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Sound transmission into a thick hollow cylinder with the fixed-end boundary condition
Applied Mathematical Modelling 33 (2009) 1656–1673
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Sound transmission into a thick hollow cylinder with the fixed-end boundary condition H. Hosseini-Toudeshky a,*, M.R. Mofakhami b, Sh. Hosseini Hashemi c a b c
Aerospace Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran 158754413, Iran Smart Structural Design Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom School of Mechanical Engineering, Iran University of Science and Technology Narmak, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 19 September 2006 Received in revised form 27 February 2008 Accepted 4 March 2008 Available online 13 March 2008
Keywords: Noise reduction Fixed-end cylinder Uniform wave Monopole source Dipole source
a b s t r a c t In this paper sound transmission through the air filled finite thick cylinders exposed to the different incident acoustic wave is studied. The effect of end boundary conditions on the noise reduction of finite cylinders is evaluated. The uniform incident wave and the wave radiated from monopole and dipole sources are used in this study. Three positions are considered for the dipole source. Every position for the dipole source causes symmetric or antisymmetric pressure distributions on the external surface of the cylinder in tangential or axial direction. For the purpose of sound transmission analysis the linear three-dimensional theory of elasticity utilizing the technique of variables separation for the infinite circular cylinders is used to analyze the vibration of finite circular cylinder. In these analyzes the stress continuity condition on the inner and outer surfaces of the cylinder is satisfied using orthogonalization technique and velocity continuity condition is exactly satisfied on the interfacial surfaces. The sound transmission evaluation is carried out for cylinders with various half-length to outer-radius ratios. The results show that in the case of the fixed-end cylinder, the effect of boundary conditions on the noise reduction can be neglected for the half-length to outer-radius ratio of more than 10. Comparing between the obtained results from different acoustic sources shows that the obtained noise reductions from the uniform acoustic wave are less than those obtained from the monopole and dipole sources. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Finite length hollow cylinders are essential in the industries such as marine structures, gas pipes and aerospace structures, necessitating the comprehensive understanding of their vibration with the fixed-end boundary conditions. These thoughtful could be used to evaluate the sound transmission through the single and multilayered finite cylinders. Most of the analytical solutions of sound transmission into a cylinder are performed on the infinite cylinder. Some analyses were already performed using the theory of shells to model the cylindrical structures [1–4] and in some works three-dimensional theory of elasticity have been used [5,6]. In the above mentioned investigations the internal acoustic field due to the plane wave excitation was obtained using continuity conditions between the structures and acoustic fields. Chang and Vaicaitis [7] studied the sound transmission into semi-cylindrical enclosures. They used Galerkin-like method to solve the acoustic wave equation for the interior acoustic field. This solution was coupled with the vibration of the stiffened panel and the response characteristics of the panels were determined using the finite element-strip method. A flexible
* Corresponding author. Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. E-mail addresses: [email protected] (H. Hosseini-Toudeshky), m.r.mofakhami@sheffield.ac.uk (M.R. Mofakhami), [email protected] (Sh. Hosseini Hashemi). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.002
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acoustic radiator model is implemented by Fyfe and Ismail [8] that can encompass a wide range of finite cylinder geometries subjected to the various modes of vibration using boundary element method which is applied to an integral form of the Helmholtz equation. Bofilios and Lyrintzis [9] analyzed noise transmission into a two layered composite cylinder due to the stochastic loads using modal decomposition and a Galerkin-like approach. In this study the skins were modeled according to the thin shell theory and the end caps taken as double-wall circular isotropic plates and no coupling between cylindrical shell and end plate was considered. Cheng [10] analyzed a coupled plate-ended cylinder using the variational principle via finding extremum of Hamilton’s function over a subspace of displacement trial functions. The excitation was considered as a harmonic unit point load applied on the structure. Niezrecki and Cudney [11] investigated internal acoustic response of a simply supported cylinder having a spatially harmonic surface vibration using Kirchhoff–Helmholtz integral. In this study the structural response for a cylinder was approximated by a single mode vibration at a particular frequency. Lee and Kim [12] analyzed the noise reduction of a periodically stiffened infinite cylinder using the principle of virtual work for the coupled system and Love equations for the motion of cylindrical shell. In the case of finite cylinders most of the investigations have been performed using the numerical methods. The authors of this paper performed a semi-analytical solution for free vibration of a finite cylinder [13]. The semi-analytical solution was developed and accuracy of the method was studied in terms of satisfying boundary conditions and calculation of natural frequencies and also some principle mode shapes were also extracted. In the present paper, the sound transmission into the fixed-end thick cylinder is investigated. For this purpose, a solution for finite cylinder vibrations with fixed-end boundary conditions has been adopted to the coupled structural-acoustic system in cylindrical shape, showing the ability of the presented method in solving the coupled problems (a finite cylinder which is coupled to acoustical media through the inside and the outside surfaces) under different types of acoustic loading. The wave propagation through the finite cylindrical structure is obtained using the technique of variables separation on the basis of linear three-dimensional theory of elasticity. The expression for the noise reduction is obtained using the interfacial continuity of pressure and radial acceleration and appropriate acoustic impedance on the inner and outer surfaces. The acoustical excitations are considered as uniform harmonic incident wave and the waves radiated from monopole and dipole sources. Analysis of the sound transmission through the thick finite cylinders with acoustical excitations has not been performed yet to the authors’ knowledge. Therefore the sound transmission through a finite thick cylinder subjected to a uniform wave and the waves radiated from monopole and dipole sources using the three-dimensional theory of elasticity is the originality of the present work. In addition the velocity continuity condition is exactly satisfied between the structure and air at the interfacial surfaces. 2. Governing equation The geometry of a typical hollow circular cylinder is shown in Fig. 1. An orthogonal cylindrical coordinate (r, h, z) system is considered as shown in this figure. The corresponding components of the displacement vector u at a point are defined as ur, uh and uz in the r, h and z directions respectively. The displacement equations governing the motion of an isotropic media are: q
o2 u ¼ l$2 u þ ðk þ lÞ$ð$ uÞ; ot 2
ð1Þ
where q is the density, k and l are the Lame constants, and $2 is the three-dimensional Laplacian operator. The most general solution of Eq. (1) may be obtained using Helmholtz decomposition as follows: u ¼ $u þ $ H;
ð2Þ
Fig. 1. A half hollow circular cylinder configuration.
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where u and H are scalar and vector potential functions respectively. Substituting u from Eq. (2) into Eq. (1) results the following wave equations: C 21 $2 u ¼
o2 u ; ot 2
C 22 $2 H ¼
o2 H ; ot2
ð3Þ
where C1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l ; q
C2 ¼
rffiffiffi l : q
ð4Þ
The constants C1 and C2 are the propagation velocity of dilatational and distortional waves in an infinite medium, respectively. Utilizing the technique of variables separation the following general solution in cylindrical coordinates is obtained [13]: " ( )# cosðd1 zÞ cosðnhÞ ixt cosð dzÞ þ R1 ð a1 rÞ e ; uðr; h; z; tÞ ¼ R1 ðarÞ sinðnhÞ sinðd1 zÞ sinðdzÞ " ( ) # sinðd2 zÞ sinðnhÞ sinðdzÞ eixt ; þ R2 ð a23 rÞ Hr ðr; h; z; tÞ ¼ R2 ðarÞ cosðnhÞ cosðd2 zÞ cosð dzÞ " ( )# ð5Þ cosðnhÞ ixt sinðd2 zÞ sinð dzÞ þ R3 ð a23 rÞ e ; Hh ðr; h; z; tÞ ¼ R3 ðarÞ sinðnhÞ cosðd2 zÞ cosð dzÞ " ( )# cosðd2 zÞ sinðnhÞ cosðdzÞ þ R4 ð a4 rÞ Hz ðr; h; z; tÞ ¼ R4 ðarÞ eixt ; sinðd2 zÞ cosðnhÞ sinð dzÞ where R1 ðarÞ ¼ F 1 J n ðarÞ þ G1 Y n ðarÞ n ½ðF 3 F 2 ÞJ n ðarÞ þ ðG3 G2 ÞY n ðarÞ; ar n R3 ðarÞ ¼ F 3 J nþ1 ðarÞ þ G3 Y nþ1 ðarÞ þ ½ðF 2 F 3 ÞJ n ðarÞ þ ðG2 G3 ÞY n ðarÞ; ar R4 ðarÞ ¼ F 4 J n ðarÞ þ G4 Y n ðarÞ;
R2 ðarÞ ¼ F 2 J nþ1 ðarÞ þ G2 Y nþ1 ðarÞ þ
R1 ð a1 rÞ ¼ F 1 J n ð a1 rÞ þ G1 Y n ð a1 rÞ; n ½ðF 3 F 2 ÞJ n ð a2 rÞ þ ðG3 G2 ÞY n ð a2 rÞ; 2 r a n ½ðF 2 F 3 ÞJ n ð R3 ð a2 rÞ ¼ F 3 J nþ1 ð a2 rÞ þ G3 Y nþ1 ð a2 rÞ þ a2 rÞ þ ðG2 G3 ÞY n ð a2 rÞ; a2 r R4 ð a2 rÞ ¼ F 4 J n ð a2 rÞ þ G4 Y n ð a2 rÞ; ! ! 2 x x2 2 2 2 2 ; d d2 þ a21 ¼ þ a ¼ d þ a ¼ d21 þ a2 ¼ 2 2 C 21 C 22 R2 ð a2 rÞ ¼ F 2 J nþ1 ð a2 rÞ þ G2 Y nþ1 ð a2 rÞ þ
ð6Þ
and Jn and Yn denote the Bessel and Neumann functions for real arguments, or will be replaced by In and Kn for imaginary arguments, n is a integer number, Fk and Gk (k = 1, 2, 3, 4) are the constants and x is the circular frequency. The second terms in Eq. (5) are added to satisfy the end boundary conditions. This general solution of potential functions may be used to analyze the wave propagation in infinite or finite circular cylinders [13]. The fixed-end hollow circular cylinder is more applicable in design and analysis of industrial problems such as the coupled structural-acoustic models. The obtained results may be affected by scattering and transmission of sound through the open-end as provided by Lee and Kim [4]. To avoid this, they used the fixed-end boundary conditions in their experiments. In this paper the fixed-end hollow circular cylinder is also considered. If an isotropic elastic circular cylinder with the inner and outer radii of a and b and finite length of 2l is considered, Substituting Eq. (6) into Eq. (5), and then the resultant into Eq. (2) leads to the following displacement components: ur ¼
h n
n i cosðd zÞ 1 J n ðarÞ aJ nþ1 ðarÞ A1 þ Y n ðarÞ aY nþ1 ðarÞ B1 r r sinðd1 zÞ nn o 1 cosðd zÞ n 2 þ d2 J nþ1 ðarÞA2 þ d2 Y nþ1 ðarÞB2 þ J n ðarÞA3 þ Y n ðarÞB3 r r sinðd2 zÞ 1 n hn a1 rÞ a1 J nþ1 ð a1 rÞ A4 þ a1 rÞ a1 Y nþ1 ð a1 rÞ B4 þ dJ nþ1 ð a2 rÞA5 J ð Y n ð þ r n r )) ( nn o 1 cosðnhÞ ixt cosð dzÞ n a2 rÞB5 þ a2 rÞA6 þ Y n ð a2 rÞB6 J n ð e ; þ dY nþ1 ð r r 1 sinðnhÞ sinðdzÞ
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
) " ( )( ( ) h n i 1 cosðd1 zÞ 1 n þ ½d2 J nþ1 ðarÞA2 þ d2 Y nþ1 ðarÞB2 J n ðarÞA1 Y n ðarÞB1 r r 1 sinðd1 zÞ 1 ( ) n n i cosðd2 zÞ Y n ðarÞ aY nþ1 ðarÞ B3 J n ðarÞ aJ nþ1 ðarÞ A3 r r sinðd2 zÞ " ( ) n n o 1 n þ J n ð a1 rÞA4 Y n ð a1 rÞB4 þ dJ ð a rÞA þ dY ð a rÞB 5 nþ1 2 5 nþ1 2 r r 1 ( ))( ) n n i cosð sinðnhÞ dzÞ J n ð a2 rÞ a2 J nþ1 ð a2 rÞ A6 a2 rÞ a2 Y nþ1 ð a2 rÞ B6 Y n ð eixt ; r r cosðnhÞ sinðdzÞ ( ( ) ( ) sinðd1 zÞ sinðd2 zÞ þ ½aJ n ðarÞA2 þ aY n ðarÞB2 uz ¼ ½d1 J n ðarÞA1 þ d1 Y n ðarÞB1 cosðd1 zÞ cosðd2 zÞ ( ))( )( ) 1 cosðnhÞ sinð dzÞ þ½dJ n ð a1 rÞA4 þ dY n ð a1 rÞB4 þ a2 J n ð a2 rÞA5 þ a2 Y n ð a2 rÞB5 eixt 1 sinðnhÞ cosð dzÞ
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(
uh ¼
ð7Þ
where A1 ¼ F 1 ;
A2 ¼ F 3 ;
A4 ¼ F 1 ;
A5 ¼ F 3 ;
1 d2 ðF 2 F 3 Þ ; a 1 1 d A6 ¼ F 4 þ ðF 2 F 3 Þ ; a2 1 A3 ¼ F 4 þ
