# On the programming of cubic acoustic resonator with periodic time-dependent coefficient

Maxime Morell, Aurélie Labetoulle, Emmanuel Gourdon, Emanuele De Bono,
Manuel Collet, Alireza Ture Savadkoohi

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Maxime Morell, Aurélie Labetoulle, Emmanuel Gourdon, Emanuele De Bono, Manuel Collet, et al.. On the programming of cubic acoustic resonator with periodic time-dependent coefficient. Communications in Nonlinear Science and Numerical Simulation, 2025, 150, pp.108972. 10.1016/j.cnsns.2025.108972. hal-05113171

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Submitted on 14 Jun 2025

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Research paper

# On the programming of cubic acoustic resonator with periodic time-dependent coefficient

Maxime Morell<sup>a ID*</sup>, Aurélie Labetoulle<sup>a</sup>, Emmanuel Gourdon<sup>a</sup>, Emanuele De Bono<sup>b</sup>, Manuel Collet<sup>c</sup>, Alireza Ture Savadkoohi<sup>a</sup>

## ARTICLE INFO

**Keywords:**

Time-dependent coefficient

Programmable

Low amplitude

Electroacoustic resonator

## ABSTRACT

In this article, we investigate an electroacoustic resonator with a programmed cubic nonlinearity with constant and time-dependent coefficients. We design an experiment coupling an acoustic mode of a tube with a digitally programmable nonlinear resonator. Both simulations and experiments are carried out to compare optimized points for the resonator with programmed cubic nonlinearity with constant and time-dependent coefficients. We show that the frequency bandwidth of efficiency and the noise reduction are greater with varying nonlinearity than with constant nonlinearity.

## 1. Introduction

The propagation of waves through a medium is an issue tackled across many fields of physics [1] and engineering [2,3]. Acoustic waves over a wide range of frequencies and amplitudes may create dangerous situations in the short or long term. Efficient vibration control requires a broad frequency response, achievable by passive systems like acoustic foams [1,4] or active noise reduction devices [5–7] that produce waves in phase opposition. However, these systems are either poorly performing at low frequencies, or across the entire considered space. A hybrid solution consists of using electrical current to adapt the acoustic impedance of a surface using controlled electroactive loudspeakers [8–10]. This approach does not produce phase-opposition waves, and may be efficient at low frequencies [11]. Although nonlinearities offer benefits for reducing vibrations, impedance control methods have largely remained linear until recent developments [12,13]. However, in other fields such as mechanics, nonlinear phenomena have been extensively studied to broaden the frequency range of efficiency of passive systems [14]. A key advantage is the ability of a nonlinear resonator to tune its resonance frequency to align with a primary oscillator’s resonance frequency [15]. A further benefit is the targeted energy transfer from the primary oscillator to the auxiliary nonlinear oscillator once the nonlinear activation threshold is reached [16,17]. Furthermore, many types of nonlinear resonators aiming to mitigate vibrations have been studied such as vibroimpact resonators [18,19], the Duffing-type resonator [20–22] or piecewise oscillators [23–27]. Until recently, in acoustics, nonlinear absorption has only been achieved by high activation thresholds, limiting its application to extremely noisy environments [28–30]. Nevertheless, the active impedance control approach by electroacoustic resonators [31–34] has recently been extended to achieve nonlinear behaviors at low excitation levels [12,13,35,36]. Moreover, the control strategy proposed in [12,13,35] is also suitable for achieve parametric resonators, so far largely overlooked as a means to reduce noise and vibrations.

<footnote><sup>a</sup> ENTPE, Ecole Centrale de Lyon, CNRS, LTDS, UMR5513, 3 rue Maurice Audin, Vaulx-en-Velin, 69518, France</footnote>

<footnote><sup>b</sup> Ecole Centrale de Lyon, CNRS, ENTPE, LTDS, UMR5513, 36 av. Guy de Collongue, Ecully, 69130, France</footnote>

<footnote><sup>c</sup> CNRS, Ecole Centrale de Lyon, ENTPE, LTDS, UMR5513, 36 av. Guy de Collongue, Ecully, 69130, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: maxime.morell@entpe.fr (M. Morell).</footnote>
The combination of nonlinear and parametric characteristics could lead to effective absorption across a wide frequency range. However, its potential is yet to be demonstrated.

