A unified modeling method for the rotary enclosed acoustic cavity

Applied Acoustics 163 (2020) 107230

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A unified modeling method for the rotary enclosed acoustic cavity Hong Zhang a, Rupeng Zhu a, Dongyan Shi b, Qingshan Wang c,⇑ a

National Key Laboratory of Science and Technology on Helicopter Transmission, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, 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 11 March 2019 Accepted 19 January 2020

Keywords: Unified analytical model Rotary enclosed cavity Acoustic characteristics Impedance walls 3D modified Fourier series method Rayleigh-Ritz energy method

a b s t r a c t The rotary enclosed acoustic cavity has three common forms, which are conical, cylindrical and spherical cavity. They are widely used in practical production and belong to the category of rotary room acoustics. Therefore, the study of their acoustic characteristics is of great engineering significance. In this paper, a unified analytical model for acoustic characteristics of rotary enclosed cavity with various impedance walls is first established. A three-dimensional (3D) modified Fourier series method is proposed to construct the admissible function of sound pressure. Specifically, the sound pressure function is invariably expressed as a 3D trigonometric series superposition, which includes the multiplication of three cosine functions and six complementary polynomials. The introduction of complementary polynomials can effectively solve the impedance acoustic wall. Based on Rayleigh-Ritz energy method, the acoustic characteristics of the unified analysis model can be obtained. The sound pressure response of the cavity under the influence of different impedance walls is further studied by placing a point sound source inside the acoustic cavity. The accuracy of the unified model is verified by comparing the present results with those obtained by finite element method (FEM) and experiment, and the effect of important parameters on the acoustic characteristics is systematically studied. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction With the continuous development of economy and society, people’s requirements for vibration and noise control are becoming more and more stringent in both military and civil fields. Enclosed sound field, as a classical problem in room acoustics, has attracted much attention. Its acoustic field distribution and acoustic response characteristics can provide a good theoretical basis for complex acoustic space and noise control. Up to now, based on the analysis model, the enclosed sound field can be divided into three categories: rectangular cavity, irregular rectangular-like cavity and rotary cavity. Based on the wall boundary condition, the enclosed sound field can be divided into dissipative acoustic wall and non-dissipative acoustic wall. Rigid walled rectangular acoustic cavity, as an idealized indoor acoustic research model, its development provides a necessary basis for the in-depth study of enclosed sound field. However, as a more practical and instructive analytical model, dissipative walled rectangular acoustic cavity has gradually attracted more

⇑ Corresponding author. E-mail addresses: [email protected] (H. Zhang), [email protected] (R. Zhu), [email protected] (D. Shi), [email protected] (Q. Wang). https://doi.org/10.1016/j.apacoust.2020.107230 0003-682X/Ó 2020 Elsevier Ltd. All rights reserved.

and more attention. Based on Newton iteration method and homotopy continuation method, Bistafa and Morrissey [1] developed two numerical procedures to find the acoustic eigenvalues in the rectangular room with arbitrary (uniform) wall impedances. Then, Naka et al. [2] adopted the interval Newton/generalized bisection (IN/GB) method for estimating the modal acoustic responses of rectangular rooms with finite wall impedances. This method could solve the problem of initial value selection and root leaking in the traditional Newton iteration method. Du et al. [3] presented a Fourier series method to analyze the acoustic characteristics of a rectangular cavity with general impedance boundary conditions. Combining the energy principle with Rayleigh-Ritz technique, the modal parameters of the acoustic cavity can be easily obtained by solving a standard matrix eigenvalue problem rather than iteratively solving a nonlinear transcendental equation. Jin et al. [4] proposed a general Chebyshev-Lagrangian method to obtain the analytical solution for a rectangular acoustic cavity with arbitrary impedance boundary conditions. The sound pressure was expressed uniformly as Chebyshev polynomials in three directions, and the series solution of Chebyshev polynomials could be obtained by Rayleigh-Ritz process. On the basis of rectangular acoustic cavity, irregular rectangular-like acoustic cavity is derived. According to the

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H. Zhang et al. / Applied Acoustics 163 (2020) 107230

coupling between rigid-walled modes of a rectangular cavity, Sum and Pan [5] obtained the shapes and resonance frequencies of rigid-walled modes of a trapezoidal cavity with an inclined rigid wall. Missaoui and Cheng [6] presented an integro-modal approach to compute the acoustic properties of irregular-shaped cavities. In this method, the whole cavity was discretized into a series of subcavities, whose sound pressure was decomposed on the modal basis of regular sub-cavity or irregular shape boundary subcavity. Comparisons with other existing methods showed that good accuracy of cavity modal calculation can be obtained by applying limited number of sub-cavities. Xie et al. [7] proposed a weak variational principle based approach to study the enclosures with inclined walls and impedance boundary conditions. The convergence, stability and accuracy of the proposed method were verified by comparing the results with those obtained by analytical solutions or FEM. Chen et al. [8] presented a domain decomposition method to predict the acoustic characteristics of an arbitrary enclosure made up of any number of sub-spaces. The influence of coupling parameters between sub-spaces on the natural frequencies and mode shapes of the overall enclosure was studied. In summary, the research on rectangular enclosed cavity with dissipative acoustic wall and non-dissipative acoustic wall has achieved fruitful results. On this basis, the research of various irregular rectangular-like cavities has been carried out, which greatly compensates for the monotony of rectangular cavity. At the same time, based on the idea of dividing the whole acoustic domain into finite subspaces, the modeling method of irregular rectangular-like cavity has certain universality. Nevertheless, when it is extended to solve the acoustic characteristics of rotary enclosed space such as aircraft, rocket and submarine, the rectangular cavity modeling method is no longer applicable. Therefore, it is of equal importance to study the rotary acoustic cavity. Based on Neumann and Dirichlet boundary conditions, Williams [9] presented a set of Green’s functions for the Helmholtz equation applied to the interior of a cylindrical cavity, and the acoustic characteristics of the cylindrical cavity were studied in detail. Chen et al. [10] proposed a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF). The eigen-solutions of circular and rectangular cavities were obtained, which validated the correctness of the proposed method. Choi et al. [11] developed a theoretical approach to obtain the natural characteristics of annular cavities which have locally non-uniform media. Parametric studies were carried out to reveal the relation between the acoustic characteristics and the local deviation of the media. Based on Green’s functions, Shao and Mechefske [12] presented an approximate acoustic model of a finite cylindrical duct. In addition to some minor differences in the area near the open ends, the results of the acoustic field calculated by this model were similar to those obtained by the boundary element method (BEM) model, but the calculation efficiency of this model was greatly improved. Through solving the homogeneous wave equation in elliptical cylindrical co-ordinates, Hong and Kim [13] achieved the analytical solutions for natural characteristics of elliptical cylindrical acoustic cavities. Lee [14,15] presented a semi-analytical method to study the eigenproblem of a 2D acoustic cavity with multiple elliptical boundaries. The acoustic characteristics of an elliptical cavity, a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders were studied in detail. On this basis, he [16] also studied the acoustic characteristics of 3D elliptical resonators with multiple elliptical through-hole. In conclusion, compared with rectangular enclosed sound field, there are relatively few studies on rotary enclosed sound field, especially on the condition of dissipative acoustic wall. At the same time, the research on the rotary enclosed sound field is mainly focused on the cylindrical acoustic cavity, but few on the conical and spherical acoustic cavity. In addition, most of the analytical

