Shock and Vibration

Shock and Vibration / 2019 / Article

Research Article | Open Access

Volume 2019 |Article ID 4894610 | 17 pages | https://doi.org/10.1155/2019/4894610

Analysis of Acoustic Characteristics of Arbitrary Triangular Prism and Quadrangular Prism Acoustic Cavities

Academic Editor: Enrico Zappino
Received10 Jun 2019
Accepted17 Sep 2019
Published26 Dec 2019

Abstract

This paper proposes a method for the analysis of acoustic modals and steady-state responses of arbitrary triangular prism and quadrangular prism acoustic cavities based on the three-dimensional improved Fourier series. First, the geometric models of arbitrary triangular prism and quadrangular prism acoustic cavities are established. To facilitate the calculation, the bottom and top surfaces of the irregular cavity are converted into the unit square domain by a coordinate transformation. Internal sound pressure-admissible functions are constructed, and energy expressions are derived after coordinate transformation based on the three-dimensional improved Fourier series. The acoustic modals of arbitrary triangular prism and quadrangular prism acoustic cavities are obtained by the Rayleigh–Ritz technique. At the same time, a point sound source excitation is introduced into the cavity to further study the steady-state responses of prismatic acoustic cavities with different acoustic impedance boundary conditions. The reliability and universality of the method are verified by comparing with the finite element results. The method and results can provide some references and benchmarks for future research and application.

1. Introduction

In recent years, prismatic acoustic cavities with the complex acoustic walls are widely used in practical engineering, such as cabins, buildings, and automobiles. With the development of society and the improvement of people’s requirements for life quality, people’s requirements for vibration and noise control are also increasingly higher. At present, the research studies on this kind of acoustic field are a focus of topic. However, most research is carried out on the rectangular acoustic cavity, and there are still few studies on irregular acoustic cavities. It is difficult to meet the needs of complex acoustic space design and noise research in actual production and life. Therefore, it is of great theoretical significance and practical value to deeply study the acoustic modals and steady-state responses of arbitrary triangular prism and quadrangular prism acoustic cavities under impedance boundary conditions.

After many theoretical research studies and experiments, people have put forward many accurate and reliable methods for the study of acoustic cavities, for example, the finite element method [17], mixed numerical method [8, 9], boundary element method [1012], statistical energy method [13, 14], and so on.

In practice, dissipative acoustic wall surfaces’ rectangular cavities are a common model, which have been paid much attention by many researchers. According to the first-order shear deformation theory, Zhang et al. [15] studied vibration characteristics of a coupled system consisting of rectangular laminated plates under elastic boundary conditions and cavities under impedance walls. Franzoni and Dowell [16] put forward a modal coupling theory for solving acoustic cavity problems with acoustic absorbing walls. Pan [17] demonstrated that the method (modal analysis) used to describe the acoustic characteristics of an acoustic cavity with the sound absorption boundary is wrong when predicting the sound intensity distribution. Pan [18] further studied the convergence of rigid wall modal models with or without modal coupling in sound intensity domain predictions. On this basis, Pan [19] discussed a new method for sound pressure expansion using the extended modal shape function as the basic function of the sound pressure mode expansion. Martin and Bodrero [20] proposed a method to solve the position optimization problem of acoustic wall impedance with rigid walls. Their goal was to reduce the sound pressure level generated by the internal velocity source of acoustic cavities. Bistafa and Morrissey [21] studied two numerical methods for solving acoustic eigenvalues of rectangular acoustic cavities with arbitrary impedance walls. Naka et al. [22] studied the modal and acoustic response of rectangular acoustic cavities with arbitrary acoustic impedance boundaries by the interval Newton’s generalized dichotomy.

On the basis of the study of rectangular cavities, people gradually pay attention to irregular quasirectangular cavities. Sum and Pan [23] obtained shapes and resonance frequencies of trapezoidal cavities with rigid walls by the coupling of the rectangular cavity. They developed a method for identifying trapezoidal cavity modals. Chen et al. [24] proposed a general analysis method for the inherent characteristics and acoustic vibration characteristics of rectangular plates with arbitrary constraints supported by irregular acoustic cavities. Chen et al. [25] also proposed a domain decomposition method to predict the acoustic characteristics of an arbitrary shell composed of any number of subspaces. Missaoui and Cheng [26] put forward an integral modal method for calculating the acoustic characteristics of irregular cavities. The cavity was discretized into a series of subcavities whose sound pressure is decomposed based on regular or irregular modals of boundary cavities. Xie et al. [27] presented a method based on the weak variational principle to study the irregular cavity under arbitrary dip angle and impedance boundaries. Wang et al. [28] introduced the meshless method into the small-scale acoustic space cavity and derived the numerical calculation model for the acoustic characteristics of the small-scale acoustic space cavity with arbitrary shapes.