In this article, a digitally programmed cubic resonator with time-dependent coefficient to control a tube acoustic mode at low excitation levels is used. By combining numerical and experimental approaches, it is shown that nonlinear auto-parametric acoustic resonators are more efficient than classical Duffing oscillators in controlling a mode. To properly demonstrate this result, an optimization of both programmed cubic nonlinearities with constant and time-dependent coefficients is conducted through both simulations and experiments.

## 2. The control strategy

The impedance control strategy used in this paper is detailed in De Bono et al. [13,35], which is able to enforce the desired acoustic response of an electroacoustic resonator (ER). The flexibility of such control algorithm allows to implement nonlinear behaviors with time-varying parameters, without requiring high excitation levels. An ER is composed of a 5-cm wide loudspeaker collocated with a microphone, and linked to a controller: a dSPACE MicroLabBox DS1202 device working at 50 kHz. The microphone retrieves the pressure close to the membrane of the loudspeaker. Based upon real-time integration [13], the controller computes the electrical current to be sent into the loudspeaker's coil using the measured pressure, and a prediction of the displacement, velocity, and acceleration of the loudspeaker's membrane. At low excitation frequencies and amplitudes, the loudspeaker can be described as a single-degree-of-freedom system as in Eq. (1a). The control equations needed to synthesize the control algorithm are provided:

$$M_0\ddot{x}(t) + R_0\dot{x}(t) + K_0x(t) = S_d p(t) - Bli(t) \quad (1a)$$

$$i(t) = \frac{1}{B\dot{l}}(-M_0\ddot{x}_d(t) - R_0\dot{x}_d(t) - K_0x_d(t) + S_d p(t)) \quad (1b)$$

$$M_d\ddot{x}_d(t) + R_d\dot{x}_d(t) + K_d x_d(t) + F_{NL}(x_d, t) = Sp(t) \quad (1c)$$

where $x$ and $x_d$ stand for the displacement and desired displacement of the loudspeaker membrane, and $S_d$ denotes its effective area. The parameters $M_0$, $R_0$, $K_0$ represent the modal characteristics of the loudspeaker, when no electrical current is sent into the coil. The damping parameter $R_0$ includes all types of damping means. The parameters $M_d$, $R_d$, $K_d$ correspond to the desired modal parameters of the first mode of the loudspeaker, and $F_{NL}$ is a nonlinear stiffness force function, which can be chosen digitally in the programming of the control [13,35]. The variable $p$ denotes the measured pressure at the surface of the loudspeaker's membrane. $B\dot{l}$ is the Laplace force factor corresponding to the magnet's magnetic field $B$ and to the length of the solenoid $l$. The electrical current sent into the solenoid is represented by $i$ [Eq. (1b)] and is obtained by computing Eq. (1b) based on the estimation of the targeted state given by Eq. (1c). The controller employs a numerical scheme to integrate the differential equation of the desired behavior [Eq. (1c)] using the measured pressure, which yields the prediction of the displacement $x_d$, velocity $\dot{x}_d$ and acceleration $\ddot{x}_d$ of the loudspeaker's membrane. The Eq. (1c) is solved using a specific numerical scheme taking into account its impact on the experimental implementation complexity that could induce numerical latency. In this study, the chosen numerical scheme is a fourth order Runge-Kutta like algorithm. While the displacement $x_d$ is computed at each increment of the scheme $t$, $t+h/2$ and $t+h$ with $h$ the time step, the pressure is not measured at each increment, which would necessitate to dissociate the acquisition rate from the calculation rate. The pressure is considered constant during the time step, which is reasonable considering the high sampling rate of 50 kHz compared to the excitation frequencies (800 Hz maximum). The predictions obtained through the numerical resolution of Eq. (1c) are introduced in the control parameter equation [Eq. (1b)] to compute the electrical current $i$ at each time step [13]. This method permits to digitally program a linear or nonlinear behavior of a transducer at low excitation amplitudes, where the transducer is within its linear regime.