models mentioned above aim at a specific shape and lack generality. Based on the limitations of current research, it is of great theoretical value and practical significance to establish a unified analysis model for acoustic field characteristics of a rotary enclosed acoustic cavity with arbitrary impedance boundary conditions. 2. Theoretical formulations 2.1. Description of the unified rotary cavity The emphasis of this paper is to establish a unified analysis model of sound field characteristics for three rotary enclosed acoustic cavities. They are conical cavity, cylindrical cavity and spherical cavity, respectively. These three kinds of acoustic cavities can be converted from each other in geometric relations and coordinate relations, which is also the basis of establishing a unified acoustic analysis model. Fig. 1 shows the basic unit model of the rotary enclosed cavity. Besides, Fig. 2 gives the geometric parameters and coordinate system of the conical, cylindrical and spherical cavities, respectively. It is not difficult to see from these two figures that the conical, cylindrical and spherical enclosed cavities can be regarded as a special form of the basic unit acoustic model. Specifically, the bottom of the model (z = 0) is used as the reference surface to establish the orthogonal coordinate system, as shown in Fig. 1. Here, Ra and Rb represent the reference curvature radius along the a-axis and b-axis. La, Lb and Lz are the dimensions of length, width and height. In addition, a monopole point source Q is introduced to analyze the sound pressure response of the cavity. The local coordinate system and corresponding geometric parameters of rotary cavities are given in Fig. 2. In Fig. 2(a), the local coordinate system of the conical cavity is (o-r, h, s). Small end radius, large end radius, apex angle, length and thickness are represented as R1, R2, c, L, and H, separately. It should be pointed out that the distance of point q to the axis of rotation can be 

expressed as: Rq ðr; sÞ ¼ rcosc þ ssinc þ R1 and Rq ¼ Rq =cosc. In Fig. 2(b), the local coordinate system of the cylindrical cavity is (o-r, h, s). The inner radius, outer radius, thickness and height are expressed by R1, R2, H and L, respectively. In Fig. 2(c), the local coordinate system of the spherical cavity is (o-r, h, u). The inner radius, outer radius, thickness and azimuthal angle are expressed by R1, R2, H and / (/ ¼ /1  /2 ), respectively. Fig. 3 shows the shapes of the three kinds of cavities with different rotation angles #. It is easy to see that the fully closed-loop rotary cavities can be obtained when # ¼ 2p. In order to more clearly explain the conversion relationship between conical cavity, cylindrical cavity and spherical cavity, the relationship between their local coordinate system and the coordinate system of the unified analysis model for rotary acoustic cavity is given below. Besides, the Lame coefficients are also given.

Fig. 1. Geometry and coordinate system of the basic unit for the rotary acoustic cavities.

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H. Zhang et al. / Applied Acoustics 163 (2020) 107230

to construct the sound pressure function expression of the rotary cavity [17,18]:

pða; b; z; tÞ ¼ ejxt PX ða; b; zÞ þ

6 X

!

PSi ða; b; zÞ Amnl

ð4Þ

i¼1

Fig. 2. Geometry and coordinate system of the cross sections of rotary conical, cylindrical and spherical acoustic cavities.

in which PX represents the expression of sound pressure inside the cavity, and PiS represents supplementary polynomials in the i’th wall. The addition of these complementary polynomials can effectively maintain the continuity of the sound pressure function at the impedance wall. Besides, Amnl is unknown coefficient matrix. Next, the specific expressions are given as follow:

(

X

P ða;b;zÞ ¼

L

L

L

L

)

coskL0a acosk0b bcoskL0z z;  ;coskL0a acosk0b bcoskLLz z;  ; coskL0a acoskNb bcoskLLz z;  ;coskLMa acoskNb bcoskLLz z

ð5:aÞ ( PS1 ða;b;zÞ ¼

L

L

L

L

a sinkL2 acosk0b bcoskL0z z;  ;sinkL2a acosk0b bcoskLLz z;  ;

)

a acoskNb bcoskLLz z;  ;sinkL1a acoskNb bcoskLLz z sinkL2

ð5:bÞ ( PS2 ða;b;zÞ ¼

L

L

L

L

b b coskL0a asink2 bcoskL0z z;  ;coskL0a asink2 bcoskLLz z;  ;

)

b b bcoskLLz z;  ;coskLMa asink1 bcoskLLz z coskL0a asink2

ð5:cÞ ( PS3 ð

a;b;zÞ ¼

L

L

z z coskL0a acosk0b bsinkL2 z;coskL0a acosk0b bsinkL1 z;  ;

L

)

L

z z z;  ;coskLMa acoskNb bsinkL1 z coskL0a acoskNb bsinkL2

ð5:dÞ ( PS4 ð

a;b;zÞ ¼

L

L

L

L

a sinkL2 asink2b bcoskL0z z;  ;sinkL2a asink2b bcoskLLz z;  ;

)

a asink1b bcoskLLz z;  ;sinkL1a asink1b bcoskLLz z sinkL2

ð5:eÞ ( Fig. 3. The unified model of rotary conical, cylindrical and spherical acoustic cavities.

PS5 ða; b; zÞ ¼

L

L

a sinkL2 acosk0b bsinkL2z z; sinkL2a acosk0b bsinkL1z z;   ;

L

)

L

a acoskNb bsinkL2z z;   ; sinkL1a acoskNb bsinkL1z z sinkL2

ð5:fÞ (a) Conical cavity:

a ¼ s; b ¼ h; z ¼ r; 0 6 s 6 L; 0 6 h 6 #; 0 6 r 6 H Ha ¼ Hs ¼ 1; Hb ¼ Hh ¼ Rðr; sÞ; Hz ¼ Hr ¼ 1

( ð1Þ

ð2Þ

(c) Spherical cavity:

a ¼ u; b ¼ h; z ¼ r; 0 6 u 6 /; 0 6 h 6 #; 0 6 r 6 H Ha ¼ Hu ¼ r; Hb ¼ Hh ¼ rsinu; Hz ¼ Hr ¼ 1

L

)

L

b b z z coskL0a asink2 bsinkL2 z; coskL0a asink2 bsinkL1 z;

L

L

b b z z bsinkL2 z;   ; coskLMa asink1 bsinkL1 z coskL0a asink1

ð5:gÞ

(b) Cylindrical cavity:

a ¼ s; b ¼ h; z ¼ r; 0 6 s 6 L; 0 6 h 6 #; 0 6 r 6 H Ha ¼ Hs ¼ 1; Hb ¼ Hh ¼ r; Hz ¼ Hr ¼ 1

PS6 ða; b; zÞ ¼

8 1 9T A0;0;0 ;  ;A10;0;l ;  ;A10;0;L ;  ;A10;N;L ;  ;A1m;n;l ;  ;A1M;N;L ; > > > > > > > > > > 2 2 2 2 2 2 > > A2;0;0 ;  ;A2;0;l ;  ;A2;0;L ;  ;A2;n;l ;  ;A1;n;l ;  ;A1;N;L ; > > > > > > > > > > 3 3 3 3 3 3 > > > A0;2;0 ;  ;A0;2;l ;  ;A0;2;L ;  ;Am;2;l ;  ;Am;1;l ;  ;AM;1;L ; > > > < = 4 4 4 4 4 4 Amnl ¼ A0;0;2 ;A0;0;1 ;  ;A0;N;2 ;  ;Am;n;2 ;  ;Am;n;1 ;  ;AM;N;1 ; > > > > > > 5 5 5 5 5 > A5 > > 2;2;0 ;  ;A2;2;l ;  ;A2;2;L ;  ;A2;1;l ;  ;A2;1;L ;  ;A1;1;L ; > > > > > > > > > 6 6 6 6 6 6 > > > A2;0;2 ;A2;0;1 ;  ;A2;n;2 ;  ;A2;N;2 ;  ;A1;n;L ;  ;A1;N;1 ; > > > > > > > : 7 ; A0;2;2 ;A70;2;1 ;  ;A7m;2;2 ;  ;A7m;1;1 ;  ;A7M;2;2 ;  ;A7M;1;1

ð6Þ L

ð3Þ

2.2. Admissible function of sound pressure The purpose of this paper is to study the sound field characteristics of rotary acoustic cavity by establishing a unified analysis model. In this paper, an improved Fourier series method is used

where kLma ¼ mp=La , knb ¼ np=Lb and kLl z ¼ lp=Lz . It can be found that the sound pressure expressions based on the three-dimensional improved Fourier series method ensure that the first derivative is continuous at any point in the whole solution domain. Therefore, the unified analysis acoustic model can be easily extended from the rigid-walled conditions to the impedance-walled conditions, including uniform and nonuniform impedance-walled conditions. Next section, from the perspective of three-dimensional solid mechanics, the specific energy