B 1 ¼ G1 ;
B2 ¼ G3 ;
B 4 ¼ G1 ;
B5 ¼ G3 ;
1 d2 ðG2 G3 Þ ; a 1 1 d B6 ¼ G4 þ ðG2 G3 Þ : a2 1 B 3 ¼ G4 þ
ð8Þ
In light of the Eqs. (7) and (8), two forms of symmetric and antisymmetric solutions are obtained. To analyze the sound transmission due to acoustical sources, the proper solution form or combination of them should be considered in the solution. Utilizing the strain–displacement and stress–strain relations, the relevant stress components can be obtained in a similar form as the displacement components obtained, as follows: ("
! !# ! ! nðn 1Þ ða2 þ d22 Þ a nðn 1Þ ða2 þ d22 Þ a 2 2 þ d ðarÞ þ ðarÞ þ B þ d ðarÞ þ ðarÞ J J Y Y 1 n nþ1 n 1 1 r2 2 r nþ1 r2 2 r
cosðd1 zÞ ðn þ 1Þ ðn þ 1Þ þ A2 d2 aJ n ðarÞ J nþ1 ðarÞ þ B2 d2 aY n ðarÞ Y nþ1 ðarÞ r r sinðd1 zÞ
1 cosðd2 zÞ n ðn 1Þ n ðn 1Þ þ A3 J n ðarÞ aJ nþ1 ðarÞ þ B3 Y n ðarÞ aY nþ1 ðarÞ r r r r sinðd2 zÞ 1
2 2 a1 a1 nðn 1Þ d a22 nðn 1Þ d a22 J n ð Y n ð þ a1 rÞ þ J nþ1 ð a1 rÞ þ B4 þ a1 rÞ þ Y nþ1 ð a1 rÞ þ A4 2 2 2 r 2 r r r
ðn þ 1Þ ðn þ 1Þ d a2 J n ð d a2 Y n ð þA5 a2 rÞ a2 rÞ þ B5 a2 rÞ a2 rÞ J nþ1 ð Y nþ1 ð r r
n ðn 1Þ n ðn 1Þ a2 rÞ a2 J nþ1 ð a2 rÞ þ B6 a2 rÞ a2 Y nþ1 ð a2 rÞ J n ð Y n ð þ A6 r r r r )) ( 1 cosðnhÞ ixt cosðdzÞ e ; 1 sinðnhÞ sinð dzÞ
1 cosðd1 zÞ 2n ð1 nÞ 2n ð1 nÞ J n ðarÞ þ aJ nþ1 ðarÞ þ B1 Y n ðarÞ þ aY nþ1 ðarÞ rrh ¼ l A1 r r r r sinðd1 zÞ 1
1 2ðn þ 1Þ 2ðn þ 1Þ þ A2 d2 aJ n ðarÞ J nþ1 ðarÞ þ B2 d2 aY n ðarÞ Y nþ1 ðarÞ r r 1
cosðd2 zÞ 2nðn 1Þ 2a 2nðn 1Þ 2a 2 J Y a2 ðarÞ ðarÞ þ B a ðarÞ ðarÞ J Y þA3 3 n nþ1 n nþ1 r2 r r2 r sinðd2 zÞ
2n ð1 nÞ 2n ð1 nÞ þ A4 a1 rÞ þ a1 J nþ1 ð a1 rÞ þ B4 a1 rÞ þ a1 Y nþ1 ð a1 rÞ J n ð Y n ð r r r r
1 2ðn þ 1Þ 2ðn þ 1Þ d a2 J n ð d a2 Y n ð a2 rÞ a2 rÞ þB5 a2 rÞ a2 rÞ J nþ1 ð Y nþ1 ð þ A5 r r 1
2nðn 1Þ 2 a2 2 J n ð J ð a2 rÞ a2 rÞ þ A6 a2 r nþ1 r2 ))
( sinðnhÞ cosð dzÞ 2nðn 1Þ 2 a2 2 Y n ð a2 Y nþ1 ð a2 rÞ a2 rÞ eixt ; þB6 2 r r cosðnhÞ sinðdzÞ rrr ¼ 2l
A1
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("
) ( )#( n n n o 1 sinðd1 zÞ rrz ¼ l 2A1 d1 J n ðarÞ þ aJ nþ1 ðarÞ þ 2B1 d1 Y n ðarÞ þ dY nþ1 ðarÞ r r 1 cosðd1 zÞ " ( ) n na na o 1 2 2 2 2 þ A2 J n ðarÞ þ ða d2 ÞJ nþ1 ðarÞ þ B2 Y n ðarÞ þ ða d2 ÞY nþ1 ðarÞ r r 1 ) ( hn n n sinðd2 zÞ nd2 nd2 þ 2A4 d J n ð d Y n ð J ðarÞ B3 Y n ðarÞ a1 rÞ þ a1 J nþ1 ð a1 rÞ þ 2B4 a1 rÞ a1 Y nþ1 ð a1 rÞ A3 r n r r r cosðd2 zÞ )
( 1 n a2 n a2 þA5 J n ð Y n ð a2 rÞ þ ð a22 d2 ÞJ nþ1 ð a2 rÞ þ B5 a2 rÞ þ ð a22 d2 ÞY nþ1 ð a2 rÞ r r 1 ( ))( ) cosðnhÞ sinð dzÞ nd nd a2 rÞB6 a2 rÞ J ð Y n ð A6 eixt ; r n r sinðnhÞ cosð dzÞ ( ( ) h na na sinðd1 zÞ 2nd1 ðA1 J n ðarÞ þ B1 Y n ðarÞÞ þ A2 J n ðarÞ d22 J nþ1 ðarÞ þ B2 Y n ðarÞ d22 Y nþ1 ðarÞ rhz ¼ l r r r cosðd1 zÞ ) ( )#( n n n o 1 sinðd2 zÞ þ A3 d2 J n ðarÞ aJ nþ1 ðarÞ þ B3 d2 Y n ðarÞ aY nþ1 ðarÞ r r 1 cosðd2 zÞ
2nd n a2 n a2 þ J n ð Y n ð ðA4 J n ð a1 rÞ þ B4 Y n ð a1 rÞÞ þ A5 a2 rÞ d2 J nþ1 ð a2 rÞ þ B5 a2 rÞ d2 Y nþ1 ð a2 rÞ r r r ( )#( ))( ) n n n o 1 sinðnhÞ sinð dzÞ d J n ð d Y n ð a2 rÞ a2 J nþ1 ð a2 rÞ þ B6 a2 rÞ a2 Y nþ1 ð a2 rÞ þ A6 eixt ; r r 1 cosðnhÞ cosð dzÞ ( ( ) ( ) cosðd1 zÞ cosðd2 zÞ ða2 d22 Þ ad2 ðA2 J n ðarÞ þ B2 Y n ðarÞÞ ðA1 J n ðarÞ þ B1 Y n ðarÞÞ rzz ¼ 2l 2 sinðd1 zÞ sinðd2 zÞ ( ))( )
2 2 cosðnhÞ cosð dzÞ d þ a2 2 þ a1 ðA4 J n ð a1 rÞ þ B4 Y n ð a1 rÞÞ a2 dðA5 J n ð a2 rÞ þ B5 Y n ð a2 rÞÞ ð9Þ eixt : 2 sinðnhÞ sinð dzÞ The aforementioned displacement and stress fields satisfy the equilibrium equations, which confirm the validity of the derived solution. In what follows, the solutions of the hollow circular cylinders with the fixed-end boundary conditions are performed. For this purpose, at first the internal and external acoustic fields’ equations are obtained, then utilizing the continuity conditions and implementing the acoustic pressure relationships, the noise reduction equations are derived. In the investigated cases in this part, the excitations are assumed to be mechanical loading instead of the acoustic waves. The radial boundary conditions at the interfaces which imply the continuity of the radial velocities and stresses should also be considered. The continuity conditions between the radial pressure gradient and acceleration of the structure are represented as oPt o2 ur ¼ qa 2 at r ¼ a; b; ð10Þ or ot where Pt is the total pressure on the structure surface, qa indicates the related medium density, ur is radial structure displacement at the interfaces of cylinder and acoustic medium respectively. Eq. (10) denotes the continuity conditions between the radial pressure gradient and acceleration of the structure. To calculate the pressure in the exterior field, the pressure field due to the incident wave, Pi, and scattered wave of the cylinder, Ps1, must be superimposed. In the case of interior field the scattered wave is defined as Ps2. Since in the far field the outward scattered wave must tend to the form of a plane wave, the pressure in the exterior field due to the scattering of the cylinder which has the density of qa1 and sound speed of Ca1 satisfy the scalar Helmholtz equation and can be obtained as 1 X 1 X n 2 iðxtkzÞ P s1 ; ð11Þ Ps1 ðr; h; zÞ ¼ nk en ðiÞ Hn ðkr rÞ cosðnhÞe n¼0 k¼0 2 where P s1 nk are coefficients, Hn ðkr rÞ is the second kind Hankel function and kr is radial wave number and also 1; n ¼ 0; en ¼ 2; n–0:
ð12Þ
Interior acoustic field is considered with the density of qa2 and sound speed of Ca2. Scattered sound pressure of the interior field similar to that obtained for the exterior field can be represented as Ps2 ðr; h; zÞ ¼
1 X 1 X
n iðxtkzÞ P s2 ; nk en ðiÞ F n ðks2 rÞ cosðnhÞe
n¼0 k¼0 x where P s2 nk are coefficients, ks2 ¼ C a2 and
ð13Þ
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F n ðks2 rÞ ¼ J n ðks2 rÞ þ fY n ðks2 rÞ:
ð14Þ
In the above representation f = i, denotes the transmitted sound pressure and f = 0 denotes the interior sound field considering the effect of the interior acoustic resonance. Similar definition had been presented by Koval [14]. Using the values of 0 < jfj<1, the sound absorption percentage of the interior walls can be modeled which may be scaled with the architectural absorption coefficient. This type of noise transmission definition is used to separate the effects of structural resonances from the effects of cavity resonance on the noise reduction. In this study, f = i is assumed to obtain the transmitted sound and the noise reduction values. The boundary conditions of the finite cylinder considering the fixed-end condition are defined as rrr þ P t ¼ rrh ¼ rrz ¼ 0 at r ¼ a; b; ur ¼ uh ¼ uz ¼ 0 at z ¼ l; l:
ð15Þ
In the solution procedure, the following boundary conditions are satisfied exactly. rrz ¼ 0
at r ¼ a; b;
ur ¼ uh ¼ 0
ð16Þ
at z ¼ l; l:
And the remaining boundary conditions are satisfied using the orthogonalization technique. rrr þ P t ¼ rrh ¼ 0 uz ¼ 0
at r ¼ a; b;
ð17Þ
at z ¼ l; l:
For the first form of the solution which is a symmetric mode solution, the first type of the boundary conditions in (16) is employed with the following assumptions: A2 ¼ A1
a cosðd1 lÞ ; d2 cosðd2 lÞ
B2 ¼ B1
a cosðd1 lÞ ; d2 cosðd2 lÞ
A5i ¼ K 1i A4i þ K 2i B4i þ K 3i A6i þ K 4i B6i ; ð2i 1Þp di ¼ ; 2l
A3 ¼ A1
cosðd1 lÞ ; cosðd2 lÞ
B3 ¼ B1
cosðd1 lÞ ; cosðd2 lÞ ð18Þ
B5i ¼ K1i A4i þ K2i B4i þ K3i A6i þ K4i B6i ;
i ¼ 1; 2; 3; . . . ;
where Kki and Kki (k = 1, 2, 3, 4) are the same as those presented in Appendix A. And for the second form of the solution ‘‘Antisymmetric mode”, the boundary conditions are employed assuming that: A2 ¼ A1
a sinðd1 lÞ ; d2 sinðd2 lÞ
di ¼ ði 1Þp;
B2 ¼ B1
a sinðd1 lÞ ; d2 sinðd2 lÞ
A3 ¼ A1
sinðd1 lÞ ; sinðd2 lÞ
B3 ¼ B1
sinðd1 lÞ ; sinðd2 lÞ
ð19Þ
i ¼ 1; 2; 3; . . .
In this case the relationship between A5i and B5i with A4i, B4i, A6i and B6i are the same as those presented in (18). To apply the orthogonality on the second type of the boundary conditions in (17), the following condition is considered: J 0n ðaj bÞA1j ¼ Y 0n ðaj bÞB1j ;
ð20Þ
where prime denotes differentiation with respect to the relevant argument, aj (j = 1, 2, 3, . . .) is the root of orthogonal function Pn(aj r) is defined as Pn ðaj rÞ ¼ Y 0n ðaj bÞJ n ðaj rÞ J 0n ðaj bÞY n ðaj rÞ:
P 0n ðaj aÞ
and the ð21Þ
And the orthogonality is presented as follows: 8 Z b < 0 Pn ðaj rÞP n ðak rÞrdr ¼ 1 n2 ðP 2n ðaj aÞP 2n ðaj bÞÞ 2 2 : ðb a Þ þ a 2 a2
for
j ¼ k;
for
j–k:
ð22Þ
j
Therefore, the boundary conditions in (17) are satisfied using the orthogonalization technique as Z b uz ðr; h; lÞPn ðaj rÞr dr ¼ 0; a ( ) Z l cosð di zÞ dz ¼ 0 at r ¼ a; b; ðrrr ðr; h; zÞ þ Pt Þ sinð di zÞ 0 ( ) Z l cosð di zÞ dz ¼ 0 at r ¼ a; b: rrh ðr; h; zÞ sinð di zÞ 0
ð23Þ
The displacement and stress fields are written in the series form with the indices i and j which are truncated with N1 and N2 terms respectively. By combining the continuity conditions (10) and the first boundary conditions of radial stress in (17), the unknown scattered coefficients can be explained versus the coefficients of the displacements and stresses presented in Eqs. (18) and (19). Also the incident wave can be explained by Fourier series in the h and z directions as follows: 1 X 1 X kpz Enk ðrÞ cosðnhÞeiðxt l Þ ; ð24Þ Pi ðr; h; zÞ ¼ k¼0 n¼0
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where Enk(r) is the coefficient of Fourier series of z with the argument of kpz . It is noted that ur in Eq. (7) is a cosine or sinusoid l function of z with different arguments. Since, all waves are considered to have the same relation with the z coordinate, the displacement functions in the r direction, ur, should be explained by the unique orthogonal functions along the z direction. . For the sake Therefore, satisfying the continuity Eq. (10) deserves to Fourier series of ur in terms of harmonic function of kpz l of brevity the fist form solution ‘‘symmetric mode” is presented here only. ur ¼
( N1 1 1 n X X 1X ek A1j pn ðaj rÞ aj qnþ1 ðaj rÞ l k¼0 n¼0 j¼1 r kp !! sin l d1j l sin kpl þ d1j l sin kpl þ d2j l cosðd1j lÞ sin kpl d2j l kp þ kp kp þ kp cosðd2j lÞ 2 l d1j 2 l þ d1j 2 l d2j 2 l þ d2j n a1i rÞ a1i J nþ1 ð a1i rÞ A4l þ a1i rÞ a1i Y nþ1 ð a1i rÞ B4l þ di J nþ1 ð a2i rÞA5i þ di Y nþ1 ð a2i rÞB5i J n ð Y n ð r r i¼1 !)
sin kp d i l i l sin kpl þ d n n kp kpl þ kp a2i rÞA6i þ Y n ð a2i rÞB6i þ J n ð z cosðnhÞeixt ; cos i i r r l 2 l d 2 l þd þ
N2 X n
ð25Þ
where ek ¼
1;
k ¼ 0;
2;
k–0:
ð26Þ
Then, using Eq. (10) the condition of continuity between the acceleration of structure and radial pressure gradient can be exactly satisfied. 3. Linear algebraic system of equations The equations of the boundary conditions satisfied by orthogonalization are used to establish a system of equations which these equations are presented in Eq. (23). By substituting Fourier series of ur (Eq. (25)) in Eq. (10) and the resultant in the second Eq. (23), and also using equations of uz, rrr and rrh from Eqs. (7) and (9) into Eq. (23), the following linear algebraic system of equations (N1 + 4N2) (N1 + 4N2) is obtained: ½M st 55 ½Dt 51 ¼ ½F s 51 :
ð27Þ
The components of matrix and vectors in (27) are:
N1
s ¼ 1;
N2
s–1;
Mst ¼ ½M st;ij sg ;
s¼
D1 ¼ fA1j gN1 1 ;
D2 ¼ fA4j gN2 1 ;
F 1 ¼ f0gN1 1 ;
g¼
N1
t ¼ 1;
N2
t–1;
D3 ¼ fB4j gN2 1 ;
F 2 ¼ F 4 ¼ F 4 ¼ F 5 ¼ f0gN2 1
D4 ¼ fA6j gN2 1 ;
F 3 ¼ ffs gN2 1 ;
D5 ¼ fB6j gN2 1 ;
ð28Þ
s ¼ 1; 2; . . . ; N 2 ;
where the components of the matrix Mst are given in Appendix B and fs is the coefficient in the orthogonal expansion of acoustical wave in cylindrical coordinate which can be calculated for each acoustical wave individually. This will be followed in the next section. 4. Acoustical wave excitation In this section the relations of the acoustical pressure of some sources are presented in the cylindrical coordinate definition. 4.1. Uniform wave To investigate the acoustical pressure field due to a uniform incident wave distributed from zp to zp as shown in Fig. 2, the coefficient of Eq. (24) is calculated as follows: Enk ðrÞ ¼
2P0 en ðiÞn kpzp J n ðkr rÞ sin : ek kp l
ð29Þ
Substituting Eqs. (24) and (25) into (10), the unknown coefficient of Eq. (11) can be obtained as: Ps1 nk ¼
qa1 x2 c1nk 2P0 kpzp 0 sin J ðk bÞ ; r n n kp l H2n ðkr bÞ en ðiÞ kr 1
0
ð30Þ
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Pi
zp
x
l
Support
Fig. 2. Fixed-end cylinder under uniform acoustical wave excitation.