The presented control presents some limitations that are defined by the needed assumptions. The control strategy uses a model of the loudspeaker in order to make a feedforward controller. The model of the loudspeaker assumes that the ER response is dominated by its first mode, with low excitation frequencies and amplitudes. In a case where these assumptions are no longer fulfilled, the controller might lead to instabilities. As a result, the loudspeaker should remain in its linear domain, and should work at frequencies close to its linear first mode. Moreover, a loss in the acoustical passivity of the ER can be observed at high frequencies (more than 3 kHz) due to the time delay in the digital implementation of the controller [33].

## 3. The cubic nonlinearity with time-dependent coefficient

The desired behavior Eq. (1c) displays the function $F_{NL}$ containing the time-varying nonlinear stiffness. A general periodic form as a Fourier series for the nonlinear time-varying stiffness is assumed [37,38]:

$$F_{NL}(x_d, t) = K_d K(t) x_d^3(t) = K_d \left( \sum_{n=-\infty}^{n=\infty} \bar{K}_n e^{j\omega_0 nt} \right) x_d^3(t) \quad (2)$$

where $\omega$ is the excitation frequency, and $\bar{K}_n \in \mathbb{C}$. To consider only the first harmonic, a Galerkin procedure is applied to the nonlinear function. Supposing that $x_d$ is as follows:

$$x_d(t) = A_{-1} e^{-j\omega_0 t} + A_1 e^{j\omega_0 t} \quad (3)$$
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 1. (a)-(b) Plots of the nonlinear time-varying restoring force with respect to the time and to the displacement of the ER's membrane with $\tilde{K}_0 = 6 \times 10^9$ m⁻² and $\tilde{K}_{2R} = 4 \times 10^8$ m⁻² and $\omega = 2\pi \times 530$ rad.s⁻¹.</figcaption></figure>

The Galerkin procedure writes:

$$F_{NL,k} = \frac{\omega}{2\pi} \int_{0}^{\frac{2\pi}{\omega}} F_{NL}(t) e^{-j\omega t} dt = \frac{\omega}{2\pi} \int_{0}^{\frac{2\pi}{\omega}} K_d \left( \sum_{n=-\infty}^{n=+\infty} \tilde{K}_n e^{j\omega t} \right) (A_{-1} e^{-j\omega t} + A_1 e^{j\omega t})^3 e^{-j\omega t} dt \quad (4)$$

with $F_{NL,k}$ the kth Fourier harmonic of $F_{NL}$. In the case of $k=1$, it yields non-zero terms corresponding to the following terms of $K(t)$:

$$K(t) = \tilde{K}_4 e^{4j\omega t} + \tilde{K}_2 e^{2j\omega t} + \tilde{K}_{-2} e^{-2j\omega t} + \tilde{K}_0 \quad (5)$$

By developing the complex coefficients $\tilde{K}_n = \tilde{K}_{nR} + j\tilde{K}_{nI}$, it gives:

$$K(t) = \tilde{K}_0 + 2\tilde{K}_{2R} \cos(2\omega t) - 2\tilde{K}_{2I} \sin(2\omega t) + f(\tilde{K}_{4R}, \tilde{K}_{4I}, 4\omega t) \quad (6)$$

with $f$ a function of $\tilde{K}_{4R}$, $\tilde{K}_{4I}$, and $t$. To simplify the programming of the control, the coefficients $\tilde{K}_{2I}$, $\tilde{K}_{4I}$ and $\tilde{K}_{4R}$ are set to zero. Setting $\tilde{K}_{2I}$ to 0 imposes the phase of the time modulation. In the following, the phase of the excitation signal is chosen to ensure that the resonator stays acoustically passive. However, the choice of setting $\tilde{K}_{2I} = 0$ has been made to facilitate the experimental implementation of the control, and to allow the realization of an experimental optimization. The optimization is needed to be able to compare the cubic nonlinearity using constant coefficient with the cubic nonlinearity including time-dependent coefficient. The nonlinear varying stiffness is shown in Fig. 1(a) and its varying coefficient becomes:

$$K(t) = \tilde{K}_0 + 2\tilde{K}_{2R} \cos(2\omega t) \quad (7)$$

By setting $\tilde{K}_{2R} = 0$, the varying nonlinearity becomes the classic Duffing equation. As an example, $\tilde{K}_0$ and $\tilde{K}_{2R}$ are chosen in a manner that $F_{NL}$ is always positive with a positive displacement, for having a hardening behavior. The Fig. 1(b) pictures the restoring force in the $(x, F_{NL})$ plan of Fig. 1(a), with respect to the displacement $x$. It reveals that the time modulation of the nonlinearity contributes to create a hybrid behavior in which the nonlinearity exhibits softening behavior at specific times, while maintaining an overall hardening characteristic due to the constant term. However, it is essential to select carefully the pair of parameters $(\tilde{K}_0, \tilde{K}_{2R})$ to prevent numerical instabilities during programming. The condition $\forall t \in \mathbb{R}_+, K(t) > 0$ effectively excludes cases of negative stiffness. Considering the extreme possible values for the cosine function, the following condition is identified:

$$-\frac{\tilde{K}_0}{2} < \tilde{K}_{2R} < \frac{\tilde{K}_0}{2} \quad (8)$$

When the condition is fulfilled, the global restoring force is odd, and $\forall t \in \mathbb{R}_+, \forall x \in \mathbb{R}_+, K_d x_d(t) + F_{NL}(x, t) < 0$. The oddity of the function and the preceding condition ensures that the created force is in the opposite direction of the mass displacement [Fig. 1(b)].

# 4. Method

## 4.1. The experimental bench

An experimental bench that couples the nonlinear resonator to a cavity mode of a tube is used [28,29,35]. The experiment is composed of a tube with a reduced section area $S_t$ and a length $L_t$ connected at one end to an acoustic source (AS) and at the other end to a coupling box of length $L_{cb}$ and radius $r_{cb}$. The coupling box is terminated by the nonlinear resonator. The waveguide is pictured in Fig. 2(b). The experimental set-up includes a fixed microphone placed inside the tube with a reduced section for data analysis of the cavity acoustic mode of the tube. A second microphone is positioned in the coupling box to measure the pressure on the membrane of the ER. The experimental set-up is pictured in Fig. 2(a). Assuming that the programmed control works perfectly:
<figure><img src="image_6.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 2. (a) Experimental set-up. (b) Scheme of the tube experiment.</figcaption></figure>

<table><caption>Table 1
Dimensions and parameters of the experiment and simulation.</caption><thead><tr><th></th><th>Dimension</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td rowspan="3">Tube</td><td>r<sub>t</sub></td><td>1.45 &times; 10<sup>-2</sup></td><td>m</td></tr><tr><td>S<sub>t</sub></td><td>6.61 &times; 10<sup>-4</sup></td><td>m<sup>2</sup></td></tr><tr><td>L<sub>t</sub></td><td>0.204</td><td>m</td></tr><tr><td rowspan="3">Coupling box</td><td>r<sub>cb</sub></td><td>0.05</td><td>m</td></tr><tr><td>L<sub>cb</sub></td><td>0.125</td><td>m</td></tr><tr><td>&gamma;</td><td>125.54</td><td>kg s<sup>-2</sup></td></tr><tr><td rowspan="4">ER</td><td>S<sub>d</sub></td><td>1.3 &times; 10<sup>-3</sup></td><td>m<sup>2</sup></td></tr><tr><td>M<sub>d</sub></td><td>3.89 &times; 10<sup>-4</sup></td><td>kg</td></tr><tr><td>R<sub>d</sub></td><td>3.29 &times; 10<sup>-2</sup></td><td>kg s<sup>-1</sup></td></tr><tr><td>K<sub>d</sub></td><td>4.34 &times; 10<sup>3</sup></td><td>kg s<sup>-2</sup></td></tr><tr><td rowspan="3">Acoustic mode</td><td>m</td><td>8.22 &times; 10<sup>-5</sup></td><td>kg</td></tr><tr><td>c</td><td>3.96 &times; 10<sup>-2</sup></td><td>kg s<sup>-1</sup></td></tr><tr><td>k</td><td>1.38 &times; 10<sup>3</sup></td><td>kg s<sup>-2</sup></td></tr></tbody></table>

