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 (Ahmadian and Jeric, 2001). 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 (Ahmadian et al., 2001).

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; Li et al., 2021; 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; Li et al., 2021; 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.

To explain the ABH effect, consider a wedge whose variable thickness follows a power law profile as h x = ε x 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. τ = ∫ 0 x k x d x (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. k x = 12 1 4 k p ε x m − 1 / 2 (2)

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 F ≈ V (3)

Eq. 3 can be derived as m d 2 x d t 2 + b d x d t + k x ≈ L d i d t + R i + 1 C ∫ i d t (4)

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. C i j = C max a i − n (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. a i = 1 + i − 1 r (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 P = ∬ Γ I ⇀ . n ⇀   d Γ (7) 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) I ⇀ = 1 2 R E ( p . v ⇀ ) (8) 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 v s , n = v a i r = p ρ a i r c a i r (9) 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 (Kim et al., 2019) E R P = 1 2 ρ a i r c a i r ∫ Γ v s , n 2 d Γ (10) E R P L = 10 ⁡ log E R P W r e f   d B   w h e r e   W r e f = 10 − 12 W (11)

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 C i j = C max 1 + i − 1 r − n (12)

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. Min   U U = ∫ f min f max E R P   d f ,   U min = U C o p t f min + f max = 2 f r e s o n a n c e (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. C s e m i − a c t i v e = ∑ i = 1 N C o p t i H * + C S H C 1 − H * (14) Where H ^* is a distribution function defined by Heaviside functions as given by Eq. 15. H * = H e a v i s i d e f − f min + H e a v i s i d e f − f max (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 C s e m i − a c t i v e = ∑ i = 1 N C o p t i H * (16)

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.

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.