where c1nk ¼
(X N1 n sin kp d h sin kp þ d l cosðd lÞ 2 1j 1j 1j l þ l A1j pn ðaj bÞ aj qnþ1 ðaj bÞ l b cosðd2j lÞ 2 kpl d1j 2 kpl þ d1j j¼1 " #! N 2 n X sin kpl d2j l sin kpl þ d2j l n kp þ kp a1i bÞ a1i J nþ1 ð a1i bÞ A4i þ a1i bÞ a1i Y nþ1 ð a1i bÞ B4i þ J n ð Y n ð b b 2 l d2j 2 l þ d2j i¼1 !) sin kp di l sin kpl þ di l n n l þ : þ di J nþ1 ð a2i bÞA5i þ di Y nþ1 ð a2i bÞB5i þ J n ð a2i bÞA6i þ Y n ð a2i bÞB6i b b di di 2 kpl 2 kpl þ
In the same way from the continuity condition, the unknown coefficient of Eq. (13) can be calculated as
1 qa2 x2 c2nk Ps2 ; n nk ¼ 0 F n ðks2 aÞ en ðiÞ ks2
ð31Þ
ð32Þ
where c2nk is calculated from Eq. (31) with replacing a by b. Using the stress continuity condition, the components of matrix Mst as presented in Eq. (27) can be obtained and fs is calculated in the series form with the index k which is truncated with N3 terms, as follows: !( !)
N3 sin ðkp ds Þh sin kp þ ds h H2n ðkr bÞ 0 zp ð1Þs1 X 2 kpzp h h fs ¼ en P 0 ðiÞn ðk bÞ J ðk bÞ þ þ : ð33Þ sin J r r n n h h kp ds ds þ ds 2 kp 2 kp H20 h h k¼1 n ðkr bÞ 4.2. Monopole source The sound field generated in the free field by a point source should be spherically symmetric around the source; consequently the wave form will depend on the distance from the center of the source and will be independent of the spherical angle. The pressure field of a monopole source with the strength of Q0 located at (r0, h0, z0) in the system coordinate of (r, h, z) is defined as Pi ðr; h; zÞ ¼
P0 iðxtkr Rs Þ e ; Rs
ð34Þ
where P0 ¼ ixQ 0 =4p
ð35Þ
R2s ¼ r 20 þ r2 2rr0 cosðh h0 Þ þ ðz z0 Þ2 :
ð36Þ
and
When point source is located in the coordinate (r, h, 0) as shown in Fig. 3 and the cylindrical expansion of acoustical pressure as presented in Eq. (24) can be explained as follows:
1 X 1 X kpz Enk ðrÞ cos ð37Þ cosðnhÞeixt ; Pi ðr; h; zÞ ¼ l n¼0 k¼0 where
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z
Point Source s y r
θ0 θ
z0
α
r0 x
Fig. 3. Representation of point source location in cylindrical coordinate.
kpz0 l
P0 ek en ei Enk ðrÞ ¼ 2pl
Z
p 0
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
ei
pffiffiffiffiffiffiffiffiffi ffi 2
kr Rs þkp l
pffiffiffiffiffiffiffiffiffi ffi 2 kp 2 þ ei kr Rs þ l Rs a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs R2s a2
Rs a2
a
ð38Þ
and a2 ¼ r 20 þ r 2 2rr0 cosðh h0 Þ:
ð39Þ
From the continuity condition between the gradient of the pressure in radial direction and structural acceleration at the outer surface of the cylinder as presented in Eq. (10), the unknown coefficient of the Eq. (11) can be obtained as Ps1 nk ¼
1 pffiffiffiffiffiffiffi n qa1 x2 c1nk Gnk ; ek kr H20 ðk bÞe ð 1 Þ r n n
ð40Þ
where kp
P 0 ek en ei l z0 Gnk ðrÞ ¼ 2pl
Z
p 0
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi ffi 2 2 kp kp 2 2 ei kr Rs þ l Rs a þ ei kr Rs þ l Rs a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðikr Rs 1Þðr r0 CosðhÞÞ cosðnhÞdh dRs R2s R2s a2
and c1nk is the same as that presented in Eq. (31). In this case fs is presented as ! ! N3 X sin kp ds h sin kp þ ds h 1 H2n ðkr bÞ h h fs ¼ E þ : G nk nk e kr H20 ds þ ds 2 kp 2 kp h h k¼0 k n ðkr bÞ
ð41Þ
ð42Þ
4.3. Dipole source The sound field generated in free field by a dipole source as shown in Fig. 4 is defined as
P0 i cosð#Þeiðxtkr Rs Þ ; 1þ Pi ðr; h; zÞ ¼ Rs kr Rs
Field point
Rs
R1
R2
υ -Ps
ds
Ps
x
Fig. 4. Acoustic pressure field due to a dipole source.
ð43Þ
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y 3
x
2
Fig. 5. Three positions for dipole source excitations.
where RS is defined in Eq. (36) and kr is the wave number. Considering the three positions of dipole sources illustrated in Fig. 5, cos(#) and unknown coefficients of series in cylindrical coordinate can be calculated for each case individually. In the case of dipole source in position 1, cos(#) is explained as cosð#Þ ¼
r cosðhÞ r 0 cosðh0 Þ Rs
ð44Þ
and P0 ek en Enk ðrÞ ¼ 2pl
Z
p
pffiffiffiffiffiffiffiffiffi
Z
0
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp R2s a2 R2s a2 l i e þ ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð#Þ 1 þ cosðnhÞdh dRs kr Rs R2s a2
and Gnk is obtained from the continuity condition (10) as follows: pffiffiffiffiffiffiffiffiffi ! Z Z l2 þa2 " P 0 ek en p ikr 3ði þ kr Rs Þ ðr r 0 cosðh h0 ÞÞ cosð#Þ Gnk ðrÞ ¼ pl Rs kr R3s 0 a pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kp kp R2s a2 R2s a2 ei kr Rs þ l þ ei kr Rs l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs : R2s a2
i þ kr Rs
!
ð45Þ
# cosðhÞ
kr R2s
ð46Þ
To calculate the acoustic pressure field for the dipole source in the position 2, h and h0 should be shifted for p/2 and then Enk is calculated as follows: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp Z Z l2 þa2 R2s a2 R2s a2 l P0 ek en p i e þ ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Enk ðrÞ ¼ cosð#Þ 1 þ ð47Þ sinðnhÞdh dRs 2pl kr Rs 0 a R2s a2 and Gnk ðrÞ ¼
pffiffiffiffiffiffiffiffiffi ! Z Z l2 þa2 " P 0 ek en p ikr 3ði þ kr Rs Þ ðr r 0 cosðh h0 ÞÞ cosð#Þ pl Rs kr R3s 0 a pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kp kp R2s a2 R2s a2 ei kr Rs þ l þ ei kr Rs l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðnhÞdh dRs ; R2s a2
i þ kr Rs kr R2s
!
# sinðhÞ
ð48Þ
where cosð#Þ ¼
r sinðhÞ r0 sinðh0 Þ : Rs
When dipole source is located at the position 3 and z0 = 0 in Eq. (43), cos(#) can be explained as z cosð#Þ ¼ : Rs
ð49Þ
ð50Þ
In this case pressure distribution of Eq. (43) is in the antisymmetric form in the z direction, and then Eq. (37) changes to the following form:
1 X 1 X kpz Enk ðrÞ sin ð51Þ cosðnhÞeixt ; Pi ðr; h; zÞ ¼ l n¼0 k¼0 where P0 ek en Enk ðrÞ ¼ 2ipl
Z 0
p
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp R2s a2 R2s a2 l i e ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs : cosð#Þ 1 þ kr Rs R2s a2
ð52Þ
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Considering the antisymmetric form of Eq. (11), the related unknown coefficient is obtained as: 1 Ps1 qa1 x2 c1nk Gnk ; nk ¼ n 20 en kr ðiÞ Hn ðkr bÞ
ð53Þ
where c1nk ¼
(X N1 n 1 A1j pn ðaj bÞ aj qnþ1 ðaj bÞ l b j¼1 " kp #! sin l d1j h sin kpl þ d1j l sin kpl þ d2j l sinðd1j lÞ sin kpl d2j l kp kp kp kp sinðd2j lÞ 2 l d1j 2 l þ d1j 2 l d2j 2 l þ d2j N 2 X n
n a1i bÞ a1i J nþ1 ð a1i bÞ A4i þ a1i bÞ a1i Y nþ1 ð a1i bÞ B4i þ di J nþ1 ð a2i bÞA5i J n ð Y n ð b i¼1 !) sin kp di l sin kpl þ di l n n l a2i bÞB5i J n ð a2i bÞA6i Y n ð a2i bÞB6i þ di Y nþ1 ð b b di di 2 kpl 2 kpl þ
þ
b
ð54Þ
and P 0 ek en Gnk ðrÞ ¼ ipl
Z
p 0
pffiffiffiffiffiffiffiffiffi "
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! # kp kp R2s a2 R2s a2 ikr 3ði þ kr Rs Þ ei kr Rs þ l ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr r 0 cosðh h0 ÞÞ cosð#Þ 3 Rs kr Rs R2s a2
cosðnhÞdh dRs : ð55Þ Using stress continuity condition in the radial direction as presented in Eq. (15), fs is calculated as follows: ! ! N3 X sin kp ds h sin kp þ ds h H2n ðkr bÞ h h fs ¼ : Gnk Enk ds þ ds 2 kp 2 kp kr H20 h h k¼1 n ðkr bÞ
ð56Þ
In this section the linear algebraic system of equations which can be used to calculate the scattered acoustic pressure in the exterior and interior fields, for different types of acoustical harmonic excitations were obtained. In the following section the noise reduction definition used in the calculation procedure is presented. 5. Noise reduction definition In this work the noise reduction procedure is performed using the mean square of sound pressure on the interior and exterior surfaces of the cylinder which is calculated as Z Z Z P2 ¼ P 2 ðr; h; z; tÞds ¼ rP 2 ðr; h; z; tÞdh dz; ð57Þ S
S
where P is the internal or external sound pressure on the body surfaces and S is the related surface. The noise reduction (NR) is calculated as NR ¼ 10log10
P 2out P2in
;
ð58Þ
where P 2out is the time and surface average of the external pressure on the external surface of the cylinder and P2in is the time and surface average of the internal scattered pressure on the internal surface of the cylinder. The above noise reduction prediction is simple and applicable to measure the sound reductions in the experiments. 6. Case study and verification In this section, case study of noise reduction due to different types of acoustical excitations has been performed based on the explained method of this paper and using MATLAB based codes. To validate the noise reduction results obtained from the presented method, a finite fixed-end cylinder made of steel material with a/b = 0.95 and l/b = 10 is analyzed. The mechanical properties of the material are presented in Table 1. With increasing the length of the finite cylinder, the effects of the fixedend become ignorable and the obtained results from the finite cylinder and those obtained from the two-dimensional analysis of the infinite cylinder should merge to the almost same value. In this analysis the first four circumferential wave numbers, n, are considered to evaluate the monostatic back scattering noise reduction of the mentioned cylinder under a uniform wave excitation. The obtained results are compared with those obtained from the infinite cylinder under the plane wave excitation in [5] as shown in Fig. 6. The agreement between the results presented in this figure shows the validity of the calculations procedure. It is noted that for this case the interior acoustic resonance f = 0 is considered in the calculations.
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Table 1 Mechanical properties of material Material
qs (kg/m3)
E (Pa)
m
Steel Aluminum
7800 2700
2.1e11 7.1e10
0.31 0.33
Fig. 6. Comparison of the noise reduction obtained from Ref. [5] with those obtained from the present work for the case of monostatic back scattering of the steel cylinder.
Infinite cylinder l/b= 1 l/b= 5 l/b=10
120
NR (dB)
100
80
60
40
2
4
Ω
6
8
Fig. 7. Finite fixed-end cylinder exposed to the uniform incident wave.
In what follows noise reduction of the aluminum finite cylinder with the fixed-end boundary condition and a/b = 0.9 and zp/l = 0.9 (Fig. 2), under uniform wave excitation is calculated using Eq. (58). In the performed calculations the acoustic fields are considered to be air with the density of 1.21 kg/m3 and the sound speed of 343 m/s. The obtained noise reductions of the finite cylinders with various values of l/b are shown in Fig. 7. This figure shows that with increasing the distance between the end boundary conditions the obtained noise reductions are decreased. It also shows that with increasing the ratio of l/b the results tend to the noise reductions obtained from an infinite cylinder under plane wave excitation presented in [6]. The noise reduction of the finite fixed-end cylinder under monopole acoustical source excitation is shown in Fig. 8. Noise reductions for different values of distance ratios, d/l, are dedicated in this figure. By increasing the monopole source distance from the outer surface of the cylinder the results tend to those obtained from the plane wave excitation.
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120 UW d/l 0.1 1.0 10 100
NR(dB)
100
80
60
40
20
0
2
4
Ω
6
8
10
Fig. 8. Noise reduction of finite cylinder under acoustical wave from monopole source (UW: Uniform wave excitation, d = r0 b).
0.6 σ |
rr r=b
Pressure
0.4
P
-P
s1
i
0.2
0
-0.2
-0.4
0
2
4
z/b
6
8
10
Fig. 9. Comparison between acoustical pressure distribution from monopole source and radial stress on outer surface of finite cylinder at h = 0 with r0 = 2 and X = 1.