For $\forall t \in \mathbb{R}_+$, $x(t) = x_d(t)$, and with the assumptions of plane waves and linear acoustics without flow, the model of the experiment writes [28,35]:

$$ \left\{ \begin{array}{l} m\ddot{u}(t) + c\dot{u}(t) + ku(t) + \frac{\gamma}{\alpha}(u(t) - \alpha x(t)) = -S_t p_{as}(t) \\ M_d\ddot{x}(t) + R_d\dot{x}(t) + K_d x(t) + K_d K(t)x(t)^3 + \gamma(\alpha x(t) - u(t)) = 0 \end{array} \right. \qquad (9) $$

where the variables $u$ stand for the modal coordinates of the first acoustic mode of the tube. The parameters $m$, $c$, and $k$ are the modal parameters of the tube's acoustic first mode, and $\alpha$ is the ratio between the areas of the loudspeaker membrane and of the reduced section tube $\alpha = S_d/S_t$. The $\gamma$ parameter represents an equivalent stiffness of the coupling box, and $p_{as}$ the pressure induced by the acoustic source. The system is excited with 60 s long increasing and decreasing frequency sweeps between 350 Hz and 800 Hz. The frequency $\omega$ of the time-dependent coefficient of the cubic stiffness is chosen equal to the frequency of the excitation.

The model of the experiment is solved using a 4-th order Runge-Kutta numerical scheme in the different cases encountered in the following. The simulation parameters from the model and from the experiment are detailed in Table 1.

The modal parameters of the ER are chosen identical to the eigen modal parameters of the loudspeaker for the modal mass and stiffness such as:

$$ \left\{ \begin{array}{l} M_d = M_0 \\ K_d = K_0 \\ R_d = R_0/8 \end{array} \right. \qquad (10) $$

The modal damping is lowered to permit the existence of the bifurcation. Indeed, analytical results have provided a condition of the existence of the bifurcations on the damping of a nonlinear resonator coupled to a linear oscillator [39]. This condition states that there exists a critical value of damping and linear stiffness beyond which there are no bifurcations. The choice of $R_d = R_0/8$ ensures that the bifurcation exists in this system for the linear stiffness $K_d = K_0$.

The choice of keeping $K_d = K_0$ and $M_d = M_0$ is due to the tuning of the nonlinear resonance of the ER to the tube acoustic mode. In this case, the ER's linear resonance is slightly below the tube acoustic mode, which allows the programmed nonlinear hardening resonance to oscillate at the same frequencies as the tube acoustic resonance.
<figure><img src="image_8.png" /><figcaption>Fig. 3. Envelope of the pressure inside the reduced section tube with a rigid termination.</figcaption></figure>

### 4.1.1. The reference measure

To understand resonators' efficiency with the cubic nonlinearity both using a constant and time-varying coefficient, a reference measure is realized by introducing a rigid termination instead of the resonator. The replacement of the resonator with a rigid termination permits to identify the modal parameters of the acoustic mode of the tube with a single measure by curvefitting the calculated response to the envelope of the time signal obtained using a frequency sine sweep. Fig. 3 shows the envelope of the pressure time signal. The envelope is obtained by computing the modulus of the Hilbert transform of the time signal. The obtained parameters are detailed in Table 1. The method for identifying the parameters is extensively described in [39].

### 4.2. The metric for realizing an optimization

For the purpose of comparing the classical cubic nonlinearity with the cubic nonlinearity with a time-varying coefficient, a metric is set:

$$ \mathbb{A}_k^l = \frac{\int_{t_1}^{t_2} |H(p_k^l)(t)| dt}{\int_{t_1}^{t_2} |H(p_{ref})(t)| dt} = \frac{A_k^l}{A_{ref}} \qquad (11) $$

where $H$ stands for the Hilbert transform of the pressure $p$ inside the reduced section tube. The bounds $t_1$ and $t_2$ represent the instants corresponding to the frequencies 500 Hz and 650 Hz during the frequency sweeps. The indices $k = \{\text{num, exp}\}$ denote numeric and experiment, and $l = \{c, v\}$ constant and varying. The variable $p_{ref}$ corresponds to the pressure inside the tube when the ER is replaced with a rigid termination. This metric is the ratio between the areas of the time signal envelops of a considered measure $A_k^l$ and the reference measure $A_{ref}$ [Fig 3] done with a rigid termination. The rigid termination measure and its surface area $A_{ref}$ is described in Fig 3. Lower metric values correspond to better absorption cases.

