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A three-dimensional modeling method for the trapezoidal cavity and multi-coupled cavity with various impedance boundary conditions
Applied Acoustics 154 (2019) 213–225
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Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
A three-dimensional modeling method for the trapezoidal cavity and multi-coupled cavity with various impedance boundary conditions Dongyan Shi a, Gai Liu a, Hong Zhang b,⇑, Wenhui Ren a, Qingshan Wang c,⇑ a
College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, PR China College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China c State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, PR China b
a r t i c l e
i n f o
Article history: Received 5 December 2018 Received in revised form 10 March 2019 Accepted 1 May 2019
Keywords: Modified Fourier series Acoustic characteristics Various impedance boundary conditions Multi-coupled cavity Trapezoidal cavity
a b s t r a c t A three-dimensional modified Fourier series method is applied to study the acoustic characteristics of the trapezoidal cavity and coupled cavity with various impedance boundary conditions. The formulation is constructed to describe the cavity systems based on the energy principle. The admissible sound pressure function is set to a Fourier cosine series and supplementary functions. These supplementary functions can eliminate the discontinuous or jumping phenomenon in the boundaries. The trapezoidal cavity can be transformed into regular rectangular cavity by coordinate transformation. The triangular cavity can be obtained by changing the dimensions of the trapezoidal cavity. The coupled cavity consists of two irregular triangular cavities and the coupling interface is regarded as a non-dissipative surface. The accuracy and convergence of the present results are verified by comparison with the results obtained by finite element calculation and experiment. The natural frequencies and mode shapes of the trapezoidal cavity and coupled cavity are studied. Besides, the sound pressure response under monopole sound source excitation is obtained. What’s more, some results are obtained in the basis of effects of various factors on acoustic characteristics, such as shapes, geometric parameters and impedance values. These results provide a benchmark for the future researches. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction As a basic enclosure, trapezoidal cavity with various impedance boundary conditions has a large proportion in practical applications, such as used in workshops, ship compartments and civil buildings. Besides, some polygonal prismatic cavities can be regarded as the coupled cavities composed by trapezoidal cavity or triangular cavity. Therefore, the study of trapezoidal cavity and coupled cavity has a good guiding significance to the practical engineering, like noise control and room designs. The indoor sound field simulation can be traced back to the 1950s [1,2]. Then, some methods are proposed to analyze the acoustic cavity, such as decomposition method [3,4], the finite element method [5,6], boundary element method [7,8], Rayleigh-Ritz method [9,10], etc. Based on these, the acoustic characteristics of trapezoidal cavity, triangular cavity and triangular coupled cavity with various impedance boundary conditions are studied in the paper.
⇑ Corresponding authors. E-mail addresses: [email protected] (H. Zhang), [email protected] (Q. Wang). https://doi.org/10.1016/j.apacoust.2019.05.001 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
At present, there are many literatures that have studied the single-cavity. Sharma and Bhat [11] analyzed the resonant frequency of isosceles triangle microstrip resonator by using the full wave formula of spectrum domain technique. Xiujuan and Weikang [12] presented a method to calculate the acoustic absorption coefficient of the microperforated panel absorbers and measured the acoustic absorption coefficients of three kinds of triangular prism absorbers. Tam [13] studied the acoustic modes of twodimensional rectangular resonators. Besides, he discussed the physical significance of strong radiation damping on cavity resonance caused by external flow at low Mach number. Missaoui and Cheng [14] calculated the natural frequencies of irregularshaped cavities by discretizing the cavity into a series of sub cavities. The acoustic pressure was decomposed into the modal basis of regular sub cavities. Xie [3] divided the trapezoidal acoustic domain into several sub-cavities with trapezoidal and rectangular faces. Then, the acoustic enclosures with walls in various inclination and impedance boundary conditions were studied by coupling each sub-cavity with adjacent sub-cavities. Sum and Pan [15] used the coupling between rigid-walled modes of a rectangular cavity to obtain the shapes and resonance frequencies of rigid-walled modes of a trapezoidal cavity. Besides, they studied how the free vibration
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characteristics of trapezoidal cavity modes change with the inclination and what determines these changes. Lee [16] proposed a semi-analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries. The sound pressure expression of this method was formulated in terms of angular and radial Mathieu functions. Ortiz [17] used a fast and efficient time domain method to study the time response of any point in a three-dimensional open rectangular cavity. Sumbatyan et al. [18] applied a system of Galerkin’s basis functions which are orthogonal in a unit cube to analyze the effect of the inclination of slanted boundary planes on the low eigenfrequencies for rectangular parallelepiped rooms. In summary, many results for different shapes of the single-cavity are obtained. However, few studies have been done on the analysis of the steady-state response of trapezoidal cavity and triangular cavity with various impedance boundary conditions. Most of the models being used in engineering are irregular straight-sided cavities. These models can be regarded as multicoupled cavities. As the practical enclosure, there are some literatures for the multi-coupled cavity. Harris and Feshbach [19] studied the problem of coupled rooms from the ‘‘wave” point of view and treated the coupled rooms as a boundary value problem to obtain the approximate solution. Jin [20] proposed a modeling method for the analysis of a rectangular opened enclosure coupled with a semi-infinite external fields. It was been used to study the effect of sizes and positions of the opening and wall impedance on the behaviors of opened enclosure system. Valeau et al. [21] obtained the three-dimensional diffusion equation by using a finite-element solver and simulated the spatial variation of sound intensity level with aperture coupling. Meissner [22] studied the acoustical properties of two rooms by a system of eigenmodes, and summarized the laws of frequency and acoustic energy losses with dimensions. Billon [23] proposed a numerical model to predict the reverberant sound field in two cavities coupled through an open aperture. This method can be used to predict the acoustic energy distribution and attenuation in a single room. Meissner [24–26] studied the acoustical properties of an irregularly shaped room consisting of two coupled rectangular sub-rooms, and predicted the distribution of the sound pressure and the reverberation time of the coupled system. Through these literatures, it can be found that most of the studied models are rectangular coupled cavities. However, there are few studies on the non-rectangular acoustic cavities.
Due to the restrictions of the cavity in the current researches, a modified three-dimensional Fourier method [27] is used in this paper to study the natural frequencies and steady-state response of trapezoidal cavity, triangular cavity and coupled cavity, and various impedance boundary conditions are considered. It has to be emphasized that although the trapezoidal cavity is studied both in Ref. [3] and in the present paper, the basic formulas are different. In addition, the trapezoidal cavity is studied by transforming its coordinates into rectangular cavity’s coordinates rather the decomposition method. The triangular cavity can be achieved from the trapezoidal cavity. In this paper, the three-dimensional Fourier cosine function and complementary functions are employed for the modeling of systems. The coupling interface of the coupled cavity is a non-dissipative surface. The accuracy and convergence of the results are verified by comparing the current results with the data obtained by finite element calculation and experiment. 2. Theoretical formulations 2.1. Description of the model As shown in Fig. 1, the model to be analyzed is given. For a trapezoidal cavity, its shape can be determined by the bottom shape and height h of the cavity. The bottom shape of the trapezoidal cavity depends on the length a, b and the angle a, b. The trapezoidal cavity of different shapes can be achieved by changing the control variables. For example, if a = b – 0, a = b = 0, a = 0 and b – 0, the cavity can be an isosceles trapezoidal cavity, a rectangular cavity and a right-angled trapezoidal cavity respectively. In addition, when b = 0, we can get the model of triangular cavity as depicted in Fig. 1(b). Based on the above research, a quadrilateral cavity can be achieved by coupling two triangular cavities on a plane which parallels to yz, it is shown in Fig. 2. A global rectangular coordinate system (o-xyz) and local-coordinate systems (o1-x1y1z1, o2-x2y2z2) of the coupled model are built. Q is the monopole sound source of the coupled cavity. Similarly, different shapes can be got by changing the angle ai, bi, and the dimension ai, hi of the cavities A and B. If a1 = a2 = 0 or b1 = b2 = 0, then right triangle cavity, acute triangle cavity and obtuse triangle cavity are respectively achieved by changing the corresponding bi or ai. If a1 + b1 = 90° and a2 + b2 = 90°, a1 = b2 and a2 = b1, the enclosure is a rectangle cavity and a parallelogram cavity respectively. What’s more, when the
Fig. 1. Geometry and coordinate system of the triangular cavity and trapezoidal cavity with various impedance boundary conditions.
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length (o-ngf). The formula of coordinate transformation is as follows:
8 9 8 9 > > 8 < xi > = X = y ¼ Ni ðn; g; fÞ yi > > > : ; i¼1 : > ; z zi
ð3Þ
in which xi, yi and zi are the coordinates of the ith node in the trapezoidal cavity. Ni (n, g, f) is the shape function of the ith node:
N1 ¼ ð1 nÞð1 gÞð1 fÞ; N2 ¼ ð1 nÞgð1 fÞ; N3 ¼ ngð1 fÞ; N4 ¼ nð1 gÞð1 fÞ; N5 ¼ ð1 nÞð1 gÞf; N6 ¼ ð1 nÞgf; N7 ¼ ngf; N8 ¼ nð1 gÞf ð4Þ The relationship between the first derivative of the coordinate system (o-xyz) and (o-ngf) is:
Fig. 2. Geometry and coordinate system of the two triangular coupled cavity with various impedance boundary conditions.