At the same time, many great achievements have been made in the study of rotary acoustic cavities. Chen et al. [29] put forward a new calculation method combining the Helmholtz internal integral equation to study the acoustic characteristics of circular and rectangular acoustic cavities. Choi et al. [30] proposed a theoretical method for studying natural frequencies and modal shapes of the annular cavity in local heterogeneous media. By solving the homogeneous wave equation in the elliptical cylindrical coordinate system, Hong and Kim [31] obtained the analytical solution of natural frequencies and mode shapes of elliptical cylinder acoustic cavities. Shao and Mechefske [32] proposed an approximate acoustic model of finite length cylindrical pipe for studying the acoustic radiation of MRI scanning. The sound field obtained by the analytical model was similar to that obtained by the boundary element method model. Lee [33] proposed a semianalytical method for solving the cavity eigenvalue problem with multiple elliptic boundaries. They gave the multipole expansion of sound pressure in the form of angular and radial Mathieu functions. Lee [34] proposed a semianalytical method for solving the eigenvalue problem of two-dimensional smooth variable boundary cavity. They employed the Bessel function and Hankel function to express the multipole expansion of sound pressure. And on that basis, Lee [35] solved acoustic characteristics of three-dimensional cylindrical cavities with multiple elliptical through-holes by a three-dimensional semianalytical formula.

In this paper, on the basis of existing research and limitations, the three-dimensional improved Fourier series method for analyzing acoustic field characteristics of arbitrary triangular prism and quadrangular prism acoustic cavities is proposed. First, the bottom and top surfaces of the irregular cavity are converted into the unit square domain. A three-dimensional improved Fourier series is used to construct acoustic pressure-admissible functions of the cavity and derive energy expressions. The modal characteristics of acoustic cavities are obtained by the Rayleigh–Ritz technique. At the same time, a monopole point sound source is introduced into the cavity to further study the steady-state response of cavities. Compared with the FEM, it is found that this method can accurately calculate acoustic modal characteristics and steady-state responses of prismatic acoustic cavities.

2. Theory and Methods

2.1. Description of the Model

As shown in Figures 1 and 2, acoustic field analysis models of arbitrary triangular prism and quadrangular prism acoustic cavities are established. Figure 1(a) shows the triangular prism cavity model. The bottom surface of the cavity is a straight triangle, and its side lengths are a, b, and c. α and β are two angles. Figure 2(a) shows the quadrangular prism cavity model, in which the bottom surface of the cavity is a straight quadrilateral. The four sides are denoted by a, b, c, and d. The three internal angles are represented by α, β, and γ, respectively. The depth of cavities is expressed by hc. The coordinate system of the whole cavity is established with the bottom surface of the cavity (z = 0) as the reference surface, and its height direction corresponds to the z-direction. Figures 1(b) and 2(b) show the shapes of the acoustic cavity after a coordinate transformation. In this paper, the acoustic field propagation medium inside cavities is air or water. In addition, a point sound source excitation Q is introduced into the acoustic cavity to analyze the steady-state response of acoustic pressure in the acoustic field.

In engineering application, many acoustic wall conditions will be formed due to the difference of actual conditions. When acoustic pressure propagates in different acoustic walls, different energy dissipation situations will occur. Based on the current research results, some common definitions of acoustic wall conditions are given below:(1)Pressure release acoustic walls (also known as Dirichlet acoustic walls): these acoustic walls do not produce energy dissipation or energy reflection:(2)Rigid acoustic walls (also known as Neumann acoustic walls): specifically, the sound velocity or displacement of wall S1 is 0, and the sound wave is completely reflected at the acoustic wall:(3)Impedance acoustic walls: there is a certain impedance relation at the acoustic walls:where P is the sound pressure, the normal orientation of the acoustic wall is denoted by n, j represents the imaginary unit, denotes the mass density, and Zi is the wall impedance.

2.2. Coordinate Transformation

In the present research, the bottom and top surfaces of arbitrary triangular prism and quadrangular prism acoustic cavities are irregular shapes. In order to facilitate the integral calculation, these two surfaces need to be transformed into the unit square by a coordinate transformation. The coordinate transformation does not involve the z-axis direction, so the depth of the cavity remains the same before and after the coordinate transformation. The transformations are shown in Figures 1 and 2.

The equation of coordinate transformation iswhere x-y is the coordinate system before the transformation, ε-η is the coordinate system after the transformation, and xi and yi (i = 1, 2, 3, 4) represent the coordinate value of the i-th vertex in the original coordinate system. In addition, since the triangle has only three sides, the third vertex needs to be mapped twice. Ni (ε, η) is the coordinate transformation form function:

According to the aforementioned coordinate transformation equation and the chained derivation rule, the transformation of the first derivative between the original domain and the transformation domain is the following:

The specific expression of J iswhere

|J−1| is the determinant of the Jacobian for the coordinate transformation.

For the sake of brevity, the above equation can be simplified as

2.3. Acoustic Pressure-Admissible Functions

In order to eliminate the discontinuity or jump problem under impedance boundary conditions, an improved three-dimensional Fourier series is proposed in the new coordinate system to construct the internal acoustic pressure-admissible functions of acoustic cavities under impedance walls:

These acoustic pressure-admissible functions can ensure the first-order continuous derivability of the arbitrary point in the solution domain of the whole acoustic field, where P represents the acoustic pressure expression inside prismatic acoustic cavities, Pi represents the supplementary polynomial, and Amnl is the unknown three-dimensional Fourier coefficient matrix. Their specific expressions arewhere λm=, λn=, and λl=/Lz and is the unknown Fourier series.