To validate the calculations at h = 0 the stress continuity condition is checked at the outer surface of the cylinder as shown in Fig. 9 for r0 = 2 and X = 1. In this figure the radial stresses obtained from vibration analysis are compared with the obtained pressure distribution on the cylinder from Eq. (34). The results show that the stress continuity condition in radial direction is satisfied accurately. In the following part the noise reduction obtained from the analyses of the fixed-end cylinder with three considered positions for the dipole source excitation (Fig. 5) are presented. To validate the calculation in each case the stress continuity condition is checked as illustrated in Fig. 10. The results in this figure show that the stress continuity condition in radial direction at the selected positions on the exterior surface of the cylinder is satisfied properly. Considering ds = 1/kr, the maximum pressure radiated in the far field and near field of the dipole source located in position 1 can be compared with the pressure radiated from monopole source as illustrated in Fig. 11. This figure shows that for the wave numbers of more than one the pressure along the dipole axis for the distance ratio of r/b = 10 is almost equal to the pressure radiated from the monopole source. Fig. 12 shows that the noise reductions obtained for monopole source and dipole source located in position 1 for the distance ratio of 10 and nondimensional frequency of more than one have almost the same values. Also it can be seen that, for different values of distance ratio the obtained results for monopole and dipole sources located in position 1 are close to each other.
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0.5 σ | -P rr r=b s1 P i
Pressure
0.4
0.3
0.2
0.1 -4
-3
0.2
-2
-1
0 θ (rad)
1
2
3
4
σ | -P rr r=b s1 P i
Pressure
0.1
0
-0.1
-0.2 -3
-2
-1
0 θ (rad)
1
2
3
0.6 σ | -P rr r=b s1 P
Pressure
0.4
i
0.2
0
-0.2
-0.4 0
2
4
6
8
10
z/b Fig. 10. Comparison between acoustical pressure distribution from monopole source and radial stress on outer surface of finite cylinder, X = 1, d/l = 0.1. (a) At Position 1: z = 0. (b) At Position 2: z = 0. (c) At Position 3: h = 0.
The obtained noise reductions for fixed-end cylinder with three different positions of dipole source are presented in Fig. 13. This figure also shows that the noise reduction obtained for the dipole position 1 is less than the values obtained
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Pressure Amplitude (dB)
50
(M) (D) k =0.1 r (D) k =1.0 r
0
-50 0 10
1
10
Distance (r/b) Fig. 11. Maximum pressure variations radiated from dipole source along the axis of dipole (M: monopole source, D: dipole source).
d/l 0.1 1.0 10 0.1 1.0 10
NR (dB)
100
80
} }
Monopole Source Dipole Source
60
40 0
2
4
6
8
10
Ω Fig. 12. Comparison between the noise reductions obtained from monopole and dipole sources for fixed-end cylinder.
Source Position
NR(dB)
100
1 2 3
80
60
40
0
2
4
Ω
8
10
Fig. 13. The noise reduction obtained for finite cylinder with a/b = 0.8 and l/b = 10 with various dipole source positions. (The dipole sources are located at d/ l = 10).
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for the other two dipole positions. The obtained pressure distribution for the dipole source position 1 is very similar to that obtained for the monopole source. Lower symmetric modes can be excited by this type of acoustical excitation; therefore the obtained noise reduction for this dipole position should be lower than those obtained from the dipole positions of 2 and 3. These positions produce antisymmetric pressure distribution on the cylinder surface in tangential and axial directions. 7. Conclusions Sound transmission through the finite thick cylinder with the fixed-end boundary conditions was studied in this paper. For this purpose, the vibration of a thick cylinder was analyzed using a semi-analytical solution based on the three-dimensional theory of elasticity. Various acoustical excitations of uniform wave and waves radiated from monopole and dipole sources were considered in different positions outside of the cylinder. By increasing the ratio of the finite cylinder length to the outer radius, the noise reduction obtained from the infinite cylinder under uniform wave was comparable with those obtained from the plane wave excitation. The results also showed that, the effect of boundary conditions on the noise reduction can be neglected for the half-length outer-radius ratio of more than 10. The obtained results for the monopole and dipole sources verified the continuity condition of radial stress at the interface of acoustic medium and structure on the outer radius of cylinder. When the dipole source axis was located perpendicular to the cylinder axis, the obtained noise reductions in the far field were almost close to those obtained from the monopole source. The noise reductions obtained for the cylinders with three different dipole positions showed that the minimum noise reduction can be obtained by locating the dipole source such that causes symmetric pressure distribution on the cylinder. Also the antisymmetric pressure distribution in the axial and tangential directions leads to increase the noise reduction. Acknowledgement The authors wish to thank Dr. R.T. Faal for useful discussions and advice concerning this research. The coefficients Kki and Kki (k = 1, 2, 3, 4, 5) of the expression of A5i and B5i versus A4i, B4i, A6i, B6i as
Appendix A.
K 1i ¼ ðC1i ðaÞW2i ðbÞ C1i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:1Þ
K 2i ¼ ðW1i ðaÞW2i ðbÞ W1i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:2Þ
K 3i ¼ ðC3i ðaÞW2i ðbÞ C3i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:3Þ
K 4i ¼ ðW3i ðaÞW2i ðbÞ W3i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:4Þ
K1i ¼ ðC1i ðaÞC2i ðbÞ þ C1i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:5Þ
K2i ¼ ðW1i ðaÞC2i ðbÞ þ W1i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:6Þ
K3i ¼ ðC3i ðaÞC2i ðbÞ þ C3i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ; K4i ¼ ðW3i ðaÞC2i ðbÞ þ W3i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:7Þ ðA:8Þ
n n di J n ð a1i rÞ þ a1i J nþ1 ð a1i rÞ ; W1i ðrÞ ¼ 2 di Y n ð a1i rÞ þ a1i Y nþ1 ð a1i rÞ ; C1i ðrÞ ¼ 2 r r
2 n a2i n a2i J n ð Y n ð C2i ðrÞ ¼ a2i rÞ þ di a22i J nþ1 ð a2i rÞ ; W2i ðrÞ ¼ a2i rÞ þ ð d2i a22i ÞY nþ1 ð a2i rÞ ; r r n di n di J ð Y n ð a2i rÞ; W3i ðrÞ ¼ a2i rÞ C3i ðrÞ ¼ r n r
ðA:9Þ
where
and x2i
a21i ¼ d2i þ
C 21
! ;
d2i þ a22i ¼
x2i C 22
! ðA:10Þ
:
The components of Mst for the first form solution of the Fixed-end circular cylinder:
Appendix B.
a2j
M11;ij ¼
d2j
M12;ij ¼
! cosðd1j lÞ tanðd2j lÞ þ d1j sinðd1j lÞ
! 1 n2 ðP 2n ðaj aÞ P2n ðaj bÞÞ 2 ; ðb a2 Þ þ 2 a2j
i ¼ 1; 2; 3; . . . ; N 2 ; j ¼ 1; 2; 3; . . . ; N 1 ;
N2 X ð1Þi1 ½ di PJ ðaj ; a1i Þ þ K 1i a2i Þ þ K1i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ; i¼1
ðB:1Þ ðB:2Þ
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H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
M 13;ij ¼
N2 X ð1Þi1 ½ di PY ðaj ; a1i Þ þ K 2i a2i Þ þ K2i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:3Þ
i¼1
M14;ij ¼
N2 X ð1Þi1 ½K 3i a2i Þ þ K3i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:4Þ
i¼1
M15;ij ¼
N2 X ð1Þi1 ½K 4i a2i Þ þ K4i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:5Þ
i¼1
where 0
bJ n ðbbÞPn ðabÞ aJ 0n ðbaÞP n ðaaÞ ; 2 a2 b 0
bY n ðbbÞPn ðabÞ aY 0n ðbaÞP n ðaaÞ PY ða; bÞ ¼ b ; 2 a2 b PJ ða; bÞ ¼ b
M21;ij ¼
"
! !
a2j þ d22j nðn 1Þ aj sinðð di d1j ÞlÞ sinðð di þ d1j ÞlÞ 2 þ P Q þ d ða aÞ þ ða aÞ n j j 1j 2 r n r2 2ð di d1j Þ 2ð di þ d1j Þ j¼1
cosðd1j lÞ nðn 1Þ a sinðð d d ÞlÞ sinðð d þ d j i 2j i 2j ÞlÞ P þ a2j Q þ ; ða aÞ ða aÞ n j j n r cosðd2j lÞ r2 2ð di d2j Þ 2ð di þ d2j Þ N1 X
ðB:6Þ
ðB:7Þ
where Q n ðaj rÞ ¼ Y 0n ðaj bÞJ nþ1 ðaj rÞ J 0n ðaj bÞY nþ1 ðaj rÞ M22;ij ¼
M23;ij ¼
M24;ij ¼
M25;ij ¼
M31;ij ¼
M32;ij ¼
M33;ij ¼
d2i a22i a1i nðn 1Þ ðn þ 1Þ d J J þ ð a aÞ þ ð a aÞ þ K J ð a aÞ ð a aÞ J a 1i 1i 1i i 2i 2i 2i n n nþ1 2 a nþ1 a2 a
e ¼ 1 for a –0; ðn þ 1Þ el i di a2i aÞ a2i aÞ Y nþ1 ð ; a2i Y n ð þ K1i a 2 e ¼ 2 for ai ¼ 0;
d2i a22i nðn 1Þ ðn þ 1Þ a1i a d Y Y þ ð a aÞ þ ð a aÞ þ K J ð a aÞ ð a aÞ J n 1i nþ1 1i 2i i 2i 2i 2i n nþ1 2 a r2 a
ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; þ K2i a2i Y n ð a 2
nðn 1Þ n a2i ðn þ 1Þ d J J ð a aÞ ð a aÞ þ K J ð a aÞ ð a aÞ J a 2i 3i i 2i n 2i n 2i nþ1 2i r nþ1 a2 a
ðn þ 1Þ el a2i J n ð di a2i aÞ a2i aÞ J nþ1 ð ; þ K3i a 2
nðn 1Þ n a2i ðn þ 1Þ di Y nþ1 ð Y n ð a2i aÞ a2i aÞ þ K 4i a2i aÞ a2i aÞ J nþ1 ð a2i J n ð 2 a a a
ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; a2i Y n ð þ K4i a 2
sinðð di d1j ÞlÞ sinðð di þ d1j ÞlÞ þ Pn ðaj aÞ aj Q n ðaj aÞ a a 2ðdi d1j Þ 2ðdi þ d1j Þ j¼1
cosðd1j lÞ sinððdi d2j ÞlÞ sinððdi þ d2j ÞlÞ þ ; cosðd2j lÞ 2ð di d2j Þ 2ð di þ d2j Þ
ðB:8Þ
ðB:9Þ
ðB:10Þ
ðB:11Þ
ðB:12Þ
N1 X 2n ðn 1Þ
2n ðn 1Þ 2ðn þ 1Þ di a2i aÞ a2i aÞ J n ðdi aÞ di J nþ1 ðdi aÞ þ K 1i J nþ1 ð a2i J n ð a a a
2ðn þ 1Þ el di a2i Y n ð a2i aÞ a2i aÞ Y nþ1 ð ; þ K1i a 2
ðB:13Þ
2n ðn 1Þ 2ðn þ 1Þ a2i J n ð di a1i aÞ a1i Y nþ1 ð a1i aÞ þ K 2i a2i aÞ a2i aÞ Y n ð J nþ1 ð a a a
2ðn þ 1Þ el a2i aÞ a2i aÞ Y nþ1 ð ; þ K2i di a2i Y n ð a 2
ðB:14Þ
ðB:15Þ
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
M 34;ij ¼
M35;ij ¼
2nðn 1Þ 2 a2i 2ðn þ 1Þ di J n ð J nþ1 ð a22i a2i rÞ a2i aÞ þ K 3i a2i aÞ a2i aÞ J nþ1 ð a2i J n ð 2 a a a
2ðn þ 1Þ el a2i Y n ð di a2i aÞ a2i aÞ Y nþ1 ð ; þ K3i a 2
2nðn 1Þ 2 a2i 2ðn þ 1Þ d Y Y a22i ð a aÞ ð a aÞ þ K J ð a aÞ ð a aÞ J a n 2i nþ1 2i 4i i 2i 2i 2i n nþ1 a a2 a
2ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; þ K4i a2i Y n ð a 2
1673
ðB:16Þ
ðB:17Þ
M4k,ij and M5k,ij (k = 1, 2, 3, 4, 5) can be calculated by replacing a with b in M2k,ij and M3k,ij, respectively. References [1] [2] [3] [4] [5]
H. Pritchard, White, Sound transmission through a finite, closed, cylindrical shell, J. Acoust. Soc. Am. 40 (1966) 1124–1130. L.R. Koval, On sound transmission into a thin cylindrical shell under flight conditions, J. Sound Vibration 48 (1976) 265–275. E.A. Skelton, J.H. James, Acoustics of an anisotropic layered cylinder, J. Sound Vibration 161 (1993) 251–264. J.H. Lee, J. Kim, Analysis and measurement of sound transmission through a double-walled cylindrical shell, J. Sound Vibration 251 (2002) 631–649. J.S. Sastry, M.L. Munjal, Response of a multi-layered infinite cylinder to two-dimensional pressure excitation by means of transfer matrices, J. Sound Vibration 209 (1998) 123–142. [6] M.R. Mofakhami, Vibrations of a Cylindrical Coupled Structure-acoustic Field, Ph.D. Dissertation, Amirkabir University of Technology, Tehran Polytechnic, 2006. [7] M.T. Chang, R. Vaicaitis, Noise transmission into semi-cylindrical enclosures through discretely stiffened curved panels, J. Sound Vibration 85 (1982) 71–83. [8] K.R. Fyfe, F. Ismail, An investigation of the acoustic properties of vibrating finite cylinders, J. Sound Vibration 128 (1989) 361–375. [9] D.A. Bofilios, C.S. Lyrintzis, Structure borne noise transmission into cylindrical enclosures of finite extent, AIAA J. 29 (1991) 1193–1201. [10] L. Cheng, Fluid–structural coupling of a plate-ended cylindrical shell: vibration and internal sound field, J. Sound Vibration 174 (1994) 641–654. [11] C. Niezrecki, H.H. Cudney, Internal acoustic response of a simply-supported cylinder with single mode structural vibration, J. Sound Vibration 229 (2000) 183–193. [12] J.H. Lee, J. Kim, Sound transmission through periodically stiffened cylindrical shell, J. Sound Vibration 251 (2002) 431–456. [13] M.R. Mofakhami, H. Hosseini Toudeshky, Sh. Hosseini Hashemi, Finite cylinder vibrations with different end boundary conditions, J. Sound Vibration 297 (2006) 293–314. [14] L.R. Koval, Effects of cavity resonances on sound transmission into a thin cylindrical shell, J. Sound Vibration 59 (1978) 22–33.