## 5. Results

### 5.1. The cubic nonlinearity with a constant coefficient

Fig. 4(a) displays the experimental pressure within the reduced section tube and the predicted desired displacement $x_d$ of the loudspeaker's membrane for a cubic nonlinearity with a constant coefficient. A nonlinear hardening behavior is observed in the displacement of the ER. At the frequencies of the nonlinear resonance, which coincide within the frequencies of the bifurcations A and B, a significant decrease in the pressure appears. Indeed, the nonlinear ER oscillates with higher amplitudes resulting in better energy pumping. Broadening the frequency range of the nonlinear resonance, along with raising its amplitude is of great interest to improve the absorption capabilities of the nonlinear ER. Simulations are realized with the parameters of Table 1 and are plotted in Figs. 4(b) and 4(c). Figures 4(b) and 4(c) compare the numerical simulations with the experimental results. The pressure inside the reduced section tube is plotted in Fig. 4(b) for an increasing frequency sweep and in Fig 4(c) for a decreasing frequency sweep. One can see a good agreement between experimental data and simulations, despite minor shifts in the frequencies of the bifurcations. This inaccuracy of the model is attributed to parameter uncertainties [35].
<figure><img src="image_9.png" /><figcaption>Fig. 4. (a) Envelopes of the pressure inside the reduced section tube and computed displacement of the ER's membrane for increasing and decreasing frequency sweeps with $\tilde{K}_0 = 6 \times 10^9 \text{ m}^{-2}$ and $\tilde{K}_{2R} = 0 \text{ m}^{-2}$ (nonlinearity with a constant coefficient). (b)–(c) Comparison of the envelopes of the pressure inside the reduced section tube for experiment and simulation in the case of increasing (b) and decreasing (c) frequency sweeps in the case of a constant coefficient.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 5. (a) Value of the optimization metric with respect to the optimization parameter for both simulations and experiments in the case of the cubic nonlinearity with a constant coefficient. (b)–(c) Envelopes of the pressure inside the reduced section tube for the numerical optimal point (b) and experimental optimal point (c) in the case of the cubic nonlinearity with a constant coefficient.</figcaption></figure>

It is worth noticing that the main difference between the ER including a constant coefficient cubic nonlinearity and a Nonlinear Energy Sink (NES) is that a NES is purely nonlinear. The considered system looks like a grounded NES. However, classical NES systems are pure nonlinear without any linear part, e.g. pure cubic nonlinearity. In our considered system, a linear part exists due to the linear mode of the ER [35], looking like more Roberson-type oscillator [22] than a NES. As the control is feedforward, the linear mode of the ER cannot be deleted and it is not possible to create a purely nonlinear system [40]. However, the use of a feedback controller may allow the suppression of the linear mode, which could be eventually coupled to the presented feedforward controller [40]. Another main dissimilarity with the NES lies in the non-reciprocity of the coupling between the ER and the acoustic mode of the tube seen in Eqs. (9). Nonetheless, the acoustic formulation realized in [35] demonstrates that the coupling can be considered reciprocal. Finally, theoretical studies carried out by authors [37,39] show that if the system faces fold singularities or if its equilibrium point is placed in the unstable zone of the slow invariant manifold, then it will present quasi-periodic responses. This response has been experimentally spotted by the authors as well where they looked in detail at time responses of the system for fixed frequencies of the excitation [41]. However, the authors have demonstrated that the particular system studied here rarely induces quasi-periodic oscillations for a cubic nonlinearity with a constant coefficient [39].

The previously presented metric is considered to assess the efficiency of a given nonlinear ER. The optimized values of $\tilde{K}_0$ and $\tilde{K}_{2R}$ are searched for the case of cubic nonlinearities with both constant coefficient and time-dependent coefficients. The case of increasing frequency sweeps of the AS is arbitrarily chosen for the optimization, which is the most favorable case for a hardening behavior. Indeed, the ER equilibrium point depends on the initial conditions.

Regarding the cubic nonlinearity with constant coefficient, the parameter $\tilde{K}_{2R}$ is set to 0, and simulations and experiments are realized with $\tilde{K}_0 \in [10^5, 10^{12}] \text{ m}^{-2}$ to find a global optimum. As such experimental campaign is spread over multiple weeks, the experimental results might present slight discrepancies due to changes in temperature, humidity, or other physical parameter that can change the modal parameters of the tube acoustic mode. In this context, multiple measures with the same parameters spread over multiple days have been realized. The shaded gray area in Fig. 5(a) contains the experimental results and is bounded by the maximum and minimum results obtained through the experimental campaign.

In Fig. 5(a), the metric A is represented with respect to the optimization parameter $\tilde{K}_0$ for both experimental and numerical results. A numerical optimal point $A_{num}^c = 0.907$ is observed for the parameter $\tilde{K}_0 = 9.88 \times 10^9 \text{ m}^{-2}$ which is dissociated from the lowest experimental optimal point $A_{exp}^c = 0.856$ at $\tilde{K}_0 = 7 \times 10^9 \text{ m}^{-2}$. This difference can be attributed to the previously noted discrepancies between the experimental results and model simulations. Moreover, the numerical metric values are higher than for the experiment. Fig. 5(b) represents the pressure envelope for the simulations optimal point. The pressure level is higher than the experimental optimal point presented in Fig. 5(c), confirming that the simulation metric values are higher than the experimental
<figure><img src="image_11.png" /><figcaption>Fig. 6. (a)–(b) Value of the optimization metric with respect to the optimization parameter for both simulations (a) with its zoom (b) and experiments (c) in the case of the cubic nonlinearity with a time-varying coefficient. (d) Envelopes of the pressure inside the reduced section tube for the experimental optimal point in the case of the cubic nonlinearity with a time-varying coefficient.</figcaption></figure>

ones. However, the model reveals the existence of a global optimum and provides insight into the parameter regions where the experimental optimum is likely to be found.

## 5.2. The cubic nonlinearity with a time-dependent coefficient

A similar process is applied to the cubic nonlinearity with time-varying coefficient with $\tilde{K}_0 \in [10^5, 10^{12}]$ m$^{-2}$ and $\tilde{K}_{2R} \in [0, 10^9]$ m$^{-2}$. A search for optima with the simulations is done through random shooting in the optimization parameters $\tilde{K}_0$ and $\tilde{K}_{2R}$ [Fig. 6(a)]. A hyperplane is then extrapolated from the simulation points to map the metric values in the plan $(\tilde{K}_0, \tilde{K}_{2R})$. This hyperplane is the surface fitting the simulation points. The plan is not computed in the zones where the model poorly fits the experiment, even if the stability condition is fulfilled [Eq. (8)]. This first sampling permits to identify the zone where the optimum might be, which is situated in the red rectangle. A new simulation study with random shooting in the red rectangle is done to improve the precision of the numerical optimization [Fig. 6(b)]. The lower metric point $A_{num}^v = 0.805$ is obtained for $\tilde{K}_0 = 7.29 \times 10^9$ m$^{-2}$ and $\tilde{K}_{2R} = 5.14 \times 10^8$ m$^{-2}$. As the experimental map of the metric value is realized using a random shooting, one cannot ensure that this point is the optimum. Moreover, the zone where the model poorly fits the experiment has not been investigated, which does not ensure that the point is the optimal value.

An experimental optimization is realized in the previously defined reduced zone investigated through the numerical simulations [Fig. 6(c)]. The measurements were mainly conducted in the zones identified with the better metric values during the numerical simulations. One can see that the optimization plan extrapolated from the measures is similar to the numerical optimization plan. The represented plane is the surface fitting the experimental points. However, the metric values are lower for the experimental results than for the numerical one, in the same manner as the cubic nonlinearity with constant coefficient. This phenomenon can be explained by the simulation assumption that the control works perfectly, which leads to neglect the remaining influence of the linear mode of the loudspeaker. The lower metric points are situated close to the defined frontier in both numerical and experimental optimizations. The lower metric point $A_{exp}^v = 0.733$ is obtained for $\tilde{K}_0 = 5 \times 10^9$ m$^{-2}$ and $\tilde{K}_{2R} = 3.5 \times 10^8$ m$^{-2}$. Fig. 6(d) presents the measurement, highlighting a substantial reduction in pressure levels within the reduced section tube over a broad frequency range.

The comparison between cubic nonlinearities with constant and time-varying coefficients reveals that the classical cubic nonlinearity optimal point has a higher metric value than many of the non-optimal points of the cubic nonlinearity time-varying coefficient. As a result, the cubic nonlinearity with a time-dependent coefficient is generally more efficient than the classical duffing nonlinearity. The optimization of the other parameters, and the higher harmonics of the periodic stiffness may also improve the performances of the ER. Moreover, the zone closer to the stability condition, where the model does not fit the experimental results, may reveal lower metric values. The phenomena involved in this zone have to be investigated.

As previously described, if the nonlinear resonance can be broadened while increasing the amplitudes of the oscillations, the acoustic absorption capabilities should be improved. In Fig. 7(a), the experimental results of the cubic nonlinearity with a constant coefficient and with a time-dependent coefficient are compared for increasing frequency sweeps. The time-dependent coefficient case provides a broadened resonance with higher amplitude oscillations seen in the displacement curves, leading to better absorption capabilities for a larger frequency bandwidth, which can be seen in the pressure curves. However, one may notice that the maximal pressure is unchanged, while the minimum pressure is improved for both signals. This phenomenon is due to the linear part of the stiffness of the ER. The ER linear resonance facilitates the apparition of the nonlinear resonance at low excitation amplitudes, which builds upon the ER linear resonance. This linear resonance also limitates the absorption capabilities of the ER at its resonance frequency.

## 6. Conclusion

To summarize, the cubic nonlinearity with a time-varying coefficient has demonstrated greater potential than the classical cubic nonlinearity in mitigating noise and vibrations. The control method to program a cubic nonlinearity with a time-varying coefficient
<figure><img src="image_12.png" /><figcaption>**Fig. 7.** Comparison of the experimental envelopes of the pressure inside the reduced section tube and computed displacement of the ER's membrane for increasing frequency sweeps between the cubic nonlinearity with a constant coefficient $\bar{K}_0 = 6 \times 10^9 \text{ m}^{-2}$, $\bar{K}_{2R} = 0 \text{ m}^{-2}$ and the cubic nonlinearity with a time-dependent coefficient $\bar{K}_0 = 5 \times 10^9 \text{ m}^{-2}$, $\bar{K}_{2R} = 3.5 \times 10^8 \text{ m}^{-2}$.</figcaption></figure>

within a loudspeaker is presented and adapted from the control method to create a constant coefficient. The programmed resonator with a time-dependent nonlinear coefficient efficiently controlled the acoustic cavity mode of a tube. The obtained results indicate that a large set of parameters {$\bar{K}_0, \bar{K}_{2R}$} provides more efficient varying nonlinearity than the optimized cubic nonlinearity with a constant coefficient. However, further research is needed to investigate the unknown zone, and the phenomena at stake. Moreover, many parameters have not been optimized such as the stiffness of the sine part of the periodic stiffness $\bar{K}_{2l}$ or its modulating frequency $\omega$. The optimization of a large number of parameters in complex nonlinear problems on a large frequency bandwidth would require advanced optimization techniques, and should be investigated in future studies. Such nonlinearity has a significant potential by its ability to mitigate vibrations on a large frequency bandwidth thanks to its nonlinear parametric nature. Furthermore, the use of this nonlinearity is not limited to acoustics, and should be investigated in other fields.

**CRediT authorship contribution statement**

**Maxime Morell:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Aurélie Labetoulle:** Writing – review & editing, Validation, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Emmanuel Gourdon:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition, Conceptualization. **Emanuele De Bono:** Writing – review & editing, Supervision, Software, Conceptualization. **Manuel Collet:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition. **Alireza Ture Savadkoohi:** Writing – review & editing, Validation, Supervision, Resources, Project administration, Funding acquisition, Conceptualization.

**Declaration of competing 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.

**Acknowledgments**

The authors thank Abdelhakim Ezzerouki for providing helpful assistance in the electronic realization of the experiment. The authors would like to gratefully thank the following organizations for supporting this research: (i) The “Ministère de la transition écologique” and (ii) LABEX CELYA (ANR-10-LABX-0060) of the “Université de Lyon” within the program “Investissement d'Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

**Data availability**

No data was used for the research described in the article.
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