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H. Zhang et al. / Applied Acoustics 163 (2020) 107230

equation of the unified rotary cavity is established to solve the acoustic characteristics. 2.3. Energy equation of the unified rotary cavity



Based on the acoustic principle, a unified solution equation for the acoustic characteristics of rotary cavity could be established, which satisfies both the Helmholtz equation and the varying impedance walled conditions. First, the Lagrange energy equation of the unified analysis model should be given. Then the sound pressure admissible function is substituted into the energy expression of the acoustic cavity system. Finally, Rayleigh-Ritz energy method is used to solve the acoustic characteristics. The Lagrange energy equation for the rotary cavity is given here:

LC ¼ T C  U C  W Coupling þ W Wall þ W S

UC ¼ ¼

1 2qc2

R



V

R La R Lb R Lz 0

0

0



X

P ða; b; zÞ þ

6 P i¼1

 PSi ð

TC ¼

2qx2

R

W 3Wall ¼  

S3

W 4Wall ¼  

a; b; zÞ Amnl

W 5Wall ¼ 

PSi ða; 0; zÞ Amnl

i¼1

#2 9 = ;

dS3 jb¼0

:

X



P a; Lb ; z þ

6 X

S

Pi a; Lb ; z



!

i¼1

1 2jxZ 5 8 Z <"

W 6Wall ¼ 

:

X

P ða; b; 0Þ þ

6 X

#2 9 = a; b; 0Þ Amnl dS j ; 5 z¼0 !

PSi ð

i¼1

:

X

P ða; b; Lz Þ þ

6 X i¼1

#2 9 = a; b; Lz Þ Amnl dS j ; 6 z¼Lz !

PSi ð

ð17Þ where j is the pure imaginary number. Zi (i = 1–6) is the impedance value corresponding to the impedance wall, and it can be a constant or a function which represents the change of impedance value. In order to qualitatively describe the effect of impedance value on sound pressure response, Zi is set as a constant value in the following numerical examples. When # ¼ 2p, two acoustic walls of the rotary cavity are coupled together to form a complete closed loop. At the coupling interface, two continuity conditions need to be satisfied at the same time, which are p(a, 0, z) = p(a, 2p, z) and @pða; 0; zÞ=@b ¼ @pða; 2p; zÞ=@b. Based on this, the coupling potential energy WCoupling is written as:

ð10Þ

si

W Coupling ¼

W Wall ¼ W 1Wall þ W 2Wall þ W 3Wall þ W 4Wall þ W 5Wall þ W 6Wall

1 2jxZ 6 8 Z <" S6

where the density and sound velocity of the acoustic medium in the cavity are represented by symbols q and c. The circular frequency of the rotary cavity is expressed as x. Besides, gradp (a, b, z) is the gradient function of sound pressure. As shown in Fig. 3, the rotary cavity has six acoustic walls when the maximum value of rotation angle is less than 2p. Therefore, when there are six impedance walls, the specific expression of the dissipation energy WWall can be written as: 6 Z 1X p2 ða; b; zÞ ¼ dSi 2 i¼1 jxZ i

#2 9 = Amnl dS j ; 4 b¼Lb

ð16Þ



ð9Þ

ð11Þ

Z

1

qx

2

0

La

8   9 6 P > > @PSi > > @PX > > ða;0;zÞ þ Hb @b ða;0;zÞ > > Hb @b > > > > i¼1 > > > > > > 2 3 >   Z Lz > = < 6 P X S A2mnl Ha Hz dadz P ða;0;zÞ þ Pi ða;0;zÞ 6 7 > 6 7 0 > i¼1 > > > > 6 7 > > 6  > 7> > > > > 6 > > > 4  PX a;L ;z þ P PS a;L ;z 5 > > > b b ; : i i¼1

1

:

!

ð15Þ

fgradpða;b;zÞg dV

S1

PX ða; 0; zÞ þ

6 X

Ha Hb Hz dadbdz

i¼1



:

1 2jxZ 4 8 Z <" S4

2

2jxZ 1 8 Z <"

#2 9 = Amnl dS j ; 2 a¼La

ð14Þ

V

W 1Wall ¼ 

P ðLa ; b; zÞ þ

! PSi ðLa ; b; zÞ

i¼1

1 2jxZ 3 8 Z <"

S5

8   2 9 6 P > > @PSi ða;b;zÞ @PX ða;b;zÞ > > > > þ Amnl > > Ha @ a Ha @ a > > > > i¼1 > > > > > >    2 < = 6 R R R S X P L L L a z @P ð a ;b;z Þ b 1 @P ða;b;zÞ i ¼ 2qx Ha Hb Hz dadbdz þ þ A 2 0 mnl 0 0 > Hb @b Hb @b > > > i¼1 > > > > >   2 > > > 6 > > P > > @PSi ða;b;zÞ @PX ða;b;zÞ > > : ; þ Amnl Hz @z Hz @z

W Wall

:

6 X

X

ð13Þ

2

ð8Þ 1

S2



p ða; b; zÞ dV 2

1 2jxZ 2 8 Z <"

ð7Þ

in which TC and UC are the total kinetic energy and potential energy in the rotary cavity. As mentioned earlier, when the rotation angle is 2p, two acoustic walls will be coupled together. The coupling potential energy (WCoupling) is used to guarantee the continuity condition of sound pressure and particle motion. Besides, the dissipative energy induced by impedance wall is expressed by WWall, and the work done by the point sound source inside the cavity is marked as WS. The total potential energy UC and the total kinetic energy TC in the rotary enclosed cavity are respectively expressed as: 1 2qc2

W 2Wall ¼ 

ð18Þ

#2 9 ! = 6 X X S dS j P ð0; b; zÞ þ Pi ð0; b; zÞ Amnl ; 1 a¼0 i¼1 ð12Þ

The effect of varying impedance walled conditions on the acoustic characteristics of the rotary cavity is further studied by placing a monopole point sound source inside the cavity. The point source is made of a pulsating ball, whose incident pressure can be expressed as:

5

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

ejkR0 R0

ppoint ¼ A

ð19Þ

in which symbol A is the amplitude (kg/s2). R0 is the sound source radiation distance. In addition, the acoustic wave number is expressed as k, and its relationship with acoustic wave velocity and sound field circular frequency is k = x/c. Thus, the sound intensity of the monopole point source Qpoint (m3/s) can be obtained:

Q point ¼

4p A jqck

Z

Lz



Z

1 jx (

La

Z

0

Lb

0

X

P ða; b; zÞ þ

0

6 X

!

)

PSi ða; b; zÞ Amnl Q Ha Hb Hz dadbdz

2

The energy Eqs. (8)–(22) in the rotary enclosed cavity are substituted into the total Lagrange energy Eq. (7). On the basis of Rayleigh-Ritz energy technique, the extreme value of total energy function against the unknown Fourier coefficient matrix, and the following relation can be obtained:

The Eq. (23) is given in the form of matrix:

ð24Þ

where K, C, Z, M and F represents the global stiffness matrix, acoustic coupling matrix, sound pressure dissipation matrix, mass matrix and force vector. Their specific expressions can be written as follow:

2

K11

6 KT 6 12 6 T 6K 6 13 6 T K¼6 6 K14 6 T 6 K15 6 6 KT 4 16 KT17 K11 ¼

1

q

Z

La

0

Z  0

Z

K12

K13

K14

K15

K16

K22

K23

K24

K25

K26

KT23

K33

K34

K35

K36

KT24

KT34

K44

K45

K46

KT25 KT26 KT27

KT35

KT45

K55

K56

KT36

KT46

KT56

K66

KT37

KT47

KT57

KT67

q

K27 7 7 7 K37 7 7 7 K47 7 7 7 K57 7 7 K67 7 5 K77

C17

3

C12

C13

C14

C15

C16

C22

C23

C24

C25

C26

CT23 CT24 CT25 CT26 CT27

C33

C34

C35

C36

CT34 CT35 CT36 CT37

C44

C45

C46

CT45

C55

C56

CT46

CT56

C66

C27 7 7 7 C37 7 7 7 C47 7 7 7 C57 7 7 C67 7 5

CT47

CT57

CT67

C77

ð29Þ

La

0

Z

Lz

(

0

)  @PX ða;0;zÞ X @PX ða;0;zÞ X

P ða;0;zÞ  P a;Lb ;z Ha Hz dadz Hb @b Hb @b ð30Þ

8 9 @PSf 1 ða;0;zÞ X > Z Z > @PX ða;0;zÞ S = 1 La Lz < Hb @b Pf 1 ða;0;zÞ þ P ða;0;zÞ Hb @b Ha Hz dadz C1f ¼ S   @Pf 1 ða;0;zÞ > > @PX ða;0;zÞ S

q 0 0 : X

;  Hb @b Pf 1 a;Lb ;z  P a;Lb ;z Hb @b ð31Þ

Cef ¼

1

q

Z

La

0

Z

Lz

(

0

)  @PSe1 ða;0;zÞ S @PS ða;0;zÞ S

Pf 1 ða;0;zÞ  e1 Pf 1 a;Lb ;z Hb @b Hb @b ð32Þ

Ha Hz dadz

2

Z11

6 ZT 6 12 6 T 6Z 6 13 6 T Z¼6 6 Z14 6 T 6 Z15 6 6 ZT 4 16 ZT17 8 > > > > > > > > <

ð25Þ Z11

Z12

Z13

Z14

Z15

Z16

Z22

Z23

Z24

Z25

Z26

ZT23 ZT24 ZT25 ZT26 ZT27

Z33

Z34

Z35

Z36

ZT34 ZT35 ZT36 ZT37

Z44

Z45

Z46

ZT45

Z55

Z56

ZT46

ZT56

Z66

ZT47

ZT57

ZT67

Z17

3

Z27 7 7 7 Z37 7 7 7 Z47 7 7 7 Z57 7 7 Z67 7 5 Z77

ð33Þ

9 i2 i2 Rh X > PX ð0; b; zÞ dS1 ja¼0 þ jZ12 P ðLa ; b; zÞ dS2 ja¼La > > > > S1 S2 > > > h i h i = 2 2

 R R X X 1 1 P ða; 0; zÞ dS3 jb¼0 þ jZ4 P a; Lb ; z dS4 jb¼Lb ¼ þ jZ3 > > S3 S4 > > > > > > h i2 h i2 > > R R > > X X > 1 1 > > P ða; b; 0Þ dS5 jz¼0 þ jZ P ða; b; Lz Þ dS6 jz¼Lz > : þ jZ ; 1 jZ 1

Rh

5

S5

6

S6

ð34Þ

8" # < @PX ða;b;zÞ 2 :

3

z

Lb

0

Lz

K17

Z



ð23Þ

 K þ C þ xZ  x2 M Amnl ¼ F

C11

1

C11 ¼

ð22Þ

2.4. Solution equation of the unified rotary cavity

Lb

0

6 CT 6 12 6 T 6C 6 13 6 T C¼6 6 C14 6 T 6 C15 6 6 CT 4 16

where Q represents the volume velocity amplitude of the monopole point source. d is a Dirac function. (a0, b0, z0) is the specific location of the point source.



0

CT17 ð21Þ

@W coupling @U C @T C @W Wall @W S þ   ¼ @Amnl @Amnl @Amnl @Amnl @Amnl

q

Z

La

ð28Þ

i¼1

Q ¼ Q point dða  a0 Þdðb  b0 Þdðz  z0 Þ

Z

1

z

ð20Þ

Then, the work done by the point sound source WS can be written as follow:

WS ¼ 

Kef ¼

9 8  S S @Pe1 ða;b;zÞ @Pf 1 ða;b;zÞ > > > > > > Ha @ a Ha @ a > > > > > > >  Z Lz > < = S S @Pe1 ða;b;zÞ @Pf 1 ða;b;zÞ þ Ha Hb Hz dadbdz Hb @b Hb @b > > 0 > > >  > > > > @PS ða;b;zÞ @PS ða;b;zÞ > > > f 1 > > : þ e1 ; H @z H @z

Ha @ a

þ

" #2 " #2 9 @PX ða;b;zÞ @PX ða;b;zÞ = þ H H H dadbdz ; a b z Hb @b Hz @z

ð26Þ

9 8  S @PX ða;b;zÞ @Pf 1 ða;b;zÞ > > > > > > Ha @ a Ha @ a > > > > > > >  Z La Z Lb Z Lz > < = S X @P ð a ;b;z Þ 1 @P ða;b;zÞ f 1 K1f ¼ þ Hb @b Ha Hb Hz dadbdz Hb @b > q 0 0 0 > > > > >  > > > > X @PS ða;b;zÞ > > > > : þ @P Hða@z;b;zÞ f 1 ; H @z z

z

ð27Þ

i 9 8 Rh X S 1 > > > > > > jZ 1 P ð0;b;zÞPf 1 ð0;b;zÞ dS1 ja¼0 > > S1 > > > > h i > > R > > X S > > 1 > > þ P ð L ;b;z ÞP ð L ;b;z Þ dS j > a a 2 a ¼L f 1 jZ 2 a > > > > > S2 > > > > h i > > R > > X S > > 1 > > þ jZ P ð a ;0;z ÞP ð a ;0;z Þ dS j > > 3 b¼0 f 1 = < 3 S3 h i Z1f ¼  S

 R X

> 1 > > þ P a;Lb ;z Pf 1 a;Lb ;z dS4 jb¼Lb > > > > > > > jZ 4 S4 > > > > h i > > R > > X S > >þ 1 > > P ð a ;b;0 ÞP ð a ;b;0 Þ dS j 5 z¼0 > > f 1 jZ 5 > > > > S5 > > > > h i > > R > > X S > >þ 1 > P ð a ;b;L ÞP ð a ;b;L Þ dS j z z 6 z¼Lz > ; : jZ6 f 1 S6

ð35Þ

6

Zef

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

i 8 Rh 9 1 > > PSe1 ð0; b; zÞPSf 1 ð0; b; zÞ dS1 ja¼0 > > jZ 1 > > > > S1 > > > > > > h i > > R S > > S 1 > > þ P ð L ; b; z ÞP ð L ; b; z Þ dS j > 2 a¼La > e1 a f 1 a > > jZ 2 > > > > S 2 > > > > h i > > R > 1 > S S > > > > þ P ð a ; 0; z ÞP ð a ; 0; z Þ dS j 3 > > b¼0 e1 f 1 < jZ3 S = 3 h i ¼



 R >þ 1 > > PSe1 a; Lb ; z PSf 1 a; Lb ; z dS4 jb¼Lb > > > > jZ4 > > > S4 > > > > > > h i > 1 R S > > > S > þ jZ5 Pe1 ða; b; 0ÞPf 1 ðs; h; 0Þ dS5 jz¼0 > > > > > > > > > S5 > > > > h i > > > 1 R S > S > > > þ P ð a ; b; L ÞP ð a ; b; L Þ dS j z z 6 z¼Lz > : jZ6 ; e1 f 1

ð36Þ

S6

2

M11 6 MT 6 12 6 T 6M 6 13 6 T M¼6 6 M14 6 T 6 M15 6 6 MT 4 16 MT17 Z

1 qc

M11 ¼

M13

M14

M15

M16

M22

M23

M24

M25

M26

MT23 MT24 MT25 MT26 MT27

M33

M34

M35

M36

MT34

M44

M45

M46

MT35

MT45

M55

M56

MT36

MT46

MT56

M66

MT37

MT47

MT57

MT67

Z 0 Lz

0

Z

1 qc0 Z 

Mef ¼

0 Lz

nh

0

F ¼ ½ F1 F1 ¼

F2

q

La

0

Z

Lz



n

Lz

0

0

0

i o PSe1 ða; b; zÞPSf 1 ða; b; zÞ Ha Hb Hz dadbdz F4

Z

F5

F6

F7 T

ð40Þ ð41Þ

Lb

0

o P ða; b; zÞdða  a0 Þdðb  b0 Þdðz  z0 ÞHa Hb Hz dadbdz X

0

ð42Þ Fe ¼

Z

4p A

q

Z



La

Z

0 Lz

0

Lb

0

n o PSe1 ða; b; zÞdða  a0 Þdðb  b0 Þdðz  z0 ÞHa Hb Hz dadbdz ð43Þ

where e = f = 2, 3, 4, 5, 6, 7. The natural frequencies and mode shapes can of the rotary cavity can be obtained by setting the right side of the Eq. (24) to zero. However, the equation has both the first term and the square term of the circular frequency, which makes the equation a non-linear problem and is not easy to solve directly. Further transformation is needed here to obtain a linear equation about frequency.

