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

Abstract

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.

Appendices

Appendix A: RL-shunt response and optimization: reference solution

Fig. 24

a Typical FRF of the RL-shunt in the optimal case. b Attenuation A as a function of k for different mechanical damping factor $\xi _i$

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., $\beta =0\; \Rightarrow \; \bar{r}_i=r_i, \bar{k}_i=k_i$), which reads:

$$\begin{aligned}&\ddot{\tilde{q}}_i+2{\xi }_i\dot{\tilde{q}}_i+\tilde{q}_i+k_ir_i\tilde{Q}={f}_i\cos {\tilde{\varOmega } \tilde{t}}, \end{aligned}$$

(A.1a)

$$\begin{aligned}&\ddot{\tilde{Q}}+2{\xi _e}r_i\dot{\tilde{Q}}+r_i^2\tilde{Q}+k_ir_i\tilde{q}_i=0. \end{aligned}$$

(A.1b)

The mechanical FRF is thus:

$$\begin{aligned}{} & {} H_q(\tilde{\varOmega })=\hat{\omega }_i^2 m_i\frac{\mathring{q}}{{F}}\nonumber \\{} & {} \quad =\frac{r_{i}^{2} - \tilde{\varOmega }^{2} + 2 j \tilde{\varOmega } \xi _e r_{i}}{\left( 1 - \tilde{\varOmega }^{2} + 2 j \tilde{\varOmega } \xi _i \right) \left( r_{i}^{2} - \tilde{\varOmega }^{2} + 2 j \tilde{\varOmega } \xi _e r_{i}\right) - k_{i}^{2} r_{i}^{2} },\nonumber \\ \end{aligned}$$

(A.2)

where $\mathring{q}_i(\tilde{\varOmega })$ is the Fourier transform of $q_i(t)$.

As a classical result (see, e.g., [2]), it is possible to choose the electrical frequency $\omega _e$ and the damping ratio $\xi _e$ 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 $\omega _e$ and $\xi _e$ are

$$\begin{aligned} \omega _e^{\text {op}} =\hat{\omega }_i \quad \Rightarrow \quad r_i^{\text {op}}=1;\quad \xi _e^{\text {op}}=\frac{\sqrt{6} k_{i}}{4}. \end{aligned}$$

(A.3)

Then, as in [2], a performance indicator of the RL-shunt 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 $\varOmega =\hat{\omega }_i$ ($\tilde{\varOmega }=1$). The optimal attenuation $A_{\text {dB}}$ is then estimated with Eq. (A.2) as:

$$\begin{aligned} A_{\text {dB}}= & {} 20 \log _{10} \frac{H_q|_{{k}_i=0} (\tilde{\varOmega }=1)}{H_q|_{{\xi }_e={\xi }_e^{\text {op}},{r}_i=1} (\tilde{\varOmega }=1)}\nonumber \\{} & {} =20 \log _{10}\left( 1+\frac{{k}_i}{\xi _i \sqrt{6}}\right) \end{aligned}$$

(A.4)

Equation (A.4) suggests that the attenuation is solely function of the mechanical damping factor $\xi _i$ and the piezoelectric coupling $k_i$, as shown in Fig. 24b. Note that the above expression for $A_{\text {dB}}$ 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:

$$\begin{aligned} y=|V| \quad \implies \quad {\left\{ \begin{array}{ll} y=-V &{}\text {if}\quad V<0,\\ y=V &{}\text {if}\quad V>0. \end{array}\right. } \end{aligned}$$

(B.1)

Then, the regularization is obtained through the following equation:

$$\begin{aligned} (y-V)(y+V)=\delta \quad \implies \quad y^2-V^2-\delta =0, \nonumber \\ \end{aligned}$$

(B.2)

where $\delta \in \mathbb {R}$ is the regularization parameter. As shown in Fig. 25, if $\delta =0$, the solution is equivalent to $y=|V|$. For a small value of $\delta$, 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 $\delta$ approaches zero.

Fig. 25

Graph of the function $y=|V|$ and its regularized versions for different values of $\delta$

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]:

where Eqs. (B.3a,b,c,d) are the main equations and Eqs. (B.3e,f) are the auxiliary equations introduced to obtain the voltage V and the regularization variable y.

Appendix C: Tuning ratio for the 2:1 tuning

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:

$$\begin{aligned} \omega _1^2=\frac{1+\bar{r}_i^2-\sqrt{\varDelta }}{2},\quad \omega _2^2=\frac{1+\bar{r}_i^2+\sqrt{\varDelta }}{2}, \end{aligned}$$

(B.4)

with $\varDelta =(1-\bar{r}_i^2)^2+4\bar{k}_i^2\bar{r}_i^2$. Enforcing $\omega _2=2\omega _1$, one arrives to the following order 2 polynomial in $\bar{r}_i^2$:

$$\begin{aligned} 4\bar{r}_i^4 +(25\bar{k}_i^2-17)\bar{r}_i^2+4=0, \end{aligned}$$

(B.5)

that have two roots. $r_i^*$ of Eq. (33) is the one which is smaller than 1, close to 0.5.