8 @ðÞ 9 > > > < @n > =
8 @ðÞ 9 2 @x > > > < @x > = 6 @n @ðÞ @ðÞ @x ¼ J ¼6 @g > @y > 4 @g > > > > > > : @ðÞ ; : @ðÞ ; @x @f
angle and the dimensions are various, it is an irregular quadrilateral. Impedance boundary conditions of the cavity can be expressed by assigning corresponding impedance value on each of the surfaces, that is:
@pðx; y; zÞ jq xpðx; y; zÞ ¼ m @n Zi
ð1Þ
where n is the outward normal to the boundary and j is the imaginary number (j2 = 1). qm is the mass density of the acoustic medium. x is the angular frequency and p(x, y, z) denotes the admissible sound pressure function. Zi is the specific acoustic impedance value of the ith surface. The coupling interface of the coupled cavity is a non-dissipative surface. It does not lose energy when the sound pressure passes through the coupling interface. The sound pressure and the velocity are continuous at the coupling interface, so the boundary conditions of the coupling interface are:
p1 jx1 ¼a1 ¼ p2 jx2 ¼a2 ;
@p1 @p ¼ 2 @x x1 ¼a1 @x x2 ¼a2
ð2Þ
2.2. Transformation of coordinate In order to facilitate the calculation of trapezoidal cavity, it is necessary to change trapezoidal cavity into rectangular cavity. As shown in Fig. 3, the coordinate system of the trapezoidal cavity (o-xyz) with various impedance boundary conditions transformed into the coordinate system of the rectangular cavity with unit
@z
@f
@y @n @y @g @y @f
38 @ðÞ 9 > > @x > > 7< @ðÞ = @z 7 @g 5 @y > > > > @z : @ðÞ ; @z @n
@f
ð5Þ
@z
where:
@I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þgf þ ðI1 I2 þ I3 I4 Þg @n þ ðI1 I4 I5 þ I8 Þf I1 þ I4 ð6aÞ @I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þnf þ ðI1 I2 þ I3 I4 Þn @g ð6bÞ þ ðI1 I2 I5 þ I6 Þf I1 þ I2 @I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þng þ ðI1 I4 I5 þ I8 Þn @f þ ðI1 I2 I5 þ I6 Þg I1 þ I5 ð6cÞ in which I can be x, y or z. J is the Jacobin matrix. 2.3. Energy expressions The acoustic modes and the steady-state response of the cavity with various impedance boundary conditions are studied mainly. Thus, the acoustic characteristics have been studied based on the Rayleigh-Ritz energy method. By transforming the threedimensional acoustic problem into a three-dimensional solid problem, a modified three-dimensional Fourier series method is proposed to express the admissible sound pressure functions of the cavity. There are many literatures describing different enclosures with Lagrangian function [28,29], and their development is mature. Therefore, the Lagrangian function is used to describe the energy equation of the proposed cavity.
Fig. 3. Schematic diagram of geometric and coordinate transformation.
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The Lagrangian function (L) for the coupled cavity with various impedance boundary conditions can be written as:
L1 ¼ U 1cavity T 1cavity P W f 1 W p
ð7aÞ
L2 ¼ U 2cavity T 2cavity P W f 2
ð7bÞ
where L1 and L2 represent the Lagrangian function of each coupled cavity respectively. Uicavity is the total acoustic potential energy of the ith coupled cavity. Ticavity denotes the total kinetic energy for the ith coupled cavity, Wfi expresses the dissipated acoustic energy on impedance surfaces, Wp is the work done by the monopole sound source. P represents the coupling potential energy of the coupled cavity. The total potential energy (Uicavity) and kinetic energy (Ticavity) which are stored in the ith cavity can be given as:
U icavity ¼ ¼
Z
1
V
Z
1 2q
2 m cm
0
and
T icavity ¼ ¼
Z
1
1
Z
0
1 0
Z
1
p2i ðn; g; fÞjJi jdndgdf
ð8Þ
2
ðgradpÞ dV
2qm x2
V
Z
1 2qm x2
Z
1
0
1
Z
0
0
1
For the triangular cavity, the coordinate of the node 1 and 2 in the Fig. 3 should be equal, as does node 5 and 6. Then the energy expressions can be obtained by using Eqs. (7)–(14). 2.4. Admissible sound pressure function and system equations
p2 ðx; y; zÞdV
2qm c2m
The coupling potential energy of the coupled cavity (P) is: Z 1 @ðp1 ðx; y;zÞ p2 ðx;y;zÞÞ dS P¼ p ðx;y; zÞ @x q m x2 S 1 9 8 @y1 @z1 > > > > > > @p1 ðn1 ;g1 ;f1 Þ @n1 @ g1 @ g1 > dgdfÞj n ¼ 1 > ðp1 ðn1 ; g1 ;f1 Þ > > > > @n1 @x1 @y1 @z1 > > 1 > > @f1 @f1 > Z 1 Z 1> < n2 ¼ 1 = 1 ¼ @y2 @z2 > q m x2 0 0 > > > > > > > > > ðp1 ðn1 ; g1 ;f1 Þ @p2 ðn2 ;g2 ;f2 Þ @n2 @ g2 @ g2 dgdfÞj > @n2 @x2 @y2 @z2 > > n1 ¼ 1 > > > @f2 @f2 > > ; : n2 ¼ 1 ð14Þ
2 h i 2 h i 2 3 J1 OTi þ J1 OTi i i 6 7 1 2 4 5jJi jdndgdf h i 2 T 1 þ Ji Oi 3
As the factor affecting the convergence and accuracy of the calculation, the selection of the sound pressure function is the most critical part of the paper. A modified Fourier series for the various impedance boundary conditions is used in the paper to express the pressure function in the coupled cavity [30]. The series solution of pressure distribution function of the unit cube can be shown as:
pðn; g; fÞ ¼ pa ðn; g; fÞ þ pb ðn; g; fÞ þ1 P þ1 P þ1 P Amnl cosðpmnÞcosðpngÞcosðplfÞ pa ðn; g; fÞ ¼ m¼0 n¼0 l¼0
ð9Þ in which [Ji1]j is the jth line of the Jacobian inverse matrix for the ith coupled cavity, cm is the sound speed of the acoustic medium. Oi expresses the first derivative of the admissible sound pressure functions versus the coordinate system after the transformation:
@pi ðn; g; fÞ @pi ðn; g; fÞ @pi ðn; g; fÞ Oi ¼ @n @g @f
ð10Þ
Since each coupled cavity has 5 impedance surfaces, the dissipated energy of the ith cavity (Wfi) can be shown as: W fi ¼ W fa þ W fb þ W fc 8 @y
9 @z = @g dgdfÞj n¼0 @z ; @f @x @z R 1 R 1 p2i ðn;g;fÞ @n @n dndfÞj þ ð g¼0 0 0 jxZ 4 @x @z @f @f @x @y R R 2 @n dndgÞj þ 1 1 ðpi ðn;g;fÞ @n f¼0 0 0 jxZ 6 @x @y @g @g
Z 0
1
Z 0
1
Z 0
1
p1 ðn; g; fÞQ jJjdndgdf jx
4p A dðx x0 Þdðy y0 Þdðz z0 Þ jqm cm k
þ
þ1 P
A1mnl sinðpmnÞcosðpngÞcosðplfÞ
1 þ1 P P
A2mnl cosðpmnÞsinðpngÞcosðplfÞ
m¼0 n¼2 l¼0
þ
þ1 P þ1 P
1 P
m¼0 n¼0 l¼2
þ
1 P
A3mnl cosðpmnÞcosðpngÞsinðplfÞ
1 þ 1 P P
m¼2 n¼2 l¼0
þ
@z @n
)
dndfÞjg¼1 @z @f 9 @y = @n dndgÞj f¼1 @y ; @g
ð12Þ
where Q expresses the distribution function of the source intensity, it can be shown as:
Q¼
m¼2 n¼0 l¼0
1 þ1 1 P P P m¼2 n¼0 l¼2
in which Wfa, Wfb and Wfc represents the dissipated energy of the impedance surfaces in the direction n, g, f respectively. The work done by the monopole sound source (Wp) is given as:
1 2
1 þ 1 þ 1 P P P
þ
ð11Þ
Wp ¼
pb ðn; g; fÞ ¼
ð13Þ
in which d is 3-D Dirac function, A represents the amplitude of sound source, k is the wave number. The monopole sound source is located at (x0, y0, z0).
þ 1 P
1 1 P P
m¼0 n¼2 l¼2
ð15Þ
A4mnl sinðpmnÞsinðpngÞcosðplfÞ A5mnl sinðpmnÞcosðpngÞsinðplfÞ A6mnl cosðpmnÞsinðpngÞsinðplfÞ
in which Amnl ; A1mnl ; A2mnl ; A3mnl ; A4mnl ; A5mnl and A6mnl are the threedimensional Fourier coefficients of the trigonometric series. pa is the traditional Fourier series, and pb is the supplementary function which can eliminate the discontinuous or jumping phenomenon in the boundaries. In practical computations, the result can be acquired by truncating the infinite series to the numerical value M, N and L. The coupled cavity is composed of two sub-cavities, so different Fourier coefficients are used. Aimnl is used for the cavity in which the monopole sound source is located, and Bimnl is for another. Let the partial derivation of Lagrangian equation Li with the respect to the Fourier coefficient be zero, Eq. (7) can be expressed as follows:
@U 1cavity @T 1cavity @P @W f 1 @W p ¼ @A @A @A @A @A
ð16aÞ
@U 2cavity @T 2cavity @P @W f 2 ¼0 @B @B @B @B
ð16bÞ
Substituting Eqs. (6)–(15) into Eq. (16). According to the Rayleigh-Ritz technology [31,32], the equations can be expressed in a matrix form:
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ðK 1 C 11 xZ 1 x2 M 1 ÞA þ C 12 B ¼ Q
ð17Þ
C 21 A þ ðK 2 xZ 2 x2 M 2 ÞB ¼ 0
in which Ki, Mi and Zi are the stiffness matrix, mass matrix and impedance matrix of the ith coupled cavity respectively. C11 and C12 are coupling matrices generated by the cavity in which the monopole sound source is located. C21 is the coupling matrix created by another cavity. A and B are the matrices of the Fourier coefficients Aimnl andBimnl respectively. When the modal characteristics of closed space are studied, we only need to set the right side of the Eq. (17) to zero. For convenience of the calculation, it is necessary to get a linear equation. Through the transformation, Eq. (17) can be shown as follows [30]:
ðR xSÞG ¼ 0
ð18Þ
where:
2
0 K 1 C 11 6K C Z 1 1 11 6 R¼6 4 0 0 C 21 0 2 0 0 K 1 C 11 0 6 0 M 0 0 1 6 S¼6 4 0 0 K2 0 0
0
0
0 C 12 0 K2 3
3
0
0 7 7 7; K2 5 Z 2 2
A
ð19Þ
3
7 6 xA 7 7 7 6 7 7; G ¼ 6 5 4 B 5
M2
xB
We can get the natural frequencies and the matrix of Fourier coefficients (A and B) by solving the eigenvalues and eigenvectors of the Eq. (19). The sound pressure distribution in the sub-cavities can be obtained by substituting the Fourier series into the Eq. (15). Besides, the sound pressure response equation under the effect of monopole sound source can be achieved by transforming the Eq. (18) into:
1 1 A ¼ K 1 C 11 xZ 1 x2 M 1 C 12 K 2 xZ 2 x2 M 2 C 21 Q 1 2 B ¼ K 2 xZ 2 x M 2 C 21 A ð20Þ The curve of the steady-state response can be achieved by sweeping the frequency. When the model to be analyzed is a single cavity, the corresponding Lagrangian formula is Eq. 7(a), but the coupling potential energy P need to be removed. The acoustic characteristics and steady-state response of the single cavity under the effect of monopole sound source can be obtained by Eqs. (7)–(20). 3. Numerical results and discussion Some numerical examples are used to study the acoustic characteristics and steady-state responses of trapezoidal cavity, triangular cavity and coupled cavity. Besides, the results verified the