ðR  xSÞG ¼ 0  R¼

0

K

K

Z

ð44Þ

 ;S ¼

In the last section, a unified analysis model of the rotary acoustic cavity is constructed based on fundamental acoustic principles and 3D modified Fourier-Ritz method. In this section, the conical, cylindrical and spherical acoustic cavities will be studied in detail by using the established unified analytical model. It mainly includes the following four parts: model validation including convergence and correctness analysis, acoustic natural frequency analysis, acoustic response analysis and experimental analysis of completely enclosed cylindrical cavity. The two fluid media used for filling the acoustic cavity are air and water, whose physical parameters are defined as qair = 1.225 kg/m3, cair = 344 m/s and qwater = 1000 kg/m3, cwater = 1480 m/s, respectively. 3.1. Model validation

Lb

F3

Z

4pA

ð37Þ

Lb

Z

La

3. Numerical results and discussions

ð39Þ

Z

nh

M27 7 7 7 M37 7 7 7 M47 7 7 7 M57 7 7 M67 7 5 M77

i o PX ða; b; zÞPSf 1 ða; b; zÞ Ha Hb Hz dadbdz

Z

0 La

3

ð38Þ

Lb

h

M17

 i2 PX ða; b; zÞ Ha Hb Hz dadbdz

Z

La

0

1 qc0 Z 

M1f ¼

M12

It is easy to find from Eq. (44) that the circular frequencies and unknown Fourier coefficients of the rotary cavity are the generalized eigenvalues and eigenvectors of this equation, separately. In addition, it should be pointed out that when the acoustic wall of the cavity is a complex impedance wall, the generalized eigenvalues and generalized eigenvectors obtained from the solution will also be complex, which belongs to the problem of complex acoustic modes. The real part of the generalized eigenvalue is the natural frequency, while the imaginary part is the attenuation coefficient of the acoustic modes, which determines the attenuation speed of the acoustic wave in the acoustic cavity. Therefore, the introduction of impedance wall does not change the natural frequency of the cavity. In order to simplify the process of solving the acoustic natural frequencies of the rotary cavity, all the acoustic walls can be set as rigid walls.

K

0

0

M



;G ¼

Amnl

xAmnl

 ð45Þ

In this section, the feasibility of the unified analysis model of the rotary acoustic cavity will be verified. First, Table 1 gives the convergence and accuracy analysis of the first seven natural frequencies of a conical acoustic cavity. The geometric parameters of the conical cavity are defined as: R1 = 0.5 m, R2 = 1 m, H = 0.5 m, # = 90° and c = 30°. It can be found from Table 1 that the present method basically converges when the truncated value is M  N  Q = 5  5  5. Whether the acoustic medium is air or water, the maximum difference between the calculated results under different truncated values does not exceed 0.001%. At the same time, we can see that the results obtained by the present method and those obtained by FEM are in good agreement, and the maximum difference is less than 0.001%. Moreover, Fig. 4 shows the first four order sound pressure distributions of the conical cavity obtained by the present method and FEM, which further validates the applicability of the established analysis model for predicting the acoustic characteristics of conical acoustic cavity. Next, Tables 2 and 3 give the convergence and accuracy analysis of the first seven natural frequencies of a cylindrical and spherical acoustic cavity respectively. Figs. 5 and 6 show the first four order sound pressure distributions of the cylindrical and spherical acoustic cavity respectively. In Table 2, the geometric parameters of the 

cylindrical cavity are chosen as: R1 = 0.5 m, R2 = 1 m, # ¼ 90 and L = 2 m. And in Table 3, the geometric parameters of the spherical cavity are chosen as: R1 = 0.5 m, R2 = 1 m, # = 90°, /1 = 30° and /2 = 150°. It can be seen from Tables 2 and 3 that the unified analytical model established in this paper has good convergence and high accuracy in dealing with cylindrical and spherical acoustic cavities, while the pressure distribution maps in Figs. 5 and 6 give us a more intuitive understanding. As mentioned earlier, two walls are reduced to form a completely enclosed rotary cavity when # = 360°. In Table 4, the accu-

7

H. Zhang et al. / Applied Acoustics 163 (2020) 107230 Table 1 Convergence and accuracy of the first seven natural frequencies (Hz) for a conical acoustic cavity. Medium

MNQ

1

2

3

4

5

6

7

Air

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

112.954 112.954 112.954 112.954 112.954 112.954 112.954

173.775 173.774 173.774 173.774 173.774 173.774 173.774

212.908 212.907 212.907 212.907 212.907 212.907 212.908

214.885 214.885 214.885 214.885 214.885 214.885 214.885

301.181 301.177 301.176 301.176 301.176 301.176 301.177

308.492 308.491 308.491 308.491 308.490 308.490 308.492

344.814 344.813 344.811 344.811 344.811 344.811 344.814

Water

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

485.965 485.964 485.963 485.963 485.963 485.963 485.963

747.635 747.633 747.633 747.633 747.633 747.633 747.633

915.998 915.997 915.997 915.997 915.997 915.997 915.998

924.505 924.505 924.504 924.504 924.503 924.503 924.505

1295.779 1295.762 1295.759 1295.757 1295.756 1295.756 1295.763

1327.232 1327.229 1327.227 1327.227 1327.226 1327.226 1327.234

1483.501 1483.497 1483.490 1483.490 1483.489 1483.489 1483.503

Mode number

Fig. 4. Nephogram for sound pressure distribution of the conical acoustic cavity.

Table 2 Convergence and accuracy of the first seven natural frequencies (Hz) for a cylindrical acoustic cavity. Medium

MNQ

1

2

3

4

5

6

7

Air

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

86.000 86.000 86.000 86.000 86.000 86.000 86.000

146.795 146.794 146.794 146.794 146.794 146.794 146.794

170.131 170.131 170.131 170.131 170.131 170.131 170.131

172.000 172.000 172.000 172.000 172.000 172.000 172.000

226.125 226.125 226.125 226.125 226.125 226.125 226.126

258.000 258.000 258.000 258.000 258.000 258.000 258.001

283.341 283.341 283.340 283.340 283.340 283.340 283.343

Water

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

370.000 370.000 370.000 370.000 370.000 370.000 370.000

631.558 631.557 631.556 631.556 631.556 631.556 631.557

731.960 731.959 731.958 731.958 731.958 731.958 731.960

740.000 740.000 740.000 740.000 740.000 740.000 740.001

972.865 972.864 972.863 972.863 972.863 972.863 972.867

1110.000 1110.000 1110.000 1110.000 1110.000 1110.000 1110.005

1219.027 1219.024 1219.022 1219.022 1219.022 1219.022 1219.033

Mode number

racy of the first seven natural frequencies for the rotary cavity is given. Except for the rotation angle, the geometric parameters used in Table 4 are consistent with those of the conical, cylindrical and spherical acoustic cavities corresponding to Table 1, Table 2 and Table 3. The results obtained by the present method match very well with those obtained by FEM, which shows that this method is still accurate in dealing with the completely closed loop rotary cavities. Similarly, in order to have a more intuitive understanding, Fig. 7 shows the sound pressure distribution maps of the completely closed loop rotary cavities obtained by the present method and FEM.

