Helmholtz resonator is a common component for controlling noise propagation in pipes, and the study of its acoustic characteristics has always been an active topic in the field of noise control [1–3]. Many scholars have done a lot of efforts on this topic and have achieved fruitful results. The effects of cavity shape and geometric parameters [4–9], neck shape and position [10–16], series-parallel connection form of the resonators [17–19], resonator array configurations [20–26], internal absorption materials [27,28] and other factors on the resonant frequency and the STL have been extensively investigated. From the study of the Khairuddin et al. [29] systematically summarizes the influences of resonator types, geometry, modifications, and arrangements on its attenuation capabilities, which provide helpful references for scholars studying Helmholtz resonators.

In most of the existing literature, the wall of Helmholtz resonant cavity is assumed to be rigid, i.e., the effect of cavity wall elasticity on the acoustic properties is ignored. However, in many practical situations, the wall elasticity cannot be ignored [30–34]. The investigation on the effect of wall elasticity on the acoustic properties of a Helmholtz resonator can be traced back to the work of Photiadis [30,31], who found that the wall elasticity has a significant influence on the resonant frequency, radiation impedance and acoustic scattering of spherical Helmholtz resonators in water medium. Zhou et al. [32] studied the STL of a water-filled cylindrical Helmholtz resonator with elastic wall and confirmed that the coupling between elastic wall and heavy fluid has a remarkable influence on lowering the resonant frequency. Later, Nudehi et al. [33,34] demonstrated that the coupling between elastic wall and light fluid produces multiple peak frequencies of the STL instead of a single resonant frequency and also lowers the first resonant frequency. Both of them indicated that the bottom plate elasticity of the cylindrical Helmholtz resonator has a greater effect on the noise reduction than the side-wall elasticity. In this study, we find that the elasticity of the resonator wall also allows the STL of Helmholtz resonator to change from a single resonant frequency to two resonant frequencies in the frequency range of interest, which has important implication for the accurate design of Helmholtz resonator.

In this paper, a simplified one-dimensional lumped-parameter analytical model was developed to accurately characterize the structure-acoustic coupling and STL of a Helmholtz resonator with an elastic bottom plate. The effects of the equivalent stiffness, mass and damping of the elastic bottom plate on the peak frequency and its corresponding amplitude of the STL are analyzed, and then a design criterion to circumvent the effect of wall elasticity of Helmholtz resonators is proposed. Namely, the structural natural frequency of the wall should be greater than three times the resonant frequency of the resonator to avoid the adverse effects of wall elasticity of Helmholtz resonators. These findings are validated by one-dimensional analysis and three-dimensional finite element simulations as well, which provide guidance for the rapid and effective design of Helmholtz resonators.

To facilitate mathematical modeling, the following assumptions and simplifications are made. The cross section of the main pipe is a square with a side length of a . Both the cavity and the neck of Helmholtz resonator are cylindrical pipes. The cross-sectional radius and length of the neck are r1 and l1 , respectively, and those of the cavity are r2 and l2 , respectively. The bottom plate of the Helmholtz resonator cavity is elastic and is reduced to a SDOF vibration system, while the other walls of cavity are assumed to be rigid. Thus, a simplified model based on plane wave propagation is shown in Fig. 1, where M , K and C are equivalent mass, stiffness and damping of the SDOF system, respectively.

As shown in Fig. 1, at the entrance of the neck (i.e., x1=0 ), the complex sound pressure of the incident wave with angular frequency ω is denoted as A+ (where “+” indicates propagation along the positive direction of the axis x1 ), then the complex sound pressure of the incident wave at any position, x1 can be expressed as(1) p+(x1)=A+e−jkx1 where j=−1 is the imaginary unit, k=ω/c0 is the wavenumber of the plane sound wave, and c0 is the speed of sound. According to the relationship between sound pressure and particle velocity, the particle velocity corresponding to the incident sound wave can be expressed as(2) u+(x1)=1ρ0c0A+e−jkx1 where ρ0 is the air density.

Similarly, at the entrance of the neck (i.e., x1=0 ), the complex sound pressure of the reflect sound waves with angular frequency ω is denoted as A− (where “ − ” indicates propagation along the negative direction of the axis x1 ), then the complex sound pressure and particle velocity of the reflect wave at any position x1 can be expressed as(3) p−(x1)=A−e+jkx1 (4) u−(x1)=−1ρ0c0A−e+jkx1

Then, the particle velocity up(x1) at an arbitrary position x1 is(5) up(x1)=u+(x1)+u−(x1)=1ρ0c0(A+e−jkx1−A−e+jkx1)

Since the sound pressure is continuous and the mass flux of particles is conserved, there is an equality relationship at x1=l1 (i.e., x2=0 ) as follows:(6) A+e(−jkl1)+A−e+jkl1=B++B− (7) (A+e−jkl1−A−e+jkl1)SC=(B+−B−)SV where B+ and B− are the complex sound pressures of the forward and reverse sound waves propagating along the axis x2 in the cavity, SC is the cross-sectional area of the neck, and SV is the cross-sectional area of the cavity, respectively.