convergence, accuracy and effectiveness of the three-dimensional modified Fourier method. On these bases, the effects of acoustic medium, boundary conditions and dimensions on the cavities are studied. The dimensions of the examples used in this paper have been listed in the Table 1. 3.1. Study of trapezoidal and triangular cavity 3.1.1. Convergence study and modal analysis Convergence and accuracy of the method are verified by comparing the obtained natural frequencies and mode shapes with the finite element results. Besides, the effects of dimensions on the natural frequencies of the cavities are studied. The geometric dimensions of the trapezoidal cavity used in this part are a = b = h = 1 m, and the inclination angles are a = b = 30°. The sizes of the triangular cavity are a = h = 1 m, the inclination angles are a = 30°, b = 60°. The medium in the enclosure is air, the mass density and speed of sound propagation in the air are qm = 1.21 kg/m3 and cm = 340 m/s respectively. The boundary conditions of the comparison models are the rigid walls, it can be achieved by setting the impedance value as a large pure imaginary number (such as j1010) in the numerical calculation and simulation. The frequencies of the trapezoidal cavity and triangular cavity are listed in Table 2. Since the zero frequency represents the rigid body mode, it is not studied in the following examples. Because the infinite series are truncated to the numerical value M, N and L, the convergence of the method can be approved by comparing the corresponding frequencies when the numbers of the M, N and L are selected different values. In addition, the accuracy of the present method is verified by comparing the convergent results with the frequencies obtained by FEM. It can be seen from the Table 2 that the natural frequencies converge rapidly with the increase of the truncated values M, N and L. Particularly, the frequency convergence to a certain value when M = N = L = 6. The max error between convergent frequencies and results obtained by finite element is no more than 0.003%. Moreover, the difference between the frequency (M N L = 3 3 3) and convergence frequency (M N L = 8 8 8) is less than 0.02%. Therefore, the present method has a good convergence and accuracy in dealing with trapezoidal cavity and triangular cavity problems. For illustrative purposes, the 2nd, 3rd, and 4th mode shapes for the triangular cavity and the trapezoidal cavity are drawn in Fig. 4. The mode shapes can represent the distribution of sound pressure for the cavities directly. From the figures, we can find that the mode shapes for isosceles trapezoidal cavity has a perfect symmetry. The more increase of the frequency, the more deformation of the mode shapes. Besides, if the vertices of the trapezoidal mode shapes numbered as the form of Fig. 3, then an equivalent triangular can be achieved by compressing the vertex 5 to the line formed by the vertices 8 and 6. It can be found that there is no difference
Table 1 Dimensions of the examples. Section
Shape
3.1.1 3.1.2 3.1.3
Triangle Trapezoid Triangle Trapezoid Triangle 1 Triangle 2 Triangle 1 Triangle 2 Triangle Trapezoid
Model Size a(m)
3.2
4
b(m)
1 – 1 1 0.8 – 0.8 1 0.866 – 2 1 – Changing with the size a (Fig. 10(a)) 0.47 – 0.38 0.28
h(m)
a(°)
b(°)
1 1 1.5 1.5 1 1 1
30 30 40 40 45 14 45
60 30 30 30 30 23.413 30
0.47 0.39
0 0
45 53
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Table 2 Convergence and accuracy of natural frequencies (Hz) for the cavities with the rigid walls. Shape
MNL
Mode Number 2
3
4
5
6
7
8
Triangle
333 444 555 666 777 888 FEM Error (%)
102.142 102.142 102.138 102.138 102.138 102.138 102.136 0.002
158.089 158.078 158.077 158.075 158.075 158.075 158.072 0.002
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
198.325 198.325 198.323 198.323 198.323 198.323 198.323 0.001
206.298 206.267 206.267 206.264 206.264 206.264 206.261 0.001
206.333 206.298 206.292 206.292 206.292 206.292 206.292 0.000
232.147 232.139 232.138 232.137 232.137 232.137 232.135 0.001
Trapezoid
333 444 555 666 777 888 FEM Error (%)
98.156 98.152 98.151 98.150 98.150 98.150 98.152 0.002
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
170.020 170.004 170.000 170.000 170.000 170.000 170.000 0.000
196.308 196.306 196.302 196.299 196.299 196.299 196.300 0.000
196.354 196.310 196.304 196.300 196.300 196.300 196.306 0.003
240.430 240.420 240.417 240.417 240.417 240.417 240.420 0.001
259.721 259.688 259.684 259.680 259.680 259.680 259.684 0.002
Fig. 4. Mode shapes of the trapezoidal cavity and triangular cavity.
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between the mode shapes of the theoretical triangular cavity and the equivalent triangular cavity. At the Fig. 4, we also give the FEM results, which shows good agreement with the mode shapes of the present method. The advantage of this method is that the parametric study can be performed efficiently. It can be realized simply by changing the relevant parameters. Thus, the effect of the different angles on the natural frequencies of the cavity is studied. As shown in the Table 3, the first eight natural frequencies of the triangular cavity and the trapezoidal cavity with different angles are listed. The medium of the cavity is air, and the truncated values are M = N = L = 6. From the Table 3, it can be concluded that the natural frequencies decrease with the increase of the angle b. Thus, it is conducive to vibration reduction when the cavity with smaller angle is adopted. 3.1.2. The sound pressure response analysis As described in the introduction of model, the present method can be utilized for the condition of rigid walls and impedance boundaries. However, only the rigid walls are considered in the previous models. Therefore, some examples with impedance boundaries are added. In this section, the steady-state response is studied. Differ from the mode shapes, the curve of response can show the trend of sound pressure at any point in the monopole sound source excitation cavity. In addition, the effect of impedance walls and medium on the sound pressure can be viewed by the curve. In the following part of this section, numerical calculations of the steady-state response are performed. The steady-state response of trapezoidal cavity is analyzed. It should be pointed that the triangular cavity is obtained by changing b of the trapezoidal cavity to zero, so the study of trapezoidal cavity is more universal. The response curves of the trapezoidal cavity under the condition of different media and impedance walls are predicted and compared with finite element simulation. The propagation media of the cavity are water and air, and the boundary conditions can be rigid and impedance walls. The mass density and sound propagation velocity of water are 1000 kg/m3 and 1480 m/s. In the studied models, the impedance walls are modeled by specifying the impedance value Z = qmcm(100-j) to all walls. The improved Fourier series is truncated to M = N = L = 6. In the cases below, the amplitude of the monopole sound source is given directly (A = 1 kg/s2) and no special explanation will be given. The obtained sound pressure responses are given in Fig. 5. Fig. 5 (a) shows the sound pressure responses in the different media.
Fig. 5(b) is the responses of rigid boundary and impedance wall. The step of computation is 1 Hz. In the calculation, the positions of the monopole sound source and observation point are (0.20, 0.40, 0.20) and (0.70, 0.58, 0.50) respectively. It is obvious that the response curve provided by present method coincides well with those by using finite element method. Because the frequency of analysis is limited by calculation step, this comparison suggests that the accuracy of the present method for the response can be guaranteed even at the high frequencies. What’s more, from Fig. 5(a), it can be concluded that the change of the medium only affects the natural frequencies of the cavity rather than the trend of curves. Comparing (a) and (b) of the Fig. 5, we can find that the impedance walls can reduce the resonant peaks effectively. Therefore, the impedance boundary is more practical in reducing the vibration and noise. As described in the Fig. 5, the present method is capable of solving the acoustic problems for rigid and impedance cavity. Therefore, some steady-state responses of various impedance boundary conditions are considered to further discussion. The medium used of this section is air. A trapezoidal cavity of 0.8 m 1 m 1.5 m and a = 40°, b = 30° with different boundary conditions is established. The dimensions of the triangular cavity are a = 0.8 m, h = 1.5 m and a = 40°, b = 30°. Fig. 6 shows that the effect of impedance values on the sound pressure response at the observation, in which the two values of impedance are assigned to z = 0 and z = h, they are Z1 = qmcm(50-j) and Z2 = qmcm(10-j). The monopole sound source is located at (0.15, 0.01, 0.20), and the observation point are selected at (0.80, 0.20, 0.50). From the figure we can find that the impedance value only affects the response amplitudes at resonant frequencies. Besides, the smaller of impedance value, the smoother of the curve. What’s more, comparing (a) and (b) in the Fig. 6, it can be concluded that the response curve becomes complex with the increase of the volume. These results are meaningful for the reduce of vibration and noise. In addition, it provides a practical method for prediction the steady-state response of the trapezoidal and triangular cavity with various impedance boundary conditions. Fig. 7 shows the sound pressure response inside the cavities with the various number of the impedance wall. Three cases have been studied, including all rigid walls, two impedance walls (z = 0 and z = h), and all impedance walls. The dimensions of the model are the same as those of the previous one. In order to exclude the occasional conditions, the location of the observation point has been redefined. The observation point is located at (0.80,
Table 3 The natural frequencies (Hz) for the different cavities with various angle a and b. Shape
Triangle
a (°) 60
b (°)
30
30
60
Trapezoid
60
30
30
60
Method
Mode Number 2
3
4
5
6
7
8
Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)
125.358 125.367 0.007 98.150 98.152 0.002 72.375 72.376 0.001
170.000 170.000 0.000 170.000 170.000 0.000 113.333 113.336 0.003
196.300 196.325 0.012 170.000 170.000 0.000 165.670 165.674 0.002
211.222 211.227 0.002 196.299 196.300 0.001 170.000 170.000 0.000
259.680 259.699 0.007 196.300 196.306 0.004 184.765 184.765 0.000
286.948 287.030 0.028 240.417 240.420 0.002 196.299 196.301 0.001
333.525 333.596 0.021 259.680 259.684 0.002 204.314 204.316 0.001
Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)
82.915 82.913 0.002 69.042 69.041 0.001 55.283 55.281 0.004
143.203 143.199 0.003 118.693 118.695 0.002 90.588 90.588 0.000
170.000 170.000 0.000 162.240 162.243 0.002 126.846 126.847 0.001
187.922 187.921 0.001 170.000 170.000 0.000 164.633 164.632 0.001
189.143 189.142 0.001 183.485 183.485 0.000 170.000 170.000 0.000
206.588 206.581 0.003 190.136 190.140 0.002 178.763 178.763 0.000
222.277 222.274 0.001 203.616 203.613 0.002 185.713 185.710 0.002
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Fig. 5. Sound pressure response of trapezoidal cavity with different acoustic walls.