3.2. Acoustic natural frequency analysis In the previous section, the convergence and correctness of the unified analysis model for sound field characteristics in a rotary acoustic cavity have been verified. In this section, the acoustic natural frequencies of the conical, cylindrical and spherical acoustic cavity will be analyzed by using the established unified analytical model. And the effects of important structural parameters on the acoustic field will be discussed in detail. First, Table 5 gives the first seven natural frequencies of the conical acoustic cavity with various radius ratios and rotation angles. Other geometric parameters

8

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

Table 3 Convergence and accuracy of the first seven natural frequencies (Hz) for a spherical acoustic cavity. Medium

MNQ

Mode number 1

2

3

4

5

6

7

Air

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

120.791 120.791 120.790 120.790 120.789 120.789 120.789

166.952 166.908 166.908 166.900 166.900 166.898 166.896

218.719 218.661 218.660 218.654 218.654 218.653 218.653

225.339 225.338 225.291 225.290 225.280 225.280 225.275

288.156 287.763 287.762 287.729 287.729 287.722 287.721

304.710 304.707 304.707 304.706 304.706 304.705 304.707

313.323 313.322 313.226 313.226 313.214 313.214 313.212

Water

33 44 55 66 77 88 FEM

     

3 4 5 6 7 8

519.684 519.682 519.677 519.676 519.675 519.675 519.675

718.281 718.095 718.094 718.058 718.057 718.047 718.043

940.999 940.750 940.749 940.721 940.721 940.715 940.714

969.480 969.476 969.273 969.273 969.228 969.227 969.208

1239.741 1238.049 1238.048 1237.905 1237.905 1237.876 1237.868

1310.961 1310.950 1310.948 1310.944 1310.943 1310.941 1310.950

1348.018 1348.015 1347.600 1347.600 1347.548 1347.548 1347.541

Fig. 5. Nephogram for sound pressure distribution of the cylindrical acoustic cavity.

that remain unchanged are R2 = 1 m, H = 0.5 m and c = 30°. From Table 5, it is known that the natural frequency of the conical cavity decreases with the increase of rotation angle. In fact, the conical cavity becomes a completely closed loop rotary cavity when # = 360°. In addition, it is found that the influence of radius ratio R1/R2 on natural frequency is more complex and no longer shows obvious monotonicity. Next, the first seven natural frequencies of a cylindrical acoustic cavity with various geometric parameters are calculated in Table 6. Other geometric parameters include: R2 = 1 m and L = 2 m. Table 7 continues to give the first seven natural frequencies of a spherical acoustic cavity with different geometric parameters. Other geometric parameters include: R2 = 1 m, /1 = 30° and /2 = 150°. The influence of rotation angle and radius ratio on cylindrical cavity and spherical cavity is consistent with that on conical cavity, which further illustrates that conical cavity, cylindrical cavity and spherical cavity are three basic forms of rotary acoustic cavity, and their sound field characteristics are inherently consistent. In addition, it can also be found that the ratio of the natural frequency of the air

cavity to the natural frequency of the water cavity is exactly the ratio of the sound velocity: cair/cwater. The new results given in these tables can be served as the benchmark for further research in this field. Next, parametric research results of some important geometric parameters of rotary acoustic cavity are given. For conical cavity, the apex angle c is another important structural parameter. Fig. 8 shows the frequency variation of conical acoustic cavity with various apex angles. The invariant geometric parameters used in Fig. 8 are chosen as: R1 = 0.5 m, R2 = 1 m and # = 90°. It can be seen that the natural frequency of the conical acoustic cavity increases continuously with the increase of the apex angle regardless of the thickness H. It should be noted that when the apex angle c increases to 90°, the conical acoustic cavity has evolved into a cylindrical acoustic cavity. In addition, the left ordinate represents the frequency of air-filled cavity, and the right ordinate represents the frequency of water-filled cavity. It can be seen that the airfilled cavity and the water-filled cavity are only different in frequency values, but the curves of the two are exactly the same.

9

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

Fig. 6. Nephogram for sound pressure distribution of the spherical acoustic cavity.

Table 4 Accuracy of the first seven natural frequencies (Hz) for the rotary closed acoustic cavity. Medium

Type

Method

1

2

3

4

5

6

7

Air

Conical

Present FEM Present FEM Present FEM

57.486 57.486 74.167 74.168 86.959 86.959

57.486 57.486 74.167 74.168 86.959 86.959

112.953 112.954 86.000 86.000 120.789 120.789

112.953 112.954 113.564 113.564 155.744 155.745

165.342 165.343 113.564 113.564 155.744 155.745

165.342 165.343 146.794 146.795 166.896 166.897

173.774 173.774 146.794 146.795 166.896 166.897

Present FEM Present FEM Present FEM

247.325 247.325 319.092 319.093 374.127 374.128

247.325 247.325 319.092 319.093 374.127 374.128

485.962 485.963 370.000 370.000 519.675 519.675

485.962 485.964 488.590 488.591 670.062 670.064

711.357 711.357 488.590 488.591 670.062 670.064

711.357 711.357 631.556 631.560 718.042 718.043

747.633 747.634 631.556 631.560 718.042 718.044

Cylindrical Spherical Water

Conical Cylindrical Spherical

Mode number

For cylindrical acoustic cavity, the cavity depth L has an important influence on the sound field characteristics. Fig. 9 gives the frequency variation of cylindrical cavity with various cavity depths. The value of cavity depth L is 0.02–2 m and R2 = 1 m. Comparing with Fig. 9(a) and (b), it can be found that the frequency variation with various cavity depths is different when the radius ratio and rotation angle are different. However, the overall trend is consistent. Specifically, when the cavity depth is small, the frequency remains unchanged, and the lower the order, the larger the cavity depth required for frequency change. When the cavity depth exceeds a certain value, the frequencies decrease with the increase of cavity depth. Fig. 10 shows the frequency variation of spherical acoustic cavity with various azimuthal angles. Among them, /1 selects three groups of 0°, 90° and 180°, while /2 changes from 1° to 179°. Other geometric parameters are: R1 = 0.5 m, R2 = 1 m and # = 90°. When /1 = 0°, the natural frequency of spherical cavity decreases with the 

increase of /2 , and the opposite is true when /1 ¼ 180 . In fact, the two sets of curves are symmetric about /2 = 90°, because the spherical cavity model corresponding to the two sets of curves is actually the same. When /1 = 90°, the change curve is also sym-

metric about /2 = 90°. Similarly, the air-filled cavity has a consistent variation with the water-filled cavity. 3.3. Acoustic response analysis On the basis of the free vibration analysis of the rotary acoustic cavity, the forced response analysis under the excitation of a point sound source will be carried out in this section. Firstly, the sound pressure response curves of a conical cavity excited by a unit point sound source are given in Fig. 11. The geometrical parameters used in Fig. 11 are consistent with those in Tables 1 and 4, and the acoustic walls are all rigid walls. The point sound source is applied at (0.156 m, 59.40°, 0.159 m). The sound pressure observation point is (0.117 m, 64.79°, 0.139 m) when # = 90°, and the sound pressure observation point is (0.001 m, 62.08°, 0.330 m) when # = 360°. It can be seen that the sound pressure response curves predicted by the present method and FEM are in good agreement with each other, whether it is a conical cavity with the angle of rotation of 90° or 360°. It shows that the unified analysis model of the rotary acoustic cavity could accurately predict the forced response of the conical cavity. It is not difficult to find that the

10

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

Fig. 7. Nephogram for sound pressure distribution of the closed rotary acoustic cavity.

sound pressure response curves of the air-filled cavity in the frequency range of 0–250 Hz are consistent with those of the water-filled cavity in the frequency range of 0–1000 Hz. The difference is that the natural frequency of the water-filled cavity is larger than that of the air-filled cavity, which makes the sound pressure response waveform of the water-filled cavity stretch in

the frequency domain. It shows that although the physical properties of air and water are quite different, they still show some obvious common laws when they are used as two common media to fill the acoustic cavity. From Fig. 11, we can see that the sound pressure response curves have obvious resonance at the natural frequency. In order

11

H. Zhang et al. / Applied Acoustics 163 (2020) 107230 Table 5 The first seven natural frequencies (Hz) of the conical acoustic cavity with various geometric parameters. Medium