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Sound transmission into a thick hollow cylinder with the fixed-end boundary condition H. Hosseini-Toudeshky a,*, M.R. Mofakhami b, Sh. Hosseini Hashemi c a b c
Aerospace Engineering Department, Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Avenue, Tehran 158754413, Iran Smart Structural Design Group, Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S1 3JD, United Kingdom School of Mechanical Engineering, Iran University of Science and Technology Narmak, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 19 September 2006 Received in revised form 27 February 2008 Accepted 4 March 2008 Available online 13 March 2008
Keywords: Noise reduction Fixed-end cylinder Uniform wave Monopole source Dipole source
a b s t r a c t In this paper sound transmission through the air filled finite thick cylinders exposed to the different incident acoustic wave is studied. The effect of end boundary conditions on the noise reduction of finite cylinders is evaluated. The uniform incident wave and the wave radiated from monopole and dipole sources are used in this study. Three positions are considered for the dipole source. Every position for the dipole source causes symmetric or antisymmetric pressure distributions on the external surface of the cylinder in tangential or axial direction. For the purpose of sound transmission analysis the linear three-dimensional theory of elasticity utilizing the technique of variables separation for the infinite circular cylinders is used to analyze the vibration of finite circular cylinder. In these analyzes the stress continuity condition on the inner and outer surfaces of the cylinder is satisfied using orthogonalization technique and velocity continuity condition is exactly satisfied on the interfacial surfaces. The sound transmission evaluation is carried out for cylinders with various half-length to outer-radius ratios. The results show that in the case of the fixed-end cylinder, the effect of boundary conditions on the noise reduction can be neglected for the half-length to outer-radius ratio of more than 10. Comparing between the obtained results from different acoustic sources shows that the obtained noise reductions from the uniform acoustic wave are less than those obtained from the monopole and dipole sources. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Finite length hollow cylinders are essential in the industries such as marine structures, gas pipes and aerospace structures, necessitating the comprehensive understanding of their vibration with the fixed-end boundary conditions. These thoughtful could be used to evaluate the sound transmission through the single and multilayered finite cylinders. Most of the analytical solutions of sound transmission into a cylinder are performed on the infinite cylinder. Some analyses were already performed using the theory of shells to model the cylindrical structures [1–4] and in some works three-dimensional theory of elasticity have been used [5,6]. In the above mentioned investigations the internal acoustic field due to the plane wave excitation was obtained using continuity conditions between the structures and acoustic fields. Chang and Vaicaitis [7] studied the sound transmission into semi-cylindrical enclosures. They used Galerkin-like method to solve the acoustic wave equation for the interior acoustic field. This solution was coupled with the vibration of the stiffened panel and the response characteristics of the panels were determined using the finite element-strip method. A flexible
* Corresponding author. Tel.: +98 21 6454 3224; fax: +98 21 6640 4885. E-mail addresses: [email protected] (H. Hosseini-Toudeshky), m.r.mofakhami@sheffield.ac.uk (M.R. Mofakhami), [email protected] (Sh. Hosseini Hashemi). S0307-904X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2008.03.002
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
1657
acoustic radiator model is implemented by Fyfe and Ismail [8] that can encompass a wide range of finite cylinder geometries subjected to the various modes of vibration using boundary element method which is applied to an integral form of the Helmholtz equation. Bofilios and Lyrintzis [9] analyzed noise transmission into a two layered composite cylinder due to the stochastic loads using modal decomposition and a Galerkin-like approach. In this study the skins were modeled according to the thin shell theory and the end caps taken as double-wall circular isotropic plates and no coupling between cylindrical shell and end plate was considered. Cheng [10] analyzed a coupled plate-ended cylinder using the variational principle via finding extremum of Hamilton’s function over a subspace of displacement trial functions. The excitation was considered as a harmonic unit point load applied on the structure. Niezrecki and Cudney [11] investigated internal acoustic response of a simply supported cylinder having a spatially harmonic surface vibration using Kirchhoff–Helmholtz integral. In this study the structural response for a cylinder was approximated by a single mode vibration at a particular frequency. Lee and Kim [12] analyzed the noise reduction of a periodically stiffened infinite cylinder using the principle of virtual work for the coupled system and Love equations for the motion of cylindrical shell. In the case of finite cylinders most of the investigations have been performed using the numerical methods. The authors of this paper performed a semi-analytical solution for free vibration of a finite cylinder [13]. The semi-analytical solution was developed and accuracy of the method was studied in terms of satisfying boundary conditions and calculation of natural frequencies and also some principle mode shapes were also extracted. In the present paper, the sound transmission into the fixed-end thick cylinder is investigated. For this purpose, a solution for finite cylinder vibrations with fixed-end boundary conditions has been adopted to the coupled structural-acoustic system in cylindrical shape, showing the ability of the presented method in solving the coupled problems (a finite cylinder which is coupled to acoustical media through the inside and the outside surfaces) under different types of acoustic loading. The wave propagation through the finite cylindrical structure is obtained using the technique of variables separation on the basis of linear three-dimensional theory of elasticity. The expression for the noise reduction is obtained using the interfacial continuity of pressure and radial acceleration and appropriate acoustic impedance on the inner and outer surfaces. The acoustical excitations are considered as uniform harmonic incident wave and the waves radiated from monopole and dipole sources. Analysis of the sound transmission through the thick finite cylinders with acoustical excitations has not been performed yet to the authors’ knowledge. Therefore the sound transmission through a finite thick cylinder subjected to a uniform wave and the waves radiated from monopole and dipole sources using the three-dimensional theory of elasticity is the originality of the present work. In addition the velocity continuity condition is exactly satisfied between the structure and air at the interfacial surfaces. 2. Governing equation The geometry of a typical hollow circular cylinder is shown in Fig. 1. An orthogonal cylindrical coordinate (r, h, z) system is considered as shown in this figure. The corresponding components of the displacement vector u at a point are defined as ur, uh and uz in the r, h and z directions respectively. The displacement equations governing the motion of an isotropic media are: q
o2 u ¼ l$2 u þ ðk þ lÞ$ð$ uÞ; ot 2
ð1Þ
where q is the density, k and l are the Lame constants, and $2 is the three-dimensional Laplacian operator. The most general solution of Eq. (1) may be obtained using Helmholtz decomposition as follows: u ¼ $u þ $ H;
ð2Þ
Fig. 1. A half hollow circular cylinder configuration.
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H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
where u and H are scalar and vector potential functions respectively. Substituting u from Eq. (2) into Eq. (1) results the following wave equations: C 21 $2 u ¼
o2 u ; ot 2
C 22 $2 H ¼
o2 H ; ot2
ð3Þ
where C1 ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ 2l ; q
C2 ¼
rffiffiffi l : q
ð4Þ
The constants C1 and C2 are the propagation velocity of dilatational and distortional waves in an infinite medium, respectively. Utilizing the technique of variables separation the following general solution in cylindrical coordinates is obtained [13]: " ( )# cosðd1 zÞ cosðnhÞ ixt cosð dzÞ þ R1 ð a1 rÞ e ; uðr; h; z; tÞ ¼ R1 ðarÞ sinðnhÞ sinðd1 zÞ sinðdzÞ " ( ) # sinðd2 zÞ sinðnhÞ sinðdzÞ eixt ; þ R2 ð a23 rÞ Hr ðr; h; z; tÞ ¼ R2 ðarÞ cosðnhÞ cosðd2 zÞ cosð dzÞ " ( )# ð5Þ cosðnhÞ ixt sinðd2 zÞ sinð dzÞ þ R3 ð a23 rÞ e ; Hh ðr; h; z; tÞ ¼ R3 ðarÞ sinðnhÞ cosðd2 zÞ cosð dzÞ " ( )# cosðd2 zÞ sinðnhÞ cosðdzÞ þ R4 ð a4 rÞ Hz ðr; h; z; tÞ ¼ R4 ðarÞ eixt ; sinðd2 zÞ cosðnhÞ sinð dzÞ where R1 ðarÞ ¼ F 1 J n ðarÞ þ G1 Y n ðarÞ n ½ðF 3 F 2 ÞJ n ðarÞ þ ðG3 G2 ÞY n ðarÞ; ar n R3 ðarÞ ¼ F 3 J nþ1 ðarÞ þ G3 Y nþ1 ðarÞ þ ½ðF 2 F 3 ÞJ n ðarÞ þ ðG2 G3 ÞY n ðarÞ; ar R4 ðarÞ ¼ F 4 J n ðarÞ þ G4 Y n ðarÞ;
R2 ðarÞ ¼ F 2 J nþ1 ðarÞ þ G2 Y nþ1 ðarÞ þ
R1 ð a1 rÞ ¼ F 1 J n ð a1 rÞ þ G1 Y n ð a1 rÞ; n ½ðF 3 F 2 ÞJ n ð a2 rÞ þ ðG3 G2 ÞY n ð a2 rÞ; 2 r a n ½ðF 2 F 3 ÞJ n ð R3 ð a2 rÞ ¼ F 3 J nþ1 ð a2 rÞ þ G3 Y nþ1 ð a2 rÞ þ a2 rÞ þ ðG2 G3 ÞY n ð a2 rÞ; a2 r R4 ð a2 rÞ ¼ F 4 J n ð a2 rÞ þ G4 Y n ð a2 rÞ; ! ! 2 x x2 2 2 2 2 ; d d2 þ a21 ¼ þ a ¼ d þ a ¼ d21 þ a2 ¼ 2 2 C 21 C 22 R2 ð a2 rÞ ¼ F 2 J nþ1 ð a2 rÞ þ G2 Y nþ1 ð a2 rÞ þ
ð6Þ
and Jn and Yn denote the Bessel and Neumann functions for real arguments, or will be replaced by In and Kn for imaginary arguments, n is a integer number, Fk and Gk (k = 1, 2, 3, 4) are the constants and x is the circular frequency. The second terms in Eq. (5) are added to satisfy the end boundary conditions. This general solution of potential functions may be used to analyze the wave propagation in infinite or finite circular cylinders [13]. The fixed-end hollow circular cylinder is more applicable in design and analysis of industrial problems such as the coupled structural-acoustic models. The obtained results may be affected by scattering and transmission of sound through the open-end as provided by Lee and Kim [4]. To avoid this, they used the fixed-end boundary conditions in their experiments. In this paper the fixed-end hollow circular cylinder is also considered. If an isotropic elastic circular cylinder with the inner and outer radii of a and b and finite length of 2l is considered, Substituting Eq. (6) into Eq. (5), and then the resultant into Eq. (2) leads to the following displacement components: ur ¼
h n
n i cosðd zÞ 1 J n ðarÞ aJ nþ1 ðarÞ A1 þ Y n ðarÞ aY nþ1 ðarÞ B1 r r sinðd1 zÞ nn o 1 cosðd zÞ n 2 þ d2 J nþ1 ðarÞA2 þ d2 Y nþ1 ðarÞB2 þ J n ðarÞA3 þ Y n ðarÞB3 r r sinðd2 zÞ 1 n hn a1 rÞ a1 J nþ1 ð a1 rÞ A4 þ a1 rÞ a1 Y nþ1 ð a1 rÞ B4 þ dJ nþ1 ð a2 rÞA5 J ð Y n ð þ r n r )) ( nn o 1 cosðnhÞ ixt cosð dzÞ n a2 rÞB5 þ a2 rÞA6 þ Y n ð a2 rÞB6 J n ð e ; þ dY nþ1 ð r r 1 sinðnhÞ sinðdzÞ
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
) " ( )( ( ) h n i 1 cosðd1 zÞ 1 n þ ½d2 J nþ1 ðarÞA2 þ d2 Y nþ1 ðarÞB2 J n ðarÞA1 Y n ðarÞB1 r r 1 sinðd1 zÞ 1 ( ) n n i cosðd2 zÞ Y n ðarÞ aY nþ1 ðarÞ B3 J n ðarÞ aJ nþ1 ðarÞ A3 r r sinðd2 zÞ " ( ) n n o 1 n þ J n ð a1 rÞA4 Y n ð a1 rÞB4 þ dJ ð a rÞA þ dY ð a rÞB 5 nþ1 2 5 nþ1 2 r r 1 ( ))( ) n n i cosð sinðnhÞ dzÞ J n ð a2 rÞ a2 J nþ1 ð a2 rÞ A6 a2 rÞ a2 Y nþ1 ð a2 rÞ B6 Y n ð eixt ; r r cosðnhÞ sinðdzÞ ( ( ) ( ) sinðd1 zÞ sinðd2 zÞ þ ½aJ n ðarÞA2 þ aY n ðarÞB2 uz ¼ ½d1 J n ðarÞA1 þ d1 Y n ðarÞB1 cosðd1 zÞ cosðd2 zÞ ( ))( )( ) 1 cosðnhÞ sinð dzÞ þ½dJ n ð a1 rÞA4 þ dY n ð a1 rÞB4 þ a2 J n ð a2 rÞA5 þ a2 Y n ð a2 rÞB5 eixt 1 sinðnhÞ cosð dzÞ
1659
(
uh ¼
ð7Þ
where A1 ¼ F 1 ;
A2 ¼ F 3 ;
A4 ¼ F 1 ;
A5 ¼ F 3 ;
1 d2 ðF 2 F 3 Þ ; a 1 1 d A6 ¼ F 4 þ ðF 2 F 3 Þ ; a2 1 A3 ¼ F 4 þ
B 1 ¼ G1 ;
B2 ¼ G3 ;
B 4 ¼ G1 ;
B5 ¼ G3 ;