Fig. 6. Sound pressure response of the acoustic cavities under the excitation of a unit point sound source with different impedance values.
0.20, 0.80), and an impedance of Z = qmcm(100-j) is considered. From the Fig. 7, it can be concluded that the response amplitudes at resonant frequencies decrease with the increase of the impedance walls. In practical engineering applications, we can choose the materials with a suitable impedance or increase the number of impedance surfaces to reduce noise.
3.2. Study of the coupled cavity Since the effects of various boundary conditions and media on the acoustic characteristics of the cavity have been discussed in the previous section, this section studies the accuracy of the present method for the analysis of air-filled coupled cavity. By setting
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Fig. 7. Sound pressure response of the acoustic cavities under the excitation of a unit point sound source with different number of impedance walls.
the impedance value as an infinitely large and purely imaginary number, the experiment achieves rigid boundary conditions. Dimensions of the two coupled triangular cavities are a1 = 0.866 m, h1 = 1 m, a1 = 45, b1 = 30° and a2 = 2 m, h2 = 1 m, a2 = 14°, b2 = 23.413°, respectively. The accuracy of the method is verified by comparing the data received by calculations and the finite element method; and the results are shown in Table 4. It is obvious that the frequencies converge when the Fourier series is truncated to M = N = L = 6, which proves that the present method has a good convergence in coupling analysis. Besides, the percentage error (|Frequenciespresent FrequenciesFEM|/FrequenciesFEM 100%) between the convergence frequency and the FEM results does not exceed 0.063% even in the worst case. Therefore, this method has a good convergence and accuracy for the triangle coupled cavity. The fifth, sixth and seventh mode shapes of the triangular coupled cavity are drawn in Fig. 8, which represents the distribution of sound pressure in the coupled cavity under the condition of rigid walls. The results match well with the results of finite element simulation. Therefore, we can conclude that the correctness of admissible sound pressure function used for the study of coupled cavity is proved. In addition, the steady-state response of the triangle coupled cavity under the rigid walls has been compared with the finite element simulation. A monopole source is applied at the location (0.11, 0.01, 0.20), and the positions of the observation points under the local coordinates of the coupled cavities A and B are (0.52, 0.38, 0.50) and (1.03, 0.23, 0.50) respectively. Fig. 9 shows the steady-state response in the range of 0–300 Hz. It can be found that the sound pressure curve predicted by the present method coincident with the results of finite element simulation. Therefore,
the present method can be a powerful tool for the prediction of the steady-state response of the coupled cavity. The coupled cavity uses different dimensions to study the effect of parameter variation on the frequencies. Fig. 10(a) shows the coupling frequencies at the different values a2. The dimensions of the cavity A are a1 = 1 m, h1 = 1 m, and a1 = 45°, b1 = 30°, and the size of cavity B varies with a2. Since the first ten nature frequencies are too dense, the trend of even order frequencies is drawn in the Fig. 10(a). Clearly, the natural frequencies of the coupled cavity decrease with the increases of the value a2. The reason is that the natural frequencies are related to the inverse of the dimensions, thus the frequencies decrease due to the increase in the length of cavity B. In addition, Fig. 10 (b) shows the coupling frequency with the different height h. The dimensions of the analyzed cavity are the same as the model in Fig. 8. It can be concluded that the frequencies tend to a certain value as the height decreases to 0.5 m. That is because the proportion of axis-z of the cavity decrease with the reduce of h, three-dimensional model tends to two-dimensional. Thereby the natural frequencies remain relatively constants as the height below 0.5 m. 4. Experimental studies In the above content, the three-dimensional modified Fourier series method is verified by comparing the results of numerical calculation and the finite element method. In this section, experiments are carried out to measure the sound pressure response of the cavity. Besides, the effectiveness of the present method is further evidenced by comparing the experimental results with the theoretical calculation results.
Table 4 Convergence and accuracy of natural frequencies (Hz) for the triangle coupled cavity with the rigid wall. MNL
333 444 555 666 777 888 FEM Error (%)
Mode Number 2
3
4
5
6
7
8
9
83.333 83.333 83.332 83.332 83.332 83.332 83.332 0.000
143.242 143.231 143.226 143.223 143.223 143.223 143.204 0.013
161.481 161.419 161.394 161.371 161.371 161.371 161.269 0.063
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
189.326 189.326 189.326 189.325 189.325 189.325 189.326 0.000
197.338 197.333 197.330 197.329 197.329 197.329 197.319 0.005
222.303 222.295 222.293 222.289 222.289 222.289 222.278 0.005
234.512 234.435 234.430 234.421 234.421 234.421 234.323 0.042
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Fig. 8. Mode shapes of the two triangular coupled cavities.
Fig. 9. Sound pressure response in two triangular coupled cavities under the condition of rigid walls.
Fig. 10. Variation of frequency of the triangular coupled cavities with different geometric dimensions.
The test system shown in Fig. 11(a) is the experimental figure of trapezoidal cavity and Fig. 11(b) is of triangular cavity. The geometrical dimensions of the trapezoidal cavity are a = 0.38 m, b = 0.28 m, h = 0.39 m, and a = 0°, b = 53° and the dimensions of
the triangular cavity are a = 0.47 m, h = 0.47 m, and a = 0°, b = 45°. Reference value of sound pressure is 2 105 Pa. The rigid boundary conditions are simulated by plexiglass plates with a thickness of 10 mm. White noise is played by the noise excitation
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1. Power 2. Dynamic signal test analysis system 3. Computer
1. Power 2. Dynamic signal test analysis system 3. Computer 1 Noise excitation source
1
2
2
3
cavity
Noise excitation source
3
cavity
Microphone
Microphone
(a) The trapezoidal cavity
(b) The triangular cavity Fig. 11. Experimental setup.
Fig. 12. Schematic diagram of the experiment.
source to trigger more frequencies, and the sound pressure values of the sound source and the measuring point are measured by two microphones (MPA201). Then, the curve of the sound pressure response can be obtained by inputting the data to a dynamic signal test analysis system (DH5922N). The schematic diagram of the experiment is shown in Fig. 12. The comparison of experimental data and current prediction are plotted in Figs. 13 and 14. The truncated values in the theoretical
calculation are M = N = L = 6. From the comparison, it can be gathered that the trend of the curves is consistent, which validates the accuracy of the three-dimensional modified Fourier series method. The reason why the values are different in the graph is that the boundary conditions used in the experiment are not the theoretical rigid walls that absorb sound pressure, which results in differences in the resonant frequency. Due to the influence of the test instrument inside the cavity, the curve shows a peak value around 150 Hz.
Fig. 13. Comparison of experimental data with current prediction of the trapezoidal cavity while sound source located in (a) (0.11, 0.05, 0); (b) (0.08, 0.26, 0); (c) (0.09, 0.19, 0).
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Fig. 14. Comparison of experimental data with current prediction of the triangular cavity while sound source located in (a) (0.22, 0.02, 0); (b) (0.03, 0.04, 0); (c) (0.14, 0.22, 0).
5. Conclusions In this paper, a modified three-dimensional Fourier series method is applied to study the acoustic characteristics of cavities with various impedance boundary conditions. The admissible sound pressure function consists of a three-dimensional Fourier cosine series and a complementary function. The unknown coefficients of the sound pressure expression can be obtained by performing the Rayleigh-Ritz procedure. The acoustic characteristics of the irregular acoustic model are analyzed by transforming its coordinates into Cartesian coordinates with unit length. The agreement of both the mode shapes and steady-state responses for the present cavities with rigid or impedance boundary conditions between the present results and those obtained from finite element validates the reliability of the present method. What’s more, three important conclusions about acoustic characteristics of cavity are obtained, which are as follows: (1) The change of the medium only affects the frequency of the model and does not change the trend of the sound pressure response curve. The natural frequency ratio of different media is equal to the sound speed ratio in the medium. (2) Reducing the impedance value of the impedance surface and increasing the number of impedance walls have a marked effect on reducing the resonance amplitude of the sound pressure response. (3) The change of geometric parameters has a great effect on the natural frequency of the coupled cavities. For example, the natural frequencies could be reduced by increasing the length or the height of the cavity.
Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 51705537 and 51679056) the Natural Science Foundation of Hunan Province of China (2018JJ3661) and Innovation Driven Program of Central South University (Grant number: 2019CX006). The authors also gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2018-11). References [1] Maa DY. Non-uniform acoustical boundaries in rectangular rooms. J Acoust Soc Am 1940;12:465. [2] Morse PM, Bolt RH. Sound waves in rooms. Rev Mod Phys 1944;16:70–150.
[3] Xie X, Yang H, Zheng H. A weak formulation for interior acoustic analysis of enclosures with inclined walls and impedance boundary. Wave Motion 2016;65:175–86. [4] Mi Y, Zheng H, Lee HP. A domain decomposition method for stochastic analysis of acoustic fields with hybrid and localized uncertainties. Wave Motion 2018;83:121–33. [5] Kopuz S, Lalor N. Analysis of interior acoustic fields using the finite element method and the boundary element method. Appl Acoust 1995;45:193–210. [6] Joppa P, Fyfe I. A finite element analysis of the impedance properties of irregular shaped cavities with absorptive boundaries. J Sound Vib 1978;56:61–9. [7] Chen Z, Hofstetter G, Mang H. A 3d boundary element method for determination of acoustic eigenfrequencies considering admittance boundary conditions. J Comput Acoust 1993;1:455–68. [8] Seybert A, Cheng C. Application of the boundary element method to acoustic cavity response and muffler analysis. J Vib Acoust Stress Reliab Des 1987;109:15–21. [9] Du J, Liu Y, Wang Y, Wang G. Vibro-acoustic analysis of an elastically restrained plate duct silencer backed by irregular acoustical cavity. Appl Acoust 2018;138:60–71. [10] Du JT, Li WL, Liu ZG, Xu HA, Ji ZL. Acoustic analysis of a rectangular cavity with general impedance boundary conditions. J Acoust Soc Am 2011;130: 807–17. [11] Sharma AK, Bhat B. Analysis of triangular microstrip resonators. IEEE Trans Microw Theory Tech 1982;30:2029–31. [12] Xiujuan L, Weikang J. Acoustic characteristics of micro-perforated panel acoustic absorbers with variable cavity depth. Mech Sci Technol Aerosp Eng 2010:6. [13] Tam CKW. The acoustic modes of a two-dimensional rectangular cavity. J Sound Vib 1976;49:353–64. [14] Missaoui J, Cheng L. A combined integro-modal approach for predicting acoustic properties of irregular-shaped cavities. J Acoust Soc Am 1997;101:3313–21. [15] Sum K, Pan J. Effects of the inclination of a rigid wall on the free vibration characteristics of acoustic modes in a trapezoidal cavity. J Acoust Soc Am 2006;119:2201–10. [16] Lee WM. Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method. J Sound Vib 2011;330:4915–29. [17] Ortiz S, Plenier CL, Cobo P. Efficient modeling and experimental validation of acoustic resonances in three-dimensional rectangular open cavities. Appl Acoust 2013;74:949–57. [18] Sumbatyan MA, Lannie MY, Zampoli V. A fast Galerkin-based method for eigenfrequencies in acoustics of small rooms with slanted boundary planes. J Sound Vib 2016;367:101–13. [19] Harris CM, Feshbach H. On the acoustics of coupled rooms. J Acoust Soc Am 1950;22:572–8. [20] Jin G, Shi S, Liu Z. Acoustic modeling of a three-dimensional rectangular opened enclosure coupled with a semi-infinite exterior field at the baffled opening. J Acoust Soc Am 2016;140:3675. [21] V. Valeau, J. Picaut, A. Sakout and A. Billon. Simulation of the acoustics of coupled rooms by numerical resolution of a diffusion equation. 2004. [22] Meissner M. Influence of room geometry on modal density, spatial distribution of modes and their damping. Arch Acoust 2005;30:205. [23] Billon A, Valeau V, Sakout A, Picaut J. On the use of a diffusion model for acoustically coupled rooms. J Acoust Soc Am 2006;120:2043–54. [24] Meissner M. Computational studies of steady-state sound field and reverberant sound decay in a system of two coupled rooms. Open Phys 2007:5. [25] Meissner M. Influence of wall absorption on low-frequency dependence of reverberation time in room of irregular shape. Appl Acoust 2008;69: 583–90. [26] Meissner M. Analytical and numerical study of acoustic intensity field in irregularly shaped room. Appl Acoust 2013;74:661–8.