2.4. Energy Expressions

In order to obtain the acoustic characteristics of prismatic acoustic cavities, energy expressions of acoustic cavities should be deduced first according to the acoustic principle, and then acoustic pressure functions should be substituted into energy expressions. Finally, acoustic characteristics and steady-state responses of the acoustic field are solved by the Rayleigh–Ritz method.

The Lagrangian energy expression of the acoustic cavity is shown below:where UC is the total acoustic potential energy of the acoustic field, TC is the total kinetic energy of the acoustic cavity, Wwall is the dissipated energy of the impedance walls, and Wext represents the work done by the point sound source. Their specific expressions arewhere ρ represents the density of the acoustic medium inside the acoustic cavity and c represents the propagation speed of the sound wave in the medium.where ω represents the circular frequency of the acoustic cavity and gradp represents the internal acoustic pressure gradient function within the acoustic cavity.

The triangular prism acoustic cavity has five impedance walls, and its dissipation energy can be expressed as

The quadrangular prism acoustic cavity has one more wall than the triangular prism cavity, so the dissipative energy of its impedance wall iswhere Zi is the impedance function of the i-th boundary wall surface. When Zi is a pure imaginary number of the infinity, the rigid wall surface can be obtained. When Zi is a pure imaginary number of the infinitesimal, the pressure release acoustic wall surface can be obtained.where Q represents the volumetric velocity amplitude of the point sound source, A is the amplitude of point sound source (kg/s2), δ is the three-dimensional Dirac delta function, k is the wave number of the sound wave and k=ω/c, and (ε0, η0, z0) represents the position of the point sound source inside the acoustic cavity under the new coordinate system.

2.5. Solution Procedure

The energy expressions of acoustic cavities are substituted into Lagrangian energy equation (15). Setting the partial derivative of the Lagrangian equation LC with respect to three-dimensional unknown Fourier coefficient vectors as zero by the Rayleigh–Ritz method,

The acoustic pressure-admissible functions are substituted into the above formula and written in the matrix form:where K is the stiffness matrix of acoustic cavities, Z is the acoustic pressure dissipation energy matrix caused by impedance walls, M is the mass matrix, and F is the force vector of the internal point sound source. Their specific expressions can be written as

If the point sound source is not placed (F = 0), the free modal characteristics of acoustic cavities can be obtained. But, in this case, the equation contains both the primary terms and the square terms of frequencies at the same time. This is a nonlinear problem, which is not easy to solve directly.

Therefore, the equation is transformed into the following eigenvalue equation by introducing E = ω Amnl:

By solving generalized eigenvalues and eigenvectors of equation (29), the frequency parameters of acoustic cavities and the three-dimensional Fourier unknown coefficient Amnl of acoustic pressure functions can be obtained. It is worth noting that generalized eigenvalues and eigenvectors under the condition of impedance walls are complex numbers. At this time, the real part of the generalized eigenvalue corresponds to the natural frequency of the acoustic cavity, while the imaginary part is the attenuation coefficient of the acoustic mode. In other words, the introduction of the impedance wall does not change the modal characteristics of the acoustic field in the acoustic cavity.

3. Numerical Results

A series of calculating examples of arbitrary triangular prism and quadrangular prism acoustic cavities are presented in this section. The accuracy of the method is verified by comparing with FEM. The FEM model is established in ABAQUS v6.14, the element type adopts the three-dimensional solid element, and the mesh size is set to 0.02 × 0.02 m. In this section, air and water are selected as the acoustic field mediums. The mass density of air and water are defined as ρair = 1.21 kg/m3 and ρwater = 1000 kg/m3. The speed of sound propagation in air and water are set as cair = 340 m/s and cwater = 1480 m/s. In addition, if no special instructions are given, the cavity depth is hc = 0.5 m.

3.1. Convergence Analysis

In Table 1, the convergence and precision of the first eight frequencies of the triangular prism acoustic cavity are studied. Geometric parameters are b/a = 1 and α = 75°. The truncation values of the Fourier series are expressed in terms of M, N, and L, which are used to compare and check the convergence of the solution. The truncated values are from 4 × 4 × 4 to 8 × 8 × 8. It can be seen from the table that the method has great convergence and precision. Under two different mediums, when the truncation value is M × N × L = 8 × 8 × 8, the data have been basically convergent. At the same time, in order to verify the accuracy of the calculation, the results obtained by FEM are also listed in the table. The maximum error between the calculated results and the FEM is not more than 0.032%. And the maximum deviation of calculated results under different truncation values is not more than 0.008%.