Similarly, the following relationship exists at the elastic bottom plate (i.e., x2=l2 ) of the cavity(8) 1ρ0c0(B+e(−jkl2)−B−e+jkl2)SV=vPSV where vP is the normal vibration velocity of the elastic bottom plate, which satisfies(9) vP=HVFP

Here, FP is the force of the sound pressure inside the cavity on the elastic bottom plate, HV  is the vibration velocity transfer function of the SDOF system. FP and HV [35] can be expressed as(10) FP=(B+e−jkl2+B−e+jkl2)SV (11) HV=jωK−Mω2+jCω

At the interface between the main pipe and the resonator, the following equation also exists(12) C++C−=A++A−=D+ (13)  (C+−C−)SP=(A+−A−)SC+D+SP where C+ and C− are the complex sound pressure of the incident wave and reflected wave in main pipe upstream of Helmholtz resonator, D+ is the complex sound pressure of the transmitted wave in main pipe downstream of Helmholtz resonator, and SP is the cross-sectional area of the main pipe, respectively.

According to the Eq. (5), at the entrance of the neck (i.e., x1=0 ), up=(A+−A−)/ρ0c0 . Let ρ0c0up=1 and combine Eqs. (6)–(13), we obtain(14) [e−jkl1SCe−jkl1001SC1e+jkl1−SCe+jkl1001−SC−1−1−SV(1−ρ0c0SVHV)e−jkl20000−1SV−(1+ρ0c0SVHV)e+jkl2000000010−SP000010SP0000−1−1SP0][A+A−B+B−C+C−D+]=[0000001]

Further, we can obtain STL as follows:(15) STL=20 log10|C+D+|

It should be mentioned that in this paper, we corrected the neck length from l1 to l1+0.85r1(1−1.25r1/r2) by referring to the study of the Selamet et al. [5].

Table 1 lists the structural dimensions and medium properties of the model shown in Fig. 1, which are consistent with the data in the study of the Selamet et al. [10]. To better investigate the coupling between the elastic plate vibration and the sound pressure inside the cavity of Helmholtz resonator, the structural natural frequency f of the elastic bottom plate remains the same as the resonant frequency f₀ of the Helmholtz resonator. The material of the elastic bottom plate is Polyvinyl Chloride (PVC) whose parameters are also listed in Table 1. In the one-dimensional analytical calculation, when considering the resonator bottom plate elasticity, the equivalent mass M and stiffness K parameters of the SDOF vibration system on the elastic bottom plate are obtained as 0.06036 kg and 26,623.8 N/m, respectively, according to the kinetic energy equivalence principle. The equivalent damping C of the bottom plate is set to 1.6036 Ns/m (i.e., the damping ratio is 0.02). When the bottom plate of the resonator degrades to rigid, B+e−jkl2−B−ejkl2 and HV are set to 0.

The three-dimensional finite element model (FEM) of Helmholtz resonator with elastic bottom plate is established corresponding to the parameters in Table 1, as shown in Fig. 2. All the walls are set to be rigid when the elasticity of the bottom plate is not considered, and the rest of the walls of the resonator other than the bottom plate are set to be rigid when considered. The damping ratio of elastic bottom plate is also set to 0.02. The air domain is discretized using tetrahedral elements, while the structural domain is discretized using shell elements and triangular mesh. Table 2 shows the mesh irrelevance test and the resonant frequencies of the Helmholtz resonator for different grid size. When the maximum grid size in the computational domain is less than 0.03 m, the change of the resonant frequencies is less than 1%, which meets the requirements for mesh irrelevance. In order to balance calculation accuracy and efficiency, the maximum mesh size is set to 0.02 m.

Fig. 3a compares the STL curves of the resonator with rigid bottom plate obtained using Eq. (15), from the study of the Selamet et al. [10], and three-dimensional FEM. It shows that the STL curves obtained by the three methods are almost consistent, and the difference of the resonant frequencies between three methods is less than 1.5 Hz, indicating that the simplified theoretical model with rigid bottom plate in this paper is correct. Fig. 3b compares the STL curves of the resonator with elastic bottom plate obtained using Eq. (15) and three-dimensional FEM. It shows that the results obtained by the two methods are almost consistent, indicating that the established one-dimensional lumped-parameter analytical model considering structure-acoustic coupling in this paper is also correct.

It can be seen from comparing the STL curves in Fig. 3a,b that the elasticity of the bottom plate makes the STL peak of resonator split into two peaks, due to the strong coupling between the structural vibration of the elastic bottom plate and the acoustic pressure inside the resonator cavity. The influence of the dynamic parameters (stiffness, mass, damping) of the elastic bottom plate on the STL of the Helmholtz resonator is analyzed using one-dimensional lumped-parameter analytical model and discussed in the following sections.

In this paper, a one-dimensional lumped-parameter analytical model is derived to analyze the STL of a Helmholtz resonator with an elastic bottom plate, which takes into account the structure-acoustic coupling between the structural vibrations of the resonator wall and the sound pressure inside the resonator cavity. The effect of the dynamic parameters of the elastic bottom plate on the STL is then investigated. The following conclusions were obtained. (1) If the structural natural frequency of the elastic wall is close to the resonant frequency of the resonator (i.e., there exists strong coupling), the wall elasticity has a significant effect on the STL of a Helmholtz resonator, and the STL peak of the resonator will be split into two peaks. (2) The vibration of the elastic wall and the acoustic pressure in the Helmholtz resonator cavity can be effectively decoupled by increasing the stiffness and thickness of elastic wall. (3) When the natural frequency of the elastic wall reaches three times the acoustic resonant frequency of the resonator, the effect of the wall elasticity on the STL can be ignored. These conclusions provide guidance for the rapid and effective design of Helmholtz resonators.