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Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
A three-dimensional modeling method for the trapezoidal cavity and multi-coupled cavity with various impedance boundary conditions Dongyan Shi a, Gai Liu a, Hong Zhang b,⇑, Wenhui Ren a, Qingshan Wang c,⇑ a
College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, PR China College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China c State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, PR China b
a r t i c l e
i n f o
Article history: Received 5 December 2018 Received in revised form 10 March 2019 Accepted 1 May 2019
Keywords: Modified Fourier series Acoustic characteristics Various impedance boundary conditions Multi-coupled cavity Trapezoidal cavity
a b s t r a c t A three-dimensional modified Fourier series method is applied to study the acoustic characteristics of the trapezoidal cavity and coupled cavity with various impedance boundary conditions. The formulation is constructed to describe the cavity systems based on the energy principle. The admissible sound pressure function is set to a Fourier cosine series and supplementary functions. These supplementary functions can eliminate the discontinuous or jumping phenomenon in the boundaries. The trapezoidal cavity can be transformed into regular rectangular cavity by coordinate transformation. The triangular cavity can be obtained by changing the dimensions of the trapezoidal cavity. The coupled cavity consists of two irregular triangular cavities and the coupling interface is regarded as a non-dissipative surface. The accuracy and convergence of the present results are verified by comparison with the results obtained by finite element calculation and experiment. The natural frequencies and mode shapes of the trapezoidal cavity and coupled cavity are studied. Besides, the sound pressure response under monopole sound source excitation is obtained. What’s more, some results are obtained in the basis of effects of various factors on acoustic characteristics, such as shapes, geometric parameters and impedance values. These results provide a benchmark for the future researches. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction As a basic enclosure, trapezoidal cavity with various impedance boundary conditions has a large proportion in practical applications, such as used in workshops, ship compartments and civil buildings. Besides, some polygonal prismatic cavities can be regarded as the coupled cavities composed by trapezoidal cavity or triangular cavity. Therefore, the study of trapezoidal cavity and coupled cavity has a good guiding significance to the practical engineering, like noise control and room designs. The indoor sound field simulation can be traced back to the 1950s [1,2]. Then, some methods are proposed to analyze the acoustic cavity, such as decomposition method [3,4], the finite element method [5,6], boundary element method [7,8], Rayleigh-Ritz method [9,10], etc. Based on these, the acoustic characteristics of trapezoidal cavity, triangular cavity and triangular coupled cavity with various impedance boundary conditions are studied in the paper.
⇑ Corresponding authors. E-mail addresses: [email protected] (H. Zhang), [email protected] (Q. Wang). https://doi.org/10.1016/j.apacoust.2019.05.001 0003-682X/Ó 2019 Elsevier Ltd. All rights reserved.
At present, there are many literatures that have studied the single-cavity. Sharma and Bhat [11] analyzed the resonant frequency of isosceles triangle microstrip resonator by using the full wave formula of spectrum domain technique. Xiujuan and Weikang [12] presented a method to calculate the acoustic absorption coefficient of the microperforated panel absorbers and measured the acoustic absorption coefficients of three kinds of triangular prism absorbers. Tam [13] studied the acoustic modes of twodimensional rectangular resonators. Besides, he discussed the physical significance of strong radiation damping on cavity resonance caused by external flow at low Mach number. Missaoui and Cheng [14] calculated the natural frequencies of irregularshaped cavities by discretizing the cavity into a series of sub cavities. The acoustic pressure was decomposed into the modal basis of regular sub cavities. Xie [3] divided the trapezoidal acoustic domain into several sub-cavities with trapezoidal and rectangular faces. Then, the acoustic enclosures with walls in various inclination and impedance boundary conditions were studied by coupling each sub-cavity with adjacent sub-cavities. Sum and Pan [15] used the coupling between rigid-walled modes of a rectangular cavity to obtain the shapes and resonance frequencies of rigid-walled modes of a trapezoidal cavity. Besides, they studied how the free vibration
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characteristics of trapezoidal cavity modes change with the inclination and what determines these changes. Lee [16] proposed a semi-analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries. The sound pressure expression of this method was formulated in terms of angular and radial Mathieu functions. Ortiz [17] used a fast and efficient time domain method to study the time response of any point in a three-dimensional open rectangular cavity. Sumbatyan et al. [18] applied a system of Galerkin’s basis functions which are orthogonal in a unit cube to analyze the effect of the inclination of slanted boundary planes on the low eigenfrequencies for rectangular parallelepiped rooms. In summary, many results for different shapes of the single-cavity are obtained. However, few studies have been done on the analysis of the steady-state response of trapezoidal cavity and triangular cavity with various impedance boundary conditions. Most of the models being used in engineering are irregular straight-sided cavities. These models can be regarded as multicoupled cavities. As the practical enclosure, there are some literatures for the multi-coupled cavity. Harris and Feshbach [19] studied the problem of coupled rooms from the ‘‘wave” point of view and treated the coupled rooms as a boundary value problem to obtain the approximate solution. Jin [20] proposed a modeling method for the analysis of a rectangular opened enclosure coupled with a semi-infinite external fields. It was been used to study the effect of sizes and positions of the opening and wall impedance on the behaviors of opened enclosure system. Valeau et al. [21] obtained the three-dimensional diffusion equation by using a finite-element solver and simulated the spatial variation of sound intensity level with aperture coupling. Meissner [22] studied the acoustical properties of two rooms by a system of eigenmodes, and summarized the laws of frequency and acoustic energy losses with dimensions. Billon [23] proposed a numerical model to predict the reverberant sound field in two cavities coupled through an open aperture. This method can be used to predict the acoustic energy distribution and attenuation in a single room. Meissner [24–26] studied the acoustical properties of an irregularly shaped room consisting of two coupled rectangular sub-rooms, and predicted the distribution of the sound pressure and the reverberation time of the coupled system. Through these literatures, it can be found that most of the studied models are rectangular coupled cavities. However, there are few studies on the non-rectangular acoustic cavities.
Due to the restrictions of the cavity in the current researches, a modified three-dimensional Fourier method [27] is used in this paper to study the natural frequencies and steady-state response of trapezoidal cavity, triangular cavity and coupled cavity, and various impedance boundary conditions are considered. It has to be emphasized that although the trapezoidal cavity is studied both in Ref. [3] and in the present paper, the basic formulas are different. In addition, the trapezoidal cavity is studied by transforming its coordinates into rectangular cavity’s coordinates rather the decomposition method. The triangular cavity can be achieved from the trapezoidal cavity. In this paper, the three-dimensional Fourier cosine function and complementary functions are employed for the modeling of systems. The coupling interface of the coupled cavity is a non-dissipative surface. The accuracy and convergence of the results are verified by comparing the current results with the data obtained by finite element calculation and experiment. 2. Theoretical formulations 2.1. Description of the model As shown in Fig. 1, the model to be analyzed is given. For a trapezoidal cavity, its shape can be determined by the bottom shape and height h of the cavity. The bottom shape of the trapezoidal cavity depends on the length a, b and the angle a, b. The trapezoidal cavity of different shapes can be achieved by changing the control variables. For example, if a = b – 0, a = b = 0, a = 0 and b – 0, the cavity can be an isosceles trapezoidal cavity, a rectangular cavity and a right-angled trapezoidal cavity respectively. In addition, when b = 0, we can get the model of triangular cavity as depicted in Fig. 1(b). Based on the above research, a quadrilateral cavity can be achieved by coupling two triangular cavities on a plane which parallels to yz, it is shown in Fig. 2. A global rectangular coordinate system (o-xyz) and local-coordinate systems (o1-x1y1z1, o2-x2y2z2) of the coupled model are built. Q is the monopole sound source of the coupled cavity. Similarly, different shapes can be got by changing the angle ai, bi, and the dimension ai, hi of the cavities A and B. If a1 = a2 = 0 or b1 = b2 = 0, then right triangle cavity, acute triangle cavity and obtuse triangle cavity are respectively achieved by changing the corresponding bi or ai. If a1 + b1 = 90° and a2 + b2 = 90°, a1 = b2 and a2 = b1, the enclosure is a rectangle cavity and a parallelogram cavity respectively. What’s more, when the
Fig. 1. Geometry and coordinate system of the triangular cavity and trapezoidal cavity with various impedance boundary conditions.
