Issue
Mechanics & Industry
Volume 26, 2025
Recent advances in vibrations, noise, and their use for machine monitoring
Article Number 2
Number of page(s) 18
DOI https://doi.org/10.1051/meca/2024036
Published online 17 January 2025

© M. Morell et al., Published by EDP Sciences, 2025

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

The control of sound and vibrations has been studied for years in various domains, such as aeronautics [1]. In the field of acoustics, multiple kinds of devices have been considered, from acoustic foams to active noise-cancellation devices. Passive absorbing materials are the most commonly used way to reduce sound levels as these materials are very efficient for frequencies greater than 1000 Hz [2], while they struggle at lower frequencies [3]. The use of active devices consists of creating a wave in phase opposition to the incident wave, resulting in the cancellation of wave energy [4,5]. Nevertheless, this concept requires large amounts of energy, as the created wave amplitude should be equal to the incident wave one [69]. Moreover, in more than 1 dimension, the energy canceling due to the wave superposition is efficient only in local spatial zones. Passive resonators do not use additional energy to be efficient but are specifically designed to reduce sound pressure at their resonance frequency. For low frequencies, their design can be massive, and they can not be adjusted to the frequency of incident waves such as the Helmholtz resonator. To create resonators that can be tuned, the use of loudspeakers controlled by electrical current has been found as being a solution by Olson and May [10]. This method named Impedance Control (IC) consists of adapting the apparent acoustic impedance of the loudspeaker thanks to microphones [1113]. Indeed, an electrical current is sent to the loudspeaker coil based on the measured pressure and on calculations done with a processor to change the loudspeaker parameters [14,15]. In the acoustic field, these resonators remained in their linear domain, and the advantages of nonlinearities [16] have not been fully employed yet due to the high activation threshold of the nonlinearity, unlike the field of mechanics [17]. Nevertheless, Bellet [18] excited a viscoelastic membrane at very high amplitudes (above 148 dB Sound Pressure Level (SPL)) to control a tube acoustic mode. Additionally, Gourdon [19] and Alamo Vargas [20] activated the nonlinear regimes of a Helmholtz resonator above 138 dB SPL. Guo [21] carried out the creation of a polynomial nonlinear electro-acoustic resonator at low excitation amplitudes (close to 95 dB) using an additional microphone placed in the back cavity of the loudspeaker. The additional microphone permits to add a polynomial nonlinear current to a frequency-based linear current. In this study, a nonlinear electroacoustic resonator is created at low excitation amplitudes using the real-time-based method presented and validated in [22], which does not need additional microphones. This method can create polynomial and non-polynomial behaviors. This research features the programming of the control that generates a cubic restoring force function to create a duffing resonator. The nonlinear electroacoustic resonator is coupled to an acoustic mode of a tube. In Section 2, the system under consideration is presented and the programming of the control strategy is briefly introduced. The analytical model presented in Section 2 is studied thanks to analytical methods in Section 3. The limits of the hypothesis of such methods are highlighted. Experimental results are presented in Section 4, with comparisons to the analytical results. The concordance between measured quantities and predictions by the analytical modelling is highlighted.

2 System under consideration

2.1 The nonlinear electroacoustic resonator

An electroacoustic resonator is composed of an actuator (a loudspeaker) collocated to sensors (one or many microphones) and equipped with a processor running an algorithm that takes the measured pressure as an input and gives the electrical current to send into the loudspeaker coil as an output. Nonlinear responses in resonators are commonly initiated at high amplitudes of motion. This research suggests employing the method presented and detailed by De Bono et al. [22] to create a nonlinear electroacoustic resonator (ER) behaving as a duffing oscillator within the displacement and pressure amplitude ranges corresponding to the linear regime of the loudspeaker. It consists of adding a programmed additional force to the passive resonator in order to drive its behavior from its inherent behavior to a desired behavior. This means that the resonator can be digitally programmed.

The experimental application of the algorithm assumes that the operational frequencies are close to the frequencies of the loudspeaker’s first mode. Additionally, it presupposes that the operational amplitudes remain within the linear regime range of the loudspeaker. Given these assumptions, along with the consideration that the frequencies and amplitudes are sufficiently low to treat the membrane as rigid, the loudspeaker can be approximated using the classical mass-spring-damper model described by the following equation (1): M0u¨(t)+R0u˙(t)+K0u(t)=p(t)SdBli(t)(1)

where M0 , R0 , and K0 represent the modal mass, damping and stiffness parameters of the loudspeaker first mode when the electrical current i is set to zero. Bl is the force factor of the solenoid of the loudspeaker, with B the magnetic field produced by the permanent magnet, and l the length of the solenoid. As a result, Bli describes the Laplace force through which the control is applied. u stands for the relative displacement of the membrane regarding its resting position, and ˙ indicates the time derivative of the considered variable. p denotes the pressure of the incident wave applied on the membrane of the loudspeaker, and Sd the effective area of the membrane. Equation (1) describes the inherent behavior of the loudspeaker.

The goal of the electrical current is to introduce the Laplace force in order to modify the intrinsic response of the loudspeaker into a desired response specified in equation (2): Mtu¨t(t)+Rtu˙t(t)+Ktut(t)+F(t,ut(t),u˙t(t),u¨t(t))=p(t)Sd(2)

where Mt , Rt , and Kt represent the targeted eigen parameters of the loudspeaker first mode. F is a nonlinear function, such as a mass, damper, or stiffness force. These parameters can be chosen in the programming of the control. Indeed, the programming of the control is the stage anterior to the experiment in which the modal parameters and the nonlinearities are chosen and programmed. The variable ut stands for the targeted displacement and is induced by the measured pressure p and by the choice of these parameters through the numerical resolution of equation (2) at each time step.

Once the numerical resolution of equation (2) yields the targeted displacement ut , velocity u˙t , and acceleration u¨t , the driving parameter of the Laplace force i can be calculated utilizing equation (3), assuming u = ut: i(t)=SdBl(p(t)(M0Sdu¨t(t)+R0Sdu˙t(t)+K0Sdut(t))).(3)

The variable implementing the control is the electrical current i and is calculated at each time step. Upon closer examination of this variable and substituting i from equation (3) into equation (1), it yields: M0(u¨u¨t)+R0(u˙u˙t)+K0(uut)=0.(4)

Equation (4) highlights that the force induced with i is applied to enforce the behavior carried out in the calculations of ut, u˙t and u¨t . The objective of the algorithm is to drive the left-hand side of equation (4) to zero. When this occurs, the targeted behavior is attained. It can be seen that the Laplace force introduces an additional force applying on the loudspeaker membrane to modify its displacement to become the targeted displacement, which is a duffing behavior displacement when FNL(ut)=±KtβNLut3(t).

The method of measurement of the internal parameters of the loudspeaker is presented in Appendix A. The values are presented in Table 1:

Table 1

Dimensions and parameters of the experiment and simulation.

2.2 The experiment

2.2.1 The model of the experiment

The objective is to control an acoustic mode of a tube with a nonlinear resonator, ensuring that the amplitudes are sufficiently low for the air propagation to be linear and the loudspeaker to operate within its linear regime.

The presented experiment is built upon the framework introduced by Bellet [18] and Gourdon [19]. The model for this experiment has been thoroughly explored in earlier works [18,19,23], and its detailed development is not done in this study.

The experiment is composed of a circular tube of length Lt and radius rt . At one of its ends, an external loudspeaker is placed as an acoustic source (AS) for excitation of the system, modeled by an incident pressure pls . To be able to model the coupling between the ER and the acoustic mode of the tube, a circular coupling box of dimensions specified by length Lcb and radius rcb is positioned at the other end of the tube. A coupling box consists of connecting a wider tube to the considered tube, permitting the establishment of a relation between the pressure and the volume variations caused by the displacement of the air particles of the reduced section tube and the displacement of the membrane of the ER. Let us set the axis passing through the center of the cross sections of the tube associated with the variable x ∈ [0, Lt]. The dimensions of the experiment are presented in Table 2. A schematic representation of the experiment is depicted in Figure 1.

The experiment has been designed in order to be modeled using a two-degree-of-freedom system. Considering that the variables of time t and space x of the solution can be decoupled, the governing equations of the system are: { mau¨a(t)+cau˙a(t)+kaua(t)+γα(ua(t)αum(t))=Stpls(t)Mtu¨m(t)+Rtu˙m(t)+Ktum(t)+FNL(um(t))+γ(αum(t)ua(t))=0 (5)

where α is defined as the ratio of the sections of the tube and of the coupling box α = Sd/St . The coefficients ma , ca and ka stand for the modal parameters of the first acoustic mode of the tube, linked to ua the modal coordinates of the first acoustic mode of the tube. These coefficients are obtained from the variational formulation of the one-dimensional acoustic propagation equation, and from employing a Rayleigh-Ritz method: uair(x,t)=ua(t)Φ1(x)=ua(t)(cosπxLt)(6)

where Φ1 represents the shape function (which is also the shape mode in this particular case). The parameter γ stands for the stiffness of the spring coupling the acoustic mode of the tube and the ER: γ=kbStSd(7)

with kb=ρ0c02Vcb,0. The parameters ρ0 and c0 denote the density of air and sound speed in air under standard conditions, while Vcb,0 stands for the volume of the coupling box cavity at its resting state. One can notice that the coupling between the two equations of system (5) representing the motion of the acoustic first mode of the tube and the motion of the ER is non-reciprocal. This nonreciprocity is due to the distinct areas of the tube and the ER. The parameters in the system of equations (5) are either directly measured or estimated using the model. The estimated parameters are those that do not require experimental measurements, they are calculated based on the dimensions of the experiment using the expressions: { ma=ρ0c0Lt2kb=ρ0c02Vcb,0 .(8)

Their values are presented in Table 3. The measured parameters are directly estimated using experimental measurements. The method of measurement of the modal parameters of the acoustic first mode of the tube is presented in Appendix B. These parameters can rely on calculated parameters as the modal mass ma and are presented in Table 4.

The coupled equations (5) allow an analytical study to be led using a perturbation approach.

Table 2

Values of the dimension of the experiment.

thumbnail Fig. 1

Scheme of the experimental set-up.

Table 3

Parameters obtained by calculations.

Table 4

Parameters obtained by measurements.

2.2.2 The experimental setup

The experimental setup is pictured in Figure 2. The dimensions and parameters have been presented in Tables 1, 2, 3 and 4. The digital programming of the control of the ER is done using a d-Space MicroLabBox DS1202, which is also used as the acquisition system working at 50 kHz. The electrical current is not amplified between the controlling device and the ER as the calculated electrical current should be equal to the electrical current to be injected into the loudspeaker coil. The ER is power-supplied using a 10 V amplitude tension. The excitation is provided by an external loudspeaker linked to the D-Space device through an amplifier. Reference measures for all considered excitation signals are made using a rigid termination instead of the ER.

thumbnail Fig. 2

Experimental set-up.

3 Method and analytical study

The analytical method chosen to study such a system is the Multiple Scale Method (MSM), coupled to the complex variables of Manevitch [24]. This method has been extensively employed for the study of nonlinear dynamical systems [25-30].

3.1 The standard form system

The system under consideration is governed by system of equations (5). This is a two-degree of freedom system. Let us consider the system rearranged as in [27]: { ua+εξaua+ua+εγ0(uaαum)=εfsin(ντ)ε(um+ξmum+KLum+KNLum3+γ0μ(αumua))=0 (9)

where ε << 1 and it has been set that =τ is a time differential operator regarding to the time τ defined as: τ=ω0t=kamat(10)

and with ν the normalized pulsation of the excitation pulsation Ω defined as depending on σ the detuning parameter, meaning that the solution is studied around the resonance of the system: ν=Ωω0=1+σε.(11)

The MSM approach is introduced here by the decomposition of the time τ in new time scales as: Tk=εkτ;k[[0;N]].(12)

Let us introduce the complex variables of Manevitch (φa , φm ) ∈ C2 and are functions of new time scales Tk ∈ ℝ [24], with i2 = −1: { ua+iνua=φa(T0,T1,T2,)eiντum+iνum=φm(T0,T1,T2,)eiντ. (13)

Additionally, a Galerkin method is used to reduce the order of the problem to the first harmonic of the solution, assuming that the solution may be decomposed as a Fourier series, i.e. a sum of mono-harmonic waves. Let us consider a general function ɀ : ℂn → ℂn , with n a natural finite number, and depending of a vector w: z(w)=kZk(w)eikνt(14)

where: Zk(w)=ν2π0ν2πz(w)eikνtdt.(15)

Naturally, the Galerkin procedure to truncate the function ɀ to its first harmonic consists of applying the scalar product of ℒ2(ℂ), which is the following function G : ℂn → ℂn to project onto the first harmonic: G(z(w))=ν2π02πνz(w)eiνtdt.(16)

By introducing the complex variables of Manevitch, and applying the Galerkin procedure supposing that φa, φm and their complex conjugate (denoted by •*) φa*,φm* are independent of T0 = τ (this hypothesis is verified after the fact), the system of equations (9) reads: { dφa dτ+iνφa2+εξaφa2+φa2iν+εγ02iν(φaαφm)=εf2iε(  dφm dτ+iνφm2+ξmφm2+KLφm2iνKNL(3i8ν3φm2φm*) +γ0μ2iν(αφmφa) )=0. (17)

Equation (12) yields the definition of the time derivative regarding τ: d dτ=k=0NεkTk.(18)

System of equations (17) yields equations at each order of ε. In our study, the equations corresponding to the different orders O(εk) with k ∈ 〚0 ; N〛 are considered. The orders correspond to the time scale considered and as a result to the fast and slow system dynamics.

3.2 Fast system dynamics

The system at O(ε0) is written with the consideration that ν = 1 + εσ: O(ε0):{ φaT0+iφa2+φa2i=0φmT0+iφm2+ξmφm2+KLφm2iKNL(3i8φm2φm*)+γ0μ2i(αφmφa)=0. (19)

The first equation of the system yields that: φaT0=0(20)

which confirms the previously assumed hypothesis “φa independent of T0”. The second equation of the system can be studied in the case of an asymptotic state, when T0 → ∞. It gives that: φmT00.(21)

As a result, the system of equations reads: iφm2(1KL34KNL| φm |2αγ0μiξm)+iγ0μφa2=0(22)

We define the map ℋ as it follows: (φa,φa*,φm,φm*)=iφm2( 1KL34KNL| φm |2 αγ0μiξm )+iγ0μφa2.(23)

The manifold (φa,φa*,φm,φm*)=0 is called the slow invariant manifold (SIM). This manifold is defined as the fixed points of the dynamical system thanks to equation (21). As a result, it is the topological space representing the asymptotic solutions of the system.

The SIM can be expressed as: φa=φmγ0μ(1KLαγ0μ34KNL| φm |2iξm).(24)

The complex variables of Manevitch can be expressed in the polar domain: { φa=Naeiδaφm=Naeiδm. (25)

Replacing the polar domain form of the complex variables of Manevitch in equation (24), and calculating the modulus of the SIM using the real and imaginary parts of the equation reads: Na=Nmγ0μ(1KLαγ0μ34KNLNm2)2+ξm2.(26)

This equation gives the magnitude of the SIM. In this particular case, the SIM is not a function of phases δa and δm .

3.3 Unstable zones of the SIM

Let us consider the second equation of system (19) and its complex conjugate: φmT0+iφm2(1KL34KNL| φm |2αγ0μiξm)+iγ0μφa2=0.(27)

Knowing that φa is independent of T0 , one can search the boundary of unstable zones only perturbing φm: { φmφm+Δφmφm*φm*+Δφm*. (28)

It gives the following system which can be presented under a matrix form in equation (29), as we decide to neglect the terms of order O(Δφm2) and superior orders: (ΔφmT0*ΔφmT0)=i2M(ΔφmΔφm*)(29)

where the matrix M ∈ 𝕄2,2 is set as: M=(1KLαγ0μ32KNL| φm |2iξm34KNLφm234KNLφm*2(1KLαγ0μ32KNL| φm |2iξm)).(30)

Let us denote the terms of the matrix Mkl with the indices (k,l) Є {1,2}, where k is the indice of lines and l of columns. The characteristic polynomial χ of the matrix can be calculated in order to get the eigenvalues: χ(λ)=(M11λ)(M22λ)M12M21(31)

which yields: χ(λ)=λ2+bλ+c(32)

with b = − tr(M) = −(M11 + M22) and c = det(M) = M11M22M12M21.

We notice that (b, c) Є ℝ2, and that b = ξm. Being a quadratic equation, up to two eigenvalues of the matrix can be obtained, named λ1 and λ2: { λ1=bb24c2λ2=b+b24c2. (33)

The unstable and stable manifolds are determined by the sign of the real part of the eigenvalues. Multiple cases arise, depending on the value of the discriminant of the quadratic equation.

If b2 − 4c < 0

In this case, (λ1, λ2) Є ℂ2: { λ1=bi4cb22λ2=b+i4cb22. (34)

It gives that (λ1)=(λ2)=b2=ξm2. As the aim of the study is not to inject energy into the system, the damping ξm of the ER is taken positive. As a result, the real part of the eigenvalue is negative. This case defines a manifold where the fixed points are stable.

If b2 − 4c = 0

In this case, (λ1, λ2) Є ℝ2 as: λ1=λ2=b2=ξm2.(35)

This case defines a manifold where the fixed points are stable.

If b2 − 4c > 0

In this case, (λ1, λ2) Є ℝ2: { λ1=bb24c2λ2=b+b24c2. (36)

It gives the condition λ1 > 0 ≡ c< 0. If c< 0, it defines the unstable manifold, and c > 0 defines the stable manifold. The condition c = 0 gives the centre manifold and as a result the boundary between the stable and unstable manifolds.

So we can solve the equation c < 0 to determine the unstable manifold: 14(ξm2+(1KLαγ0μ32KNL| φm |2)2)(38KNL)2| φm |4<0(37)

which can be organized as: (2716KNL2)X23KNL(1KLαγ0μ)X+(ξm2+(1KLαγ0μ)2)<0(38)

where X=| φm |2=Nm2. It is a quadratic equation, which solutions X1 and X2 are: { X1=(1KLαγ0μ)+12(1KLαγ0μ)23ξm298KNLX2=(1KLαγ0μ)12(1KLαγ0μ)23ξm298KNL. (39)

Each solution Xk, k = {1,2} gives two solution for the equation Xk=Nm2. However, only the positive solutions are chosen, as Nm ∈ ℝ+ . We can observe that in order to get an unstable manifold, the condition b2 − 4c > 0 is needed, which means that X1 and X2 would be two dissociated real solutions. It gives the following condition on the parameters KL and ξm: (1KLαγ0μ)23ξm2>0.(40)

This condition in the literature is well known and usually defines a critical value of the damping ξ2,c value. In our study, a critical value of the linear stiffness KL,c can be defined as well. However, the possible range of the value of the linear stiffness is limited by the hypothesis of the loudspeaker being solicited on its linear first mode.

3.4 Slow system dynamics

The slow time scale corresponds to the equations at O(ε). The first equation of (17) at order O(ε) gives: φaT1+iφa2(2σγ0iξa)+iαγ02φm=if2.(41)

We can set a manifold defined by ϵ(φa,φa*,φm,φm*)=0, with: ϵ(φa,φa*,φm,φm*)=iφa2(2σγ0iξa)+iαγ02φmif2.(42)

Additionally, the evolution of the SIM at time scale T1 can be calculated using the chain rule: { d dT1=φaφaT1+φmφmT1+φa*φa*T1+φm*φm*T1 d* dT1=*φaφaT1+*φmφmT1+*φa*φa*T1+*φm*φm*T1 (43)

which can be rearranged as: (φmφm**φm*φm*)(φmT1φm*T1)=(φaφa**φa*φa*)(φaT1φa*T1)(44)

Let A stand for the jacobian matrix of (ℌ, ℌ*) regarding the variables φm and φm*: A=(φmφm**φm*φm*)(45)

The method defines the equilibrium and singular points as being the intersection of the manifolds defined by є = 0 and = 0, combined with a condition about the invertibility of A. The non-invertibility of A gives the extremums of the SIM regarding φm and φm* in the time scale T1 . As a result, the singular points are defined as: { ϵ(φa,φm,φa*,φm*)=0(φa,φm,φa*,φm*)=0det(A)=0 (46)

And equilibrium points are defined as: { ϵ(φa,φm,φa*,φm*)=0(φa,φm,φa*,φm*)=0det(A)0 (47)

3.4.1 Singular points

The matrix A is: A=(i2(1KLαγ0μ32KNL| φm |2iξm)3i8KNLφm23i8KNLφm*2i2(1KLαγ0μ32KNL| φm |2+iξm).)(48)

And its determinant is: det(A)=14((1KLαγ0μ32KNL| φm |2)2+ξm2)964KNL2| φm |4.(49)

Let us solve the equation det(A) = 0: 14((1KLαγ0μ32KNL| φm |2)2+ξm2)964KNL2| φm |4=0.(50)

Equation (50) and equation (37) are the same relation, meaning that the singular points coincide with the stability borders of the SIM if the points are on the manifolds defined equation (46). In this case, the singularities define the stability of the manifold. The solutions of the quadratic equation are given in equations (39).

3.4.2 Equilibrium points

The equilibrium points are defined by the intersection of the manifolds defined by є = 0 and = 0. Let us solve the system defined by equations (47): { iφa2(2σγ0iξa)+iαγ02φmif2=0iφm2(1KL34KNL| φm |2αγ0μiξm)+iγ0μφa2=0det(A)0. (51)

As previously done in equation (24), the second equation of system (51) can be organized to express φa as a function of φm . This equation can be replaced in the first equation of (51). The resulting equation is a polynomial equation regarding φm . We express the variable in polar form φm=Nmeiδm , and we take the modulus of the equation. It becomes a polynomial expression of the third degree: aX3+bX2+cX+d=0(52)

with: { X=Nm2a=(34KNL)2((2σγ0)2+ξa2)b=(1KLγ0μα)((2σγ0)2+ξa2)αγ02μ(2σγ0)c=(1KLγ0μα)2((2σγ0)2+ξa2)+(ξaξm+αγ02μ)2        +ξm2(2σγ0)22αγ02μ(1KLγ0μα)(2σγ0)d=f2γ02μ.(53)

This equation can be solved using the Cardano method. It gives the values of Nm corresponding to the intersection of the two manifolds, which are the equilibrium points or the singularities. The values of Nm placed at the intersection of the manifold are now known, the calculations of the values of Na can be done using either of the equations defining the manifolds. The condition set by equation (50) gives the nature of the considered points.

4 Results

4.1 Analytical results

4.1.1 Formalization of the equations

In this study, the nonlinear function chosen to produce a duffing behavior FNL(um)=KtβNLum3(t). Defining the excitation as pls(t) = P sin (Ωt) with Ω its pulsation, system of equation (5) becomes: { mau¨a(t)+cau˙a(t)+kaua(t)+γα(ua(t)αum(t))=StPsin(Ωt)Mtu¨m(t)+Rtu˙m(t)+Ktum(t)+KtβNLum3(t)+γ(αum(t)ua(t))=0. (54)

If we set the eigen pulsation of the acoustic first mode of the tube: ω0=kama.(55)

And we introduce the change of variables on the time with the notation =τ: τ=ω0t.(56)

At the working pressure, we evaluate the order of displacements of the air mass in the reduced section tube and of the membrane of the loudspeaker of the ER. We re-scale the displacements of the system ua and um to a value of order O(1) by introducing the variables χ = 1 × 10−6, u1 and u2 with following change of variables: { ua(t)=χu1(t)um(t)=χu2(t). (57)

It yields the system of equations with the normalized pulsation regarding the primary system (the acoustic mode): { u1(τ)+cakamau1(τ)+u1(τ)+γαka(u1(τ)αu2(τ))=StPkaχsin(Ωω0τ)u2(τ)+RtMtmakau2(τ)+KtmakaMtu2(τ)+KtmakaMtχ2βNLu23(τ)+γαkamaαMt(αu2(τ)u1(τ))=0. (58)

In mechanics, the parameter ε is defined as the ratio between the mass of the nonlinear oscillator and the mass of the primary system. In this case, the mass of the ER is higher than the mass of the air volume. As a result, the parameter ε << 1 is defined here as an arbitrary numerical value. In order to respect the standard form of equations (9), one should set the following parameters: { cakama=εξaγαka=εγ0StPkaχ=εfRtMtmaka=ξmKtmakaMt=KLKtmakaMtχ2βNL=KNLγαkamaαMt=γ0μ (59)

Where the parameters ξa , γ0 , f , ξm , KL , KNL and γ0µ should be order O(1), and we choose ε = 10−2. However, the value of the parameters of the experiment presented in Table 2 does not allow to respect all the hypotheses. Indeed, the parameter γ0µ should be order O(1), which means that the ratio (maα)/Mt should be order O(ε−1), due to the previously defined relation γ/(αka) = εγ0. The value of (maα)/Mt = 0.42 does not respect the hypothesis made in the analytical developments due to the weak coupling. We set µ as ε(maα)/Mt = µ.

For simplification purposes, the targeted parameters of the ER are chosen by the choice of the coefficients ηM , ηR and ηK as: { Mt=ηMM0Rt=ηRR0Kt=ηKK0. (60)

The rescaled dimensionless parameters are presented in Table 5, with the choice of β = 5.0 × 1010 m−2, ηR = 1/8, and ηM = ηK = 1, and the value of the parameters M0, R0 and K0 presented in Table 1. The choice of ηR is driven by the condition of equation (40).

As a result, the system of equations is organized as the standard form equations (9), and the method presented can be applied, while one should keep in mind the limits of the hypothesis made here.

Table 5

Values of the rescaled parameters.

4.1.2 Analytical predictions

Let us study the results given by the analytical calculations. Each analytical results are supported by numerical simulations. Multiple phenomena can be anticipated using analytic and numeric approaches. The SIM presented equation (24) with the stability boundary conditions introduced equation (39) is depicted in Figure 3a with the given parameters and ηK = 0.9. The unstable fixed points whose existence indicates eventual bifurcations are indicated in green. In fact, an additional condition about ηK can be observed: if the linear frequency of the ER is not tuned with the frequency of the acoustic mode of the tube, the bifurcation can not happen. It is due to the coupling between the pressure inside the reduced-section tube and the ER, such as the pressure needs to reach the activation threshold of the nonlinear behavior of the ER. Indeed, the linear frequency of the ER should be chosen such as the nonlinear resonance is aligned with the acoustic mode of the tube. Taking ηK = 0.9 lowers the linear frequency of the ER which aligns the nonlinear resonance with the resonance of the reduced-section tube since the behavior is hardening. This effect can also be observed by taking an interest in the softening behavior. If we set βNL = −5 × 1010 m−2 and we keep ηK = 0.9, the bifurcation can not happen as presented in Figure 3c. However, if we set ηK = 1.222, the linear frequency is placed at higher frequencies than the frequency of the reduced-section tube resonance, leading to the alignment of the nonlinear resonance of the ER with the frequency of the reduced-section tube resonance. It causes the existence of unstable fixed points, resulting in possible bifurcations as depicted in Figure 3d. Additionally, the condition of existence of unstable zones given equation (40) is verified in Figure 3b. In fact, the critical damping parameter with the considered parameters is ξ2,c = 0.0444, which yields ηR = 0.2232. It can be seen that the SIM with ηR = 1/4 does not include unstable zones, while the SIMs for ηR = 1/8 and ηR = 1/6 do. Let us plot the equilibrium points of the two degrees of freedom system with both the softening and hardening restoring forces. We solve the equation (52) by finding the roots of the polynomial expression using the Cardano method. The results are plotted in Figure 4 with incident pressure P = 0.8486 Pa which is the numerical input that suits the experimental pressure input. One can observe in Figure 4 that both behaviors do not demonstrate quasi-periodic regime with initial conditions set to 0. It is needed to pre-stress to system to attain quasi-periodic regimes. Moreover, possible bifurcations can be expected from the system with hardening behavior with the given parameters due to the unstable zones of the main branch, as it can be seen in Figure 4a. Additionally, the Figure 5a predicts that a bifurcation may happen around σ = 20. Regarding the softening case in Figure 4b, we observe an isolated branch of equilibrium points that is far from the main branch. However, Figure 5c shows that the isolated branch is situated at lower energy than the main branch for the acoustic mode, and at higher energy than the main branch for the ER (Figure 5d). Indeed, the larger the amplitudes of the ER, the lower the amplitudes of the acoustic mode, as the ER can absorb more energy with large oscillations.

thumbnail Fig. 3

Figures of the SIM of the two degree of freedom system defined by equation (24), with P = 0.5 Pa.

thumbnail Fig. 4

Figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

thumbnail Fig. 5

Figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

4.2 Comparing experimental results to analytical predictions

In this section, both hardening and softening behavior are presented. For the experiments, the specified parameters for the hardening behavior are kept as ηM = ηK = 1, βNL = 5 × 1010. The targeted parameters for the softening behaviors are ηM = 1 ηK = 1.222, βNL = −5 × 1010. Multiple nonlinear phenomena will be shown, such as faster decrease in transient regime, or the condition of existence of the bifurcations. Bifurcations will be showcased, and a discussion about the potential of such system for noise reduction is realized based on the experimental results.

4.2.1 Validation of the model by comparing predicted simulations with measured quantities

This section aims at validating the analytical model. To that end, the system of equations (5) is solved using an order 4 Runge-Kutta method. The excitation signal is a frequency sweep from 350 Hz to 800 Hz that lasts 60 seconds with either increasing or decreasing frequency sweeps in both simulations and experiments. The use of such excitation signal enables to observe the equilibrium points that would require non-zero initial conditions to be reached. Additionally, a 60-seconds sweep is sufficient to assume that the response is stationary. The frequency bounds are chosen large enough to contain the first mode of the ER loudspeaker. Variations of pressure inside the reduced-section tube for both experimental and numerical results with increasing and decreasing frequency sweeps are presented in Figure 6, and compared to each other. The parameters are ηK = 1, βNL = 5 × 1010 m−2 for the hardening behavior and ηK = 1.222, βNL = −5 × 1010 m−2 for the softening behavior. Notice that these parameters are the ones used for the experiment.

It can be observed that the numerical integration of the analytical model is very efficient in predicting the behavior of both hardening and softening behaviors with either increasing or decreasing frequency sweeps. The frequency of the bifurcation is also well-predicted in all cases. Nevertheless, one can observe that the absorption of the ER for the softening behavior with the decreasing frequency sweep (Fig. 6d) is better for the experiment than for the numerical simulation. It is caused by the intrinsic resonance of the ER loudspeaker which is close to the cavity resonance to control. Indeed, the intrinsic resonance of the ER loudspeaker is not considered in the reduced order model, and it is also not deleted by the programming of the control of the ER. This improved absorption of the experimental data with respect to the numerical data has to be remembered to evaluate the efficiency of the analytical study done with the MSM coupled to the complex variables of Manevitch method.

thumbnail Fig. 6

Comparison between experimental and corresponding numerical results obtained from direct numerical integration of equation (5): variation of the pressure amplitude for the hardening behavior with (a) increasing frequency sweep, (b) decreasing frequency sweep and for the softening behavior with (c) increasing frequency sweep, (d) decreasing frequency sweep.

4.2.2 The hardening and softening cubic behavior frequency response function

In this section, the nonlinear cubic restoring force implemented in an ER is studied with multiple excitation signals. In addition to the frequency sweeps, monoharmonic signals are composed of a 5-second long sinus, while the acquisition system also records the decrease in transient regime. The amplitude of mono-harmonic signals is retrieved and is plotted on the Function Response Function (FRF). Both hardening and softening behaviors are experimentally implemented. Variations of pressure inside the reduced-section tube and electrical current injected in the ER under frequency sweeps excitation are presented in Figure 7. One can observe that bifurcations occur as brutal changes of equilibrium points can be seen. Moreover, a frequency bandwidth where two stable points of equilibrium exist is showcased by the increasing and decreasing frequency sweeps. Additionally, the pressure in the reduced-section tube is significantly reduced when the ER behaves with large oscillations, which are placed at its nonlinear resonance. The oscillations are maximized in the case of a hardening behavior for increasing frequency sweep, and in the case of a softening behavior for decreasing frequency sweep. However, it can be seen that mono-harmonic signals excite the lower branch in amplitude of the resonator, resulting in a higher pressure in the reduced-section tube. These observations mean that if the ER is not pre-stressed, i.e. has initial conditions without energy, the ER will select the lower branch of its equilibrium points. Therefore, the decreasing frequency sweep for the hardening behavior and the increasing frequency sweep for the softening behavior are equivalent to the response under mono-harmonic signals if the resonator is not pre-stressed. Nevertheless, the excitation signals that maximize the ER response, i.e. increasing frequency sweep for hardening behavior and decreasing frequency sweep for softening behavior could be obtained with pre-stressed resonators. Future studies will explore the possibility of using the higher energy branch of the resonator to maximize the energy transfer from the primary system (the acoustic mode of the tube) to the nonlinear resonator.

One can observe that the analytical results predict well the shape of the curves in the case of the hardening behavior, such as the Figure 5a which is to be compared to Figure 7a. The bifurcation is well localized (after the resonance), and towards higher amplitude. Notice that the equilibrium points obtained by analytical means describe a mono-harmonic response, and do not describe the response to the frequency sweep signal. However, the analytical results indicated that ηK = 1 does not allow unstable zones to exist. This shift can be explained by the hypothesis made during the analytical development on the scale of the coupling parameter. Additionally, the differences that we can observe between analytical and experimental data, such as the value of the parameter ηK , are probably not caused by the intrinsic resonance of the ER loudspeaker which is close to the cavity resonance to control as previously mentioned. Indeed, this discrepancy does not seem to be caused by the model as it can be seen from the numerical simulations of Section 4.2.1. The parameters used for the numerical simulations are the parameters of the experiment.

The same phenomena can be seen for the softening behavior presented in Figure 7b, as the parameter ηK = 1.222 does not describe well the experimental results.

Choosing the parameter ηK = 1.122 to compute the analytical predictions (i.e. taking the same shift of −0.1 as for the hardening behavior) gives the Figure 8. A more accurate description of the phenomena is observed in Figure 8b. Indeed, the analytical predictions provide the same behavior as the measured pressure in Figure 7b, confirming that the analytical modelling is efficient for the softening behavior as well.

thumbnail Fig. 7

Experimental results: variations of pressure inside the reduced-section tube and electrical current injected in the ER under frequency sweeps excitation.

thumbnail Fig. 8

Analytical predictions: figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

4.2.3 Condition of existence of the bifurcations

As presented in Section 3.3, the bifurcation only exists within a range of the resonator damping values. To study this condition, the system is excited using a frequency sweep with both increasing and decreasing frequency sweeps. Let us consider the hardening cubic behavior with a fixed linear stiffness such as ηK = 0.9, and multiple damping values ηR . The critical damping value calculated using the experimental data is ηR,c = 0.2232. Higher values of ηR prevent the bifurcations to occur. Variations of pressure inside the reduced section tube and electrical current injected into the loudspeaker’s coil for various values of the damping parameter ηR are presented in Figure 9. One can observe that the analytical results about the role of damping on the existence of bifurcations are accurate. Figures 9a and 9b show that bifurcations occur at lower values of damping ηR = 1/8 and ηR = 1/6 than the critical value ηR,c . Figure 9d presents a small bifurcation, even with a slightly larger value than ηR,c. Comparing the Figure 9 with Figure 3b exhibits a good agreement with the analytical predictions. Moreover, the frequency bandwidth of the existence of at least two stable equilibrium points is narrowed with higher ηR . From the analytical study, it is known that the unstable zone of the SIM becomes smaller with the rise of the damping value ηR . Nevertheless, bifurcations are smoothed when ηR approach the critical value ηR,c as seen in Figures 9c and 9d, which is due to the two stable equilibrium points getting very close in amplitudes. When the damping is superior to the critical damping, the bifurcations do not exist, and there is not a frequency bandwidth where two stable equilibrium points exist, as seen in Figures 9e and 9f. The agreement with analytical results is quite good, as the critical damping value is well predicted.

thumbnail Fig. 9

Experimental results: variations of pressure inside the reduced-section tube and electrical current injected in the ER under frequency sweeps excitation for different values of damping ηR .

4.2.4 The transient regime

To study the transient regime, two different sets of measures were carried out with two different excitation signals. The transient decrease analysis is done from a mono-harmonic signal featuring a 5-second long sinus, and from a frequency sweep signal. The frequency sweep signal starts at 350 Hz and stops on a chosen frequency for 3 seconds. This excitation signal has been designed to enable the study of the transient regime while the ER state is placed on the higher amplitude branch. The two decreases from Figure 10 can be compared in terms of shape. The transient decrease of the primary system when the ER is placed on a high energy equilibrium point is presented in Figure 10a. It highlights a first decrease, and then an increase in amplitude can be noted. It leads to an exponential decrease until an amplitude of 0 Pa. The transient decrease of the primary system when the ER is placed on a low energy equilibrium point is presented in Figure 10b. Only an exponential decrease can be observed. The different phenomena are due to either the energy level of the ER is higher or lower than the bifurcation minimum energy.

thumbnail Fig. 10

Experimental results: pressure inside the reduced-section tube at 580 Hz.

5 Conclusion

The paper focuses on the analytical method to study nonlinear systems, with an application to the presented experimental bench. The Multiple Scales Method coupled to the complex variables of Manevitch lies on multiple assumptions, which are not fully respected here. It is shown that the method is suitable to study the phenomena and have a better understanding of them. The critical damping value can be well predicted, and the behavior of the two degrees of freedom system can be qualitatively predicted. However, the inaccuracy of the prediction of the quantitative values is due to the model that is not complete. Indeed, the intrinsic resonance of the ER loudspeaker is not modeled, depite that it is not deleted by the control of the ER. As a result, despite the numerous assumptions needed by the analytical method and the reduced order model, its predictions can be used to understand the phenomena at stake.

Acknowledgements

The authors thank Abdelhakim Ezzerouki for providing helpful assistance in the electronic realization of the experiment. The authors would like to thank the following organizations for supporting this research: (i) The “Minist`ere de la transition ´ecologique” and (ii) LABEX CELYA (ANR-10-LABX-0060) of the “Universit´e de Lyon” within the program “Investisse- ment d’Avenir” (ANR-11- IDEX-0007) operated by the French National Research Agency (ANR).

Funding

This research received no external funding.

Conflicts of interest

The authors declare no competing interests.

Data availability statement

The data that supports the findings of this study are available from the corresponding author upon reasonable request.

Author contribution statement

Conceptualization: M.M., M.C., E.G., A.T.S., E.D.B.; Data curation: M.M.; Formal analysis: M.M.; Funding acquisition: M.C., E.G., A.T.S.; Investigation: M.M.; Methodology: M.M., M.C., E.G., E.D.B.; Project administration: M.C., E.G.; Software: M.M.; Resources: M.C., E.G., A.T.S.; Supervision: M.C., E.G., E.D.B.; Validation: M.M., M.C., E.G., A.T.S., E.D.B.; Visualization: M.M.; Writing - original draft: M.M.; Writing -review & editing: M.M., M.C., E.G., A.T.S., E.D.B.

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Cite this article as: M. Morell, M. Collet, E. Gourdon, A. Ture Savadkoohi, E. De Bono, Experimental validation of the analytical modelling of a digitally created duffing acoustic nonlinear oscillator at low amplitudes, Mechanics & Industry 26, 2 (2025), https://doi.org/10.1051/meca/2024036

Appendix A Estimation of the parameters of the loudspeaker’s first mode

This appendix aims at briefly describing the method of identification of the modal parameters of the first mode of the loudspeaker. The needed assumptions are:

  • Low excitation amplitudes: we consider the loudspeaker in its linear regime.

  • Low frequency amplitudes: we consider that the wavelength is large enough compared to the diameter of the loudspeaker’s membrane.

  • The loudspeaker is excited on its first mode: the operating frequencies are situated near its first mode.

The loudspeaker’s first mode can be described by equation (1): M0u¨m(t)+R0u˙m(t)+K0um(t)=p(t)SdBli(t).(A.1)

Let us apply a Laplace transform to the equation: M0sv(s)+R0v(s)+K0v(s)s=p(s)SdBli(s)(A.2)

with v=u˙m the velocity of the membrane of the loudspeaker. When no current is applied into the loudspeaker’s coil, the acoustic impedance Za0 of the loudspeaker is: Za0(s)=p(s)v(s)=M0Sds+R0Sd+K0Sd1s.(A.3)

To retrieve the acoustic modal parameters M0 /Sd , R0 /Sd and K0 /Sd , the curve-fitting of the experimental transfer function between p and v can be realized. To measure these quantities, a microphone is placed close to the loudspeaker’s membrane to retrieve the pressure, and a Laser Doppler Velocimeter (LDV) measures the velocity of the membrane. An acoustic source excites the loudspeaker with a frequency sweep. The experimental setup is presented in Figure A.1a. The transfer function is plotted in Figure A.1b.

thumbnail Fig. A.1

Experimental setup and results for the estimation of the parameters of the loudspeaker’s first mode.

Once the first measurement is done, the coefficient Bl/Sd is still unknown. To measure this coefficient, an additional measurement with a non-zero electrical current has to be realized.

Let us set the current i as being proportional to the measured pressure p: i(s)=gp(s)(A.4)

with g a constant coefficient. It gives the following acoustic impedance: Za=Za01+gBlSd.(A.5)

The expression of equation (A.5) is then curve-fitted to the transfer function obtained through experimental measurements to obtain Bl/Sd . The acoustic impedance and its curve-fitting are depicted in Figure A.1c with a gain g = 0.001. This experiment is repeated 5 times, and the average values are taken as the modal parameters of the loudspeaker.

Appendix B Estimation of the parameters of the experiment’s acoustic first mode

To obtain the parameters of the tube ’s acoustic first mode, we could also proceed with curve-fitting, in stationary or transient regime. However, the retained solution consists in calculating the parameters ma , kb and γ based on the model, and deducing the remaining parameters through simple modal analysis. We consider the experimental setup presented in Figures 1 and 2, with a rigid termination instead of the ER. In this case, the first mode of the tube is no longer coupled to the ER, and the rigid termination model can be derived: mau¨a+cau˙a+(ka+kbSt2)ua=0.(B.1)

We denote the equivalent stiffness K=ka+kbSt2. A frequency sweep excitation signal is sent by the acoustic source on a large frequency bandwidth. The pressure inside the tube is measured by a microphone, and we obtain the relation: p=φ(f)(B.2)

with φ the function which maps the envelop of the pressure amplitude with respect to the frequency. The frequency of the first mode f0 is then identified by selecting the frequency with the highest signal amplitude. Then, we find the frequencies f1 and f2 (with f > f1) which satisfy the condition: φ(f0)2=φ(f).(B.3)

Let us denote Δf = f2f1. The parameters ka and ca can be deduced: { K=(2πf0)2maka=KkbSt2ca=2πΔfma .(B.4)

This process is repeated 5 times, and the average values are retained. The pressure level sent into the tube is then obtained by curve-fitting of an analytical expression of φ onto the measured quantity. The model and the experiment are compared in Figure A.2.

thumbnail Fig. A.2

Envelop of the pressure inside the tube with a reduced section with a rigid termination.

All Tables

Table 1

Dimensions and parameters of the experiment and simulation.

Table 2

Values of the dimension of the experiment.

Table 3

Parameters obtained by calculations.

Table 4

Parameters obtained by measurements.

Table 5

Values of the rescaled parameters.

All Figures

thumbnail Fig. 1

Scheme of the experimental set-up.

In the text
thumbnail Fig. 2

Experimental set-up.

In the text
thumbnail Fig. 3

Figures of the SIM of the two degree of freedom system defined by equation (24), with P = 0.5 Pa.

In the text
thumbnail Fig. 4

Figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

In the text
thumbnail Fig. 5

Figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

In the text
thumbnail Fig. 6

Comparison between experimental and corresponding numerical results obtained from direct numerical integration of equation (5): variation of the pressure amplitude for the hardening behavior with (a) increasing frequency sweep, (b) decreasing frequency sweep and for the softening behavior with (c) increasing frequency sweep, (d) decreasing frequency sweep.

In the text
thumbnail Fig. 7

Experimental results: variations of pressure inside the reduced-section tube and electrical current injected in the ER under frequency sweeps excitation.

In the text
thumbnail Fig. 8

Analytical predictions: figures of the equilibrium points of the two degree of freedom system defined by equation (52), with P = 0.8468 Pa.

In the text
thumbnail Fig. 9

Experimental results: variations of pressure inside the reduced-section tube and electrical current injected in the ER under frequency sweeps excitation for different values of damping ηR .

In the text
thumbnail Fig. 10

Experimental results: pressure inside the reduced-section tube at 580 Hz.

In the text
thumbnail Fig. A.1

Experimental setup and results for the estimation of the parameters of the loudspeaker’s first mode.

In the text
thumbnail Fig. A.2

Envelop of the pressure inside the tube with a reduced section with a rigid termination.

In the text

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