonlinear nature of the bilinear component, a multi-harmonic response is at hand, leading to a nonlinear coupling between the mechanical and electrical modes. Two particular tunings between the electrical and the mechanical resonance frequencies are tested. The first one is one-to-one, for which the electrical resonance is tuned close to the mechanical one. It is proved to be similar to a classical linear resonant shunt, with the additional tuning ability. The second case consists in tuning the electrical circuit at half the mechanical resonance, leading to a two-to-one (2:1) internal resonance. The obtained response is also found to be similar to a classical resonant shunt near the main resonance. In either case, the shunt performances are analytically and numerically studied, leading to optimal values of the design parameters as well as an estimation of the amplitude reduction provided by the shunt. Finally, experimental validation is proposed, targeting the damping of the twisting mode of a hydrofoil structure, in which the bilinear component is realized with a diode.


Introduction
Piezoelectric shunt absorbers represent an efficient way to attenuate the vibrations of structures under external excitation, especially when the size and weight constraints are presented. Those absorbers were proposed in [1] as an extension to the mechanical absorbers by utilizing a piezoelectric (PE) transducer connected to a dedicated electronic circuit.
The simplest architectures of those shunt circuits are the resistive shunt (R-shunt, including a simple resistor) and the resonant shunt (RL-shunt, composed of a resistor and an inductor L), which are passive absorbers with a linear behavior. While the R-shunt dissipates the energy in heat form, the RL-shunt consists in coupling an electrical resonator (an RLC circuit, with C being the intrinsic capacitance of PE patch) to the host structure. It is thus an electrical analog of a tuned mass damper (TMD) which can be tuned in a similar manner to achieve an optimal attenuation. In this context, it has been shown through the optimal study of the RL-shunt in [2] that for a given mechanical damping ratio, the optimal attenuation depends only on the PE coupling factor. Thus, enhancing the absorber's performance can be done by increasing the PE coupling using, for example, a negative capacitance [3] or by optimizing the placement of the PE patch on the host structure [4]. Two main issues are encountered with the PE shunts: (i) their sensitivity to any change in the resonance frequency of the structure [5], (ii) for some applications targeting low-frequency modes, a very high inductance is required, which can be a practical issue [6,7]. In addition, the R-and RL-shunts act near a single structure mode, but they can be extended for multimodal attenuation through multiple attachments [8][9][10]. One can refer further to the following reviews [11,12] for a wider illustration regarding the optimization and different applications of the PE shunts.
PE shunt properties can be enhanced by intentionally introducing a nonlinear component in the shunt circuit to exploit some purely nonlinear features. An example of such applications is the so-called nonlinear energy sink "NES," introduced in [13], which relies on the so-called nonlinear targeted energy transfer [14,15] and proved to be efficient to attenuate the vibrations for a broad range of input frequencies [16]. The NES was realized theoretically and experimentally with piezoelectric shunt methodology in [17] and [18], respectively, by adding cubic nonlinear component in the shunt circuit. Another example is the socalled nonlinear tuned vibration absorber "NLTVA" introduced in [19] and realized with a PE shunt theoretically [20] and experimentally [21]. Recently, a new semi-passive absorber, introduced in [22] and realized experimentally in [23], exploited a two-to-one (2:1) internal resonance and a saturation phenomenon by adding a quadratic nonlinear component in a classical RL-shunt. This concept was improved to correct an unwanted detuning by adding a cubic component, as shown theoretically in [24] and experimentally in [25].
In many applications, the nonlinearities can be presented (or approximated) in a polynomial form (i.e., quadratic or/and cubic). In essence, polynomial nonlinearities naturally appear in a wide range of nonlinear models involving geometric and material nonlin-earities such as thin beam and plate models [26,27]. The free and forced response of systems involving such types of nonlinearities can be analyzed analytically (up to a certain order) using several methods such as perturbation techniques [28], elliptic functions [29], and normal form [30]. However, in some other applications involving clearance [31], vibro-impacts [32] or dry friction [33,34], nonsmooth piecewise functions naturally appear in the model, leading to strong nonlinear effects.
Utilizing nonsmooth nonlinear components for vibration reduction was firstly introduced in [35] to enhance the design of the NES to act as shock isolators. A lot of studies were then conducted to analyze the energy transfer between linear and nonsmooth NES attachments. For example, the targeted energy transfer between a single degree of freedom linear system and nonsmooth NES was studied in [36]. Also, the dynamics of a NES of non-polynomial piecewise potential was investigated in [37,38]. In addition, introducing a nonsmooth component led to the design of vibroimpact NES where detailed design rules were given in [39]. A recent theoretical and experimental study [40] proposed an enhanced NES by using a cubic spring in addition to a piecewise linear spring. In the same context, it is also worth mentioning the energy harvesting enhancement using a nonsmooth piecewise bistable vibration absorber [41,42].
To the authors' knowledge, the only attempt to use a nonsmooth component in a PE shunt methodology involved the synchronized switch damping (SSD) technique. The latter idea is proposed in [43] for piezoelectric transduction that is switched between two distinct shunt impedances, in synchronicity with the oscillations of the host structure [44]. The principle is the following: When the system is moving away from equilibrium, a voltage is applied such that the piezoelectric force resulting tends to restore the system to its equilibrium point. Various topologies have been proposed regarding the circuit connected in series with the patch via the switch: a simple resistor [43], a resistor and an inductor to increase the voltage inversion [45] or the same circuit with an added static voltage source [46,47]. In the latter case, the voltage may have to be controlled as well to avoid instabilities [48]. This technique is proved to be efficient in terms of vibration reduction, but it can become complicated to tune in practice due to the imperfection of synchronization time required [44] as well as to the creation of higher harmonics and noise [49].
The main originality of this work is to present the design of a tunable PE resonant shunt with a nonsmooth voltage component, without any required synchronization, compared to that in [44]. Thus, the nonsmooth shunt can be easily tuned and controlled in practice.
Comparing to the studies in [35][36][37]39,40] where a nonsmooth NES was realized, the main idea in this work is to exploit the nonsmooth behavior using the PE shunt methodology through particular tunings of the shunt circuit. In practice, the nonsmooth voltage component can be realized in a passive way with a halfwave rectifier circuit that includes a diode, so that the bilinear function of the voltage across the piezoelectric transducer is obtained. In addition, the proposed design of the nonsmooth component allows us to tune the resonance frequency of the shunt circuit without changing the inductance. Thus, a lower inductance can be set for the optimum design, which is an advantage when considering low-frequency modes. This paper focuses on the theoretical bases of the absorber and its experimental proof of concept, by targeting the mitigation of the first twisting resonance of a hydrofoil structure. This paper is organized as follows: In Sect. 2, the theoretical study is illustrated, showing the main governing equations with the nonsmooth term in addition to their approximation. We also show the optimal study for one-to-one and two-to-one tuning cases of the electric resonance frequency. The obtained optimal parameters, in addition to the optimal attenuation, are then numerically verified. In Sect. 4, we show the experimental proof of concept where the nonlinear shunt circuit and the measurement protocol are illustrated. In addition, we show the experimental frequency responses for both tuning cases to validate the theoretical estimations.