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length (o-ngf). The formula of coordinate transformation is as follows:
8 9 8 9 > > 8
ð3Þ
in which xi, yi and zi are the coordinates of the ith node in the trapezoidal cavity. Ni (n, g, f) is the shape function of the ith node:
N1 ¼ ð1 nÞð1 gÞð1 fÞ; N2 ¼ ð1 nÞgð1 fÞ; N3 ¼ ngð1 fÞ; N4 ¼ nð1 gÞð1 fÞ; N5 ¼ ð1 nÞð1 gÞf; N6 ¼ ð1 nÞgf; N7 ¼ ngf; N8 ¼ nð1 gÞf ð4Þ The relationship between the first derivative of the coordinate system (o-xyz) and (o-ngf) is:
Fig. 2. Geometry and coordinate system of the two triangular coupled cavity with various impedance boundary conditions.
8 @ðÞ 9 > > > < @n > =
8 @ðÞ 9 2 @x > > > < @x > = 6 @n @ðÞ @ðÞ @x ¼ J ¼6 @g > @y > 4 @g > > > > > > : @ðÞ ; : @ðÞ ; @x @f
angle and the dimensions are various, it is an irregular quadrilateral. Impedance boundary conditions of the cavity can be expressed by assigning corresponding impedance value on each of the surfaces, that is:
@pðx; y; zÞ jq xpðx; y; zÞ ¼ m @n Zi
ð1Þ
where n is the outward normal to the boundary and j is the imaginary number (j2 = 1). qm is the mass density of the acoustic medium. x is the angular frequency and p(x, y, z) denotes the admissible sound pressure function. Zi is the specific acoustic impedance value of the ith surface. The coupling interface of the coupled cavity is a non-dissipative surface. It does not lose energy when the sound pressure passes through the coupling interface. The sound pressure and the velocity are continuous at the coupling interface, so the boundary conditions of the coupling interface are:
p1 jx1 ¼a1 ¼ p2 jx2 ¼a2 ;
@p1 @p ¼ 2 @x x1 ¼a1 @x x2 ¼a2
ð2Þ
2.2. Transformation of coordinate In order to facilitate the calculation of trapezoidal cavity, it is necessary to change trapezoidal cavity into rectangular cavity. As shown in Fig. 3, the coordinate system of the trapezoidal cavity (o-xyz) with various impedance boundary conditions transformed into the coordinate system of the rectangular cavity with unit
@z
@f
@y @n @y @g @y @f
38 @ðÞ 9 > > @x > > 7< @ðÞ = @z 7 @g 5 @y > > > > @z : @ðÞ ; @z @n
@f
ð5Þ
@z
where:
@I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þgf þ ðI1 I2 þ I3 I4 Þg @n þ ðI1 I4 I5 þ I8 Þf I1 þ I4 ð6aÞ @I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þnf þ ðI1 I2 þ I3 I4 Þn @g ð6bÞ þ ðI1 I2 I5 þ I6 Þf I1 þ I2 @I ¼ ðI1 þ I2 I3 þ I4 þ I5 I6 þ I7 I8 Þng þ ðI1 I4 I5 þ I8 Þn @f þ ðI1 I2 I5 þ I6 Þg I1 þ I5 ð6cÞ in which I can be x, y or z. J is the Jacobin matrix. 2.3. Energy expressions The acoustic modes and the steady-state response of the cavity with various impedance boundary conditions are studied mainly. Thus, the acoustic characteristics have been studied based on the Rayleigh-Ritz energy method. By transforming the threedimensional acoustic problem into a three-dimensional solid problem, a modified three-dimensional Fourier series method is proposed to express the admissible sound pressure functions of the cavity. There are many literatures describing different enclosures with Lagrangian function [28,29], and their development is mature. Therefore, the Lagrangian function is used to describe the energy equation of the proposed cavity.
Fig. 3. Schematic diagram of geometric and coordinate transformation.
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The Lagrangian function (L) for the coupled cavity with various impedance boundary conditions can be written as:
L1 ¼ U 1cavity T 1cavity P W f 1 W p
ð7aÞ
L2 ¼ U 2cavity T 2cavity P W f 2
ð7bÞ
where L1 and L2 represent the Lagrangian function of each coupled cavity respectively. Uicavity is the total acoustic potential energy of the ith coupled cavity. Ticavity denotes the total kinetic energy for the ith coupled cavity, Wfi expresses the dissipated acoustic energy on impedance surfaces, Wp is the work done by the monopole sound source. P represents the coupling potential energy of the coupled cavity. The total potential energy (Uicavity) and kinetic energy (Ticavity) which are stored in the ith cavity can be given as:
U icavity ¼ ¼
Z
1
V
Z
1 2q
2 m cm
0
and
T icavity ¼ ¼
Z
1
1
Z
0
1 0
Z
1
p2i ðn; g; fÞjJi jdndgdf
ð8Þ
2
ðgradpÞ dV
2qm x2
V
Z
1 2qm x2
Z
1
0
1
Z
0
0
1
For the triangular cavity, the coordinate of the node 1 and 2 in the Fig. 3 should be equal, as does node 5 and 6. Then the energy expressions can be obtained by using Eqs. (7)–(14). 2.4. Admissible sound pressure function and system equations
p2 ðx; y; zÞdV
2qm c2m
The coupling potential energy of the coupled cavity (P) is: Z 1 @ðp1 ðx; y;zÞ p2 ðx;y;zÞÞ dS P¼ p ðx;y; zÞ @x q m x2 S 1 9 8 @y1 @z1 > > > > > > @p1 ðn1 ;g1 ;f1 Þ @n1 @ g1 @ g1 > dgdfÞj n ¼ 1 > ðp1 ðn1 ; g1 ;f1 Þ > > > > @n1 @x1 @y1 @z1 > > 1 > > @f1 @f1 > Z 1 Z 1> < n2 ¼ 1 = 1 ¼ @y2 @z2 > q m x2 0 0 > > > > > > > > > ðp1 ðn1 ; g1 ;f1 Þ @p2 ðn2 ;g2 ;f2 Þ @n2 @ g2 @ g2 dgdfÞj > @n2 @x2 @y2 @z2 > > n1 ¼ 1 > > > @f2 @f2 > > ; : n2 ¼ 1 ð14Þ
2 h i 2 h i 2 3 J1 OTi þ J1 OTi i i 6 7 1 2 4 5jJi jdndgdf h i 2 T 1 þ Ji Oi 3
As the factor affecting the convergence and accuracy of the calculation, the selection of the sound pressure function is the most critical part of the paper. A modified Fourier series for the various impedance boundary conditions is used in the paper to express the pressure function in the coupled cavity [30]. The series solution of pressure distribution function of the unit cube can be shown as:
pðn; g; fÞ ¼ pa ðn; g; fÞ þ pb ðn; g; fÞ þ1 P þ1 P þ1 P Amnl cosðpmnÞcosðpngÞcosðplfÞ pa ðn; g; fÞ ¼ m¼0 n¼0 l¼0
ð9Þ in which [Ji1]j is the jth line of the Jacobian inverse matrix for the ith coupled cavity, cm is the sound speed of the acoustic medium. Oi expresses the first derivative of the admissible sound pressure functions versus the coordinate system after the transformation:
@pi ðn; g; fÞ @pi ðn; g; fÞ @pi ðn; g; fÞ Oi ¼ @n @g @f
ð10Þ
Since each coupled cavity has 5 impedance surfaces, the dissipated energy of the ith cavity (Wfi) can be shown as: W fi ¼ W fa þ W fb þ W fc 8 @y
9 @z = @g dgdfÞj n¼0 @z ; @f @x @z R 1 R 1 p2i ðn;g;fÞ @n @n dndfÞj þ ð g¼0 0 0 jxZ 4 @x @z @f @f @x @y R R 2 @n dndgÞj þ 1 1 ðpi ðn;g;fÞ @n f¼0 0 0 jxZ 6 @x @y @g @g
Z 0
1
Z 0
1
Z 0
1
p1 ðn; g; fÞQ jJjdndgdf jx
4p A dðx x0 Þdðy y0 Þdðz z0 Þ jqm cm k
þ
þ1 P
A1mnl sinðpmnÞcosðpngÞcosðplfÞ
1 þ1 P P
A2mnl cosðpmnÞsinðpngÞcosðplfÞ
m¼0 n¼2 l¼0
þ
þ1 P þ1 P
1 P
m¼0 n¼0 l¼2
þ
1 P
A3mnl cosðpmnÞcosðpngÞsinðplfÞ
1 þ 1 P P
m¼2 n¼2 l¼0
þ
@z @n
)
dndfÞjg¼1 @z @f 9 @y = @n dndgÞj f¼1 @y ; @g
ð12Þ
where Q expresses the distribution function of the source intensity, it can be shown as:
Q¼
m¼2 n¼0 l¼0
1 þ1 1 P P P m¼2 n¼0 l¼2
in which Wfa, Wfb and Wfc represents the dissipated energy of the impedance surfaces in the direction n, g, f respectively. The work done by the monopole sound source (Wp) is given as:
1 2
1 þ 1 þ 1 P P P
þ
ð11Þ
Wp ¼
pb ðn; g; fÞ ¼
ð13Þ
in which d is 3-D Dirac function, A represents the amplitude of sound source, k is the wave number. The monopole sound source is located at (x0, y0, z0).
þ 1 P
1 1 P P
m¼0 n¼2 l¼2
ð15Þ
A4mnl sinðpmnÞsinðpngÞcosðplfÞ A5mnl sinðpmnÞcosðpngÞsinðplfÞ A6mnl cosðpmnÞsinðpngÞsinðplfÞ
in which Amnl ; A1mnl ; A2mnl ; A3mnl ; A4mnl ; A5mnl and A6mnl are the threedimensional Fourier coefficients of the trigonometric series. pa is the traditional Fourier series, and pb is the supplementary function which can eliminate the discontinuous or jumping phenomenon in the boundaries. In practical computations, the result can be acquired by truncating the infinite series to the numerical value M, N and L. The coupled cavity is composed of two sub-cavities, so different Fourier coefficients are used. Aimnl is used for the cavity in which the monopole sound source is located, and Bimnl is for another. Let the partial derivation of Lagrangian equation Li with the respect to the Fourier coefficient be zero, Eq. (7) can be expressed as follows:
@U 1cavity @T 1cavity @P @W f 1 @W p ¼ @A @A @A @A @A
ð16aÞ
@U 2cavity @T 2cavity @P @W f 2 ¼0 @B @B @B @B
ð16bÞ
Substituting Eqs. (6)–(15) into Eq. (16). According to the Rayleigh-Ritz technology [31,32], the equations can be expressed in a matrix form:
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ðK 1 C 11 xZ 1 x2 M 1 ÞA þ C 12 B ¼ Q
ð17Þ
C 21 A þ ðK 2 xZ 2 x2 M 2 ÞB ¼ 0
in which Ki, Mi and Zi are the stiffness matrix, mass matrix and impedance matrix of the ith coupled cavity respectively. C11 and C12 are coupling matrices generated by the cavity in which the monopole sound source is located. C21 is the coupling matrix created by another cavity. A and B are the matrices of the Fourier coefficients Aimnl andBimnl respectively. When the modal characteristics of closed space are studied, we only need to set the right side of the Eq. (17) to zero. For convenience of the calculation, it is necessary to get a linear equation. Through the transformation, Eq. (17) can be shown as follows [30]:
ðR xSÞG ¼ 0
ð18Þ
where:
2
0 K 1 C 11 6K C Z 1 1 11 6 R¼6 4 0 0 C 21 0 2 0 0 K 1 C 11 0 6 0 M 0 0 1 6 S¼6 4 0 0 K2 0 0
0
0
0 C 12 0 K2 3
3
0
0 7 7 7; K2 5 Z 2 2
A
ð19Þ
3
7 6 xA 7 7 7 6 7 7; G ¼ 6 5 4 B 5
M2
xB
We can get the natural frequencies and the matrix of Fourier coefficients (A and B) by solving the eigenvalues and eigenvectors of the Eq. (19). The sound pressure distribution in the sub-cavities can be obtained by substituting the Fourier series into the Eq. (15). Besides, the sound pressure response equation under the effect of monopole sound source can be achieved by transforming the Eq. (18) into:
1 1 A ¼ K 1 C 11 xZ 1 x2 M 1 C 12 K 2 xZ 2 x2 M 2 C 21 Q 1 2 B ¼ K 2 xZ 2 x M 2 C 21 A ð20Þ The curve of the steady-state response can be achieved by sweeping the frequency. When the model to be analyzed is a single cavity, the corresponding Lagrangian formula is Eq. 7(a), but the coupling potential energy P need to be removed. The acoustic characteristics and steady-state response of the single cavity under the effect of monopole sound source can be obtained by Eqs. (7)–(20). 3. Numerical results and discussion Some numerical examples are used to study the acoustic characteristics and steady-state responses of trapezoidal cavity, triangular cavity and coupled cavity. Besides, the results verified the
convergence, accuracy and effectiveness of the three-dimensional modified Fourier method. On these bases, the effects of acoustic medium, boundary conditions and dimensions on the cavities are studied. The dimensions of the examples used in this paper have been listed in the Table 1. 3.1. Study of trapezoidal and triangular cavity 3.1.1. Convergence study and modal analysis Convergence and accuracy of the method are verified by comparing the obtained natural frequencies and mode shapes with the finite element results. Besides, the effects of dimensions on the natural frequencies of the cavities are studied. The geometric dimensions of the trapezoidal cavity used in this part are a = b = h = 1 m, and the inclination angles are a = b = 30°. The sizes of the triangular cavity are a = h = 1 m, the inclination angles are a = 30°, b = 60°. The medium in the enclosure is air, the mass density and speed of sound propagation in the air are qm = 1.21 kg/m3 and cm = 340 m/s respectively. The boundary conditions of the comparison models are the rigid walls, it can be achieved by setting the impedance value as a large pure imaginary number (such as j1010) in the numerical calculation and simulation. The frequencies of the trapezoidal cavity and triangular cavity are listed in Table 2. Since the zero frequency represents the rigid body mode, it is not studied in the following examples. Because the infinite series are truncated to the numerical value M, N and L, the convergence of the method can be approved by comparing the corresponding frequencies when the numbers of the M, N and L are selected different values. In addition, the accuracy of the present method is verified by comparing the convergent results with the frequencies obtained by FEM. It can be seen from the Table 2 that the natural frequencies converge rapidly with the increase of the truncated values M, N and L. Particularly, the frequency convergence to a certain value when M = N = L = 6. The max error between convergent frequencies and results obtained by finite element is no more than 0.003%. Moreover, the difference between the frequency (M N L = 3 3 3) and convergence frequency (M N L = 8 8 8) is less than 0.02%. Therefore, the present method has a good convergence and accuracy in dealing with trapezoidal cavity and triangular cavity problems. For illustrative purposes, the 2nd, 3rd, and 4th mode shapes for the triangular cavity and the trapezoidal cavity are drawn in Fig. 4. The mode shapes can represent the distribution of sound pressure for the cavities directly. From the figures, we can find that the mode shapes for isosceles trapezoidal cavity has a perfect symmetry. The more increase of the frequency, the more deformation of the mode shapes. Besides, if the vertices of the trapezoidal mode shapes numbered as the form of Fig. 3, then an equivalent triangular can be achieved by compressing the vertex 5 to the line formed by the vertices 8 and 6. It can be found that there is no difference
Table 1 Dimensions of the examples. Section
Shape
3.1.1 3.1.2 3.1.3
Triangle Trapezoid Triangle Trapezoid Triangle 1 Triangle 2 Triangle 1 Triangle 2 Triangle Trapezoid
Model Size a(m)
3.2
4
b(m)
1 – 1 1 0.8 – 0.8 1 0.866 – 2 1 – Changing with the size a (Fig. 10(a)) 0.47 – 0.38 0.28
h(m)
a(°)
b(°)
1 1 1.5 1.5 1 1 1
30 30 40 40 45 14 45
60 30 30 30 30 23.413 30
0.47 0.39
0 0
45 53
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Table 2 Convergence and accuracy of natural frequencies (Hz) for the cavities with the rigid walls. Shape
MNL
Mode Number 2
3
4
5
6
7
8
Triangle
333 444 555 666 777 888 FEM Error (%)
102.142 102.142 102.138 102.138 102.138 102.138 102.136 0.002
158.089 158.078 158.077 158.075 158.075 158.075 158.072 0.002
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
198.325 198.325 198.323 198.323 198.323 198.323 198.323 0.001
206.298 206.267 206.267 206.264 206.264 206.264 206.261 0.001
206.333 206.298 206.292 206.292 206.292 206.292 206.292 0.000
232.147 232.139 232.138 232.137 232.137 232.137 232.135 0.001
Trapezoid
333 444 555 666 777 888 FEM Error (%)
98.156 98.152 98.151 98.150 98.150 98.150 98.152 0.002
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
170.020 170.004 170.000 170.000 170.000 170.000 170.000 0.000
196.308 196.306 196.302 196.299 196.299 196.299 196.300 0.000
196.354 196.310 196.304 196.300 196.300 196.300 196.306 0.003
240.430 240.420 240.417 240.417 240.417 240.417 240.420 0.001
259.721 259.688 259.684 259.680 259.680 259.680 259.684 0.002
Fig. 4. Mode shapes of the trapezoidal cavity and triangular cavity.
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D. Shi et al. / Applied Acoustics 154 (2019) 213–225
between the mode shapes of the theoretical triangular cavity and the equivalent triangular cavity. At the Fig. 4, we also give the FEM results, which shows good agreement with the mode shapes of the present method. The advantage of this method is that the parametric study can be performed efficiently. It can be realized simply by changing the relevant parameters. Thus, the effect of the different angles on the natural frequencies of the cavity is studied. As shown in the Table 3, the first eight natural frequencies of the triangular cavity and the trapezoidal cavity with different angles are listed. The medium of the cavity is air, and the truncated values are M = N = L = 6. From the Table 3, it can be concluded that the natural frequencies decrease with the increase of the angle b. Thus, it is conducive to vibration reduction when the cavity with smaller angle is adopted. 3.1.2. The sound pressure response analysis As described in the introduction of model, the present method can be utilized for the condition of rigid walls and impedance boundaries. However, only the rigid walls are considered in the previous models. Therefore, some examples with impedance boundaries are added. In this section, the steady-state response is studied. Differ from the mode shapes, the curve of response can show the trend of sound pressure at any point in the monopole sound source excitation cavity. In addition, the effect of impedance walls and medium on the sound pressure can be viewed by the curve. In the following part of this section, numerical calculations of the steady-state response are performed. The steady-state response of trapezoidal cavity is analyzed. It should be pointed that the triangular cavity is obtained by changing b of the trapezoidal cavity to zero, so the study of trapezoidal cavity is more universal. The response curves of the trapezoidal cavity under the condition of different media and impedance walls are predicted and compared with finite element simulation. The propagation media of the cavity are water and air, and the boundary conditions can be rigid and impedance walls. The mass density and sound propagation velocity of water are 1000 kg/m3 and 1480 m/s. In the studied models, the impedance walls are modeled by specifying the impedance value Z = qmcm(100-j) to all walls. The improved Fourier series is truncated to M = N = L = 6. In the cases below, the amplitude of the monopole sound source is given directly (A = 1 kg/s2) and no special explanation will be given. The obtained sound pressure responses are given in Fig. 5. Fig. 5 (a) shows the sound pressure responses in the different media.
Fig. 5(b) is the responses of rigid boundary and impedance wall. The step of computation is 1 Hz. In the calculation, the positions of the monopole sound source and observation point are (0.20, 0.40, 0.20) and (0.70, 0.58, 0.50) respectively. It is obvious that the response curve provided by present method coincides well with those by using finite element method. Because the frequency of analysis is limited by calculation step, this comparison suggests that the accuracy of the present method for the response can be guaranteed even at the high frequencies. What’s more, from Fig. 5(a), it can be concluded that the change of the medium only affects the natural frequencies of the cavity rather than the trend of curves. Comparing (a) and (b) of the Fig. 5, we can find that the impedance walls can reduce the resonant peaks effectively. Therefore, the impedance boundary is more practical in reducing the vibration and noise. As described in the Fig. 5, the present method is capable of solving the acoustic problems for rigid and impedance cavity. Therefore, some steady-state responses of various impedance boundary conditions are considered to further discussion. The medium used of this section is air. A trapezoidal cavity of 0.8 m 1 m 1.5 m and a = 40°, b = 30° with different boundary conditions is established. The dimensions of the triangular cavity are a = 0.8 m, h = 1.5 m and a = 40°, b = 30°. Fig. 6 shows that the effect of impedance values on the sound pressure response at the observation, in which the two values of impedance are assigned to z = 0 and z = h, they are Z1 = qmcm(50-j) and Z2 = qmcm(10-j). The monopole sound source is located at (0.15, 0.01, 0.20), and the observation point are selected at (0.80, 0.20, 0.50). From the figure we can find that the impedance value only affects the response amplitudes at resonant frequencies. Besides, the smaller of impedance value, the smoother of the curve. What’s more, comparing (a) and (b) in the Fig. 6, it can be concluded that the response curve becomes complex with the increase of the volume. These results are meaningful for the reduce of vibration and noise. In addition, it provides a practical method for prediction the steady-state response of the trapezoidal and triangular cavity with various impedance boundary conditions. Fig. 7 shows the sound pressure response inside the cavities with the various number of the impedance wall. Three cases have been studied, including all rigid walls, two impedance walls (z = 0 and z = h), and all impedance walls. The dimensions of the model are the same as those of the previous one. In order to exclude the occasional conditions, the location of the observation point has been redefined. The observation point is located at (0.80,
Table 3 The natural frequencies (Hz) for the different cavities with various angle a and b. Shape
Triangle
a (°) 60
b (°)
30
30
60
Trapezoid
60
30
30
60
Method
Mode Number 2
3
4
5
6
7
8
Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)
125.358 125.367 0.007 98.150 98.152 0.002 72.375 72.376 0.001
170.000 170.000 0.000 170.000 170.000 0.000 113.333 113.336 0.003
196.300 196.325 0.012 170.000 170.000 0.000 165.670 165.674 0.002
211.222 211.227 0.002 196.299 196.300 0.001 170.000 170.000 0.000
259.680 259.699 0.007 196.300 196.306 0.004 184.765 184.765 0.000
286.948 287.030 0.028 240.417 240.420 0.002 196.299 196.301 0.001
333.525 333.596 0.021 259.680 259.684 0.002 204.314 204.316 0.001
Present FEM Error (%) Present FEM Error (%) Present FEM Error (%)
82.915 82.913 0.002 69.042 69.041 0.001 55.283 55.281 0.004
143.203 143.199 0.003 118.693 118.695 0.002 90.588 90.588 0.000
170.000 170.000 0.000 162.240 162.243 0.002 126.846 126.847 0.001
187.922 187.921 0.001 170.000 170.000 0.000 164.633 164.632 0.001
189.143 189.142 0.001 183.485 183.485 0.000 170.000 170.000 0.000
206.588 206.581 0.003 190.136 190.140 0.002 178.763 178.763 0.000
222.277 222.274 0.001 203.616 203.613 0.002 185.713 185.710 0.002
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Fig. 5. Sound pressure response of trapezoidal cavity with different acoustic walls.
Fig. 6. Sound pressure response of the acoustic cavities under the excitation of a unit point sound source with different impedance values.
0.20, 0.80), and an impedance of Z = qmcm(100-j) is considered. From the Fig. 7, it can be concluded that the response amplitudes at resonant frequencies decrease with the increase of the impedance walls. In practical engineering applications, we can choose the materials with a suitable impedance or increase the number of impedance surfaces to reduce noise.
3.2. Study of the coupled cavity Since the effects of various boundary conditions and media on the acoustic characteristics of the cavity have been discussed in the previous section, this section studies the accuracy of the present method for the analysis of air-filled coupled cavity. By setting
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Fig. 7. Sound pressure response of the acoustic cavities under the excitation of a unit point sound source with different number of impedance walls.
the impedance value as an infinitely large and purely imaginary number, the experiment achieves rigid boundary conditions. Dimensions of the two coupled triangular cavities are a1 = 0.866 m, h1 = 1 m, a1 = 45, b1 = 30° and a2 = 2 m, h2 = 1 m, a2 = 14°, b2 = 23.413°, respectively. The accuracy of the method is verified by comparing the data received by calculations and the finite element method; and the results are shown in Table 4. It is obvious that the frequencies converge when the Fourier series is truncated to M = N = L = 6, which proves that the present method has a good convergence in coupling analysis. Besides, the percentage error (|Frequenciespresent FrequenciesFEM|/FrequenciesFEM 100%) between the convergence frequency and the FEM results does not exceed 0.063% even in the worst case. Therefore, this method has a good convergence and accuracy for the triangle coupled cavity. The fifth, sixth and seventh mode shapes of the triangular coupled cavity are drawn in Fig. 8, which represents the distribution of sound pressure in the coupled cavity under the condition of rigid walls. The results match well with the results of finite element simulation. Therefore, we can conclude that the correctness of admissible sound pressure function used for the study of coupled cavity is proved. In addition, the steady-state response of the triangle coupled cavity under the rigid walls has been compared with the finite element simulation. A monopole source is applied at the location (0.11, 0.01, 0.20), and the positions of the observation points under the local coordinates of the coupled cavities A and B are (0.52, 0.38, 0.50) and (1.03, 0.23, 0.50) respectively. Fig. 9 shows the steady-state response in the range of 0–300 Hz. It can be found that the sound pressure curve predicted by the present method coincident with the results of finite element simulation. Therefore,
the present method can be a powerful tool for the prediction of the steady-state response of the coupled cavity. The coupled cavity uses different dimensions to study the effect of parameter variation on the frequencies. Fig. 10(a) shows the coupling frequencies at the different values a2. The dimensions of the cavity A are a1 = 1 m, h1 = 1 m, and a1 = 45°, b1 = 30°, and the size of cavity B varies with a2. Since the first ten nature frequencies are too dense, the trend of even order frequencies is drawn in the Fig. 10(a). Clearly, the natural frequencies of the coupled cavity decrease with the increases of the value a2. The reason is that the natural frequencies are related to the inverse of the dimensions, thus the frequencies decrease due to the increase in the length of cavity B. In addition, Fig. 10 (b) shows the coupling frequency with the different height h. The dimensions of the analyzed cavity are the same as the model in Fig. 8. It can be concluded that the frequencies tend to a certain value as the height decreases to 0.5 m. That is because the proportion of axis-z of the cavity decrease with the reduce of h, three-dimensional model tends to two-dimensional. Thereby the natural frequencies remain relatively constants as the height below 0.5 m. 4. Experimental studies In the above content, the three-dimensional modified Fourier series method is verified by comparing the results of numerical calculation and the finite element method. In this section, experiments are carried out to measure the sound pressure response of the cavity. Besides, the effectiveness of the present method is further evidenced by comparing the experimental results with the theoretical calculation results.
Table 4 Convergence and accuracy of natural frequencies (Hz) for the triangle coupled cavity with the rigid wall. MNL
333 444 555 666 777 888 FEM Error (%)
Mode Number 2
3
4
5
6
7
8
9
83.333 83.333 83.332 83.332 83.332 83.332 83.332 0.000
143.242 143.231 143.226 143.223 143.223 143.223 143.204 0.013
161.481 161.419 161.394 161.371 161.371 161.371 161.269 0.063
170.000 170.000 170.000 170.000 170.000 170.000 170.000 0.000
189.326 189.326 189.326 189.325 189.325 189.325 189.326 0.000
197.338 197.333 197.330 197.329 197.329 197.329 197.319 0.005
222.303 222.295 222.293 222.289 222.289 222.289 222.278 0.005
234.512 234.435 234.430 234.421 234.421 234.421 234.323 0.042
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Fig. 8. Mode shapes of the two triangular coupled cavities.
Fig. 9. Sound pressure response in two triangular coupled cavities under the condition of rigid walls.
Fig. 10. Variation of frequency of the triangular coupled cavities with different geometric dimensions.
The test system shown in Fig. 11(a) is the experimental figure of trapezoidal cavity and Fig. 11(b) is of triangular cavity. The geometrical dimensions of the trapezoidal cavity are a = 0.38 m, b = 0.28 m, h = 0.39 m, and a = 0°, b = 53° and the dimensions of
the triangular cavity are a = 0.47 m, h = 0.47 m, and a = 0°, b = 45°. Reference value of sound pressure is 2 105 Pa. The rigid boundary conditions are simulated by plexiglass plates with a thickness of 10 mm. White noise is played by the noise excitation
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1. Power 2. Dynamic signal test analysis system 3. Computer
1. Power 2. Dynamic signal test analysis system 3. Computer 1 Noise excitation source
1
2
2
3
cavity
Noise excitation source
3
cavity
Microphone
Microphone
(a) The trapezoidal cavity
(b) The triangular cavity Fig. 11. Experimental setup.
Fig. 12. Schematic diagram of the experiment.
source to trigger more frequencies, and the sound pressure values of the sound source and the measuring point are measured by two microphones (MPA201). Then, the curve of the sound pressure response can be obtained by inputting the data to a dynamic signal test analysis system (DH5922N). The schematic diagram of the experiment is shown in Fig. 12. The comparison of experimental data and current prediction are plotted in Figs. 13 and 14. The truncated values in the theoretical
calculation are M = N = L = 6. From the comparison, it can be gathered that the trend of the curves is consistent, which validates the accuracy of the three-dimensional modified Fourier series method. The reason why the values are different in the graph is that the boundary conditions used in the experiment are not the theoretical rigid walls that absorb sound pressure, which results in differences in the resonant frequency. Due to the influence of the test instrument inside the cavity, the curve shows a peak value around 150 Hz.
Fig. 13. Comparison of experimental data with current prediction of the trapezoidal cavity while sound source located in (a) (0.11, 0.05, 0); (b) (0.08, 0.26, 0); (c) (0.09, 0.19, 0).
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Fig. 14. Comparison of experimental data with current prediction of the triangular cavity while sound source located in (a) (0.22, 0.02, 0); (b) (0.03, 0.04, 0); (c) (0.14, 0.22, 0).
5. Conclusions In this paper, a modified three-dimensional Fourier series method is applied to study the acoustic characteristics of cavities with various impedance boundary conditions. The admissible sound pressure function consists of a three-dimensional Fourier cosine series and a complementary function. The unknown coefficients of the sound pressure expression can be obtained by performing the Rayleigh-Ritz procedure. The acoustic characteristics of the irregular acoustic model are analyzed by transforming its coordinates into Cartesian coordinates with unit length. The agreement of both the mode shapes and steady-state responses for the present cavities with rigid or impedance boundary conditions between the present results and those obtained from finite element validates the reliability of the present method. What’s more, three important conclusions about acoustic characteristics of cavity are obtained, which are as follows: (1) The change of the medium only affects the frequency of the model and does not change the trend of the sound pressure response curve. The natural frequency ratio of different media is equal to the sound speed ratio in the medium. (2) Reducing the impedance value of the impedance surface and increasing the number of impedance walls have a marked effect on reducing the resonance amplitude of the sound pressure response. (3) The change of geometric parameters has a great effect on the natural frequency of the coupled cavities. For example, the natural frequencies could be reduced by increasing the length or the height of the cavity.
Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 51705537 and 51679056) the Natural Science Foundation of Hunan Province of China (2018JJ3661) and Innovation Driven Program of Central South University (Grant number: 2019CX006). The authors also gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2018-11). References [1] Maa DY. Non-uniform acoustical boundaries in rectangular rooms. J Acoust Soc Am 1940;12:465. [2] Morse PM, Bolt RH. Sound waves in rooms. Rev Mod Phys 1944;16:70–150.
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