MediumsM × N × LModel sequence
12345678

Air4 × 4 × 4192.21235.78340.00356.05390.57413.11413.75462.42
5 × 5 × 5192.21235.78340.00356.04390.57413.09413.75462.40
6 × 6 × 6192.21235.78340.00356.04390.57413.08413.75462.38
7 × 7 × 7192.21235.78340.00356.04390.57413.08413.75462.38
8 × 8 × 8192.21235.78340.00356.04390.57413.07413.75462.38
FEM192.20235.76339.94355.98390.46412.97413.62462.27

Water4 × 4 × 4836.701026.331480.001549.851700.141798.241801.042012.91
5 × 5 × 5836.691026.321480.001549.841700.131798.151801.042012.79
6 × 6 × 6836.691026.321480.001549.831700.131798.111801.032012.73
7 × 7 × 7836.681026.311480.001549.831700.131798.121801.032012.71
8 × 8 × 8836.681026.321480.001549.831700.131798.091801.032012.71
FEM836.641026.201479.801549.601699.701797.601800.502012.20

Next, in Table 2, the convergence and precision of the first eight frequency parameters of the quadrangular prism acoustic cavity are studied. Geometric parameters are b/a = 0.7, c/a = 0.8, α = 80°, and β = 75°. Under two different mediums, the maximum error between the calculated results and the FEM is not more than 0.039% when the truncation value is M × N × L = 8 × 8 × 8. The maximum deviation of calculated results under different truncation values is not more than 0.004%.


MediumsM × N × LModel sequence
12345678

Air4 × 4 × 4208.17235.92340.00344.78396.33398.67413.83464.79
5 × 5 × 5208.17236.07340.00341.78396.33398.66413.83464.78
6 × 6 × 6208.17235.92340.00344.78396.33398.66413.83464.77
7 × 7 × 7208.17235.92340.00344.78396.33398.66413.83464.77
8 × 8 × 8208.17235.92340.00344.78396.33398.66413.83464.77
FEM208.17235.87339.94344.76396.17398.54413.67464.61

Water4 × 4 × 4906.151026.981480.001500.811725.211735.371801.392023.21
5 × 5 × 5906.141026.951480.001500.811725.211735.361801.392023.14
6 × 6 × 6906.141026.941480.001500.801725.201735.361801.392023.13
7 × 7 × 7906.141026.941480.001500.801725.201735.361801.392023.13
8 × 8 × 8906.141026.941480.001500.801725.201735.361801.392023.12
FEM906.061026.901479.801500.501724.601734.801800.802022.50

According to the above analysis of convergence and accuracy, the truncated value of sound pressure-admissible functions of the acoustic cavity calculated in this paper is uniformly set as M × N × L = 8 × 8 × 8.

3.2. Acoustic Modal Analysis of Prismatic Acoustic Cavities

After the analysis of convergence, more modal characteristics of arbitrary triangular prism and quadrangular prism acoustic cavities are studied.

First of all, Table 3 shows the comparison results of the first eight natural frequencies of the triangular prism acoustic cavities with FEM under different shapes. The bottom shape of acoustic cavities contains the isosceles triangle (b/a = 1, α = 75°), the equilateral triangle (b/a = 1, α = 60°), the right triangle (b/a = 2, α = 90°), the obtuse triangle (b/a = 2, α = 120°), and the acute triangle (b/a = 2, α = 75°). As can be seen from the results in the table, the calculated results of this method are matched well with the FEM. This method can accurately calculate natural frequencies of triangular prism acoustic cavities with different geometrical shapes. Figure 3 shows the first three modals of acoustic pressure distribution of the positive triangular prism acoustic cavity when the medium is air. The accuracy of this method is proved again by comparison.


MediumsShapesModel sequence
12345678

AirIsosceles192.21235.78340.00356.04390.57413.07413.75462.38
192.21a235.78340.00356.05390.57413.11413.75462.42
Equilateral226.67226.67340.00392.60408.63408.63453.33453.33
226.65a226.65339.94392.50408.50408.50453.21453.21
Right angle99.54178.88212.49266.47318.68340.00353.16354.27
99.54a178.87212.47266.46318.64339.64353.14354.20
Obtuse angle89.70156.70211.57272.20322.27340.00340.99351.63
89.70a156.71211.70272.70322.17339.52341.94351.01
Acute angle106.15194.28207.14281.53328.05340.00356.19366.87
106.15a194.28207.13281.51328.01339.94356.11366.82

WaterIsosceles836.681026.321480.001549.831700.131798.091801.032012.71
836.64a1026.201479.801549.601699.701797.601800.502012.20
Equilateral986.67986.671480.001708.961778.741778.741973.341973.34
986.61a986.611479.801708.501778.201778.201972.801972.80
Right angle433.30778.64924.941159.941387.201480.001537.291542.13
433.30a778.61924.891159.901387.001479.801537.201541.80
Obtuse angle390.47682.10920.941184.871402.841480.001484.311530.64
390.46a682.12921.121185.501402.701479.401485.301529.90
Acute angle462.08845.71901.661225.491427.981480.001550.461596.96
462.08a845.67901.611225.401427.801479.801550.101596.80

aResults from finite element software ABAQUS.

