Numerical realization of a semi-active virtual acoustic black hole effect

Noise mitigation by means of the acoustic black hole (ABH) effect is a well-known engineering solution. However, the conventional method of applying ABH effect which requires modification of the structure geometry has various limitations which encourage the research of virtual ABH concept. In this study, the effect of ABH was applied through introducing virtual stiffness by a shunt circuit. According to the force-voltage electric analogy, stiffness has an inverse relationship with capacitance. So that the ABH effect can be virtually realized by following a power law profile using an array of independent capacitive shunts. The concept is studied through finite element simulation developing a macro code in ANSYS Parametric Design Language (APDL). To evaluate the influence of capacitance profile on the acoustic radiated power, parametric studies are conducted. Based on the results of the parametric studies, the capacitance profile is tuned for minimum radiated power. It is revealed that the virtual acoustic black hole (ABH) effect can offer 10.29%, 6.37%, and 7.47% reduction in the radiated power from the first to the last targeted mode, respectively. The virtual ABH effect introduced in this study can be used for semi-active structural noise isolation without any weight or manufacturing penalty.


Introduction
The increasing requirement for noise management in industries has led to a substantial research being conducted on noise cancellation methods (Simons and Waters, 2004;Wang, 2010;Towers et al., 2021). Sound emission can be generally classified to structure-borne and air-borne noise. Structure-born noise occurs when a mechanism vibrates due to direct mechanical contact with the vibration source. Air-borne noise is produced by a source which radiates directly to the air (Berendt et al., 1967). Usually, the structure-borne noise requires significant attention since it is a low-frequency tonal noise which cannot be controlled by conventional barrier or insulation methods (Ngai and Ng, 2003;Fathiah Waziralilah et al., 2018). Passive techniques are often adopted to dampen the structure-borne noise despite a significant weight penalty often associated.
The effectiveness of electronic damping as a low-cost and lightweight passive technique is examined for its vibroacoustic attenuation effect by many authors (Park and Inman, 2003;Zhao et al., 2016;Suryakant et al., 2022). A case study revealed 7 dB reduction in sound transmission achieved by a shunt circuit for a clamped plate excited by a sound source in the frequency range of 10-1,000 Hz . Another study revealed at least a 57% reduction in radiated noise for a clamped plate excited by a shaker in the frequency range of 0-245 Hz . Another efficient solution for noise cancellation is the acoustic black hole (ABH) effect, a passive, lightweight, and economic technique. According to the ABH effect theory, the vibration wave can be focalized and diminished to zero due to an exponential reduction in the host structure thickness (Li and Ding, 2018;Hook et al., 2019;Mousavi et al., 2022). The thickness exponential function and position of ABH are determinative factors for the amount of energy trapped (Rothe et al., 2016;Liang et al., 2022). There exist several restrictions in the introduction of the ABH effect to the geometry. The dependency of ABH profile position on its efficiency limits its isolation effect to a few modes of resonance (Zhang et al., 2019;Gao et al., 2022). Another shortcoming of the ABH effect is its inadequacy at low frequencies (Denis et al., 2014;Liang et al., 2021). In one study, the effective frequency range of ABH is broadened by integrating shunt damping and transferring the energy from low to high frequencies (Zhang et al., 2022). An inevitable condition for ABH geometries is that the host structure must be large enough to allow carving out the power law profile which is a limitation in vehicle and aero applications (Ji et al., 2018;Park et al., 2019). This constraint is overcome partially by using ABH-inspired active electronic dampening in conjunction with the force-current electric analogy by Maugan et al. (2019).
The use of semi-active systems, also known as adaptive systems, for noise isolation has received attention in the last decade (Bein et al., 2008;Zhu et al., 2017;Wrona et al., 2021a). With the help of semi-active systems, certain resonant peaks can be attenuated while also adverse impacts of vibration absorbers on non-target modes may be prevented. Using the shunt technique in an adaptive manner, one may absorb noise and vibration at various modes by proper adjustment of the shunt circuit impedances (Soong and Spencer, 2000;Corr and Clark, 2001;Niederberger et al., 2003;Bein et al., 2008). Optimizing tuning variable which determines the control states is a common task for semi-active shunt approach (Gonzalez-Buelga et al., 2014). The control states can consist of ON and OFF modes (Wrona et al., 2021b), or it can take several states as reported by Li and Zhu (2021) using a rheostat to introduce numerous resistance levels.
Limitations associated with physical ABH execution such as low-frequency inefficiency (Denis et al., 2014;Liang et al., 2021), space occupation (Ji et al., 2018;Park et al., 2019), and tunability to narrow frequency ranges (Zhang et al., 2019;Gao et al., 2022) motivates the research to apply the ABH effect virtually through a semi-active strategy.
In the present study, a beam structure is covered by an array of piezoelectric elements shunted by capacitors to impose the power law stiffness profile virtually. To attenuate the three resonant peaks of the beam which contribute in the radiated noise, the shunt circuit is tuned in terms of power law function. The present approach can be applied to ultra-thin structures where physically applying the ABH effect is impossible. To the best of our knowledge, this is the first study to introduce the concept of virtual ABH effect integrated with force-voltage analogy as a semi-active noise mitigation approach.

Materials and methods
To explain the ABH effect, consider a wedge whose variable thickness follows a power law profile as h(x) εx m (ε and m > 0 ), shown by Figure 1. If k(x) is the local wave number of a flexural wave, the total wave transit time through the wedge can be derived by Eq. 1.
Considering k p as the wave number of quasi-longitudinal waves, the local wave number can be expressed by Eq. 2 for a wedge with a power-law profile.
By substitution of Eq. 2 into Eq. 1, it can be seen that the integral diverges for m ≥ 2. It means that if the wedge is designed ideally, the wave never reaches the edge. This perfect absorption is known as ABH effect which is imposed by geometry (Krylov and Tilman, 2004).
In the present study it is of interest to introduce the ABH effect imposed not by geometry but by applying the electric analogy. In this regard, consider a dynamic system consisting of a mass (m), a spring (k), and a damper (b) subjected to the excitation f as given by Figure 2A. Moreover, take a basic electric circuit with resistance (R), inductance (I), capacitance (C), subjected to a voltage V as given by Figure 2B.
The force-voltage analogy can be established as Eq. 3 can be derived as Hence, Eq. 4 implies that capacitance is relevant to the reverse of stiffness (Darleux et al., 2022).
In this regard, the ABH effect imposed by geometry (see Figure 3A) is virtually defined by considering i number of piezoelectric elements with equal distances attached on a beam surface (see Figure 3B). Each piezoelectric element is shunted independently by a capacitor. The ideal ABH stiffness, ideal capacitance, and real capacitance for the beam structure are well noted by k, C i , and C, respectively (see Figure 3C). Here, two points should be emphasized for a better understanding of the problem. First, the lower limit of real and ideal capacitance is always non-zero which means an artificial thickness is added along the beam. Second, the maximum value of capacitance is infinite for an ideal ABH (C i ) shown by Figure 3C. However, the present modeling cannot simulate an ideal ABH for two reasons: (a) the beam initial thickness cannot be set to zero, and (b) there is a geometry constraint that corresponds to the distance between piezoelectric patches. So that the maximum capacitance is not infinite but as big as super capacitors. By tuning the capacitive shunt technique, maximum capacitance should be defined at the beam antinode and effective noise isolation is expected, consequently. Since the antinode position is variable from mode to mode, a semi-active scheme can be adopted to change the power law profile, and attenuate several modes. So that the capacitance values can be switched among various states as depicted by Figure 4. To absorb the j th mode of resonance, a series of i number of capacitors can be considered as {C 1j , C 2j , ..., C ij }, tunable to the corresponding mode according to power law as indicated by Eq. 5.
Where C max is the maximum value of capacitance which can be taken by one or more than one capacitors depending on the target mode to be tuned. Moreover, a i represents the element of an arithmetic progression, the first term of which is 1 and the difference between terms is r as expressed by Eq. 6.
The effectiveness of the proposed semi-active scheme can be evaluated by solving the harmonic analysis of the cantilever beam. The vibrating beam excites the surrounding air and emits noise. The sound power corresponding to the radiated noise can be represented as the integral of the acoustic intensity along the normal direction on a given surface Γ as where I is the acoustic intensity, n . is the normal vector, and Γ is the structure-air interface. The power carried by the acoustic wave per unit area in the normal surface direction is known as the acoustic intensity given by (Hambric and Taylor, 1994) where p and v refer to the radiated acoustic pressure, and particle velocity vector of air, respectively. By establishing continuity of velocity at the solid/air interface, the structural velocity will be equaled to air particle velocity as where v s,n , v air , ρ air (1.2041 kg/m3), c air , and p correspond to the structure normal velocity, air particle velocity, air density, sound speed in air, and acoustic pressure, respectively. The equivalent sound power released from the vibrating structure can be represented as a function of the structure's vibration velocity by substitution of Eq. 9 into Eq. 8. As employed in this study, the equivalent radiated power (ERP) and its level (ERPL) can be obtained as ) The present study proposes a finite element solution using APDL 1 to solve the harmonic problem. By finite element discretization of the integral in Eq. 10, the power radiated all over the 3D beam faces can be calculated.
To minimize the radiated power, the power law profile of capacitance is required to be optimally tuned. By substitution of Eq. 6 into Eq. 5, the capacitance profile can be expressed as So that optimization of parameters r and n can offer minimum radiated noise. The area under frequency reponse curve which has been used as a performance measure in some studies (Joshi et al., 2010;Sarigul et al., 2018), is considered as the optimization objective function. The optimization algorithm is provided by Eq. 13.
Where C opt can be obtained based on optimum values of r and n. Thus, for semi-active noise isolation, the optimization must be conducted for frequency intervals centered by the resonance frequency. Then, the semi-active treatment function for a frequency range including N number of eigenfrequencies can be expressed as follows.
Where H* is a distribution function defined by Heaviside functions as given by Eq. 15.
The semi-active function (Eq. 14) allows to select the frequency intervals for isolation based on the optimal capacitance profile, and deselect the frequency intervals where the isolator is not effective by short-circuiting the piezo-patches. Accordingly, the short circuit capacitance equals to zero (C SHC ). Therefore, the semi-active treatment function can be expressed as

Results
At first, a validation study is carried out to check the accuracy of present numerical modeling. Next, the numerical model for realization of ABH effect is introduced and considered for parametric studies. Based on optimization of parametric studies results, a semi-active noise treatment is applied.

Validation of numerical modeling
The present numerical model has been validated by comparison with numerical findings reported by Larbi and Deü (Larbi and Deü, 2019). Using solid elements for the host structure and Circu94 elements for the piezoelectric patch, calculations have been made for the eigenanalysis of a cantilever steel beam with a piezoelectric PIC 151 ceramic patch (see Figure 5) which properties are given by Table 1. Both open-and short-circuit conditions are considered to obtain the results for the eigenanalysis. The discrepancy of results reported by Table 2 attributes to different finite element modeling techniques as well as different elements type and number.

FIGURE 4
Capacitive shunts apply semi-active scheme applied to shunted piezoelectric elements.

ANSYS Parametric Design Language.
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Numerical realization of virtual ABH effect
For the numerical realization of virtual ABH effect, a steel cantilever beam with L b = 11.19 cm length, W b = 5 mm width, and T b = 1 mm thickness is assumed. The beam is covered by 20 piezoelectric elements with L p = 5 mm edge length, and T p = 0.3T b thickness positioned at equal distances to each other. The beam is excited by harmonic out-ofplane displacement applied at the free end. A structural damping coefficient of 0.01 is adopted (Devasia et al., 1993).
The ERPL response for the beam structure with shorted shunt circuit is presented by Figure 6. As a result, the three eigenmodes dominant in the ERPL response are targeted for attenuation.

Parametric study
Parametric studies are carried out to figure out the influence of power law formulation constants (n and r) described by Eq. 12, on the structure response. In this regard, particular capacitance profiles assigned for the first three modes of resonance which contribute to the overall radiated noise are given by Figure 7. The capacitance profiles are minimum and maximum at nodes and antinodes of vibration displacement mode shape, respectively. Between each node and antinode, the capacitance varies according to power law formulation.
The results of parametric studies are reported by Figures 8-13. It is well revealed that the capacitance profile constants (n and r) are the tuning parameters responsible for the radiated power response. Compared to the short-circuit case, the application of virtual ABH effect results a heavily fluctuant response due to trapping the elastic wave.

Semi-active noise isolation
The results of parametric studies reported by Figures 8-13 are considered for optimization. So that the optimal values of r and n are chosen for each mode of resonance. The vibrating beam is treated semi-actively considering optimal parameters of (r, n) taken as (0.02, 5), (0.04, 5), and (0.05, 4), respectively for the first, the second and the third targeted modes of resonance. The treatment is applied at the vicinity of resonance with a radius of 10 Hz. The semi-active treatment function is expressed as follows.
The results of this semi-active treatment is represented by Figure 14. The radiated power is attenuated by the rates of 10.29%, 6.37%, and 7.47% from the first to the last targeted modes, respectively.
In order to assess the influence of loading, harmonic excitations with u 0 , and 2u 0 amplitudes are imposed. The results reported by Figure 15 indicate the system is linear for all studied resonances with and without treatment.

Conclusion
A numerical model is developed to realize the ABH effect virtually with the help of capacitive shunt circuits. The main objective is to attenuate structure-born noise for several resonance modes by proposing a semi-active approach. Theoretically, virtually imposing the ABH effect is feasible through the electrical analogy which equalizes stiffness with capacitance.
A noteworthy advantage of virtual ABH effect is the potential for semi-active use which is obviously impossible with the ABH effect imposed by geometry. In addition, the geometrical restrictions related to the physical manifestation of the ABH effect do not apply to the virtual method.
For two main reasons the virtual ABH effect deviates from the ideal ABH effect: 1. Since the energy conversion rate in piezoelectric material is not 100%, the ABH effect can never be idealized. 2. The capacitance profile is discrete along the beam due to the distance between piezoelectric elements which is unavoidable.
The results of present numerical modeling for the first eigenmodes which contribute in overall radiated noise revealed at least 6.37% improvement for noise radiation. The attenuation is well achieved by breaking down each peak to many more peaks which results an effective absorption effect. Although the present approach targets the first three eigenmodes, expanding the effective frequency range requires an increasing of the number of piezo-patches to tune the shunt impedance with respect to the eigenmode shapes.
Finally, due to the weight and space limitation in vehicle and aero applications, the structures are too thin to be carved out for an ABH geometric profile. Thus, the virtual ABH effect can be considered as a solution not only to overcome the mentioned restriction, but also to apply a multi-mode structural noise isolation.

Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions
SS conceived and formulated the problem, developed the macrocode, analyzed the results, and wrote the paper. GP, FF, and SD