R1/R2

#

Mode number 1

2

3

4

5

6

7

Air

0

90° 180° 360° 90° 180° 360° 90° 180° 360°

93.490 69.058 69.054 116.727 60.634 60.634 98.605 49.391 49.391

120.141 93.490 69.054 145.737 116.727 60.634 195.845 98.605 49.391

176.914 120.141 93.487 199.572 145.737 116.726 290.623 147.475 98.605

208.339 148.535 120.141 216.209 160.678 116.726 345.979 266.302 98.605

216.527 168.947 120.141 287.903 167.896 145.736 360.718 330.231 147.474

261.864 176.914 148.477 304.949 199.572 160.678 402.003 345.979 147.474

279.241 208.339 148.477 308.771 250.712 160.678 430.545 349.716 195.845

90° 180° 360° 90° 180° 360° 90° 180° 360°

402.225 297.109 297.093 502.196 260.867 260.865 424.232 212.495 212.495

516.888 402.225 297.093 627.009 502.196 260.865 842.591 424.232 212.495

761.143 516.888 402.213 858.626 627.009 502.193 1250.354 634.483 424.232

896.344 639.047 516.884 930.201 691.290 502.193 1488.516 1145.717 424.232

931.569 726.864 516.884 1238.654 722.342 627.005 1551.926 1420.762 634.483

1126.626 761.143 638.796 1311.989 858.626 691.290 1729.549 1488.516 634.483

1201.387 896.344 638.796 1328.435 1078.645 691.290 1852.346 1504.594 842.592

0.4

0.8

Water

0

0.4

0.8

Fig. 8. Variation of frequency of conical acoustic cavity with various apex angles.

Table 6 The first seven natural frequencies (Hz) of the cylindrical acoustic cavity with various geometric parameters. Medium

R1/R2

#

1

2

3

4

5

6

7

Air

0

90° 180° 360° 90° 180° 360° 90° 180° 360°

86.000 86.000 86.000 86.000 80.032 80.032 86.000 60.956 60.956

167.217 100.804 100.804 155.620 86.000 80.032 121.897 86.000 60.956

172.000 132.505 100.804 172.000 117.478 86.000 149.181 105.412 86.000

188.036 167.217 132.504 177.802 155.620 117.478 172.000 121.897 105.412

209.783 172.000 132.504 231.952 172.000 117.478 210.815 149.181 105.412

226.727 188.036 167.217 258.000 177.802 155.619 243.674 172.000 121.897

239.887 199.363 167.217 289.191 189.708 155.619 258.000 182.482 121.897

90° 180° 360° 90° 180° 360° 90° 180° 360°

370.000 370.000 370.000 370.000 344.322 344.322 370.000 262.253 262.253

719.423 433.692 433.690 669.527 370.000 344.322 524.441 370.000 262.253

740.000 570.078 433.690 740.000 505.428 370.000 641.824 453.516 370.000

808.993 719.423 570.076 764.962 669.527 505.428 740.000 524.441 453.516

902.556 740.000 570.076 997.931 740.000 505.428 906.994 641.824 453.516

975.452 808.993 719.423 1110.000 764.962 669.526 1048.366 740.000 524.441

1032.071 857.723 719.423 1244.195 816.185 669.526 1110.000 785.096 524.441

0.4

0.8

Water

0

0.4

0.8

Mode number

to weaken the resonance peak, this paper introduces damping by 

setting complex sound velocity c ¼ c0 ð1  jgc Þ. The conversion relationship between the damping introduced in this paper and the acoustic model damping could be obtained by 



gc ¼ n=ð1  jnÞ, according to k ¼ kð1  jnÞ and k0 ¼ x= c . Fig. 12

shows the sound pressure response of the conical acoustic cavity with different damping coefficients. The geometric parameters of the conical acoustic cavity are defined as: R1 = 0.5 m, R2 = 1 m, H = 0.5 m, # = 90° and c = 30°. All acoustic walls are set to rigid walls. In air-filled cavity, the point sound source is applied at (H/8, #/8, L/8) and the sound pressure observation point is located

12

H. Zhang et al. / Applied Acoustics 163 (2020) 107230

Table 7 The first seven natural frequencies (Hz) of the spherical acoustic cavity with various geometric parameters. Medium

R1/R2

#

1

2

3

4

5

6

7

Air

0

90° 180° 360° 90° 180° 360° 90° 180° 360°

133.568 100.191 100.186 126.234 91.436 91.432 103.188 73.983 73.980

177.658 133.568 100.186 172.756 126.234 91.432 143.674 103.188 73.980

226.593 167.065 133.567 223.902 161.590 126.234 190.264 133.793 103.187

232.869 177.658 167.064 230.398 172.756 161.588 196.331 143.674 133.792

246.011 226.593 167.064 291.365 223.902 161.588 254.638 190.264 133.792

292.383 232.869 177.647 307.892 230.398 172.743 270.800 196.331 143.663

308.676 244.821 177.647 308.731 242.733 172.743 278.978 207.934 143.663

90° 180° 360° 90° 180° 360° 90° 180° 360°

574.651 431.054 431.033 543.101 393.387 393.370 443.947 318.299 318.285

764.343 574.651 431.033 743.252 543.101 393.370 618.132 443.947 318.285

974.877 718.770 574.648 963.298 695.211 543.098 818.576 575.621 443.946

1001.878 764.343 718.764 991.249 743.252 695.203 844.680 618.132 575.616

1058.421 974.877 718.764 1253.548 963.298 695.203 1095.537 818.576 575.616

1257.928 1001.878 764.293 1324.650 991.249 743.198 1165.068 844.680 618.086

1328.023 1053.301 764.293 1328.263 1044.315 743.198 1200.253 894.601 618.086

0.4

0.8

Water

0

0.4

0.8

Mode number

Fig. 9. Variation of frequency of cylindrical acoustic cavity with various cavity depths.

(H/4, #/4, L/4). And in water cavity, the point sound source is applied at (H/2, #/2, L/2) and the sound pressure observation point is located (H/4, #/4, L/4). It is not difficult to find that the introduction of damping can significantly weaken the resonance peak value and keep the other position waveform and amplitude unchanged, whether in air cavity or water cavity. With the increase of damping value, the weakening of resonance peak also increases. In fact, complex acoustic impedance can also weaken the resonance phenomenon other than damping. Therefore, taking cylindrical acoustic cavity as an example, the effect of complex acoustic impedance on the sound pressure response curve will be illustrated. Before that, Fig. 13 shows the sound pressure response curves of a cylindrical cavity excited by a unit point sound source. The geometrical parameters used in Fig. 13 are consistent with those in Tables 2 and 4. The point sound source is applied at (0.350 m, 045.10°, 0.50 m). The sound pressure observation point is (0.276 m, 23.54°, 1.699 m) when # = 90°, and the sound pressure observation point is located (0.362 m, 45.09°, 1.502 m) when # = 360°. Here, the bottom surface (s = 0) and the top surface (s = L) of the cylindrical cavity are set as impedance walls, the impedance value is Z ¼ qcð50  jÞ. The other walls are still rigid walls. It should be noted that in order to highlight the effect of the impedance wall, no damping will be installed here. It can be seen from Fig. 13 that the unified analytical model is still highly accurate and applicable when the model evolves from a conical cavity to a cylindrical cavity and the acoustic wall is extended from a rigid-walled boundary conditions to an impedance-walled boundary conditions.