Table 4 shows the comparison results of the first eight natural frequencies of the quadrangular prism acoustic cavities with different shapes. The bottom shape contains the quadrilateral (b/a = 0.7, c/a = 0.8, α = 70°, β = 75°), the square (b/a = 1, c/a = 1, α = 90°, β = 90°), the rectangle (d/a = 2, c/a = 2, α = 90°, β = 90°), the rhombus (b/a = 1, c/a = 1, α = 60°, β = 120°), and the trapezoid (b/a = 1, c/a = 1.064, α = 70°, β = 90°). Table 4 illustrates the accuracy of this method in calculating quadrangular prism acoustic cavities with different geometric shapes by comparing with the FEM. Figure 4 shows the first three modals of acoustic pressure distribution of the quadrangular prism acoustic cavity when the medium is water.


MediumsShapesModel sequence
12345678

AirQuadrilateral208.17235.92340.00344.78396.33398.66413.83464.77
208.17a235.87339.94344.76396.17398.54413.67464.61
Square170.00170.00240.42340.00340.00340.00380.13380.13
169.99a169.99240.39339.94339.94339.94380.03380.03
Rectangular85.00170.00170.00190.07240.42255.00306.47340.00
85.00a169.99169.99190.05240.39254.98306.41339.94
Rhombus144.75226.67226.67331.34340.00369.53373.46392.60
144.74a226.64226.66331.25339.94369.43373.39392.51
Trapezoidal164.29204.61277.48340.00340.90375.55377.61396.82
164.28a204.59277.44339.94340.85375.43377.51396.69

WaterQuadrilateral906.141026.941480.001500.801725.201735.361801.392023.12
906.06a1026.901479.801500.501724.601734.801800.802022.50
Square740.00740.001046.521480.001480.001480.001654.691654.69
739.97a739.971046.401479.801479.801479.801654.201654.20
Rectangular370.00740.00740.00827.351046.521110.001334.051480.00
370.00a739.97739.97827.291046.401109.901333.801479.80
Rhombus630.09986.67986.731442.291480.001608.541625.651708.09
630.05a986.54986.651441.901479.801608.101625.401708.60
Trapezoidal715.14890.641207.851480.001483.931634.761643.721727.32
715.11a890.571207.701479.801483.701634.201643.301726.80

aResults from finite element software ABAQUS.

For prismatic cavities, cavity depth is also a very important parameter. Tables 5 and 6 list the first eight frequency parameters of the triangular prism (b/a = 1, α = 75°) and the quadrangular prism (b/a = 0.7, c/a = 0.8, α = 80°, β = 75°) acoustic cavities at different cavity depths. As can be seen from two tables, the variation trend of frequency parameters is basically the same in two mediums. The first three natural frequency parameters do not change significantly when the depth is 0.2–0.4 m. Subsequently, natural frequency parameters of cavities decrease with the increase of the depth, and the higher-order frequency parameters are sensitive to the change of the depth.


MediumsDepthsModel sequence
12345678

Air0.2192.21235.78356.04413.07462.38533.42568.43645.54
0.3192.21235.78356.04413.07462.38533.42566.67568.43
0.4192.21235.78356.04413.07425.00462.38466.44486.02
0.5192.21235.78340.00356.04390.57413.07413.75462.38
0.6192.21235.78283.33342.38356.04368.60413.07455.02
0.7192.21235.78242.86309.72338.48356.04413.07430.98
0.8192.21212.50235.78286.53317.41356.04413.07414.64

Water0.2836.681026.321549.831798.092012.712321.952474.362809.99
0.3836.681026.321549.831798.092012.712321.952466.672474.36
0.4836.681026.321549.831798.091850.002012.712030.402115.61
0.5836.681026.321480.001549.831700.131798.091801.032012.71
0.6836.681026.321233.331490.351549.831604.501798.091980.68
0.7836.681026.321057.141348.181473.391549.831798.091876.04
0.8836.68925.001026.321247.261381.651549.831798.091804.88


MediumsDepthsModel sequence
12345678

Air0.2208.17235.92344.78396.33464.77523.95554.15586.23
0.3208.17235.92344.78396.33464.77523.95554.15566.67
0.4208.17235.92344.78396.33425.00464.77473.24486.09
0.5208.17235.92340.00344.78396.33398.66413.83464.77
0.6208.17235.92283.33344.78351.58368.69396.33446.26

Water0.2906.141026.941500.801725.202023.122280.722412.202551.84
0.3906.141026.941500.801725.202023.122280.722412.202466.67
0.4906.141026.941500.801725.201850.002023.122060.002115.92
0.5906.141026.941480.001500.801725.201735.361801.392023.12
0.6906.141026.941233.331500.801530.421604.911725.201942.55