1 d2 ðG2 G3 Þ ; a 1 1 d B6 ¼ G4 þ ðG2 G3 Þ : a2 1 B 3 ¼ G4 þ
ð8Þ
In light of the Eqs. (7) and (8), two forms of symmetric and antisymmetric solutions are obtained. To analyze the sound transmission due to acoustical sources, the proper solution form or combination of them should be considered in the solution. Utilizing the strain–displacement and stress–strain relations, the relevant stress components can be obtained in a similar form as the displacement components obtained, as follows: ("
! !# ! ! nðn 1Þ ða2 þ d22 Þ a nðn 1Þ ða2 þ d22 Þ a 2 2 þ d ðarÞ þ ðarÞ þ B þ d ðarÞ þ ðarÞ J J Y Y 1 n nþ1 n 1 1 r2 2 r nþ1 r2 2 r
cosðd1 zÞ ðn þ 1Þ ðn þ 1Þ þ A2 d2 aJ n ðarÞ J nþ1 ðarÞ þ B2 d2 aY n ðarÞ Y nþ1 ðarÞ r r sinðd1 zÞ
1 cosðd2 zÞ n ðn 1Þ n ðn 1Þ þ A3 J n ðarÞ aJ nþ1 ðarÞ þ B3 Y n ðarÞ aY nþ1 ðarÞ r r r r sinðd2 zÞ 1
2 2 a1 a1 nðn 1Þ d a22 nðn 1Þ d a22 J n ð Y n ð þ a1 rÞ þ J nþ1 ð a1 rÞ þ B4 þ a1 rÞ þ Y nþ1 ð a1 rÞ þ A4 2 2 2 r 2 r r r
ðn þ 1Þ ðn þ 1Þ d a2 J n ð d a2 Y n ð þA5 a2 rÞ a2 rÞ þ B5 a2 rÞ a2 rÞ J nþ1 ð Y nþ1 ð r r
n ðn 1Þ n ðn 1Þ a2 rÞ a2 J nþ1 ð a2 rÞ þ B6 a2 rÞ a2 Y nþ1 ð a2 rÞ J n ð Y n ð þ A6 r r r r )) ( 1 cosðnhÞ ixt cosðdzÞ e ; 1 sinðnhÞ sinð dzÞ
1 cosðd1 zÞ 2n ð1 nÞ 2n ð1 nÞ J n ðarÞ þ aJ nþ1 ðarÞ þ B1 Y n ðarÞ þ aY nþ1 ðarÞ rrh ¼ l A1 r r r r sinðd1 zÞ 1
1 2ðn þ 1Þ 2ðn þ 1Þ þ A2 d2 aJ n ðarÞ J nþ1 ðarÞ þ B2 d2 aY n ðarÞ Y nþ1 ðarÞ r r 1
cosðd2 zÞ 2nðn 1Þ 2a 2nðn 1Þ 2a 2 J Y a2 ðarÞ ðarÞ þ B a ðarÞ ðarÞ J Y þA3 3 n nþ1 n nþ1 r2 r r2 r sinðd2 zÞ
2n ð1 nÞ 2n ð1 nÞ þ A4 a1 rÞ þ a1 J nþ1 ð a1 rÞ þ B4 a1 rÞ þ a1 Y nþ1 ð a1 rÞ J n ð Y n ð r r r r
1 2ðn þ 1Þ 2ðn þ 1Þ d a2 J n ð d a2 Y n ð a2 rÞ a2 rÞ þB5 a2 rÞ a2 rÞ J nþ1 ð Y nþ1 ð þ A5 r r 1
2nðn 1Þ 2 a2 2 J n ð J ð a2 rÞ a2 rÞ þ A6 a2 r nþ1 r2 ))
( sinðnhÞ cosð dzÞ 2nðn 1Þ 2 a2 2 Y n ð a2 Y nþ1 ð a2 rÞ a2 rÞ eixt ; þB6 2 r r cosðnhÞ sinðdzÞ rrr ¼ 2l
A1
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("
) ( )#( n n n o 1 sinðd1 zÞ rrz ¼ l 2A1 d1 J n ðarÞ þ aJ nþ1 ðarÞ þ 2B1 d1 Y n ðarÞ þ dY nþ1 ðarÞ r r 1 cosðd1 zÞ " ( ) n na na o 1 2 2 2 2 þ A2 J n ðarÞ þ ða d2 ÞJ nþ1 ðarÞ þ B2 Y n ðarÞ þ ða d2 ÞY nþ1 ðarÞ r r 1 ) ( hn n n sinðd2 zÞ nd2 nd2 þ 2A4 d J n ð d Y n ð J ðarÞ B3 Y n ðarÞ a1 rÞ þ a1 J nþ1 ð a1 rÞ þ 2B4 a1 rÞ a1 Y nþ1 ð a1 rÞ A3 r n r r r cosðd2 zÞ )
( 1 n a2 n a2 þA5 J n ð Y n ð a2 rÞ þ ð a22 d2 ÞJ nþ1 ð a2 rÞ þ B5 a2 rÞ þ ð a22 d2 ÞY nþ1 ð a2 rÞ r r 1 ( ))( ) cosðnhÞ sinð dzÞ nd nd a2 rÞB6 a2 rÞ J ð Y n ð A6 eixt ; r n r sinðnhÞ cosð dzÞ ( ( ) h na na sinðd1 zÞ 2nd1 ðA1 J n ðarÞ þ B1 Y n ðarÞÞ þ A2 J n ðarÞ d22 J nþ1 ðarÞ þ B2 Y n ðarÞ d22 Y nþ1 ðarÞ rhz ¼ l r r r cosðd1 zÞ ) ( )#( n n n o 1 sinðd2 zÞ þ A3 d2 J n ðarÞ aJ nþ1 ðarÞ þ B3 d2 Y n ðarÞ aY nþ1 ðarÞ r r 1 cosðd2 zÞ
2nd n a2 n a2 þ J n ð Y n ð ðA4 J n ð a1 rÞ þ B4 Y n ð a1 rÞÞ þ A5 a2 rÞ d2 J nþ1 ð a2 rÞ þ B5 a2 rÞ d2 Y nþ1 ð a2 rÞ r r r ( )#( ))( ) n n n o 1 sinðnhÞ sinð dzÞ d J n ð d Y n ð a2 rÞ a2 J nþ1 ð a2 rÞ þ B6 a2 rÞ a2 Y nþ1 ð a2 rÞ þ A6 eixt ; r r 1 cosðnhÞ cosð dzÞ ( ( ) ( ) cosðd1 zÞ cosðd2 zÞ ða2 d22 Þ ad2 ðA2 J n ðarÞ þ B2 Y n ðarÞÞ ðA1 J n ðarÞ þ B1 Y n ðarÞÞ rzz ¼ 2l 2 sinðd1 zÞ sinðd2 zÞ ( ))( )
2 2 cosðnhÞ cosð dzÞ d þ a2 2 þ a1 ðA4 J n ð a1 rÞ þ B4 Y n ð a1 rÞÞ a2 dðA5 J n ð a2 rÞ þ B5 Y n ð a2 rÞÞ ð9Þ eixt : 2 sinðnhÞ sinð dzÞ The aforementioned displacement and stress fields satisfy the equilibrium equations, which confirm the validity of the derived solution. In what follows, the solutions of the hollow circular cylinders with the fixed-end boundary conditions are performed. For this purpose, at first the internal and external acoustic fields’ equations are obtained, then utilizing the continuity conditions and implementing the acoustic pressure relationships, the noise reduction equations are derived. In the investigated cases in this part, the excitations are assumed to be mechanical loading instead of the acoustic waves. The radial boundary conditions at the interfaces which imply the continuity of the radial velocities and stresses should also be considered. The continuity conditions between the radial pressure gradient and acceleration of the structure are represented as oPt o2 ur ¼ qa 2 at r ¼ a; b; ð10Þ or ot where Pt is the total pressure on the structure surface, qa indicates the related medium density, ur is radial structure displacement at the interfaces of cylinder and acoustic medium respectively. Eq. (10) denotes the continuity conditions between the radial pressure gradient and acceleration of the structure. To calculate the pressure in the exterior field, the pressure field due to the incident wave, Pi, and scattered wave of the cylinder, Ps1, must be superimposed. In the case of interior field the scattered wave is defined as Ps2. Since in the far field the outward scattered wave must tend to the form of a plane wave, the pressure in the exterior field due to the scattering of the cylinder which has the density of qa1 and sound speed of Ca1 satisfy the scalar Helmholtz equation and can be obtained as 1 X 1 X n 2 iðxtkzÞ P s1 ; ð11Þ Ps1 ðr; h; zÞ ¼ nk en ðiÞ Hn ðkr rÞ cosðnhÞe n¼0 k¼0 2 where P s1 nk are coefficients, Hn ðkr rÞ is the second kind Hankel function and kr is radial wave number and also 1; n ¼ 0; en ¼ 2; n–0:
ð12Þ
Interior acoustic field is considered with the density of qa2 and sound speed of Ca2. Scattered sound pressure of the interior field similar to that obtained for the exterior field can be represented as Ps2 ðr; h; zÞ ¼
1 X 1 X
n iðxtkzÞ P s2 ; nk en ðiÞ F n ðks2 rÞ cosðnhÞe
n¼0 k¼0 x where P s2 nk are coefficients, ks2 ¼ C a2 and
ð13Þ
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F n ðks2 rÞ ¼ J n ðks2 rÞ þ fY n ðks2 rÞ:
ð14Þ
In the above representation f = i, denotes the transmitted sound pressure and f = 0 denotes the interior sound field considering the effect of the interior acoustic resonance. Similar definition had been presented by Koval [14]. Using the values of 0 < jfj<1, the sound absorption percentage of the interior walls can be modeled which may be scaled with the architectural absorption coefficient. This type of noise transmission definition is used to separate the effects of structural resonances from the effects of cavity resonance on the noise reduction. In this study, f = i is assumed to obtain the transmitted sound and the noise reduction values. The boundary conditions of the finite cylinder considering the fixed-end condition are defined as rrr þ P t ¼ rrh ¼ rrz ¼ 0 at r ¼ a; b; ur ¼ uh ¼ uz ¼ 0 at z ¼ l; l:
ð15Þ
In the solution procedure, the following boundary conditions are satisfied exactly. rrz ¼ 0
at r ¼ a; b;
ur ¼ uh ¼ 0
ð16Þ
at z ¼ l; l:
And the remaining boundary conditions are satisfied using the orthogonalization technique. rrr þ P t ¼ rrh ¼ 0 uz ¼ 0
at r ¼ a; b;
ð17Þ
at z ¼ l; l:
For the first form of the solution which is a symmetric mode solution, the first type of the boundary conditions in (16) is employed with the following assumptions: A2 ¼ A1
a cosðd1 lÞ ; d2 cosðd2 lÞ
B2 ¼ B1
a cosðd1 lÞ ; d2 cosðd2 lÞ
A5i ¼ K 1i A4i þ K 2i B4i þ K 3i A6i þ K 4i B6i ; ð2i 1Þp di ¼ ; 2l
A3 ¼ A1
cosðd1 lÞ ; cosðd2 lÞ
B3 ¼ B1
cosðd1 lÞ ; cosðd2 lÞ ð18Þ
B5i ¼ K1i A4i þ K2i B4i þ K3i A6i þ K4i B6i ;
i ¼ 1; 2; 3; . . . ;
where Kki and Kki (k = 1, 2, 3, 4) are the same as those presented in Appendix A. And for the second form of the solution ‘‘Antisymmetric mode”, the boundary conditions are employed assuming that: A2 ¼ A1
a sinðd1 lÞ ; d2 sinðd2 lÞ
di ¼ ði 1Þp;
B2 ¼ B1
a sinðd1 lÞ ; d2 sinðd2 lÞ
A3 ¼ A1
sinðd1 lÞ ; sinðd2 lÞ
B3 ¼ B1
sinðd1 lÞ ; sinðd2 lÞ
ð19Þ
i ¼ 1; 2; 3; . . .
In this case the relationship between A5i and B5i with A4i, B4i, A6i and B6i are the same as those presented in (18). To apply the orthogonality on the second type of the boundary conditions in (17), the following condition is considered: J 0n ðaj bÞA1j ¼ Y 0n ðaj bÞB1j ;
ð20Þ
where prime denotes differentiation with respect to the relevant argument, aj (j = 1, 2, 3, . . .) is the root of orthogonal function Pn(aj r) is defined as Pn ðaj rÞ ¼ Y 0n ðaj bÞJ n ðaj rÞ J 0n ðaj bÞY n ðaj rÞ:
P 0n ðaj aÞ
and the ð21Þ
And the orthogonality is presented as follows: 8 Z b < 0 Pn ðaj rÞP n ðak rÞrdr ¼ 1 n2 ðP 2n ðaj aÞP 2n ðaj bÞÞ 2 2 : ðb a Þ þ a 2 a2
for
j ¼ k;
for
j–k:
ð22Þ
j
Therefore, the boundary conditions in (17) are satisfied using the orthogonalization technique as Z b uz ðr; h; lÞPn ðaj rÞr dr ¼ 0; a ( ) Z l cosð di zÞ dz ¼ 0 at r ¼ a; b; ðrrr ðr; h; zÞ þ Pt Þ sinð di zÞ 0 ( ) Z l cosð di zÞ dz ¼ 0 at r ¼ a; b: rrh ðr; h; zÞ sinð di zÞ 0
ð23Þ
The displacement and stress fields are written in the series form with the indices i and j which are truncated with N1 and N2 terms respectively. By combining the continuity conditions (10) and the first boundary conditions of radial stress in (17), the unknown scattered coefficients can be explained versus the coefficients of the displacements and stresses presented in Eqs. (18) and (19). Also the incident wave can be explained by Fourier series in the h and z directions as follows: 1 X 1 X kpz Enk ðrÞ cosðnhÞeiðxt l Þ ; ð24Þ Pi ðr; h; zÞ ¼ k¼0 n¼0
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where Enk(r) is the coefficient of Fourier series of z with the argument of kpz . It is noted that ur in Eq. (7) is a cosine or sinusoid l function of z with different arguments. Since, all waves are considered to have the same relation with the z coordinate, the displacement functions in the r direction, ur, should be explained by the unique orthogonal functions along the z direction. . For the sake Therefore, satisfying the continuity Eq. (10) deserves to Fourier series of ur in terms of harmonic function of kpz l of brevity the fist form solution ‘‘symmetric mode” is presented here only. ur ¼
( N1 1 1 n X X 1X ek A1j pn ðaj rÞ aj qnþ1 ðaj rÞ l k¼0 n¼0 j¼1 r kp !! sin l d1j l sin kpl þ d1j l sin kpl þ d2j l cosðd1j lÞ sin kpl d2j l kp þ kp kp þ kp cosðd2j lÞ 2 l d1j 2 l þ d1j 2 l d2j 2 l þ d2j n a1i rÞ a1i J nþ1 ð a1i rÞ A4l þ a1i rÞ a1i Y nþ1 ð a1i rÞ B4l þ di J nþ1 ð a2i rÞA5i þ di Y nþ1 ð a2i rÞB5i J n ð Y n ð r r i¼1 !)
sin kp d i l i l sin kpl þ d n n kp kpl þ kp a2i rÞA6i þ Y n ð a2i rÞB6i þ J n ð z cosðnhÞeixt ; cos i i r r l 2 l d 2 l þd þ
N2 X n
ð25Þ
where ek ¼
1;
k ¼ 0;
2;
k–0:
ð26Þ
Then, using Eq. (10) the condition of continuity between the acceleration of structure and radial pressure gradient can be exactly satisfied. 3. Linear algebraic system of equations The equations of the boundary conditions satisfied by orthogonalization are used to establish a system of equations which these equations are presented in Eq. (23). By substituting Fourier series of ur (Eq. (25)) in Eq. (10) and the resultant in the second Eq. (23), and also using equations of uz, rrr and rrh from Eqs. (7) and (9) into Eq. (23), the following linear algebraic system of equations (N1 + 4N2) (N1 + 4N2) is obtained: ½M st 55 ½Dt 51 ¼ ½F s 51 :
ð27Þ
The components of matrix and vectors in (27) are:
N1
s ¼ 1;
N2
s–1;
Mst ¼ ½M st;ij sg ;
s¼
D1 ¼ fA1j gN1 1 ;
D2 ¼ fA4j gN2 1 ;
F 1 ¼ f0gN1 1 ;
g¼
N1
t ¼ 1;
N2
t–1;
D3 ¼ fB4j gN2 1 ;
F 2 ¼ F 4 ¼ F 4 ¼ F 5 ¼ f0gN2 1
D4 ¼ fA6j gN2 1 ;
F 3 ¼ ffs gN2 1 ;
D5 ¼ fB6j gN2 1 ;
ð28Þ
s ¼ 1; 2; . . . ; N 2 ;
where the components of the matrix Mst are given in Appendix B and fs is the coefficient in the orthogonal expansion of acoustical wave in cylindrical coordinate which can be calculated for each acoustical wave individually. This will be followed in the next section. 4. Acoustical wave excitation In this section the relations of the acoustical pressure of some sources are presented in the cylindrical coordinate definition. 4.1. Uniform wave To investigate the acoustical pressure field due to a uniform incident wave distributed from zp to zp as shown in Fig. 2, the coefficient of Eq. (24) is calculated as follows: Enk ðrÞ ¼
2P0 en ðiÞn kpzp J n ðkr rÞ sin : ek kp l
ð29Þ
Substituting Eqs. (24) and (25) into (10), the unknown coefficient of Eq. (11) can be obtained as: Ps1 nk ¼
qa1 x2 c1nk 2P0 kpzp 0 sin J ðk bÞ ; r n n kp l H2n ðkr bÞ en ðiÞ kr 1
0
ð30Þ
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
1663
Pi
zp
x
l
Support
Fig. 2. Fixed-end cylinder under uniform acoustical wave excitation.
where c1nk ¼
(X N1 n sin kp d h sin kp þ d l cosðd lÞ 2 1j 1j 1j l þ l A1j pn ðaj bÞ aj qnþ1 ðaj bÞ l b cosðd2j lÞ 2 kpl d1j 2 kpl þ d1j j¼1 " #! N 2 n X sin kpl d2j l sin kpl þ d2j l n kp þ kp a1i bÞ a1i J nþ1 ð a1i bÞ A4i þ a1i bÞ a1i Y nþ1 ð a1i bÞ B4i þ J n ð Y n ð b b 2 l d2j 2 l þ d2j i¼1 !) sin kp di l sin kpl þ di l n n l þ : þ di J nþ1 ð a2i bÞA5i þ di Y nþ1 ð a2i bÞB5i þ J n ð a2i bÞA6i þ Y n ð a2i bÞB6i b b di di 2 kpl 2 kpl þ
In the same way from the continuity condition, the unknown coefficient of Eq. (13) can be calculated as
1 qa2 x2 c2nk Ps2 ; n nk ¼ 0 F n ðks2 aÞ en ðiÞ ks2
ð31Þ
ð32Þ
where c2nk is calculated from Eq. (31) with replacing a by b. Using the stress continuity condition, the components of matrix Mst as presented in Eq. (27) can be obtained and fs is calculated in the series form with the index k which is truncated with N3 terms, as follows: !( !)
N3 sin ðkp ds Þh sin kp þ ds h H2n ðkr bÞ 0 zp ð1Þs1 X 2 kpzp h h fs ¼ en P 0 ðiÞn ðk bÞ J ðk bÞ þ þ : ð33Þ sin J r r n n h h kp ds ds þ ds 2 kp 2 kp H20 h h k¼1 n ðkr bÞ 4.2. Monopole source The sound field generated in the free field by a point source should be spherically symmetric around the source; consequently the wave form will depend on the distance from the center of the source and will be independent of the spherical angle. The pressure field of a monopole source with the strength of Q0 located at (r0, h0, z0) in the system coordinate of (r, h, z) is defined as Pi ðr; h; zÞ ¼
P0 iðxtkr Rs Þ e ; Rs
ð34Þ
where P0 ¼ ixQ 0 =4p
ð35Þ
R2s ¼ r 20 þ r2 2rr0 cosðh h0 Þ þ ðz z0 Þ2 :
ð36Þ
and
When point source is located in the coordinate (r, h, 0) as shown in Fig. 3 and the cylindrical expansion of acoustical pressure as presented in Eq. (24) can be explained as follows:
1 X 1 X kpz Enk ðrÞ cos ð37Þ cosðnhÞeixt ; Pi ðr; h; zÞ ¼ l n¼0 k¼0 where
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z
Point Source s y r
θ0 θ
z0
α
r0 x
Fig. 3. Representation of point source location in cylindrical coordinate.
kpz0 l
P0 ek en ei Enk ðrÞ ¼ 2pl
Z
p 0
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
ei
pffiffiffiffiffiffiffiffiffi ffi 2
kr Rs þkp l
pffiffiffiffiffiffiffiffiffi ffi 2 kp 2 þ ei kr Rs þ l Rs a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs R2s a2
Rs a2
a
ð38Þ
and a2 ¼ r 20 þ r 2 2rr0 cosðh h0 Þ:
ð39Þ
From the continuity condition between the gradient of the pressure in radial direction and structural acceleration at the outer surface of the cylinder as presented in Eq. (10), the unknown coefficient of the Eq. (11) can be obtained as Ps1 nk ¼
1 pffiffiffiffiffiffiffi n qa1 x2 c1nk Gnk ; ek kr H20 ðk bÞe ð 1 Þ r n n
ð40Þ
where kp
P 0 ek en ei l z0 Gnk ðrÞ ¼ 2pl
Z
p 0
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffi ffi 2 2 kp kp 2 2 ei kr Rs þ l Rs a þ ei kr Rs þ l Rs a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðikr Rs 1Þðr r0 CosðhÞÞ cosðnhÞdh dRs R2s R2s a2
and c1nk is the same as that presented in Eq. (31). In this case fs is presented as ! ! N3 X sin kp ds h sin kp þ ds h 1 H2n ðkr bÞ h h fs ¼ E þ : G nk nk e kr H20 ds þ ds 2 kp 2 kp h h k¼0 k n ðkr bÞ
ð41Þ
ð42Þ
4.3. Dipole source The sound field generated in free field by a dipole source as shown in Fig. 4 is defined as
P0 i cosð#Þeiðxtkr Rs Þ ; 1þ Pi ðr; h; zÞ ¼ Rs kr Rs
Field point
Rs
R1
R2
υ -Ps
ds
Ps
x
Fig. 4. Acoustic pressure field due to a dipole source.
ð43Þ
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y 3
x
2
Fig. 5. Three positions for dipole source excitations.
where RS is defined in Eq. (36) and kr is the wave number. Considering the three positions of dipole sources illustrated in Fig. 5, cos(#) and unknown coefficients of series in cylindrical coordinate can be calculated for each case individually. In the case of dipole source in position 1, cos(#) is explained as cosð#Þ ¼
r cosðhÞ r 0 cosðh0 Þ Rs
ð44Þ
and P0 ek en Enk ðrÞ ¼ 2pl
Z
p
pffiffiffiffiffiffiffiffiffi
Z
0
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp R2s a2 R2s a2 l i e þ ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosð#Þ 1 þ cosðnhÞdh dRs kr Rs R2s a2
and Gnk is obtained from the continuity condition (10) as follows: pffiffiffiffiffiffiffiffiffi ! Z Z l2 þa2 " P 0 ek en p ikr 3ði þ kr Rs Þ ðr r 0 cosðh h0 ÞÞ cosð#Þ Gnk ðrÞ ¼ pl Rs kr R3s 0 a pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kp kp R2s a2 R2s a2 ei kr Rs þ l þ ei kr Rs l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs : R2s a2
i þ kr Rs
!
ð45Þ
# cosðhÞ
kr R2s
ð46Þ
To calculate the acoustic pressure field for the dipole source in the position 2, h and h0 should be shifted for p/2 and then Enk is calculated as follows: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp Z Z l2 þa2 R2s a2 R2s a2 l P0 ek en p i e þ ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Enk ðrÞ ¼ cosð#Þ 1 þ ð47Þ sinðnhÞdh dRs 2pl kr Rs 0 a R2s a2 and Gnk ðrÞ ¼
pffiffiffiffiffiffiffiffiffi ! Z Z l2 þa2 " P 0 ek en p ikr 3ði þ kr Rs Þ ðr r 0 cosðh h0 ÞÞ cosð#Þ pl Rs kr R3s 0 a pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi kp kp R2s a2 R2s a2 ei kr Rs þ l þ ei kr Rs l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðnhÞdh dRs ; R2s a2
i þ kr Rs kr R2s
!
# sinðhÞ
ð48Þ
where cosð#Þ ¼
r sinðhÞ r0 sinðh0 Þ : Rs
When dipole source is located at the position 3 and z0 = 0 in Eq. (43), cos(#) can be explained as z cosð#Þ ¼ : Rs
ð49Þ
ð50Þ
In this case pressure distribution of Eq. (43) is in the antisymmetric form in the z direction, and then Eq. (37) changes to the following form:
1 X 1 X kpz Enk ðrÞ sin ð51Þ cosðnhÞeixt ; Pi ðr; h; zÞ ¼ l n¼0 k¼0 where P0 ek en Enk ðrÞ ¼ 2ipl
Z 0
p
pffiffiffiffiffiffiffiffiffi
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi ffi
i kr Rs þkp pffiffiffiffiffiffiffiffiffi kp R2s a2 R2s a2 l i e ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðnhÞdh dRs : cosð#Þ 1 þ kr Rs R2s a2
ð52Þ
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Considering the antisymmetric form of Eq. (11), the related unknown coefficient is obtained as: 1 Ps1 qa1 x2 c1nk Gnk ; nk ¼ n 20 en kr ðiÞ Hn ðkr bÞ
ð53Þ
where c1nk ¼
(X N1 n 1 A1j pn ðaj bÞ aj qnþ1 ðaj bÞ l b j¼1 " kp #! sin l d1j h sin kpl þ d1j l sin kpl þ d2j l sinðd1j lÞ sin kpl d2j l kp kp kp kp sinðd2j lÞ 2 l d1j 2 l þ d1j 2 l d2j 2 l þ d2j N 2 X n
n a1i bÞ a1i J nþ1 ð a1i bÞ A4i þ a1i bÞ a1i Y nþ1 ð a1i bÞ B4i þ di J nþ1 ð a2i bÞA5i J n ð Y n ð b i¼1 !) sin kp di l sin kpl þ di l n n l a2i bÞB5i J n ð a2i bÞA6i Y n ð a2i bÞB6i þ di Y nþ1 ð b b di di 2 kpl 2 kpl þ
þ
b
ð54Þ
and P 0 ek en Gnk ðrÞ ¼ ipl
Z
p 0
pffiffiffiffiffiffiffiffiffi "
Z
l2 þa2
a
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ! # kp kp R2s a2 R2s a2 ikr 3ði þ kr Rs Þ ei kr Rs þ l ei k r R s l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr r 0 cosðh h0 ÞÞ cosð#Þ 3 Rs kr Rs R2s a2
cosðnhÞdh dRs : ð55Þ Using stress continuity condition in the radial direction as presented in Eq. (15), fs is calculated as follows: ! ! N3 X sin kp ds h sin kp þ ds h H2n ðkr bÞ h h fs ¼ : Gnk Enk ds þ ds 2 kp 2 kp kr H20 h h k¼1 n ðkr bÞ
ð56Þ
In this section the linear algebraic system of equations which can be used to calculate the scattered acoustic pressure in the exterior and interior fields, for different types of acoustical harmonic excitations were obtained. In the following section the noise reduction definition used in the calculation procedure is presented. 5. Noise reduction definition In this work the noise reduction procedure is performed using the mean square of sound pressure on the interior and exterior surfaces of the cylinder which is calculated as Z Z Z P2 ¼ P 2 ðr; h; z; tÞds ¼ rP 2 ðr; h; z; tÞdh dz; ð57Þ S
S
where P is the internal or external sound pressure on the body surfaces and S is the related surface. The noise reduction (NR) is calculated as NR ¼ 10log10
P 2out P2in
;
ð58Þ
where P 2out is the time and surface average of the external pressure on the external surface of the cylinder and P2in is the time and surface average of the internal scattered pressure on the internal surface of the cylinder. The above noise reduction prediction is simple and applicable to measure the sound reductions in the experiments. 6. Case study and verification In this section, case study of noise reduction due to different types of acoustical excitations has been performed based on the explained method of this paper and using MATLAB based codes. To validate the noise reduction results obtained from the presented method, a finite fixed-end cylinder made of steel material with a/b = 0.95 and l/b = 10 is analyzed. The mechanical properties of the material are presented in Table 1. With increasing the length of the finite cylinder, the effects of the fixedend become ignorable and the obtained results from the finite cylinder and those obtained from the two-dimensional analysis of the infinite cylinder should merge to the almost same value. In this analysis the first four circumferential wave numbers, n, are considered to evaluate the monostatic back scattering noise reduction of the mentioned cylinder under a uniform wave excitation. The obtained results are compared with those obtained from the infinite cylinder under the plane wave excitation in [5] as shown in Fig. 6. The agreement between the results presented in this figure shows the validity of the calculations procedure. It is noted that for this case the interior acoustic resonance f = 0 is considered in the calculations.
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Table 1 Mechanical properties of material Material
qs (kg/m3)
E (Pa)
m
Steel Aluminum
7800 2700
2.1e11 7.1e10
0.31 0.33
Fig. 6. Comparison of the noise reduction obtained from Ref. [5] with those obtained from the present work for the case of monostatic back scattering of the steel cylinder.
Infinite cylinder l/b= 1 l/b= 5 l/b=10
120
NR (dB)
100
80
60
40
2
4
Ω
6
8
Fig. 7. Finite fixed-end cylinder exposed to the uniform incident wave.
In what follows noise reduction of the aluminum finite cylinder with the fixed-end boundary condition and a/b = 0.9 and zp/l = 0.9 (Fig. 2), under uniform wave excitation is calculated using Eq. (58). In the performed calculations the acoustic fields are considered to be air with the density of 1.21 kg/m3 and the sound speed of 343 m/s. The obtained noise reductions of the finite cylinders with various values of l/b are shown in Fig. 7. This figure shows that with increasing the distance between the end boundary conditions the obtained noise reductions are decreased. It also shows that with increasing the ratio of l/b the results tend to the noise reductions obtained from an infinite cylinder under plane wave excitation presented in [6]. The noise reduction of the finite fixed-end cylinder under monopole acoustical source excitation is shown in Fig. 8. Noise reductions for different values of distance ratios, d/l, are dedicated in this figure. By increasing the monopole source distance from the outer surface of the cylinder the results tend to those obtained from the plane wave excitation.
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120 UW d/l 0.1 1.0 10 100
NR(dB)
100
80
60
40
20
0
2
4
Ω
6
8
10
Fig. 8. Noise reduction of finite cylinder under acoustical wave from monopole source (UW: Uniform wave excitation, d = r0 b).
0.6 σ |
rr r=b
Pressure
0.4
P
-P
s1
i
0.2
0
-0.2
-0.4
0
2
4
z/b
6
8
10
Fig. 9. Comparison between acoustical pressure distribution from monopole source and radial stress on outer surface of finite cylinder at h = 0 with r0 = 2 and X = 1.
To validate the calculations at h = 0 the stress continuity condition is checked at the outer surface of the cylinder as shown in Fig. 9 for r0 = 2 and X = 1. In this figure the radial stresses obtained from vibration analysis are compared with the obtained pressure distribution on the cylinder from Eq. (34). The results show that the stress continuity condition in radial direction is satisfied accurately. In the following part the noise reduction obtained from the analyses of the fixed-end cylinder with three considered positions for the dipole source excitation (Fig. 5) are presented. To validate the calculation in each case the stress continuity condition is checked as illustrated in Fig. 10. The results in this figure show that the stress continuity condition in radial direction at the selected positions on the exterior surface of the cylinder is satisfied properly. Considering ds = 1/kr, the maximum pressure radiated in the far field and near field of the dipole source located in position 1 can be compared with the pressure radiated from monopole source as illustrated in Fig. 11. This figure shows that for the wave numbers of more than one the pressure along the dipole axis for the distance ratio of r/b = 10 is almost equal to the pressure radiated from the monopole source. Fig. 12 shows that the noise reductions obtained for monopole source and dipole source located in position 1 for the distance ratio of 10 and nondimensional frequency of more than one have almost the same values. Also it can be seen that, for different values of distance ratio the obtained results for monopole and dipole sources located in position 1 are close to each other.
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0.5 σ | -P rr r=b s1 P i
Pressure
0.4
0.3
0.2
0.1 -4
-3
0.2
-2
-1
0 θ (rad)
1
2
3
4
σ | -P rr r=b s1 P i
Pressure
0.1
0
-0.1
-0.2 -3
-2
-1
0 θ (rad)
1
2
3
0.6 σ | -P rr r=b s1 P
Pressure
0.4
i
0.2
0
-0.2
-0.4 0
2
4
6
8
10
z/b Fig. 10. Comparison between acoustical pressure distribution from monopole source and radial stress on outer surface of finite cylinder, X = 1, d/l = 0.1. (a) At Position 1: z = 0. (b) At Position 2: z = 0. (c) At Position 3: h = 0.
The obtained noise reductions for fixed-end cylinder with three different positions of dipole source are presented in Fig. 13. This figure also shows that the noise reduction obtained for the dipole position 1 is less than the values obtained
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Pressure Amplitude (dB)
50
(M) (D) k =0.1 r (D) k =1.0 r
0
-50 0 10
1
10
Distance (r/b) Fig. 11. Maximum pressure variations radiated from dipole source along the axis of dipole (M: monopole source, D: dipole source).
d/l 0.1 1.0 10 0.1 1.0 10
NR (dB)
100
80
} }
Monopole Source Dipole Source
60
40 0
2
4
6
8
10
Ω Fig. 12. Comparison between the noise reductions obtained from monopole and dipole sources for fixed-end cylinder.
Source Position
NR(dB)
100
1 2 3
80
60
40
0
2
4
Ω
8
10
Fig. 13. The noise reduction obtained for finite cylinder with a/b = 0.8 and l/b = 10 with various dipole source positions. (The dipole sources are located at d/ l = 10).
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for the other two dipole positions. The obtained pressure distribution for the dipole source position 1 is very similar to that obtained for the monopole source. Lower symmetric modes can be excited by this type of acoustical excitation; therefore the obtained noise reduction for this dipole position should be lower than those obtained from the dipole positions of 2 and 3. These positions produce antisymmetric pressure distribution on the cylinder surface in tangential and axial directions. 7. Conclusions Sound transmission through the finite thick cylinder with the fixed-end boundary conditions was studied in this paper. For this purpose, the vibration of a thick cylinder was analyzed using a semi-analytical solution based on the three-dimensional theory of elasticity. Various acoustical excitations of uniform wave and waves radiated from monopole and dipole sources were considered in different positions outside of the cylinder. By increasing the ratio of the finite cylinder length to the outer radius, the noise reduction obtained from the infinite cylinder under uniform wave was comparable with those obtained from the plane wave excitation. The results also showed that, the effect of boundary conditions on the noise reduction can be neglected for the half-length outer-radius ratio of more than 10. The obtained results for the monopole and dipole sources verified the continuity condition of radial stress at the interface of acoustic medium and structure on the outer radius of cylinder. When the dipole source axis was located perpendicular to the cylinder axis, the obtained noise reductions in the far field were almost close to those obtained from the monopole source. The noise reductions obtained for the cylinders with three different dipole positions showed that the minimum noise reduction can be obtained by locating the dipole source such that causes symmetric pressure distribution on the cylinder. Also the antisymmetric pressure distribution in the axial and tangential directions leads to increase the noise reduction. Acknowledgement The authors wish to thank Dr. R.T. Faal for useful discussions and advice concerning this research. The coefficients Kki and Kki (k = 1, 2, 3, 4, 5) of the expression of A5i and B5i versus A4i, B4i, A6i, B6i as
Appendix A.
K 1i ¼ ðC1i ðaÞW2i ðbÞ C1i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:1Þ
K 2i ¼ ðW1i ðaÞW2i ðbÞ W1i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:2Þ
K 3i ¼ ðC3i ðaÞW2i ðbÞ C3i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:3Þ
K 4i ¼ ðW3i ðaÞW2i ðbÞ W3i ðbÞW2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:4Þ
K1i ¼ ðC1i ðaÞC2i ðbÞ þ C1i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:5Þ
K2i ¼ ðW1i ðaÞC2i ðbÞ þ W1i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:6Þ
K3i ¼ ðC3i ðaÞC2i ðbÞ þ C3i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ; K4i ¼ ðW3i ðaÞC2i ðbÞ þ W3i ðbÞC2i ðaÞÞ=ðW2i ðaÞC2i ðbÞ C2i ðaÞW2i ðbÞÞ;
ðA:7Þ ðA:8Þ
n n di J n ð a1i rÞ þ a1i J nþ1 ð a1i rÞ ; W1i ðrÞ ¼ 2 di Y n ð a1i rÞ þ a1i Y nþ1 ð a1i rÞ ; C1i ðrÞ ¼ 2 r r
2 n a2i n a2i J n ð Y n ð C2i ðrÞ ¼ a2i rÞ þ di a22i J nþ1 ð a2i rÞ ; W2i ðrÞ ¼ a2i rÞ þ ð d2i a22i ÞY nþ1 ð a2i rÞ ; r r n di n di J ð Y n ð a2i rÞ; W3i ðrÞ ¼ a2i rÞ C3i ðrÞ ¼ r n r
ðA:9Þ
where
and x2i
a21i ¼ d2i þ
C 21
! ;
d2i þ a22i ¼
x2i C 22
! ðA:10Þ
:
The components of Mst for the first form solution of the Fixed-end circular cylinder:
Appendix B.
a2j
M11;ij ¼
d2j
M12;ij ¼
! cosðd1j lÞ tanðd2j lÞ þ d1j sinðd1j lÞ
! 1 n2 ðP 2n ðaj aÞ P2n ðaj bÞÞ 2 ; ðb a2 Þ þ 2 a2j
i ¼ 1; 2; 3; . . . ; N 2 ; j ¼ 1; 2; 3; . . . ; N 1 ;
N2 X ð1Þi1 ½ di PJ ðaj ; a1i Þ þ K 1i a2i Þ þ K1i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ; i¼1
ðB:1Þ ðB:2Þ
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M 13;ij ¼
N2 X ð1Þi1 ½ di PY ðaj ; a1i Þ þ K 2i a2i Þ þ K2i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:3Þ
i¼1
M14;ij ¼
N2 X ð1Þi1 ½K 3i a2i Þ þ K3i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:4Þ
i¼1
M15;ij ¼
N2 X ð1Þi1 ½K 4i a2i Þ þ K4i a2i Þ; a2i PJ ðaj ; a2i PY ðaj ;
ðB:5Þ
i¼1
where 0
bJ n ðbbÞPn ðabÞ aJ 0n ðbaÞP n ðaaÞ ; 2 a2 b 0
bY n ðbbÞPn ðabÞ aY 0n ðbaÞP n ðaaÞ PY ða; bÞ ¼ b ; 2 a2 b PJ ða; bÞ ¼ b
M21;ij ¼
"
! !
a2j þ d22j nðn 1Þ aj sinðð di d1j ÞlÞ sinðð di þ d1j ÞlÞ 2 þ P Q þ d ða aÞ þ ða aÞ n j j 1j 2 r n r2 2ð di d1j Þ 2ð di þ d1j Þ j¼1
cosðd1j lÞ nðn 1Þ a sinðð d d ÞlÞ sinðð d þ d j i 2j i 2j ÞlÞ P þ a2j Q þ ; ða aÞ ða aÞ n j j n r cosðd2j lÞ r2 2ð di d2j Þ 2ð di þ d2j Þ N1 X
ðB:6Þ
ðB:7Þ
where Q n ðaj rÞ ¼ Y 0n ðaj bÞJ nþ1 ðaj rÞ J 0n ðaj bÞY nþ1 ðaj rÞ M22;ij ¼
M23;ij ¼
M24;ij ¼
M25;ij ¼
M31;ij ¼
M32;ij ¼
M33;ij ¼
d2i a22i a1i nðn 1Þ ðn þ 1Þ d J J þ ð a aÞ þ ð a aÞ þ K J ð a aÞ ð a aÞ J a 1i 1i 1i i 2i 2i 2i n n nþ1 2 a nþ1 a2 a
e ¼ 1 for a –0; ðn þ 1Þ el i di a2i aÞ a2i aÞ Y nþ1 ð ; a2i Y n ð þ K1i a 2 e ¼ 2 for ai ¼ 0;
d2i a22i nðn 1Þ ðn þ 1Þ a1i a d Y Y þ ð a aÞ þ ð a aÞ þ K J ð a aÞ ð a aÞ J n 1i nþ1 1i 2i i 2i 2i 2i n nþ1 2 a r2 a
ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; þ K2i a2i Y n ð a 2
nðn 1Þ n a2i ðn þ 1Þ d J J ð a aÞ ð a aÞ þ K J ð a aÞ ð a aÞ J a 2i 3i i 2i n 2i n 2i nþ1 2i r nþ1 a2 a
ðn þ 1Þ el a2i J n ð di a2i aÞ a2i aÞ J nþ1 ð ; þ K3i a 2
nðn 1Þ n a2i ðn þ 1Þ di Y nþ1 ð Y n ð a2i aÞ a2i aÞ þ K 4i a2i aÞ a2i aÞ J nþ1 ð a2i J n ð 2 a a a
ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; a2i Y n ð þ K4i a 2
sinðð di d1j ÞlÞ sinðð di þ d1j ÞlÞ þ Pn ðaj aÞ aj Q n ðaj aÞ a a 2ðdi d1j Þ 2ðdi þ d1j Þ j¼1
cosðd1j lÞ sinððdi d2j ÞlÞ sinððdi þ d2j ÞlÞ þ ; cosðd2j lÞ 2ð di d2j Þ 2ð di þ d2j Þ
ðB:8Þ
ðB:9Þ
ðB:10Þ
ðB:11Þ
ðB:12Þ
N1 X 2n ðn 1Þ
2n ðn 1Þ 2ðn þ 1Þ di a2i aÞ a2i aÞ J n ðdi aÞ di J nþ1 ðdi aÞ þ K 1i J nþ1 ð a2i J n ð a a a
2ðn þ 1Þ el di a2i Y n ð a2i aÞ a2i aÞ Y nþ1 ð ; þ K1i a 2
ðB:13Þ
2n ðn 1Þ 2ðn þ 1Þ a2i J n ð di a1i aÞ a1i Y nþ1 ð a1i aÞ þ K 2i a2i aÞ a2i aÞ Y n ð J nþ1 ð a a a
2ðn þ 1Þ el a2i aÞ a2i aÞ Y nþ1 ð ; þ K2i di a2i Y n ð a 2
ðB:14Þ
ðB:15Þ
H. Hosseini-Toudeshky et al. / Applied Mathematical Modelling 33 (2009) 1656–1673
M 34;ij ¼
M35;ij ¼
2nðn 1Þ 2 a2i 2ðn þ 1Þ di J n ð J nþ1 ð a22i a2i rÞ a2i aÞ þ K 3i a2i aÞ a2i aÞ J nþ1 ð a2i J n ð 2 a a a
2ðn þ 1Þ el a2i Y n ð di a2i aÞ a2i aÞ Y nþ1 ð ; þ K3i a 2
2nðn 1Þ 2 a2i 2ðn þ 1Þ d Y Y a22i ð a aÞ ð a aÞ þ K J ð a aÞ ð a aÞ J a n 2i nþ1 2i 4i i 2i 2i 2i n nþ1 a a2 a
2ðn þ 1Þ el di a2i aÞ a2i aÞ Y nþ1 ð ; þ K4i a2i Y n ð a 2
1673
ðB:16Þ
ðB:17Þ
M4k,ij and M5k,ij (k = 1, 2, 3, 4, 5) can be calculated by replacing a with b in M2k,ij and M3k,ij, respectively. References [1] [2] [3] [4] [5]
H. Pritchard, White, Sound transmission through a finite, closed, cylindrical shell, J. Acoust. Soc. Am. 40 (1966) 1124–1130. L.R. Koval, On sound transmission into a thin cylindrical shell under flight conditions, J. Sound Vibration 48 (1976) 265–275. E.A. Skelton, J.H. James, Acoustics of an anisotropic layered cylinder, J. Sound Vibration 161 (1993) 251–264. J.H. Lee, J. Kim, Analysis and measurement of sound transmission through a double-walled cylindrical shell, J. Sound Vibration 251 (2002) 631–649. J.S. Sastry, M.L. Munjal, Response of a multi-layered infinite cylinder to two-dimensional pressure excitation by means of transfer matrices, J. Sound Vibration 209 (1998) 123–142. [6] M.R. Mofakhami, Vibrations of a Cylindrical Coupled Structure-acoustic Field, Ph.D. Dissertation, Amirkabir University of Technology, Tehran Polytechnic, 2006. [7] M.T. Chang, R. Vaicaitis, Noise transmission into semi-cylindrical enclosures through discretely stiffened curved panels, J. Sound Vibration 85 (1982) 71–83. [8] K.R. Fyfe, F. Ismail, An investigation of the acoustic properties of vibrating finite cylinders, J. Sound Vibration 128 (1989) 361–375. [9] D.A. Bofilios, C.S. Lyrintzis, Structure borne noise transmission into cylindrical enclosures of finite extent, AIAA J. 29 (1991) 1193–1201. [10] L. Cheng, Fluid–structural coupling of a plate-ended cylindrical shell: vibration and internal sound field, J. Sound Vibration 174 (1994) 641–654. [11] C. Niezrecki, H.H. Cudney, Internal acoustic response of a simply-supported cylinder with single mode structural vibration, J. Sound Vibration 229 (2000) 183–193. [12] J.H. Lee, J. Kim, Sound transmission through periodically stiffened cylindrical shell, J. Sound Vibration 251 (2002) 431–456. [13] M.R. Mofakhami, H. Hosseini Toudeshky, Sh. Hosseini Hashemi, Finite cylinder vibrations with different end boundary conditions, J. Sound Vibration 297 (2006) 293–314. [14] L.R. Koval, Effects of cavity resonances on sound transmission into a thin cylindrical shell, J. Sound Vibration 59 (1978) 22–33.