On the premise of validating the correctness of the analysis model, Fig. 14 shows the sound pressure response curves of cylindrical acoustic cavity under different impedance-walled boundary conditions. The geometric parameters of the conical acoustic cavity are defined as: R1 = 0.5 m, R2 = 1 m, # = 90° and L = 2 m. The point sound source is applied at (H/10, #/10, L/10), and the sound pressure observation point is located (H/5, #/5, L/5). Specifically, the sound pressure response curves at different impedance values are given in Fig. 14(a) and (b). The bottom surface (s = 0) and the top surface (s = L) of the cylindrical cavity are set as impedance walls, the three impedance values are set to be rigid, Z 1 ¼ qcð100  jÞ and Z 2 ¼ qcð20  jÞ. The other walls are still rigid walls. By comparison, the resonance peak value can be effectively suppressed by introducing impedance-walled boundary conditions in both air-filled and water-filled cavities, and the attenuation effect becomes more significant with the decrease of impedance value. In addition, the sound pressure response curves with different numbers of impedance walls are given in Fig. 14(c) and (d). Whether the impedance wall is set on the bottom and top surface or on six surfaces, the impedance value is set to be Z ¼ qcð50  jÞ. Similar to the effect of reducing the impedance value, the resonance of the pressure response is significantly suppressed with increasing the number of impedance walls. Finally, taking the spherical cavity as an example, the effect of external excitation amplitude on the sound pressure response is studied. Firstly, Fig. 15 gives the sound pressure response curves of the spherical cavity obtained by the present method and FEM. The geometrical parameters used in Fig. 15 are consistent with

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Fig. 10. Variation of frequency of spherical acoustic cavity with various azimuthal angles.

Fig. 11. Sound pressure response of the conical acoustic cavity under the excitation of a unit point sound source.

those in Tables 3 and 4, and the acoustic walls are all rigid walls. The acoustic damping is set to be n ¼ 0:01. The point sound source is applied at (0.147 m, 9.05°, 42.65°). The sound pressure observation point is (0.446 m, 23.12°, 32.58°) when # = 90°, and the sound pressure observation point is (0.362 m, 31.86°, 70.06°) when # = 360°. The response curves obtained by the two methods are in good agreement, which shows that the unified analytical model can predict the pressure response in a spherical acoustic cavity correctly. On this basis, the sound pressure response curves of spherical acoustic cavity with different excitation amplitudes are given in Fig. 16. The geometric parameters of the spherical cavity are chosen as: R1 = 0.5 m, R2 = 1 m, # = 90°, /1 =30° and /2 = 150°. It is easy to find the change of the amplitude of the point sound source will make the amplitude of the sound pressure response shift longitudinally, and the amplitude of the sound pressure response will increase with the increase of the external excitation amplitude.

3.4. Experimental analysis of completely enclosed cylindrical cavity In the front part, based on the basic acoustic principle and the 3D improved Fourier series method, the acoustic characteristics of the conical, cylindrical and spherical enclosed cavities are studied in detail. This section aims to select a reasonable experimental scheme to further verify the analysis model. The sound pressure response test of an enclosed cylindrical cavity with rigid-walls under the excitation of a point sound source is carried out. The experimental schematic diagram is given in Fig. 17. Table 8 shows the main instruments used in this experiment, including the 16Channel DH5922N dynamic signal testing and analyzing system, DHDAS dynamic signal acquisition and analysis system, MPA201 microphone and a small speaker as a point sound source transmitter. The acoustic cavity with rigid wall boundary conditions is enclosed by 10 mm thick organic glass, including a cylinder and two circular glass plates. A circular hole with a diameter of

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Fig. 12. Sound pressure response of the cylindrical conical acoustic cavity under the excitation of a unit point sound source with different acoustic damping coefficients.

Fig. 13. Sound pressure response of the cylindrical acoustic cavity under the excitation of a unit point sound source.

Fig. 14. Sound pressure response of the cylindrical acoustic cavity under the excitation of a unit point sound source with different impendence boundary conditions.

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Fig. 15. Sound pressure response of the spherical acoustic cavity under the excitation of a unit point sound source.

Fig. 16. Sound pressure response of the spherical acoustic cavity under the excitation of a point sound source with different amplitudes.

Fig. 17. Schematic diagram of sound pressure test for cylindrical enclosed cavity.

20 mm is opened at a certain position of the cylinder to pass through the connecting device. In order to ensure the closure of the cylindrical cavity, sealant is applied between the cylindrical cylinder and two circular glass plates, as well as between the connecting device and the 20 mm circular hole. The effective size parameters of the cylindrical enclosed cavity are R2 = 0.2 m and L = 0.4 m.

The actual experimental device and cavity layout are shown in Fig. 18. In this test, a small speaker is used to simulate the point source approximately and emit white noise with a bit rate of 320 kbps. Microphone 1 directly receives the sound source excitation signal from the speaker and is collected into the test system as an input signal. Microphone 2 acts as the observation point and collects the sound signal in the enclosed cavity as the output

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Table 8 Experimental apparatus for sound pressure response test of cylindrical acoustic cavity. Instrument

Sound source transmitter

Microphone

Dynamic signal testing and analyzing system

Model Purpose Physical map

APS-BA202 Simulating point source excitation

MPA201 Picking up sound pressure signals

DH5922N Collecting sound pressure signals

Fig. 18. Experiment setup of sound pressure test for cylindrical enclosed cavity.

signal. Sampling frequency is set to 5 kHz. DHDAS dynamic signal acquisition and analysis system processes the two sets of signals, and the sound pressure response of the cylindrical enclosed cavity in the frequency domain could be obtained finally. Fig. 19 gives the comparison of sound pressure test for cylindrical enclosed cavity. The physical parameters of air in theoretical calculation are defined as: qair = 1.225 kg/m3 and cair = 344 m/s, respectively. It is not difficult to find that the sound pressure response curves obtained by the two methods have the same trend of change, which proves that the method proposed in this paper is completely applicable to the sound pressure prediction of rotary enclosed cavity. On the other hand, the predictions of resonance frequencies by the two methods are in good agreement, which shows that the proposed method can accurately predict the natural frequencies of rotary enclosed cavity. In conclusion, the good agreement between the theoretical and experimental results fully proves the validity of the unified analysis model for the acoustic field characteristics of the rotating enclosed cavity established in this paper. However, there are some errors between the theoretical and experimental results at the response amplitude and some resonance frequencies. And the error is especially obvious in the range of 0–450 Hz. The reason for this phenomenon is that the enclosed cavity will inevitably be disturbed by external noise during the experiment. In particular, the peak resonance at about 150 Hz is due to the background noise during the testing process. It can be found from the figure that the experimental values at the resonance peak are generally lower than the theoretical values. This is because that it is impossible to achieve the perfect rigid acoustic wall in the experiment. At the same time, the location of the point sound source, the location of the pressure observation point and

the physical parameters of air cannot be completely consistent with the values used in the theoretical calculation. The above factors will bring some errors to the experimental results. But generally speaking, the errors are acceptable. The results of this experiment achieve the purpose of supplementary verification of the method presented in this paper. 4. Conclusions In this paper, a unified analysis model for acoustic characteristics of rotary cavity is established based on the 3D modified Fourier series method and Rayleigh-Ritz energy method. The analytical model can effectively analyze the acoustic characteristics of conical, cylindrical and spherical cavities with various impedancewalled boundary conditions, and its accuracy is verified by comparing the present results with those obtained by FEM and experiment. The new results of rotary enclosed acoustic cavity presented in this paper can provide reference for further research in this field in the future. Some important conclusions are obtained through numerical example analysis: (1) The effect of various geometric parameters on the natural frequencies of conical, cylindrical and spherical cavities is correlative and consistent. For example, the natural frequencies of the three kinds of cavities decrease with the increase of the rotation angle. (2) As the most common two kinds of media, the changes rule of acoustic characteristics obtained by air-filled cavity and water-filled cavity exist consistency. There is a multiple correspondence for the natural frequencies between the

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Fig. 19. Comparison of sound pressure test for cylindrical enclosed cavity.

air-filled cavity and the water-filled cavity. Besides, the sound pressure response curve of the water-filled cavity can be regarded as stretching in the frequency domain of the sound pressure response curve of the air-filled cavity. (3) The introduction of damping and impedance walls can effectively suppress resonance and keep the waveform and amplitude of other positions unchanged. In addition, the change of the amplitude of the point sound source will cause the longitudinal shift of the sound pressure response curve.

Declaration of Competing Interest The authors declare that there is no conflict of interest regarding the publication of this article. Acknowledgements The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 51705537 and 51679056) and Natural Science Foundation of Heilongjiang Province of China (E2016024). The authors gratefully acknowledge the supports from State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2018-11).

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