On the basis of above results, the effects of geometric parameters on the natural frequencies of acoustic cavities are further studied. Figures 5 and 6 show the variation of the first three natural frequency parameters of the triangular prism and the quadrangular prism acoustic cavities under different angles (α). The bottom surface of the triangular prism is an isosceles triangle. The bottom surface of the quadrangular prism is a rhombus. It can be seen from the figure that in the two mediums, frequency change trends of cavities are similar, but the amplitude is different. For the triangular prism acoustic cavity, the first-order frequency first increases slowly with the increase of α and reaches the maximum at 60° and then decreases with the increase of the angle. The second-order frequency first decreases with the increase of α and reaches the minimum at 60° and then increases with the increase of α and decreases again after 90°. The third-order frequency first decreases with the increase of α and remains unchanged between 40° and 110° and finally decreases again. For the rhombus prism acoustic cavity, the first-order frequency increases with the increase of α. The second-order frequency first increases with the increase of α and reaches the maximum at 60° and then decreases suddenly. The third-order frequency first increases with the increase of α and reaches the maximum at 40°, then decreases with the increase of α and reaches the minimum at 60°, and then increases slowly again. At 60°, the second-order frequency and the third-order frequency are equal. We can also find that when the angle is equal to 60°, there will be a phenomenon in which the two natural frequencies are equal. Such phenomenon of the same frequency and different modes (Figure 3) due to the symmetry of the structure is called the double-mode phenomenon.

Figures 7 and 8 show the variation of the first three natural frequencies of the triangular prism and the quadrangular prism acoustic cavities under different aspect ratios (b/a). The bottom surface of the triangular prism is a right triangle. The bottom surface of the quadrangular prism is a rectangular. In two different mediums, variation trends of frequencies are very close. But, the frequency parameters of the water cavity are much higher than those of the air cavity. For the triangular acoustic cavity, the first three frequencies all decrease with the increase of b/a. For the rectangular acoustic cavity, the first-order frequency decreases with the increase of b/a. The second-order frequency remains constant between 1.0 and 2.0 and then gradually decreases. The third-order frequency first decreases with the increase of b/a, then remains the same between 2.0 and 3.0, and finally decreases again.

3.3. Steady-State Response Analysis of Prismatic Acoustic Cavities

The steady-state acoustic pressure responses of prismatic cavities under an internal point sound source are studied in this section. Figures 9 and 10 show acoustic pressure response curves of the triangular prism and the quadrangular prism cavities with rigid walls under an internal unit point sound source excitation. The bottom surface area unit is equilateral triangle and isosceles trapezoid (b/a = 1, c/a = 1, α = β = 80°). For the triangular prism acoustic cavity, the loading point is (0.5, 0.5, 0.2). Observation point 1 is (0.82, 0.219, 0.2), and observation point 2 is (0.5, 0.566, 0.4). For the quadrangular prism acoustic cavity, the loading point is (0.5, 0.492, 0.25). Observation point 1 is (0.21, 0.2, 0.1), and observation point 2 is (0.8, 0.59, 0.4). It can be seen that the difference in the physical properties of the medium does not affect the variation trend of the sound pressure response by comparing the response curve trend of two different mediums in a certain frequency band. But, the response amplitude of the water cavity is slightly larger than that of the air cavity. At the same time, the sound pressure response curves calculated by this method are consistent with the FEM basically, which proves the correctness of the acoustic field analysis model again in this paper. The reason for the difference is mainly because the error between the paper and the finite element results is greater at high frequencies. At the same time, the damping is not introduced when calculating the response, so the simulation error for values of resonance peak is larger.

Figures 11 and 12 show acoustic pressure response curves of the triangular prism and the quadrangular prism acoustic cavities with impedance walls under an internal unit point sound source excitation. The impedance value is Z=ρc (200–10j). We set the bottom surface (z = 0) and the top surface (z = hc) of prismatic cavities as impedance walls and the other surfaces as rigid walls. For the triangular cavity, the bottom surface is an isosceles right triangle. The loading point is (0.2, 0.453, 0.1), and the observation point is (0.4, 0.45, 0.3). For the quadrangular acoustic cavity, the bottom surface is a rhombus (α = 60°, β = 120°). The loading point is (0.7, 0.432, 0.25), and the observation point is (0.4, 0.173, 0.15). It can be seen from the figure that the trend of the acoustic pressure response curves calculated by this method in this paper and the FEM is still basically consistent under impedance walls. This indicates that prismatic acoustic cavity analysis models established in this paper can also accurately predict the steady-state response of the acoustic field in prismatic acoustic cavities under impedance walls.

On the basis of correctness, this section studies the effect of impedance walls on the steady-state response of the prismatic acoustic cavity. Figure 13 shows the acoustic pressure response of prismatic acoustic cavities with various impedance walls under the unit point sound source excitation. The impedance value is Z=ρc (200–10j). Since the cavity medium does not obviously affect the sound pressure response curve, the cavity medium is set as air uniformly. For the triangular cavity, the bottom surface is an isosceles triangle (α = 120°). The loading point is (0.3, 0.15, 0.2), and the observation point is (0.5, 0.2, 0.3). For the quadrangular acoustic cavity, the bottom surface is a rhombus (α = 55°, β = 125°). The loading point is (0.45, 0.35, 0.15), and the observation point is (0.8, 0.3, 0.35). It can be seen from figures that the introduction of impedance walls can suppress the sound pressure level of the steady-state response of acoustic cavities, and the inhibition effect increases with the number of impedance walls. However, the introduction of impedance walls does not affect the trend of sound pressure response.