Governing equations
We consider an arbitrary elastic structure subjected to an external excitation and connected to a shunt circuit via a piezoelectric patch, as shown in Fig. 1a. Following [2,22,23], the displacement u(x, t) at position x in the structure and time t is truncated on one linear mode such that: where ψ(x) is the mode shape and q i (t) is the ith modal displacement. Then, (Q(t), q i (t)) are the solutions of the following system: In the above equations, m i , ξ i ,ω i , θ i , and F i are, respectively, the modal mass, the mechanical damping ratio, the natural frequency in open-circuit condition (Q = 0), the piezoelectric coupling coefficient, and the forcing, all corresponding to the ith mode. C pi is the effective capacitance of the piezoelectric patch for the ith mode [3]. The electrical natural frequency ω e and electric damping ratio ξ e are defined as: where R and L are the resistance and the inductance in the shunt circuit, respectively. The PE patch voltage V (t) can be defined as a function of q i (t) and Q(t) using [2]: We also define the dimensionless electromechanical modal coupling factor (EMMCF) of the ith mode as: withω i the natural frequency in short circuit (V = 0). In this study, we consider the nonlinear voltage component V nl (t) to be bilinear in terms of V (t), as shown in Fig. 1b, and expressed as: where β is a dimensionless gain. The goal of introducing a nonsmooth nonlinearity in the form presented in Eq. (6) is first to generate even harmonics in the system, because it is not a symmetrical function of V , to possibly activate a 2:1 internal resonance, and second to use simple electrical components such as a diode or a transistor to practically realize it. By substituting Eq. (6) in Eq. (2b) and eliminating V using Eq. (4), we obtain: Two important conclusions can then be drawn at this stage: (i) The governing equations in (7a,b) are indeed nonlinear because of the nonsmooth term (|V | ⇒ |Q + θ i q i |) which only appears in Eq. (7b) and creates a harmonic distortion. (ii) Although nonlinearities are present, (7a,b) are in fact fully linear in amplitude. For example, replacing F by 2F in Eq. (7a) leads to obtain (2q i , 2Q) instead of (q i , Q) (see also Fig. 10) since the absolute value function is linear in amplitude (i.e., |2x| = 2|x| for x ∈ R). However, it is not always additive or superposable, since

1DOF system with the nonsmooth term
To give a first analysis of the behavior of the coupled system (7a,b), we consider here only Eq. (7b), we exclude the piezoelectric coupling (i.e., setting θ i = 0), and we add a direct forcing term, leading to: where P is the forcing amplitude.

Free solution
To analyze the free response, we set P = 0, and we rewrite Eq. (8) as:Q where ω e = ω e √ 1 − 2β. In the conservative case (ξ e = 0), Eq. (9) suggests that the free response is a succession of one half period of sine of period ω e and one half period of sine of frequency ω e , as illustrated Fig. 2a . Then, the oscillation period can be expressed as: whereω e is the modified angular frequency of the free response due to the nonsmooth term, which reads: where η represents the ratio between the modified (ω e ) and unmodified (ω e ) natural frequencies. Eq. (11) proves that even if the behavior is nonlinear, the free oscillation frequency of Eq. (7b) does not depend on the amplitude. Indeed, the oscillations are periodic but not harmonic since they are the succession of two half sine functions of different periods: the more β is large, the more the two periods are different, and higher the harmonics content is expected. Moreover, Eq. (11) suggests that given an electrical resonance frequency ω e (i.e., given an inductance), the modified electrical oscillation frequencyω e can be adjusted according to the value of β. However, the value of β should be less than 1/2 to avoid unstable response ( if β > 0. 5, ω e admits imaginary part). This feature is the main advantage of this nonlinear shunt circuit: by adjusting the nonlinear parameter β, one can tune the electrical circuit frequency at a particular value while fixing the inductance at a low value, which is an advantage for practical implementation. Figure 2b shows the ratio η as a function of β, illustrating that it can be lowered to zero (at β = 0.5) and increased up to a factor of 2 (for β → −∞).

Forced solution
To analyze the forced response, we rewrite Eq. (8) in a dimensionless form by introducing the following parameters: Using the dimensionless parameters in Eq. (12), one obtains: One can realize that the solution of Eq. (13) depends only on the damping ratio ξ e and the gain β. As a consequence, the shape of the resonance curve of Eq. (13) is independent of the excitation level P and can be obtained for any value of P and ω e using the scaling defined in (12). Eq. (13) is solved numerically with the continuation software MANLAB [50,51] since there is no analytical solution at hand. MANLAB is based on a harmonic balance method (HBM) to compute the periodic solutions. It requires a regularization of the nonsmooth term, which is discussed in Appendix B. Note that in all the numerical simulations with MANLAB, 40 harmonics were considered. Figure 3 shows the dimensionless frequency response, normalized with respect to the excitation level, of the zeroth (H0), first (H1), and second (H2) harmonics of the periodic solutions of Eq. (8) for different values of β. Note that only H0, H1, and H2 harmonics are considered since they represent the dominant harmonics observed in the numerical simulations, and they will be relevant in the upcoming analysis. Indeed, other harmonics with lower amplitudes are present such as the third and fourth harmonics.
Two important conclusions can be drawn from the response shown in Fig. 3. First, the nonlinear term creates a full harmonic content, with H0 being dominant in amplitude, H2 being significant but with lower amplitude, and the higher harmonics being negligible. Second, the relative amplitude of the harmonics with respect to the fundamental H1 increases with the absolute value of the nonlinear coefficient |β|: with β = 0, the response is fully linear with no harmonics, except H1. However, if β is close to 0.5 or tends to −∞, the harmonic content increases significantly. Interestingly, the amplitude of H0 exceeds that of H1 for extreme values of β (e.g., for β = 0.492 and β = −152.497).
The most important observation is that the shape of each harmonic, considered separately, seems close to the frequency response of a linear oscillator near its resonance frequency, which is obtained here in the vicinity of the free oscillations frequency, for Ω =ω e . Such property is conserved even for higher excitation, as shown in Fig. 4, in which the linearity property with respect to the excitation level is clearly observed.
On the contrary to the quasi-linear behavior described above, the nonlinear nonsmooth term can generate a strongly nonlinear superharmonic response near Ω = ω e /2. This is emphasized in Fig. 5, which shows the superharmonic responses for different values of β, of the H0, H1, and H2 harmonics. The results suggest that the superharmonic response appears to be nonlinear (hardening effect with unstable branches), and the nonlinear effect increases with |β|. This suggests that introducing the nonsmooth term can represent an option for the application involving internal resonance (as will be discussed in Sect. 3.2).

Forced response approximation with a linear oscillator
As discussed in the previous section, the nonlinear frequency response of the harmonics seems similar to the frequency response function (FRF) of a linear oscillator near the primary resonance. Thus, we seek an approximation of Eq. (8) that is defined by the following linear oscillator, for p = 0, 1, 2: where Q p is the approximated response of the pth harmonic of Q(t),ξ e is the modified damping ratio due to the nonsmooth term, and φ p is an unknown gain, corresponding to the pth harmonic, introduced to quantify the difference between the analytical and numerical solutions and to be determined by fitting the analytical frequency response of Q p with the numerical frequency response of the pth harmonic. From a phys-  ical point of view, the gain φ p enables quantifying the energy distribution at each harmonic due to the presence of the nonsmooth component.
To determine the modified damping ratioξ e , we assume (and we will verify hereafter) that the nonsmooth term has no effect on the damping coefficient (i.e., the term multiplyingQ), and thus: Then, the FRF of Q p is expressed as: To validate the estimated damping ratio and to determine the gain φ p , we setξ e according to Eq. (15) and we only vary the value of φ p to fit the numerical FRFs, as shown in Fig. 3, for each harmonic with their equivalent analytical ones, defined in Eq. (16). Examples of those fittings for different values of β are shown in Fig. 6 near the primary resonance. Notice the appearance of superharmonic responses, clearly observed in the H2 response in Fig. 6c, not taken into account in our equivalent linear model.
Considering this numerical fitting, we can draw two conclusions. The first one is that the equivalent damping ratioξ e , defined in Eq. (15), is the perfect one since the fitted equivalent linear FRFs match exactly the shape of the numerical (nonlinear) ones, by varying only the value of φ H p to reach the perfect fit.
The second conclusion is that by fitting the analytical and numerical frequency responses for H0, H1, and H2 for different values of β, the gain φ p for each of those harmonics can be estimated as a function of β (it is shown by markers in Fig. 7). Note that the estimated values of φ p are firstly plotted with respect to log 10 (1 − 2β) and then mapped to the values of β. Additionally, the results prove that increasing |β| leads to more distribution of energy to the other harmonics as both φ 0 and φ 2 increase. Compatible with the results in Fig. 3, H2 harmonic has a lower influence compared to H0 even for high values of |β|. More precisely, it appeared that the amplitude of H2 tends to a constant value as |β| increases. On the contrary, H0 harmonic admits larger influence in which φ 0 suggests higher value compared to φ 1 for extreme values of |β|.
Interestingly, those last results suggest that φ p admits a symmetric behavior with respect to log 10 (1 − 2β). Thus, to obtain a closed-form approximation of φ p , which will be used in the further analysis, we seek a polynomial expression of φ p as: where N ∈ N is the polynomial order and a m ∈ R are the polynomial coefficients. Thanks to the symmetric property, the relation φ p (y) = −φ p (y) can be used to estimate the coefficients a m only for y > 0 (i.e., β > 0) and then directly obtain the coefficients for y < 0 (i.e., β < 0) by changing the sign of the coefficients a m of the odd monomials (i.e., a m with m odd). In addition, to take into account the fact that only H1 will respond for y = 1 (i.e., β = 0), we enforce constraints on a 0 for each harmonic such that a 0 = 1 for H1 and a 0 = 0 for H0 and H2. With the constraint enforcement on a 0 , a good fitting between the polynomial expressions (solid lines in Fig. 7) and the obtained data (circle markers) required N = 4 for φ 1 , N = 5 for φ 0 , and N = 6 for φ 2 . The estimated coefficients a m for each harmonic are gathered in Table 1 only for β > 0.

Response of the full system according to the tuning of the nonsmooth component
In this section, we study the response of the full system (7a,b) with two particular tunings of the nonsmooth term. Based on the analysis in Sect. 2.2, two main points can be emphasized. First, it was shown that each harmonic of the response of the electrical oscillator could be considered separately and approximated by an equivalent linear system. This suggests using the first harmonic as an oscillator that can be tuned on one resonance of the host mechanical system to act as an RL-shunt, with the additional interest of tuning it using the nonsmooth gain β. This is investigated in Sect. 3.1, by tuning the electrical oscillator on the ith vibration mode, such thatω e ω i . Secondly, a nonlinear behavior was observed with the appearance of H0 and H2 harmonics and of a 1:2 superharmonic response of the electrical oscillator. This behavior is a characteristic of a  Fig. 1b) and suggests to investigate a 2:1 internal resonance, by tuning the electric oscillator at half the mechanical frequency (ω e ω i /2) and testing a transfer of energy from the mechanical oscillator to the electrical one, as already done using quadratic nonlinear terms in [22,23,25]. This is investigated in Sect. 3.2.

One-to-one tuning of the nonlinear shunt
(ω e =ω i )

FRF approximation
As analyzed in the previous sections, the nonlinear electrical oscillator (Eq. (7b)) oscillates with a fundamental harmonic (H1) that responds linearly around its primary resonance Ω ω e , suggesting that it can serve as a linear attachment similar to a standard RL-shunt. Thus, we consider here a tuning of β to haveω e =ω i : the modified electrical frequency is tuned to the opencircuit natural frequency of the host structure, as a classical linear resonant shunt, expecting similar behavior.
For this tuning case, we are interested in the first harmonic response behavior. Then, we replace Eq. (7b) by its linear approximation (14). Since the external forcing is only applied on Eq. (7a), the PE coupling term θ/(LC pi )q i in Eq. (7b) is considered as a forcing term, thus a multiplication by φ 1 will be enforced based on the linear approximation in Sect. 2.2.3. In addition, ω e = 1/( LC pi ) is replaced by the modified frequencyω e . Thus, we arrive at the following approximation: To simplify the writing of (18a,b), we define the following change of variables: where the dimensionless timet and frequencyΩ are different than those of Sect. 2.2.2. The two following frequency ratios are also defined: r i being related to the electrical frequency ω e andr i its modified version due to the detuning brought by the nonsmooth term. One arrives at the following system: where the derivatives are with respect to the dimensionless timet.k i is the modified dimensionless electromechanical modal coupling factor due to the presence of the nonsmooth term defined as: Note thatk i depends on β, as suggested by the expression of φ p in Eq. (17). More importantly,k i is smaller than the unmodified coupling factor k i since φ 1 < 1 for any nonzero value of β, as illustrated in Fig. 7. The physical meaning of this observation is that the nonsmooth term leads to a distribution of the energy to different harmonics, contrary to the linear case, where all the energy is transferred to the fundamental harmonic. Thus, a reduction in energy in the H1 harmonic is observed and quantified by the gain φ 1 .
The FRFs of Eq. (18a,b) reads: with i and whereq andQ are the Fourier transform of q(t) and Q(t), respectively.

Optimal tuning
To obtain the optimal response of our modified RLshunt, we follow the same optimization procedure as the one used for classical RL-shunts [2], as recalled in Appendix A. Comparing Eqs. (21) and (A.1), one observes that (ω e ,k i ,ξ e ) replace (ω e , k i , ξ e ). Considering first the frequency tuning, the optimal value for the modified electrical frequency, denoted byω op e , should satisfy: Then, by substitutingω e from Eq. (11) in Eq. (24), one arrives to the following optimal value of β, denoted by β op , as a function of the unmodified electrical tuning ratio r i = ω e /ω i : Notice that (see Fig. 2b): such that r i is necessary larger than 0.5. The above equations show that the nonsmooth component brings a new degree of freedom in the tuning of the resonant shunt. In practice, one can adjust the value of the nonlinear gain such that β = β op (Eq. (25)), bringing the modified electrical frequency close to the mechanical one, such thatω e =ω i (r i = 1). In practice, a standard RL-shunt is tuned using the value L of the inductance, such that the electrical frequency is equal to the open-circuit mechanical one: Here, adjusting the value of β is an additional degree of freedom to tune the RL-shunt. The range of tuning can be such that ω e >ω i (r i > 1): in this case β op > 0; on the contrary, one can havê ω i /2 < ω e <ω i , leading to β op < 0. This is illustrated in Fig. 8a, which shows the evolution of β op as a function of r i .
This new degree of freedom can be used in practice to finely adjust the tuning of an RL-shunt that has a nonoptimal value of the inductance L. It can also be used intentionally in the case of a low-frequency mechanical mode that would require a high value of L, often difficult to achieve in practice. In this case, one chooses a smaller L than the optimal one and corrects the tuning by adjusting β. To clarify the reduction in L, we compare it with the required inductance for the optimal condition of the linear RL-shunt (see Appendix A), denoted by L 0 . Considering that ω e = 1/ LC pi , one has: Thus, the inductance can be reduced by a factor of r 2 i compared to that of the linear RL-shunt. This quadratic law can be very advantageous in practice: for instance, a value of r i = 10 leads to a reduction in the inductance of a factor L 0 /L = 100. This is illustrated in Fig. 8b, which shows β op as a function of L 0 /L. In addition to the frequency tuning, the electrical damping must also be tuned. The optimal value of the electrical damping ratio ξ e and thus of the resistor R are obtained by: where φ Contrary to the linear RL-shunt, the optimal damping ratio ξ op e do not depend only on the coupling factor k i , but also on the linear tuning r i = ω e /ω i of the shunt, since φ op 1 depends on r i through β op (Eq. (25)): where a m are defined in Table 1 for the first harmonic and for β > 0 (r i > 1). For β < 0 (0.5 < r i < 1),

Performance
The optimal attenuation A dB of the nonlinear RL-shunt, in dB scale, can be obtained using the procedure of Appendix A by replacing k i by its modified valuek i . It reads: By substituting Eq. (29) in Eq. (30), we can conclude that in contrary to the linear resonant shunt case where only k i and ξ i are involved, the optimal attenuation depends also on the ratio r i (and thus on the inductance). In addition, one can expect a reduction in A dB as β increases due to the reduction in k i by a factor of √ φ 1 .
The dependence of A dB on (k i , ξ i , r i ) is examined in Fig. 9. Classically, one observes that A dB increases as a function of the coupling factor k i and when the mechanical damping ratio ξ i is decreased, in the same way than for a linear RL-shunt (see Fig. 24 in Appendix A and [2]). On the contrary, the effect of the nonsmooth term on A dB appears through its dependence on the tuning ratio r i through φ Figure 9 clearly shows that this dependence on r i is very weak since A dB is only slightly reduced with respect to the linear RLshunt (with β = 0 ⇔ r i = 1). This is an excellent result since the huge benefit of being able to decrease a lot the inductance L (r i = 10 ⇔ L = L 0 /100) is not counterbalanced by a significant decrease in the performance, of the order of 1 dB.

Numerical validation
In this section, we numerically validate the frequency response and the optimal parameters of our modified RL-shunt by considering the first twisting mode of the hydrofoil structure that will be tested experimentally in Sect. 4 (see Fig. 17). The modal parameters corresponding to this mode are gathered in Table 3. The numerical results are obtained by solving Eq. (7a,b) with MANLAB, considering 40 harmonics in the simulations. Then, the frequency responses of the first harmonic amplitude of the displacement u H1 , the charge Q H1 , and the PE voltage V H1 are obtained. Note that in what follows, the modal parameters correspond to this particular mode, and thus the subscript i associated with the modal parameters notations is omitted in the remaining of this section. Note also that the modal parameters are gathered in Table 3 follow the normalization of the mode shape such that ψ = 1 at the point of the structure where the displacement is studied and thus according to Eq. (1), we have u(t) = q(t).
As a first step, we choose the inductance L such that ω e = 2ω (i.e., r = 2 and L = 0.737 H ), meaning that the linear RL-shunt is overtuned by a factor  28)). The first harmonics of u(t) and Q(t) in the periodic steady state are shown in Fig. 10(a,b) as a function of the driving frequency Ω, for three values of the forcing F. Then, in Fig. 10(c,d), the same curves are plotted normalized by the forcing level F and superimposed to the FRFs H q (Ω) and H Q (Ω) (Eq. (23)) of the linear equivalent system (18a,b). One observes a perfect match, proving first that the linearity of the response with respect to the excitation level is perfect, as well as the agreement between the full nonlinear system (7a,b) and its linear approximation (18a,b). Figure 11 is analogous to Fig. 10(c,d) and shows how the frequency responses are changed when the parameters β and ξ e are changed around their optimal values β op and ξ op e . Note that we chose to plot V H1 instead of Q H1 since obtaining the order of magnitudes of the voltage will be useful for the practical realization of the shunt circuit. Those plots are similar to the response of a linear RL-shunt when ω e and ξ e are varied around their optimal value, showing again the equivalence of our modified RL-shunt to a classical linear RL-shunt.
To validate numerically the dependence of the optimal attenuation on the frequency ratio r and, thus, the inductance, as suggested in Fig. 9(a,b), Fig. 12a shows the numerical frequency response of u H1 normalized with respect to the forcing level for different values of r in which the optimal conditions are always satisfied (i.e., for each value of r we set R = R op , β = β op ⇒ω e =ω). Similar to Fig. 11, we consider the first twisting mode of the hydrofoil structure shown in Fig. 17. The results validate that tuning r to higher values, and thus lower inductance, reduces the optimal attenuation with respect to that of the RL-shunt while preserving the response shape. In addition, Fig. 12b shows the optimal attenuation A dB as a function of r for the RL-shunt with β = 0 (Eq. (A.4)) and for the shunt with the nonsmooth term Eq. (30). Compatible with the results in Fig. 9(a,b), the results clearly show that for higher values of r (lower values of inductance), the optimal attenuation is slightly decreased, compared to the linear RL-shunt, by 1dB for a reduction in the inductance by a factor of 100 for r = 10, compatible with results in Fig. 9(a,b). One can note that the attenuation predicted in Eq. (30) is numerically validated by a comparison with that obtained numerically, where an excellent fitting is obtained for different values of r . Note that the numerical attenuation, shown in circle markers in Fig. 12b, is directly deduced from Fig. 12a by the amplitude difference in dB scale between the opencircuit frequency and the optimal response at Ω =ω e .
3.2 One-to-two tuning of the nonlinear shunt (ω e = 1/2ω i ) In this section, we examine another possible tuning denoted by two-to-one (2:1) tuning such thatω e =  Table 3 and L = 0.737H ⇔ r = ω e /ω = 2 ω i /2. Due to the nonlinearities generated by the nonsmooth term, this tuning is expected to activate a nonlinear modal coupling through a 2:1 internal resonance. As discussed in [22,23], if quadratic nonlinear terms are present in V nl (t), the 2:1 internal resonance leads to create an antiresonance at which the amplitude of the host structure becomes independent of the forcing level, a so-called saturation phenomenon. In this section, we investigate the behavior of our system (which includes a nonsmooth nonlinearity and not a quadratic one) in the same internal resonance condition.

Validation of the occurrence of the 2:1 internal resonance
To validate the energy transfer through the 2:1 internal resonance, we focus on the first harmonic response of the displacement amplitude u H1 and the first subharmonic response of the charge Q H1/2 . We consider them dominant in the Fourier series like in a classical 2:1 internal resonance energy transfer (see [22]) and we express u(t) and Q(t) as:  Table 3 and L = 0.737H ⇔ r = ω e /ω = 2 where "oh" denotes other harmonics, including the contribution of the H0 and H2 harmonics due to the presence of the nonsmooth term. γ 2 is the phase angle of the first harmonic of the displacement with respect to the forcing, and γ 1 can be considered as the relative phase angle between the two oscillators, which is proved to be locked at 3π/2 at the antiresonance frequency in the case of 2:1 internal resonance [22]. Similar to the 1:1 tuning case in Sect. 3.1, the electrical frequency is tuned according to the nonsmooth gain β with no condition on the inductance value. Thus, the 2:1 tuning condition requires tuning β as: To test the ability of activating the 2:1 internal resonance, we solve numerically the system (2a,b) by setting β according to Eq. (32) with an arbitrary low electrical damping ratio and using the modal parameters in Table 3. In addition, the inductance L = 2.948 H is chosen to have r = ω e /ω = 1. The results are shown in Fig. 13, which clearly shows that a nonlinear energy transfer occurs from the first harmonics H1 of u(t) to the subharmonic H1/2 of Q(t). This energy transfer occurs after pitchfork bifurcation "PF" (or period-  Table 3 doubling bifurcation), in a frequency band centered around the mechanical resonance, at Ω ω i . However, contrary to a classical 2:1 internal resonance produced by quadratic nonlinear terms (as in [22]), the topology of the coupled response curve is completely different and does not show any saturation phenomenon. They appear with a shape very close to the response of a linear RL-shunt. This suggests to consider the second harmonic of Q(t) (which is H2 here) as an equivalent linear oscillator that can be optimized as a classical RL-shunt to mitigate the resonance of the ith mode of the host structure. Following this idea, as seen in Fig. 13a, the tuning of β according to Eq. (32) and the chosen ξ e do not lead to an optimal response, that would be characterized by a symmetric shape and minimal peaks, as observed in Fig. 11 for the 1:1 tuning case. Thus, an optimization study is necessary to obtain the optimal values of β and ξ e to achieve an optimal attenuation similar to that of a linear RL-shunt.
Another important feature concerns the relative phase γ 1 that locks at 3π/2 at the antiresonance frequency, independent from the excitation level. This feature was proved analytically and numerically in [22] with the addition of quadratic nonlinearities in the shunt circuit. As seen in Fig 13c, the same phase lockage is equally preserved, which can be exploited for tuning the system in practical implementation, similar to the experimental study in [23].

Optimization study
Similar to the case with 1:1 tuning, the optimization is established through the nonsmooth gain β to tune the electrical resonance frequency and the electrical damping ratio ξ e (thus the resistance R), to obtain a mechanical frequency response as low and flat as possible. Two points are considered for this optimization: (i) We are looking for an optimal value of β that symmetrizes the frequency response, as for a classical RL-shunt; (ii) the coupled response is obtained through a 2:1 internal resonance, as observed in the previous section.
As shown in [22], the symmetric response with a 2:1 internal resonance requires a ratio between the electrical frequencyω e and the mechanical oneω i to be slightly higher than half (i.e., contrary to the tuning in (Eq. (32)). The exact value of this ratio, denoted by r * i , can be estimated by setting ω 2 = 2ω 1 where ω 2 and ω 1 are the natural frequencies in the electromechanical basis (see Appendix C and [22]). The obtained r * i depends only on the piezoelectric coupling factor k i as follows: Given the value of r * i , the optimal value of β is then estimated as: Fig. 13 a and b show, respectively, the numerical frequency response of u H1 and Q H1/2 for increasing excitation levels. c shows the relative phase γ 1 between the two oscillators. The linear response is plotted in black. The stable and unstable responses are depicted in solid and dotted lines, respectively. The purple star denotes the pitchfork bifurcation (PF). The numerical values used are β = 0.444, R = 36Ω, L = 2.948H . The modal parameters used are the ones gathered in Table 3 ω Since it is the second harmonics of the charge Q(t) that is coupled to the primary (mechanical) oscillator, we consider for the optimization its equivalent linear oscillator, as defined by Eq. (14) in Sect. 2.2.3 with p = 2. Thus, the key point is to modify the piezoelectric coupling through the gain φ 2 . Following the same reasoning as in Sect. 3.1.1, the modified PE coupling factor for the two-to-one tuning case can be expressed as: We then express the optimal value of ξ e (and the resistance R) as follows: where φ op 2 = φ 2 (β = β op ) to satisfy the optimal conditions. Considering Eq.
where a m are defined in Table 1 for the second harmonic. The optimal attenuation can then be expressed as: A dB = 20 log 10 Similar to 1:1 tuning case, the optimal attenuation depends explicitly on r i (thus the inductance) but with a reduction in the piezoelectric coupling factor by √ φ 2 .
Thus, a further reduction in the attenuation compared to that of the linear resonant shunt is expected since φ 2 admits an order of magnitude lower than that of φ 1 , as suggested in Fig. 7. This is further illustrated in Fig. 14 by comparing A dB of the 2:1 tuning case and that of a linear RL-shunt.

Numerical validation
In order to validate the results suggested from the optimal analysis, we solve numerically the system (7a,b) to obtain the frequency responses of u H1 and V H1/2 for different values of β and R (thus ξ e ) including the optimal conditions in Eq. (34) and Eq. (36), respectively. This is shown in Fig. 15 in which we choose the inductance such that r = 1 ⇔ L = 2.948 H . Similar to the plots in Sect. 3.1.4, we consider the first twisting mode of the foil shown in Fig. 17. The results clearly suggest that setting R and β according to the obtained optimal conditions enables the recovery of the symmetric behavior and the minimization of the peaks near the resonance frequency. It is worth mentioning that the response is very sensitive to any small perturbation of the optimal value of β (observe the very close values of β used in Fig. 15(a,c)).
Regarding the optimal attenuation, one can realize that Eq. (38) shares similar behavior compared to that of the 1:1 tuning case in Eq. (30) in which the attenuation depends on the inductance. However, as observed in Fig. 7 and contrary to the behavior of φ 1 , the value of φ 2 increases with β then tends to admit a constant value as β increases. This suggests that decreasing the inductance and thus the optimal value of β leads to enhancing the optimal attenuation, contrary to the oneto-one tuning case. To validate this result, we perform a similar analysis to that in Sect. 3.1.4 (in Fig. 12) by studying the frequency response of u H1 at the optimal condition with different values of the inductance. The numerical results in Fig. 16a clearly validate the increase in the attenuation as the inductance decreases, which can also be seen in Fig. 16b, where the optimal attenuation is plotted versus the inductance value. In addition, it is realized that the attenuation increases up to a certain level where it remains almost constant as the inductance decreases. This behavior is explained by the presence of φ 2 in the optimal attenuation expression, which admits the same trend versus β as shown in Fig. 7. Moreover, comparing the numerical attenuation obtained numerically and analytically shows an excellent fitting, validating the attenuation expression in Eq. (38). It is also worth mentioning that the attenuation in this tuning case is reduced compared to the optimized RL-shunt with a narrower bandwidth due to the substantial reduction in the coupling factor suggested in Eq. (35) since φ 2 admits a very low amplitude.

Summary
The introduced nonsmooth term enables tuning the electric resonance frequency of the shunt circuit while setting a low inductance in the circuit. In addition, it suggests a multi-harmonic response dominated by the zeroth, first, and second harmonics that appeared to be similar to that of a linear oscillator. In addition, strong nonlinearities appeared in the superharmonic response. A 1:1 and 2:1 tuning cases has been studied through the approximation of the main system (7a, b) to an approximated system in (18a,b) by modifying (and reducing) the piezoelectric coupling factor by √ φ 1 and √ φ 2 for the 1:1 and 2:1 tuning, respectively. In either case, closed-form expressions of the optimal values of β and R in addition to optimal attenuation are determined and gathered in Table 2. Contrary to the linear resonant shunt, the frequency ratio r i (thus the inductance) appears explicitly in the attenuation expression. The 1:1 tuning case shows an important feature in which one can reduce the required inductance by a factor of 100 for an attenuation reduction around 1dB compared to that of a linear resonant shunt. While for the 2:1 tuning case, a substantial reduction in the attenuation is observed but with an interesting feature in which the attenuation increases with the reduction in the inductance up to a certain level where it stays almost constant. The analytical results were then validated numerically by applying the absorber to attenuate the first twisting mode of a hydrofoil structure. Considering also the first twisting mode of the hydrofoil, an experimental validation will be shown in the next section.

Experimental setup
The experimental tests were conducted on a clampedfree foil structure fixed at its base in a vice, as shown in Fig. 17. This structure is similar to the one used in [25,52]. Three PE patches are glued on cavities machined on the surfaces of the foil: on one face (denoted patches 1, 2), two P-876.A15 DuraAct polymer coated multilayer piezoelectric patches are used for transduction with the first bending mode, whereas a macro-fiber composite M8557-F1 with 45 • fiber orientation is used on the other face (denoted patch 3) for coupling with the first twisting mode. In all the tests of this article, only patch 3 is used since we focus on the first twisting mode. Consequently, patch 3 is connected to the shunt circuit, and patch 1 and 2 are shortcircuited. The three PE patches are from the PI Ceramic manufacturer. The mechanical excitation is realized by a coil interacting with a magnet fixed on the foil. The applied force is proportional to the current in the coil, i.e., F(t) = α I (t), where α = 0.6NA −1 [23], with very low distortion [53]. To properly design the nonlinear shunt, we firstly performed an experimental modal analysis, equivalent to that in [23,25], to obtain the modal characteristics of the first twisting mode of the foil, which are gathered in Table 3. Note that since we tackle only the twisting mode, the subscript i will be omitted in what follows.
Note that ψ(x m ) = 1 means that the modal parameters are estimated by scaling the mode shape at the measurement point of the structure x m to be unity, thus according to Eq. 1, u(t) = q(t) (i.e., the same scaling of the mode shape considered in the previous section).

Nonsmooth shunt circuit
The nonlinear shunt circuit used for the tests is shown in Fig. 18. It consists of three interconnected subcircuits to realize (i) a tunable synthetic inductor (outlined in blue); (ii) a voltage measurement (outlined in orange), and (iii) a nonlinear voltage generator (outlined in light red). To handle the large voltage levels that can appear, especially during transients, the    Table 3 Fig. 17 Experimental setup. Dimensions are in mm. The thickness of the PE patches is 1mm   [54] to achieve the high inductance value needed to obtain an electrical resonance frequency close to half of the mechanical resonance frequency. The inductance value is given by where L is tuned using the variable resistance R 4 . As discussed in Sect. 3, the electric resonance frequency is controlled solely with the nonsmooth gain β according to a free choice of the inductance L. In all the experimental tests in this study, and for different tunings of the shunt circuit, the value of L is always chosen to be less than 3 H . For such values, the parasitic resistance generated by the Antoniou inductor, observed in [23], is to the order of 0.1Ω and will be neglected here. Thus, the resistor R in the shunt models corresponds to a physical resistance added to tune the electrical damping. Note that the parasitic resistance is estimated with a frequency-to-bandwidth ratio of the electrical resonance, close to Ω = ω e , in linear operation, with β = 0.
The PE voltage measurement is established through two voltage dividers composed of R 5 and R 6 and two voltage followers OA 3 to OA 4 (OPA445, Texas Instrument) to ensure a large input impedance. The voltage dividers are used to attenuate the voltage to values compatible with the differential amplifier DA (INA826, Texas Instrument). Since the PE patch is not grounded, these voltages are first fed to the differential amplifier DA. The nonsmooth voltage with the absolute function, as in Eq. (6), is generated through a half-wave rectifier circuit composed of OA 5 , a diode (1N4002), and a resistor R 7 connected to the ground. The output voltage of the half-wave rectifier circuit is fed to an inverting amplifier, realized with OA 6 , to respect the assumed sign convention of the nonsmooth voltage in (2a,b) and to control the nonsmooth gain β. The follower OA 7 is used to minimize the output impedance while protecting the INA826 from excessive current. The nonsmooth gain β is controlled according to the variable resistor R 9 as: Note that changing the sign of the nonsmooth voltage can be done by flipping the polarity of the diode without using the inverting amplifier. However, for such a case, the nonsmooth gain will be controlled either by a voltage divider composed of R 5 and R 6 or by adding a new voltage divider instead of the inverting amplifier.
In the first case, and for certain values of β, one might risk achieving a voltage at the input of the DA higher than the operating levels. In the second case, a restriction on the value of β is presented. Thus, we prefer to use the inverting amplifier so that the sign convention is respected with easy controllability of the nonsmooth gain.

Measurement protocol
The experimental tests aim to verify the results suggested theoretically in Sect. 3, and thus we consider the same tuning cases, namely, the 1:1 and 2:1 tuning. It is proved that the former case leads to a fully linear response near the resonance frequency, while the latter activates the 2:1 internal resonance and thus leads to the response of the subharmonic of the charge. Since the mode shape is scaled to verify ψ(x m ) = 1, we express the time signals of the displacement u(t) and the charge Q(t) for the 1:1 tuning as: and for the 2:1 tuning case, we include the subharmonic response such that [22]: where u H1 and u H1/2 are, respectively, the amplitude of the first and subharmonic of the displacement and Q H1 and Q H1/2 are, respectively, the amplitude of the first and subharmonic of the charge. δ 1 and δ 2 are the phase angles of first harmonic of u(t) and Q(t) for 1:1 tuning case. γ 2 is the phase angle of the first harmonic of u(t) for the 2:1 tuning case. Note that δ 1 , δ 2 , and γ 2 are measured with respect to the forcing. γ 1 is the relative phase angle between the first and the subharmonic, which is estimated to be locked at 3π/2 at the antiresonance frequency associated with the 2:1 internal resonance [22,23]. "oh" denotes the other harmonics, in particular, the zeroth and the second harmonics, generated due to the nonsmooth component. In practice, the amplitude and phase angles of the harmonics of u(t) and Q(t) can be obtained by measuring the foil tip velocity v(t) with the laser vibrometer and the voltage across the PE patch terminals V (t) using the differential voltage probe (see the procedure in [23]). In what follows, we chose to show the experimental frequency response of u H1 , instead of v H1 , in both tuning cases because it will be more convenient to validate the optimization results suggested theoretically (see Table 2). In addition, we chose to show V H1 and V H1/2 in the 1:1 and 2:1 tuning cases, respectively, instead of Q H1 and Q H1/2 , since it will be useful to obtain the voltage level to ensure that it does not exceed the operating limits of the operational and differential amplifiers in the shunt circuit (see Fig. 18). Indeed, the subharmonic response is measured for the 2:1 tuning case to validate the energy transfer and, thus, the activation of the 2:1 internal resonance.
Note that in either tuning case, the frequency responses will be shown normalized with respect to the forcing, following the results in Sect. 3, which suggest that the response curves are linear with respect to the excitation. To estimate the forcing F in [N], the current I in the coil is measured using a PHILIPS PM 9355 precision current probe, then the relation F = α I is used.
To establish the amplitude and phase frequency responses, we consider two different procedures for the 1:1 and 2:1 tuning cases. In the former case, the frequency response is directly estimated by exciting the foil with a chirp signal of time-varied excitation frequency using the coil/magnet system. This method can be extended to realize the frequency response for the 1:1 tuning case since it is proved in Sect. 3.1 that the response is fully linear near the resonance frequency (i.e., no bifurcations or change of stability). In addition, we consider the first harmonics to be dominant in comparison with the zeroth and second harmonics for the range of β used in this study. Thus, the multiharmonic nature of the response at the steady state will not lead to a significant effect on the measured frequency response. In the 2:1 tuning case, the chirp signal method is not suitable due to the bifurcation and change of stability associated with the response. Thus the sine-stepped method is used instead, similar to that in [23]. Namely, a sine current with fixed intensity is fed to the coil/magnet device to generate a harmonic force F = F 0 cos Ωt. Then, the excitation frequency Ω is increased (or decreased) by steps, keeping the intensity fixed, sweeping a domain close to the resonance frequency of the first twisting mode of the foil. For each frequency, the harmonics amplitude and phases of the velocity (thus the displacement) and the PE voltage are extracted in the steady state using the demodulation technique explained in [23]. The input/output signals were synthesized/measured with National Instrument cards (NI-9234, NI-9263) driven by a MATLAB program.

Experimental validation of the theoretical results
In order to validate the numerical predictions in Sect. 3 and, in particular, the estimated optimal parameters, a comparison between the numerical and experimental results is presented in Fig. 19 for the 1:1 and 2:1 tuning cases. In the former case, the frequency responses of u H1 and V H1 in addition to δ 1 are shown. In the latter case, u H1 and V H1/2 in addition to the phase γ 1 and γ 2 are presented. Moreover, in the 1:1 tuning case, the inductance L = 0.73H is chosen such that r = ω e /ω = 2, while for the 2:1 tuning case, L = 2.94H is chosen such that r = ω e /ω = 1. In either case, the values of β and R are set according to their optimal conditions gathered in Table 2.
For completeness and comparison purposes, both the linear and the nonlinear frequency responses are depicted. The linear ones are estimated with the PE patch 3 in open and short circuit configurations. Comparing the numerical and the experimental results of the linear responses shows an excellent fitting validating the modal parameters estimated in Table 2 and the inductance tuning with the Antoniou inductor.
Regarding the nonlinear responses (i.e., with the nonsmooth component included in the shunt), the numerical and experimental frequency responses of the displacement, PE voltage, and phase angles suggest an excellent agreement for the 1:1 and 2:1 tuning cases. The following major points, suggested numerically in Sect. 3, are validated: (i) The response of u H1 for both tuning cases suggests a curve shape similar to that of the resonant shunt. (ii) The estimated optimal parameters of β and R lead to the optimal response for both tuning cases since the response curves of u H1 show a symmetric behavior with respect to the resonance frequency with the amplitude of the two peaks around primary resonance is minimized. (iii) Although an equivalent response shape compared to the RL-shunt is observed, the amplitude of u H1 in the 1:1 tuning case admits slightly higher amplitude, while a substantial increase in the amplitude is observed for 2:1 tuning case. This suggests different attenuation abilities for the two tuning cases, which is further discussed in Sect. 5.3. (iv) The response of the voltage subharmonic for the 2:1 tuning validates the energy transfer and the activation of the 2:1 internal resonance. It can be realized the substantial increase in the order of magnitude of V H1/2 compared to V H1 of the 1:1 tuning case. (v) The last major point concerns the phase angles. It appeared that in the optimal condition, the first harmonic phase angle of the displacement for both tuning cases (i.e., δ 1 and γ 2 ) is equal at the resonance frequency. Thus, if the first harmonic is dominant, the displacement signals corresponding to the two tuning cases will be in phase (see Fig. 20a). In addition, the relative phase angle γ 1 , only defined for the 2:1 tuning case, admits a value of 3π/2 at the antiresonance frequency, equivalent to the results obtained in [22,23] when quadratic nonlinearities are introduced in the shunt.  Table 3 The time signals of the velocity v(t) and the PE voltage V (t) are shown in Fig. 20 for the tuning cases in addition to the linear response in the open-circuit configuration. The signals are recorded at the open-circuit frequency (i.e., Ω =ω = 511.3 × (2π) rad/s). The signals clearly show that for the 1:1 tuning case, the switching effect between two resonance frequencies (i.e., ω e and ω e in Eq. (9)) appears clearly in PE voltage signals with a slight effect in the velocity signal. In addition, one can realize that the velocity signals of the 1:1 tuning and 2:1 tuning in Fig. 20a are in phase at the resonance frequency, as suggested in Fig. 19b. Also, one can clearly observe the presence of the subharmonic in the voltage and velocity signal in the 2:1 tuning case in Fig. 20b, following the expressions in (42a,b).

Effect of the design parameters
This section is devoted to show the tuning effect of the design parameters, namely the resistor in the shunt R (thus ξ e ) and the nonsmooth β. Thus, we performed two sets of experiments. In the first set, we considered three different values of β, including its optimal value, and we fixed the R at the optimum value. Similarly, in the second set, we considered three different values of R, including its optimal value, and we fixed β at its optimum value. We show in Fig. 21 the frequency responses of u H1 and V H1 for the 1:1 tuning case and in Fig. 22, we show the frequency responses of u H1 and V H1/2 for the 2:1 tuning case. Regarding the value of the inductance, we consider the same setting as in Fig. 19 for both tuning cases. One can realize from Fig. 21(a,c) and Fig. 22(a,c) that the response is very sensitive to β since it appears that any small perturbation of the optimal value of β in both tuning cases can break the response symmetry (i.e., the two peaks around the primary resonance admit different amplitudes) and thus shift the antiresonance from the resonance frequency. As β is varied with the resistance R 9 in the shunt circuit (see Fig. 18), the value of this resistance should be set in a precise manner. The results in Fig. 21(b,d) and Fig. 22(b,d) show that setting a resistance lower than the optimal value in the shunt leads to an increase in the peaks' amplitude around the primary resonance with a decrease in the amplitude of the antiresonance. However, setting a higher value leads to the appearance of one peak near the resonance frequency in which no antiresonance is detected. The experimental results clearly validate the numerical ones in Fig. 11 and Fig. 15 for the 1:1 and 2:1 tuning cases, respectively. In addition, it is validated that the behavior with respect to β and R is equivalent to that of the linear RL-shunt but with the main difference that the tuning parameter in our case is β instead of the inductance (see Appendix A).

Discussion about the optimal attenuation
The main advantage suggested by the absorber is the ability to choose the inductance to be lower compared to that required by the RL-shunt, in which the electric resonance frequency is tuned with the nonsmooth gain β. Thus, it is necessary to experimentally validate the effect of the inductance value on the optimum attenuation suggested with the nonsmooth component. The analytical expressions of the optimal attenuation in Table 2 show that the inductance appears explicitly through the gain φ 1 and φ 2 for the 1:1 and 2:1 tuning cases, respectively.
To validate those results, we measure the frequency response of u H1 considering the two tuning cases for different values of the inductance. The results are shown in Fig 23a and b for the 1:1 and 2:1 tuning, respectively. For each value of the inductance, we cor- The displacement and voltage responses are plotted in dB and linear scales, respectively rect β and the value of R to respect the optimal condition. The 1:1 tuning suggests that the attenuation decreases as the chosen inductance in the circuit is reduced. On the contrary, the 2:1 tuning case suggests an increase of the attenuation, accompanied by a narrower bandwidth, as the inductance is reduced up to a certain level where the attenuation appears to stay constant around 22dB. Indeed, the optimal attenuation for both tuning always admits a value lower than that of the linear RL-shunt.
This attenuation behavior for both tunings is predicted through the expressions in Table 2 as shown in Fig. 23c (see the solid and dashed lines). For clarity, the optimal attenuation is plotted versus the frequency ratio r instead of the inductance. Those two quantities are related by r = ω e /ω = 1/(ωLC p ). As the results in Figs 23a and b are shown in the dB scale, the optimal attenuation, measured experimentally, is obtained by subtracting the amplitude of the linear response from that corresponding to the nonsmooth response, both measured at the resonance frequency (similar to the plots in Fig. 12 and Fig. 16). This is repeated for different values of inductance, in which the results are then plotted in Fig. 23c (with circle and diamond markers). Clearly, for both tunings, the experimental results follow the same pattern suggested analytically with a small discrepancy, especially for the 2:1 tuning. Realize also that we were able to reach an inductance reduction by a factor of 64 compared to that required by the linear resonant shunt with a reduction in the attenua- tion around 0.8dB for the 1:1 tuning case. A further reduction in the inductance was hard to be achieved in practice since it requires a value β more closer to 0.5, which can lead to unstable response.
It is worth mentioning that not only the attenuation depends on the tuning of β but indeed the whole dynamics. For example, if we consider the case with the inductance L = 0.11H (red curve in Fig. 23(a,b)), the transition between the 1:1 tuning (i.e., fully linear behavior near the resonance frequency) to the 2:1 tuning (i.e., activation of the 2:1 internal resonance) required a change of β by around 1%. Thus, a high precision in the value of β is required to obtain the required behavior associated with each tuning case which is challenging in practice.

Conclusion
In this article, a new nonlinear piezoelectric shunt was proposed, based on the intentional use of a bilinear nonsmooth component in the electrical circuit. This nonsmooth component has three important properties. The first one is to create an apparent electrical oscillation frequencyω e with a value that can be adjusted with the gain β, thus creating a tunable oscillator. The second property is that its response is fully linear in amplitude: Despite its nonsmooth nature, no strong nonlinear behavior was observed. The nonlinearity appears through the creation of harmonics in the response of the system, mainly of even order, because of its nonsymmetric characteristic. The third property Fig. 23 a and b show the experimental frequency response of the first harmonic of the velocity u H1 for different values of the inductance for the 1:1 and 2:1 tuning, respectively. c Compari-son between the optimal analytical attenuation and that measured experimentally for both tuning cases is that the behavior of the harmonics can be approximated by independent linear oscillators. Mixing those three properties, it was shown that the behavior of our shunt is (i) equivalent to a standard RL-shunt with (ii) the noteworthy property of being tunable using the gain β.
We investigated two possible tuning of the shunt. The first (classical) one is a 1:1 tuning, for which the apparent electrical frequency is tuned equal to the mechanical resonance frequency to mitigate (ω e =ω i ). In this case, it was shown that it is possible to detune the RL-shunt by a factor r i = ω e /ω i between 0.5 and 10 (in theory up to +∞) without changing the amplitude reduction performance significantly: Only a 1 dB decrease was observed. This tuning ability can be a considerable advantage in practice since it substantially decreases the inductance L of the shunt by a factor r 2 i , i.e., up to a factor 100 if r i = 10.
The second tuning case, relying on a 2:1 internal resonance (the apparent electrical frequency is tuned equal to half the mechanical resonance frequency to mitigate, ω e ω i /2), shows a similar RL-shunt behavior though relying on a purely nonlinear process of energy transfer between two frequency bands, from the driving frequency Ω to its subharmonic Ω/2. However, the performances are decreased since they rely on the energy transfer between the oscillators, which is weaker in the case of a 2:1 tuning with respect to the 1:1. In either cases, an optimization study has been established to determine the optimal tuning of the electrical damping ration ξ e , the nonsmooth gain β, and the optimal attenuation. The optimization procedure relied on the definition of a reduced piezoelectric factor that depends on β and the first and second harmonics amplitude for the 1:1 and 2:1 tunings, respectively. The absorber was then tested to damp the first twisting mode of a hydrofoil structure in which the numerical results were validated for both tuning cases.
The main contributions of this work can then be summarized as follows: (i) The introduced piezoelectric shunt circuit enabled exploiting the nonsmooth component to enhance the resonant shunt through the significant reduction in the required inductance without losing the performance, (ii) the inherent nonlinearities can activate the 2:1 internal resonance which can be useful for applications involving energy harvesting. Indeed, due to multi-harmonic response, the nonsmooth term allows the activation of other types of internal resonance, (iii) the nonsmooth term was easily analyzed with a diode in practice, leading to a shunt circuit that can be easily controlled and tuned, and iv) the proposed absorber can pave the way for future studies involving the possibility of adding quadratic and cubic nonlinearities in combination with the nonsmooth component to activate the 2:1 internal resonance. Such a design can possibly lead to a response similar to an RL-shunt, but associated with a saturation phenomenon.
Funding The Région Hauts de France and the Carnot ARTS Institute, France, are warmly thanked for the PhD grant of the first author.

Data Availability
The datasets generated during and/or analyzed during the current study are not publicly available but are available from the corresponding author upon reasonable request.

Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A: RL-shunt response and optimization: reference solution
In this section, we recall the response of a mechanical system coupled to a RL-shunt. We thus consider Eq. (21a,b) without the nonsmooth term (i.e., β = 0 ⇒r i = r i ,k i = k i ), which reads: Q + 2ξ e r iQ + r 2 iQ + k i r iqi = 0. (A.1b) The mechanical FRF is thus: whereq i (Ω) is the Fourier transform of q i (t).
As a classical result (see, e.g., [2]), it is possible to choose the electrical frequency ω e and the damping ratio ξ e such that the amplitude of the ith resonance of the elastic structure is minimized and has a blunt shape, as shown in Fig. 24a. The optimal values of ω e and ξ e are Then, as in [2], a performance indicator of the RLshunt can be defined as the difference, in dB, between the amplitude at the resonance of the mechanical oscillator without shunt (for instance, with the shunt in open circuit) and the maximum amplitude of the blunt shunt with the optimal RL-shunt. Here, to simplify, we define the latter as the amplitude at the open-circuit frequency, i.e., for Ω =ω i (Ω = 1). The optimal attenuation A dB is then estimated with Eq. (A.2) as: = 20 log 10 1 + k i ξ i √ 6 (A.4) Equation (A.4) suggests that the attenuation is solely function of the mechanical damping factor ξ i and the piezoelectric coupling k i , as shown in Fig. 24b. Note that the above expression for A dB is taken at the open circuit frequency and not at the frequency of the fixed points as in [2] to obtain a simpler expression whose numerical value is very close.

Appendix B: Regularization of the nonsmooth term and implementation in MANLAB
The numerical results in this work were all obtained using MANLAB, which requires regularization of the nonsmooth term associated with the absolute value function of the piezoelectric voltage |V |. As a first step, we define a new variable y as: Then, the regularization is obtained through the following equation: where δ ∈ R is the regularization parameter. As shown in Fig. 25, if δ = 0, the solution is equivalent to y = |V |. For a small value of δ, the angular point at V = 0 of the graph of the absolute value function is replaced by a smooth curve that gets closer to the nonsmooth exact one as δ approaches zero. Upon obtaining the regularization, the two degree of freedom system in (2a,b) is implemented in MANLAB in the first order form considering four main variables (q, Q, v, w) and two auxiliary variables (V , y) [51]: Following the reasoning used in [22], we consider the linear system (21a, b). Since the stiffness part is not diagonal, it is possible to compute its two dimensionless frequencies, which read: with Δ = (1 −r 2 i ) 2 + 4k 2 ir 2 i . Enforcing ω 2 = 2ω 1 , one arrives to the following order 2 polynomial inr 2 i : 4r 4 i + (25k 2 i − 17)r 2 i + 4 = 0, (B.5) that have two roots. r * i of Eq. (33) is the one which is smaller than 1, close to 0.5.
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