4. Conclusions

In this paper, an acoustic field characteristics analysis method of arbitrary triangular prism and quadrangular prism acoustic cavities is firstly presented based on the three-dimensional improved Fourier series. The modal and response results of the prism acoustic cavity with various impendence walls are obtained by transforming the x-y-z coordinate system into the ε-η one. After verifying the correctness of this method, the influence of different geometric parameters on the acoustic field characteristics is analyzed.

The examples and numerical results in this study indicate some implications and conclusions:(1)It is found that this method can calculate acoustic characteristics and steady-state responses of arbitrary triangular prism and quadrangular prism acoustic cavities exactly. This method can be applied to the spatial acoustic characteristics analysis and noise control research in this study field.(2)From the theoretical part of the research process, this method does not involve any complicated theories, equations, or programs. This method has great reference values for the study of acoustic characteristics and steady-state responses of similar structures.(3)From the numerical results, it can be seen that geometric parameters, medium properties, and boundary conditions all have effects on acoustic characteristics and steady-state responses of prismatic acoustic cavities. Therefore, special attention should be paid to these factors when designing and applying cavity structures.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no potential conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (grant no. 51679056) and Natural Science Foundation of Heilongjiang Province of China (E2016024).

References

  1. S. Dhandole and S. V. Modak, “A constrained optimization based method for acoustic finite element model updating of cavities using pressure response,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 399–413, 2012. View at: Publisher Site | Google Scholar
  2. V. Easwaran and A. Craggs, “On further validation and use of the finite element method to room acoustics,” Journal of Sound and Vibration, vol. 187, no. 2, pp. 195–212, 1995. View at: Publisher Site | Google Scholar
  3. W. Larbi, J.-F. Deü, and R. Ohayon, “A new finite element formulation for internal acoustic problems with dissipative walls,” International Journal for Numerical Methods in Engineering, vol. 68, no. 3, pp. 381–399, 2006. View at: Publisher Site | Google Scholar
  4. M. Petyt, G. H. Koopmann, and R. J. Pinnington, “The acoustic modes of a rectangular cavity containing a rigid, incomplete partition,” Journal of Sound and Vibration, vol. 53, no. 1, pp. 71–82, 1977. View at: Publisher Site | Google Scholar
  5. M. Petyt, J. Lea, and G. H. Koopmann, “A finite element method for determining the acoustic modes of irregular shaped cavities,” Journal of Sound and Vibration, vol. 45, no. 4, pp. 495–502, 1976. View at: Publisher Site | Google Scholar
  6. H. Li and X. Huang, “Analysis of sound field characteristics in acoustic space with different shapes,” in Proceedings of the National Conference on Vibration and Noise Control, Baltimore, MA, USA, April 2010. View at: Google Scholar
  7. X. Shen and Y. Shen, “Study on the location of acoustic materials in a small room based on finite element method,” Journal of Acoustics, vol. 4, pp. 324–328, 2005. View at: Google Scholar
  8. R. Gao, Y. Zhang, and D. Kennedy, “A hybrid boundary element-statistical energy analysis for the mid-frequency vibration of vibro-acoustic systems,” Computers & Structures, vol. 203, pp. 34–42, 2018. View at: Publisher Site | Google Scholar
  9. S. Kopuz and N. Lalor, “Analysis of interior acoustic fields using the finite element method and the boundary element method,” Applied Acoustics, vol. 45, no. 3, pp. 193–210, 1995. View at: Publisher Site | Google Scholar
  10. A. F. Seybert and C. Y. R. Cheng, “Application of the boundary element method to acoustic cavity response and muffler analysis,” Journal of Vibration and Acoustics, vol. 109, no. 1, pp. 15–21, 1987. View at: Publisher Site | Google Scholar
  11. P. K. Banerjee, S. Ahmad, and H. C. Wang, “A new BEM formulation for the acoustic eigenfrequency analysis,” International Journal for Numerical Methods in Engineering, vol. 26, no. 6, pp. 1299–1309, 1988. View at: Publisher Site | Google Scholar
  12. G. Wu, Numerical Simulation and Analysis of Indoor Three-Dimensional Acoustic Wave Field Based on Boundary Element Method, Nanjing university of science and technology, Nanjing, China, 2013.
  13. X. Hu, “Research on acoustic optimization design based on statistical energy analysis,” Civil Aircraft Design and Research, vol. 3, pp. 9–13, 2010. View at: Google Scholar
  14. Y. Hu, K. . a. Chen, and K. Pan, “Research on cabin noise reduction based on statistical energy analysis,” Noise and Vibration Control, vol. 27, no. 2, pp. 65–68, 2007. View at: Google Scholar
  15. H. Zhang, D. Shi, S. Zha, and Q. Wang, “Vibro-acoustic analysis of the thin laminated rectangular plate-cavity coupling system,” Composite Structures, vol. 189, pp. 570–585, 2018. View at: Publisher Site | Google Scholar
  16. L. P. Franzoni and E. H. Dowell, “On the accuracy of modal analysis in reverberant acoustical systems with damping,” The Journal of the Acoustical Society of America, vol. 97, no. 1, pp. 687–690, 1995. View at: Publisher Site | Google Scholar
  17. J. Pan, “A note on the prediction of sound intensity,” The Journal of the Acoustical Society of America, vol. 93, no. 3, pp. 1641–1644, 1993. View at: Publisher Site | Google Scholar
  18. J. Pan, “A second note on the prediction of sound intensity,” The Journal of the Acoustical Society of America, vol. 97, no. 1, pp. 691–694, 1995. View at: Publisher Site | Google Scholar
  19. J. Pan, “A third note on the prediction of sound intensity,” The Journal of the Acoustical Society of America, vol. 105, no. 1, pp. 560–562, 1999. View at: Publisher Site | Google Scholar
  20. V. Martin and A. Bodrero, “An introduction to the control of sound fields by optimising impedance locations on the wall of an acoustic cavity,” Journal of Sound and Vibration, vol. 204, no. 2, pp. 331–357, 1997. View at: Publisher Site | Google Scholar
  21. S. R. Bistafa and J. W. Morrissey, “Numerical solutions of the acoustic eigenvalue equation in the rectangular room with arbitrary (uniform) wall impedances,” Journal of Sound and Vibration, vol. 263, no. 1, pp. 205–218, 2003. View at: Publisher Site | Google Scholar
  22. Y. Naka, A. A. Oberai, and B. G. Shinn-Cunningham, “Acoustic eigenvalues of rectangular rooms with arbitrary wall impedances using the interval Newton∕generalized bisection method,” The Journal of the Acoustical Society of America, vol. 118, no. 6, pp. 3662–3671, 2005. View at: Publisher Site | Google Scholar
  23. K. S. Sum and J. Pan, “Effects of the inclination of a rigid wall on the free vibration characteristics of acoustic modes in a trapezoidal cavity,” The Journal of the Acoustical Society of America, vol. 119, no. 4, pp. 2201–2210, 2006. View at: Publisher Site | Google Scholar
  24. Y. Chen, G. Jin, Z. Feng, and Z. Liu, “Modeling and vibro-acoustic analysis of elastically restrained panel backed by irregular sound space,” Journal of Sound and Vibration, vol. 409, pp. 201–216, 2017. View at: Publisher Site | Google Scholar
  25. Y. Chen, G. Jin, and Z. Liu, “A domain decomposition method for analyzing a coupling between multiple acoustical spaces (L),” The Journal of the Acoustical Society of America, vol. 141, no. 5, pp. 3018–3021, 2017. View at: Publisher Site | Google Scholar
  26. J. Missaoui and L. Cheng, “A combined integro-modal approach for predicting acoustic properties of irregular-shaped cavities,” The Journal of the Acoustical Society of America, vol. 101, no. 6, pp. 3313–3321, 1997. View at: Publisher Site | Google Scholar
  27. X. Xie, H. Yang, and H. Zheng, “A weak formulation for interior acoustic analysis of enclosures with inclined walls and impedance boundary,” Wave Motion, vol. 65, pp. 175–186, 2016. View at: Publisher Site | Google Scholar
  28. H. Wang, X. Y. Zeng, and L. Chen, “Numerical method of Galerkin sound field in small scale acoustic space without grid,” Journal of Northwest Polytechnical University, vol. 30, no. 1, pp. 102–107, 2012. View at: Google Scholar
  29. I. L. Chen, J. T. Chen, S. R. Kuo, and M. T. Liang, “A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method,” The Journal of the Acoustical Society of America, vol. 109, no. 3, pp. 982–998, 2001. View at: Publisher Site | Google Scholar
  30. H. G. Choi, S. W. Yoo, J. D. Jeong, and J. M. Lee, “Acoustic characteristics of annular cavities with locally non-uniform media,” Journal of Sound and Vibration, vol. 266, no. 5, pp. 967–980, 2003. View at: Publisher Site | Google Scholar
  31. K. Hong and J. Kim, “Natural mode analysis of hollow and annular elliptical cylindrical cavities,” Journal of Sound and Vibration, vol. 183, no. 2, pp. 327–351, 1995. View at: Publisher Site | Google Scholar
  32. W. Shao and C. K. Mechefske, “Acoustic analysis of a finite cylindrical duct based on Green’s functions,” Journal of Sound and Vibration, vol. 287, no. 4-5, pp. 979–988, 2005. View at: Publisher Site | Google Scholar
  33. W. M. Lee, “Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method,” Journal of Sound and Vibration, vol. 330, no. 20, pp. 4915–4929, 2011. View at: Publisher Site | Google Scholar
  34. W.-M. Lee, “Eigenproblems of two-dimensional acoustic cavities with smoothly varying boundaries by using the generalized multipole method,” Meccanica, vol. 49, no. 7, pp. 1617–1628, 2014. View at: Publisher Site | Google Scholar
  35. W. M. Lee, “Acoustic eigenproblems of elliptical cylindrical cavities with multiple elliptical cylinders by using the collocation multipole method,” International Journal of Mechanical Sciences, vol. 78, pp. 203–214, 2014. View at: Publisher Site | Google Scholar

Copyright © 2019 Dongyan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

135 Views | 70 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder