A unified semi-analytic method for vibro-acoustic analysis of submerged shells of revolution

Ocean Engineering 189 (2019) 106345

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A unified semi-analytic method for vibro-acoustic analysis of submerged shells of revolution Kun Xie a, Meixia Chen b, *, Linke Zhang c, d, Wencheng Li a, Wanjing Dong a a

College of Engineering, Huazhong Agricultural University, Wuhan, 430070, China School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China c Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan, 430063, China d Key Laboratory of Marine Power Engineering & Technology, Wuhan University of Technology, Wuhan, 430063, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Shells of revolution Vibro-acoustic analysis Helmholtz integral equation Semi-analytical method

A unified semi-analytic method is developed to predict vibro-acoustic responses of submerged shells of revolu­ tion. The shell is firstly decomposed to several narrow shell segments to be treated as conical shells. By employing Flügge shell theory and power series, displacements and forces at any cross-section of conical shells are expressed as eight unknown coefficients. Then, expanding surface pressure and velocity as Fourier series in circumferential direction, the surface Helmholtz integral equation is reduced to the line integral along the generator line, and the pressure is further represented as displacements of all segments through meshing the generator line to some 3-node isoparametric elements and utilizing relationships of velocities and displacements. Last, continuity conditions between adjacent segments, which include modifications introduced by acoustic pressure, and elastic boundary conditions are assembled to the finial vibro-acoustic governing equation of submerged shells of revolution. Through comparing vibro-acoustic results of present method with ones in appropriate references or calculated by FEM/BEM for independent conical, cylindrical and spherical shells and combined spherical-cylindrical-spherical and conical-cylindrical-spherical shells, rapid convergence, wide application and high accuracy of the semi-analytic model are demonstrated. Meanwhile, contributions of different circumferential modes and different compartments of the combined conical-cylindrical-spherical shell to vibro-acoustic responses are investigated.

1. Introduction Shells of revolution are very common structures in engineering and usually exist in the form of independent conical shells, cylindrical shells, spherical shells and combinations of those independent shells. In many circumstances, various kinds of dynamic loads act on the shells and generate serious vibrations, and acoustic pressure is further radiated from the vibrating shells when the shells are coupled with fluid. For most engineering structures about shells of revolution, such as cooling towers, water tanks and loudspeakers, reducing vibrations is an efficient way to improve the performance and increase the service life of structures. For some special engineering structures about shells of revolution, e.g. submarines, torpedoes, aircrafts and rockets, vibration and acoustic responses are vital and they must be controlled in accordance with the relative standards. In consequence, knowing vibro-acoustic character­ istics of shells of revolution is indispensable in design process. However,

to the authors’ knowledge, investigations about vibro-acoustic problems of shells of revolution with arbitrary shapes are rare. As the simplest forms of shells of revolution, structural-acoustic coupling problems of independent conical shells (Guo, 1995; Caresta and Kessissoglou, 2008; Chen et al., 2014; Liu et al., 2014; Xie et al., 2015), cylindrical shells (Stepanishen, 1982; Guo, 1996; Zhang, 2002; Lynch et al., 2013; Chen et al., 2016; Aslani et al., 2017; Wang et al., 2017) and spherical shells (Junger, 1952; Hayek, 1966; Ding and Chen, 1998; Choi et al., 2012; Lynch et al., 2013) were extensively studied, and only limited papers from hundreds and thousands of ones are referred above. In contrast with independent shells of revolution, studies about combined shells of revolution are relatively scant. Some researchers recently investigated vibro-acoustic characteristics of combined conical-cylindrical shells (Caresta et al., 2008; Caresta and Kessissoglou, 2010; Zhang et al., 2013), combined spherical-cylindrical-spherical shells (Maury and Filippi, 2001; Peters et al., 2014, 2015; Qu et al.,

* Corresponding author. E-mail address: [email protected] (M. Chen). https://doi.org/10.1016/j.oceaneng.2019.106345 Received 24 April 2019; Received in revised form 5 July 2019; Accepted 20 August 2019 Available online 4 September 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Ocean Engineering 189 (2019) 106345

Fig. 1. Schematic diagram and global cylindrical coordinate system of shells of revolution.

2015) and combined conical-cylindrical-spherical shells (Liu et al., 2010; Pan et al., 2011; Leader et al., 2013; Jin et al., 2018). It should be mentioned that, for the cited papers about combined conical-cylindrical-spherical shells, only Jin and coworkers (Jin et al., 2018) developed an energy-based formulation conical-cylindrical-spherical shells on the basis of vibration governing equations of combined conical-cylindrical shells (Ma et al., 2014) and combined conical-cylindrical-spherical shells (Su and Jin, 2016), and the others were experiments. However, these studies are mainly limited to one special form of shells of revolution, and they are no longer appropriate for general shells of revolution. For general shells of revolution with arbitrary shapes, the radius varies with axial coordinate and it is difficult or impossible to give an­ alytic expressions of acoustic pressure acting on the shells. Boundary element method about Helmholtz integral equation is widely adopted to solve acoustic radiation problems. How to improve the efficiency solv­ ing the integral equation of shells of revolution has received widespread attention since the nineties of last century. In these studies, by taking full advantage of the axisymmetric property of shells of revolution, re­ searchers expanded surface pressure and velocity as Fourier series with respect to the angle of revolution, and reduced the surface integral to line integral along the generator line (Juhl, 1993; Soenarko, 1993; Grannell, 1994; Kuijpers et al., 1997; Wang et al., 1997; Provatidis, 1998; Tsinopoulos et al., 1999; Liu et al., 2002; Mohsen and Ochmann, 2010; Ramesh et al., 2012; Xiang et al., 2012; Young et al., 2012). To eliminate the non-uniqueness, the combined Helmholtz integral equa­ tion (CHIEF) developed by Schenck (1968) was utilized by Mohsen and Ochmann (2010), whereas the composite Helmholtz integral equation proposed by Burton and Miller (1971) was employed in the articles (Grannell, 1994; Wang et al., 1997; Tsinopoulos et al., 1999; Liu et al., 2002; Ramesh et al., 2012). Xiang et al. (2012) introduced an approach, the wave superposition method with complex radius vector, to deal with the non-uniqueness. For singular integrals, the problem was overcome by dividing the singular integrals about the angle of revolution into non-singular and singular parts in most of the listed papers, and the singular integrals were calculated via the elliptic integral and its recurrence formulas. Liu et al. (2002) proposed the reproduction of di­ agonal terms (RDT) to avoid hyper-singular integrals. Although the ef­ ficiency about the integral equation was improved by reducing the surface integral to line integral, the fast Fourier transform was utilized to more efficiently calculate the integrals about the angle of revolution in the papers (Kuijpers et al., 1997; Provatidis, 1998; Tsinopoulos et al., 1999; Xiang et al., 2012; Young et al., 2012). In addition, the integral €m along the generator line was calculated through a high-order Nystro discretization scheme (Young et al., 2012) and the fast Fourier transform (Xiang et al., 2012), respectively. Unfortunately, only acoustic radiation problems are discussed and structural-acoustic coupling problems are not involved in. Due to the complexity in modeling and solving procedure about simultaneously considering both dynamic models of shells of revolution and acoustic models of external fluid, opened works about theoretical

approaches (analytic and semi-analytic approaches) for vibro-acoustic analysis of shells of revolution are rare. Chen and Ginsberg (1993) combined the surface variational principle and the dynamic equations of shells to investigate acoustic radiation of shells of revolution, and emphatically discussed vibro-acoustic characteristics of a spheroidal shell. Chen and Stepanishen (1994) used time-dependent in vacuo eigenvector expansions to evaluate the acoustic transient radiation from fluid-loaded shells of revolution, which were excited by axisymmetric excitations. By idealizing axisymmetric shells as two-node ring finite elements and using the method of singularities to solve the boundary equation, Berot and Peseux (1998) proposed a numerical model to study vibro-acoustic behaviors of axisymmetric shells immersed in heavy fluid. Combining the modified variational method, the multi-segment technique and the spectral Kirchhoff-Helmholtz integral formulation, Qu and Meng (2015, 2016) analyzed vibro-acoustic behaviors of multilayered and functionally graded shells of revolution in light and heavy fluids. The main purpose of present paper is to develop a unified semianalytic method to predict vibro-acoustic responses of submerged shells of revolution, which include independent shells and their com­ binations, e.g. conical shells, cylindrical shells, spherical shells, com­ bined conical-cylindrical-spherical shells and so forth. The shell is firstly divided into some shell segments and the segments are treated as conical shells. By employing Flügge shell theory and expanding displacements as power series, displacements and forces at arbitrary cross-section of conical shells can be expressed as 8 unknown coefficients. These seg­ ments are easily assembled to the overall shell through continuity con­ ditions, and corresponding vibration governing equation is established by additionally considering boundary conditions. Helmholtz integral equation is adopted to describe motions of external acoustic pressure and it is reduced to the line integral along the generator line through expanding acoustic pressure and velocity as Fourier series. Meshing the generator line into several 3-node elements and combining the surface velocities and displacements of segments, vibro-acoustic governing equation for structural-acoustic coupling problems about shells of rev­ olution is obtained by introducing acoustic pressure to continuity con­ ditions between shell segments. The proposed semi-analytic method is believed to contain following novelties. First, it provides a unified approach to analyze vibro-acoustic problems of general shells of revo­ lution, and both independent and combined shells of revolution are included. Second, as compared with traditional finite element-boundary element method, the efficiency of present method is much higher, especially for the shells with straight generator line. Last, the method can be easily extended to analyze vibro-acoustic behaviors of shells of revolution stiffened by bulkheads and rings with rectangular, L and T cross-sections as modeling stiffening members as combinations of annular plates with different semi-vertex angles and radii. 2. Theory formulas A general shell of revolution with thickness h is shown in Fig. 1, and a 2

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Ocean Engineering 189 (2019) 106345

Fig. 2. Local coordinate system, positive directions of displacements and forces of a conical shell.

global cylindrical coordinate system O rθz is introduced. To uniformly consider the shell model and external acoustic pressure, the shell is divided into some shell segments according to dashed lines in the figure. As segments are narrow enough, segments can be treated as conical shells with different semi-angles and radii, and effects of external acoustic pressure are considered by adding the pressure to continuity conditions of shell segments.

unknown coefficients are deduced, which are given in Appendix B. By using the recurrence formulas and omitting the time part e jωt , Eq. (2) further becomes ∞ h X

uðx; θÞ ¼

uCn ⋅ xCn

�T

cosðnθÞ þ uSn ⋅ xSn

�T

i sinðnθÞ

n¼0 ∞ h X

vðx; θÞ ¼

vCn ⋅ xCn

�T

sinðnθÞ þ vSn ⋅ xSn

�T

i

(3)

cosðnθÞ

n¼0

2.1. Shell structures

∞ h X

wðx; θÞ ¼

∞ X ∞ h i X aCn;m xm cosðnθÞ þ aSn;m xm sinðnθÞ e ∞ X ∞ h i X bCn;m xm sinðnθÞ þ bSn;m xm cosðnθÞ e

(1)

∞ X ∞ h X

i cCn;m xm cosðnθÞ þ cSn;m xm sinðnθÞ e

(4)

can be obtained by substituting the coefficients into Eq. (2) and merging similar items in terms with 8 unknown coefficients in xCn . Similar pro­ cedure is adopted for base functions of antisymmetric modes. Considering differences of semi-vertex angles of adjacent segments, displacements and forces in normal and merdional directions are transferred to the ones in the global coordinate system, and some new notations (u, w, N and S) are introduced.

(2)

n¼0 m¼0

wðx; θ; tÞ ¼

i sinðnθÞ

bCn;mþ2 and cCn;mþ4 of different m, the base functions of symmetric modes

jωt

jωt

�T

1 � 8 vectors uCn , vCn , wCn , uSn , vSn and wSn are base functions. It should be mentioned that, although it is difficult or impossible to give analytical expressions of the base functions, they can be easily obtained via nu­ merical calculation in MATLAB. After calculating coefficients aCn;mþ2 ,

n¼0 m¼0

vðx; θ; tÞ ¼

cosðnθÞ þ wSn ⋅ xSn

where unknown coefficient vectors xCn and xSn are o n 8 < xCn ¼ aCn;0 ; aCn;1 ; bCn;0 ; bCn;1 ; cCn;0 ; cCn;1 ; cCn;2 ; cCn;3 o n : xS ¼ aS ; aS ; bS ; bS ; cS ; cS ; cS ; cS n;0 n;1 n;0 n;1 n;2 n;3 n n;0 n;1

where differential operators Lij ði; j ¼ 1 : 3Þ are given in Appendix A. In order to solve the equation, displacements u, v and w are expanded as power series and Fourier series in meridional and circumferential di­ rections, respectively, and detailed expressions are (Tong, 1993) uðx; θ; tÞ ¼

�T

n¼0

2.1.1. Conical shell segments Fig. 2 shows a conical shell and a special local coordinate system Oc xc zc θc . The subscript ‘c’ will be omitted for simplicity hereafter. α is the semi-vertex angle. R1 and R2 are radii at two ends, and R0 is the mean radius. For thin-walled conical shells, Flügge shell theory is adopted to describe equations of motion (Leissa, 1993) L11 u þ L12 v þ L13 w ¼ 0 L21 u þ L22 v þ L23 w ¼ 0 L31 u þ L32 v þ L33 w ¼ 0

wCn ⋅ xCn

u ¼ u cos α w sin α w ¼ w cos α þ u sin α N ¼ N cos α Ssin α S ¼ Scos α þ N sin α

jωt

n¼0 m¼0

(5)

Substituting Eq. (3) into expressions of slope β ¼ ∂w=∂x and forces (given in Appendix A), displacements and forces at any cross-section of conical shells are given in matrix form as 8" 9 # # " T T N < X � � = ΤC ⋅DCn ΤS ⋅DSn T C S fu; v; w; β; N; T; S; Mg ¼ (6) ⋅ xn þ ⋅ xn : ΤC ⋅FCn ; ΤS ⋅FSn n¼0

where m, n, j ω and t denote the index, circumferential mode number, imaginary unit, circular frequency and time. Superscript ‘C’ and ‘S’ indicate symmetric and antisymmetric modes. aCn;m , bCn;m , cCn;m , aSn;m , bSn;m

and cSn;m are unknown coefficients. Substituting symmetric or antisym­

metric parts of Eq. (2) into Eq. (1), recurrence formulas about these 3

Ocean Engineering 189 (2019) 106345

K. Xie et al.

Fig. 3. Schematic diagram of field points and elements of the generator line of a shell of revolution. Table 1 Convergence and comparison of natural frequencies of a clamped cylindrical shell (Hz). Order

1 2 3 4 5 6 7 8

m

n

1 1 2 2 3 1 2 3

2 3 3 2 3 4 4 4

Present

Zhang (Zhang, 2002)

P ¼ 60

P ¼ 80

P ¼ 100

Analytic

FEM/BEM

4.93 9.06 10.73 11.25 14.69 18.60 19.06 20.30

4.91 9.02 10.69 11.23 14.63 18.51 18.97 20.19

4.91 9.00 10.66 11.21 14.60 18.45 18.91 20.13

4.95 8.95 10.66 11.54 14.73 18.26 18.71 20.00

4.92 9.06 10.71 11.24 14.70 18.68 19.14 20.37

where N is the truncated circumferential mode number. 4� 8 matrices DCn , DSn , FCn and FSn are 3 3 3 2 2 3 2 2 C S uCn uSn N N n n 7 7 7 6 6 7 6 6 6 vC 7 6 vS 7 6 TC 7 6 TS 7 7 7 7 6 6 7 6 6 DCn ¼ 6 nC 7 ; DSn ¼ 6 nS 7 FCn ¼ 6 Cn 7 ; FSn ¼ 6 Sn 7 (7) 6 wn 7 6 wn 7 6 Sn 7 6 Sn 7 5 5 5 4 4 5 4 4 βCn βSn MCn MSn 4�8

4�8

4�8

4�8

4 � 4 matrices ΤC and ΤS are ΤC ¼ diagðcosðnθÞ; sinðnθÞ; cosðnθÞ; cosðnθÞÞ ΤS ¼ diagðsinðnθÞ; cosðnθÞ; sinðnθÞ; sinðnθÞÞ

(8)

On the basis of the recurrence formulas of unknown coefficients, relationships of DCn and DSn , FCn and FSn are certain and given in Appendix B. 2.1.2. Vibration governing equation Displacements at edges of shells are restrained by artificial springs and corresponding boundary conditions are Ku u � N ¼ 0; Kv v � T ¼ 0 Kw w � S ¼ 0; Kβ β � M ¼ 0

Fig. 4. Effects of stiffness constants on frequencies of the cylindrical shell: (a) m ¼ 1, n ¼ 2 (b) m ¼ 1, n ¼ 3.

(9)

where plus and minus signs are appropriate for the left and right edges, respectively. Ku , Kv , Kw and Kβ are stiffness constants restraining axial displacement, circumferential displacement, radial displacement and

� � � � uRi �x¼lR ¼ uLiþ1 �x¼lL ; vRi �x¼lR ¼ vLiþ1 �x¼lL i i iþ1 � � � � iþ1 wR � R ¼ wL � L ; βR � R ¼ βL � L i x¼l i

slope. The unit of stiffness constants Ku , Kv and Kw is N=m2 whereas the one of stiffness constant Kβ is N=m. For the sake of simplicity, the unit of stiffness constants is omitted. By adopting appropriate constants, both elastic and classical boundary conditions can be analyzed, and it will be emphatically discussed in Sec.3.1.1. For any adjacent two segments, continuity conditions of displace­ ments and equilibrium equations of forces are

iþ1 x¼l iþ1

i x¼l i

� � N Ri �x¼lR þN Liþ1 �x¼lL ¼ 0; i iþ1 � R� L � Si �x¼lR þS2;i � L ¼ F; i

x¼liþ1

(10)

iþ1 x¼l iþ1

R� L� T i �x¼lR þT i �x¼lL ¼ 0 i iþ1 � � M R � R þM L � L ¼ 0 i x¼l i

(11)

i x¼l iþ1

where superscript ‘R’ and ‘L’ indicate the right and left ends of a segment, subscript ‘i’ indicates the ith segment. lRi and lLiþ1 are local 4

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Ocean Engineering 189 (2019) 106345

Table 2 Convergence and comparison of natural frequencies of a conical shell with different boundary conditions (Hz). n

m

F-C

C-C

Present

1 2 3

4

5

1 2 1 2 3 1 2 3 4 1 2 3 4 1 2 3 4

P ¼ 40

P ¼ 60

P ¼ 80

33.68 60.86 15.25 34.10 60.15 14.22 29.40 47.17 67.26 13.93 27.96 44.94 63.35 15.01 28.02 43.41 60.09

33.71 60.98 15.26 34.05 60.04 14.19 29.32 46.99 66.97 13.90 27.88 44.80 63.13 14.97 27.92 43.24 59.85

33.71 60.96 15.26 34.01 59.97 14.17 29.28 46.89 66.79 13.88 27.84 44.72 63.02 14.94 27.84 43.16 59.73

S-S

Caresta and Kessissoglou (2008)

Present P ¼ 40

P ¼ 60

P ¼ 80

35.1 61.7 15.6 33.9 59.1 14.3 29.6 46.7 65.4 13.9 27.7 44.5 62.4 14.8 27.6 42.7 58.9

39.39 72.00 22.62 46.77 74.51 15.16 32.92 55.12 78.21 13.97 28.14 45.52 64.75 15.01 28.02 43.42 60.14

39.46 72.28 22.60 46.76 74.56 15.12 32.86 55.02 78.09 13.93 28.06 45.39 64.55 14.97 27.92 43.26 59.90

39.46 72.31 22.58 46.73 74.52 15.11 32.82 54.96 77.99 13.92 28.02 45.31 64.43 14.94 27.87 43.18 59.77

Caresta and Kessissoglou (2008)

Present P ¼ 40

P ¼ 60

P ¼ 80

42.3 74.1 22.6 46.7 73.9 15.1 32.7 54.4 76.6 13.9 27.9 44.9 63.5 14.8 27.6 42.7 58.9

32.32 63.66 12.12 44.56 72.31 8.38 29.09 52.81 75.98 9.40 25.59 43.94 63.38 11.14 25.80 42.23 59.55

32.35 63.81 12.10 44.53 72.33 8.37 29.02 52.70 78.85 9.38 25.51 43.83 63.17 11.11 25.71 42.07 59.30

32.34 63.81 12.1 44.5 72.28 8.36 28.98 52.63 75.75 9.37 25.47 43.75 63.06 11.09 25.67 41.99 59.17

Caresta and Kessissoglou (2008)

32.9 64.7 11.8 45.2 72.0 8.4 28.7 52.0 74.6 9.3 25.3 43.3 62.2 11.1 25.3 41.4 58.3

Fig. 5. Division of the spherical shell. Table 3 Convergence and comparison of natural frequencies of a spherical shell (Hz). m

n¼0

n¼1

Present

Berot and Peseux (1998)

Present

n¼2 Berot and Peseux (1998)

Present

Berot and Peseux (1998)

Cas1

Cas2

Cas3

Cas1

Cas2

Cas3

Cas1

Cas2

Cas3

1 2 3 4 5 6 7

80.52 101.37 116.23 128.25 138.56 147.72 156.07

79.60 100.16 114.73 126.47 136.50 145.35 153.36

79.43 99.98 114.54 126.12 136.05 144.58 152.46

79.6 100.7 111.1 122.9 132.8 142.2 150.5

79.33 99.86 114.40 126.09 136.10 144.96 153.00

79.46 100.03 114.58 126.27 136.21 145.01 152.99

79.44 99.99 114.51 126.17 136.06 144.82 152.75

81.1 100.7 114.6 126.0 136.0 145.0 153.2

79.57 100.20 114.81 126.58 136.65 145.56 153.65

79.48 100.06 114.59 126.28 136.25 144.99 152.96

79.44 100.00 114.50 126.16 136.08 144.80 152.70

81.2 100.9 115.0 126.3 136.2 145.1 153.2

m

n¼3 Present Cas1

Cas2

Cas3

Berot and Peseux (1998)

n¼4 Present Cas1

Cas2

Cas3

Berot and Peseux (1998)

n¼5 Present Cas1

Cas2

Cas3

Berot and Peseux (1998)

100.25 114.86 126.63 136.69 145.60 153.79

100.05 114.60 126.29 136.23 145.02 153.05

100.00 114.51 126.17 136.07 144.82 152.82

101.0 115.2 126.6 136.4 145.3 153.5

114.86 126.65 136.72 145.61 153.69 –

114.59 126.27 136.24 145.00 152.93 –

114.50 126.15 136.08 144.80 152.70 –

115.1 126.8 136.7 145.7 153.7 –

126.63 136.70 145.61 153.66 – –

126.27 136.22 145.00 152.91 – –

126.15 136.07 144.81 152.70 – –

126.8 137.0 146.0 154.0 – –

1 2 3 4 5 6

5

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Ocean Engineering 189 (2019) 106345

coordinates of the right end of the ith segment and the left end of the (iþ1)th segment, respectively, and they can be readily obtained through the meridional length of segments. F is an external point excitation in radial direction and it is expressed as Dirac delta function F ¼ F0 δðx

xF Þδðθ

FSen;i , and the expressions are not given for simplicity.

In Eq. (20), matrices KCHn and KSHn are similar and their expressions are uniformly given as 3 2

(12)

θF Þ=RF

where F0 is the amplitude of excitation and its global coordinate is ðxF ;θF ; RF Þ. Substituting Eq. (6) into Eqs.(9)-(11) and utilizing the orthogonality of trigonometric functions, boundary and continuity conditions further become 8� � � � � �T � �T C � C � C C C > > < Ks ⋅Dn;1 �x¼lL þ Fn;1 �x¼lL ⋅ xn;1 ¼ Bn;1 ⋅ xn;1 ¼ 0 1 1 � � � � �T � � �T > � > : Ks ⋅DS �� þ FSn;1 � L ⋅ xSn;1 ¼ BSn;1 ⋅ xSn;1 ¼ 0 n;1 L x¼l1

8� � > C � > K ⋅D � > s n;P <

x¼lR

� � > � > > : Ks ⋅DSn;P �

P

(13)

x¼l1

� � FCn;P �

x¼lR

� � �T � �T ⋅ xCn;P ¼ BCn;P ⋅ xCn;P ¼ 0

� FSn;P � R x¼l

� � �T � �T ⋅ xSn;P ¼ BSn;P ⋅ xSn;P ¼ 0

P

x¼lR

n¼0:N

KHn

�

P

x¼li

DLn;2 FLn;2 DRn;2

DLn;3

FRn;2

FLn;3

::: DRn;P

1

DLn;P

FRn;P

1

FLn;P Bn;P

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

(22)

8P�8P

n¼0:N

DLn;i ,

i

(15)

n¼0:N

DRn;i ,

FLn;i

FRn;i

In above expression, and are appropriate for sym­ metric and antisymmetric modes and they are � � DLn;i ¼ Dn;i �x¼lL ; DRn;i ¼ Dn;i �x¼lR i � �i (23) FLn;i ¼ Fn;i �x¼lL ; FRn;i ¼ Fn;i �x¼lR

(14)

P

8 � � �T � �T � � C � C C C > > < Dn;i �x¼lR ⋅ xn;i ¼ Dn;iþ1 �x¼lL ⋅ xn;iþ1 i iþ1 � � �T � �T � > � > S S : DS �� ⋅ xS ¼ D ⋅ x � n;i n;iþ1 n;i n;iþ1 R L

6 6 Bn;1 6 6 R 6 Dn;1 6 6 R 6 Fn;1 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 6 6 6 4

i

In Eq. (22) and Eq. (23), superscript C and S are omitted.

x¼liþ1

8 � � �T C � C > > < Fn;i �x¼lR ⋅ xn;i i � � �T > > : FS �� ⋅ xS n;i n;i R x¼li

� � FCn;iþ1 �

� �T ⋅ xCn;iþ1 ¼ FCen;i

� � FSn;iþ1 �

�T � ⋅ xSn;iþ1 ¼ FSen;i

x¼lLiþ1

x¼lLiþ1

2.2. Acoustic pressure n¼0:N

The acoustic pressure of Point P radiated from a vibrating structure of revolution, as shown in Fig. 3, can be calculated by Helmholtz integral equation � Z � ∂GðP; QÞ ∂pðQÞ cðPÞpðPÞ ¼ GðP; QÞ dS0 ðQÞ (24) pðQÞ ∂nq ∂nq

(16)

In Eqs. (13) and (14), 4 � 4 matrix Ks is

s0

(17)

Ks ¼ diagðKu ; Kv ; Kw ; Kβ Þ FCen;i

FSen;i

In Eq. (16), vectors and 8 < FC ¼ f0; 0; εF0 cosðθF Þ=RF ; 0gT en;i : FS ¼ f0; 0; εF0 sinðθF Þ=RF ; 0gT

where Point Q is on the surface S0 . The geometric constant cðPÞ and freespace Green’s functionGðP; QÞ are � � Z ∂ 1 cðPÞ ¼ 1 þ (25) dS0 ðQÞ ∂nq 4πRðP; QÞ

are (18)

S0

en;i

where ε equals to 1 =ð2πÞ for n ¼ 0 or 1 =π for n � 1. Assuming the shell of revolution divided into P segments and assembling all continuity and boundary conditions, vibration governing equation is "

ΚCH 0

# � � ( C) Fe XC ⋅ ¼ S X FSe ΚSH 0

where 2

3 C

6 KH0 6 6 ΚCH ¼ 6 6 4

GðP; QÞ ¼

where

3

2 S

KCH1 ::: KCHN

KSH1 ::: KSHN

oT n 8 < XC ¼ xC0;1 ; xC0;2 ; :::; xC0;P ; :::; xCn;1 ; :::; xCn;P ; :::; xCN;1 ; :::; xCN;P oT : S n S X ¼ x0;1 ; xS0;2 ; :::; xS0;P ; :::; xSn;1 ; :::; xSn;P ; :::; xSN;1 ; :::; xSN;P

7 7 7 7 7 5

(26)

In above equations, RðP; QÞ is the distance between Point Q and Point P, and kf is wavenumber. From Fig. 3, RðP; QÞ in terms of the coordinates of PðrP ; θP ; xP Þ and QðrQ ; θQ ; xQ Þ is � �2 R2 ðP; QÞ ¼ r2p þ r2q 2rp rq cos θq θp þ zq zp qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ¼ R 1 λ cos2 ðθ=2Þ (27)

(19)

7 6 KH0 7 6 7 S 6 7ΚH ¼ 6 7 6 5 4

e jkf RðP;QÞ 4πRðP; QÞ

θ ¼ θq

(20)

R¼

θp

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffi 2 2 rp þ rq þ zp zq

pffiffiffiffiffiffiffiffi� λ ¼ 2 rp rq R

(28) (29) (30)

It should be mentioned that the subtraction of θq and θp is defined as θ in Eq. (28), which is the same with the circumferential coordinate of global cylindrical system. Considering the axisymmetric property of titled structures, acoustic pressure and normal velocity are expanded as Fourier series represented

(21)

Excitation vectors FCe and FSe are easily obtained by means of FCen;i and 6

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Ocean Engineering 189 (2019) 106345

Fig. 6. Schematic diagram of three kinds of excitations forced on the combined spherical-cylindrical-spherical shell.

Fig. 8. Radiated sound power of the combined spherical-cylindrical-spherical shell submerged in water: (a) transverse ring force excitation (b) axial point force.

Fig. 7. Radiated sound power of the combined spherical-cylindrical-spherical shell submerged in air: (a) axial ring force (b) transverse ring force.

in the exponential form. ∞ X

pðr; θ; zÞ ¼

pn ðr; zÞejnθ

where Z gn ¼

(31)

n¼ ∞ ∞ X

wðr; _ θ; zÞ ¼

2π

GðP; QÞejnθ dθ

(34)

∂GðP; QÞ jnθ ∂g e dθ ¼ n ∂n ∂n

(35)

0

w_ n ðr; zÞejnθ

Z

(32)

2π

hn ¼

n¼ ∞

0

Substituting Eqs.(26)-(32) into Eq. (24), the surface integral is reduced to the line integral � Z � ∂pn ðQÞ cðPÞpn ðPÞ ¼ rq dΓ hnq pn ðQÞ gn (33) ∂nq

Eq. (34) and Eq. (35) are the same for n and n. To eliminate the singularity, the integral kernels are decomposed into non-singular and singular parts. The singular part is analytically solved by using the elliptic integral and its recurrence formulas, whereas the non-singular part is directly calculated by Gaussian quadrature formulas. For the

Γ

7

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Fig. 9. Displacements and sound pressure of the combined conical-cylindrical-spherical shell: (a) PointA (b) PointB (c) FieldA (d) FieldB.

sake of simplification, derivations about the integrals gn and hn are not given and readers can refer to the paper (Wang et al., 1997) as needed. As the generator line is meshed into some 3-node isoparametric el­ ements, as shown in Fig. 3, local coordinates, pressure and velocity in one element are expressed as 3 X

r¼

z¼

i¼1 3 X

(36)

3 X

Ni ðζÞpi ; i¼1

Ni ðζÞzi ; i¼1

w_ ¼

ACHIEF ⋅ p_ n ¼ BCHIEF ⋅w_ n n n

Ni ðζÞw_ i i¼1

1 1 ζ þ ζ2 N2 ðζÞ ¼ 1 2 2

1 1 ζ2 N3 ðζÞ ¼ ζ þ ζ2 2 2

(39)

For field points outside the shell, the procedure is the same with CHIEF points excluding that the geometry constant cðPÞ becomes 1 from 0. Combining Eq. (38) and Eq. (39), the final relationship about pres­ sure and velocity becomes

where ri , zi , pi and w_ i are parameters of the nodes, and shape functions N1 ðζÞ, N2 ðζÞ and N3 ðζÞ are N1 ðζÞ ¼

(38)

For nonclosed shells, e.g. cylindrical shells without end plates, the acoustic model is sealed by rigid baffles at two ends. The CHIEF method (Schenck, 1968) is employed to eliminate the nonuniqueness. Since no singularity occurs for points inside the shell, Eq. (33) about these points is easily solved by using similar procedure of Eq. (38) and it is

3 X

Ni ðζÞri ;

p¼

An ⋅ p_ n ¼ Bn ⋅w_ n

(37)

By using Gaussian quadrature formulas, Eq. (33) is further expressed in matrix form as

An ⋅ p_ n ¼ Bn ⋅w_ n

Fig. 10. Surface displacements of the combined conical-cylindrical-spherical shell. 8

(40)

K. Xie et al.

and w_ Sn are

where " An ¼

Ocean Engineering 189 (2019) 106345

# An ACHIEF n

; Bn ¼

oT n 8 C C C
#

" Bn BCHIEF n

(41)

Eq. (40) is overdetermined and it can be solved by the least square method.

In Eq. (43), TCW and TSW are ðP þ1Þ � 8P matrices combining node n n velocity vectors and displacement functions of segments and they are 3 3 2 2

2.3. Final vibro-acoustic governing equation Acoustic pressure acting on the shell is considered through adding the pressure to continuity conditions between segments as analyzing structural-acoustic coupling problems. To uniformly analyze the shell and acoustic pressure, the discretization of generator line in acoustic pressure analysis is taken into account as dividing the shell to segments, which means that axial locations of the shell segments and nodes are the same. Accordingly, there are P =2 elements to be considered as the shell divided into P segments. By virtue of the Euler’s formula, Eq. (40) is appropriate for the two cases that surface pressure and velocity are expanded as Fourier series represented in the trigonometric form and the exponential form. Considering the form of displacements in Eq. (3), Eq. (40) is transformed to ( An ⋅p_ Cn ¼ Bn ⋅w_ Cn (42) An ⋅p_ Sn ¼ Bn ⋅w_ Sn where node velocity vectors of symmetric and antisymmetric modes

(43)

TCW n

6 CL 6 w_ 6 n;1 6 6 6 6 ¼6 6 6 6 6 6 4

where 8 > > _ CL w > n;i ¼ > > > > > > > CR > > < wn;i ¼ > > _ SL w > n;i ¼ > > > > > > > > > wSR : n;i ¼

7 6 SL 7 6 w_ 7 6 n;1 7 6 7 6 7 6 7 SW 6 ::: 7Tn ¼ 6 7 6 7 6 w_ CL n;P 7 6 6 CR 7 6 w_ n;P 7 5 4

_ CL w n;2

� � � jω wCn;i �

⋅sinαi L

x¼li

� � � jω wCn;i � R ⋅sinαi x¼l � � i � jω wSn;i � L ⋅sinαi x¼li

� � � jω wSn;i �

x¼lRi

⋅sinαi

� � uCn;i �

x¼lLi

w_ SL n;2 :::

7 7 7 7 7 7 7 7 SL 7 w_ n;P 7 7 7 7 w_ SR n;P 5

(44)

� ⋅cosαi

� � � uCn;i � R ⋅cosαi x¼l � i � � uSn;i � L ⋅cosαi x¼li � � S � un;i � R ⋅cosαi

(45)

x¼li

_ Cn w

Fig. 11. Directivity pattern of radiated sound pressure of the combined conical-cylindrical-spherical shell: (a) 1st resonant peak (b) 2nd resonant peak (c) 3rd resonant peak (d) 4th resonant peak. 9

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Ocean Engineering 189 (2019) 106345

Correspondingly, Eq. (42) becomes oT n 8 C C C C < p_ n ¼ TCP n ⋅ xn;1 ; xn;2 ; :::; xn;P n oT : S S S S p_ n ¼ TSP n ⋅ xn;1 ; xn;2 ; :::; xn;P

(46)

where ( þ CW TCP n ¼ ðAn Þ ⋅Bn ⋅Tn

(47)

þ

SW TSP n ¼ ðAn Þ ⋅Bn ⋅Tn

In Eq. (47), the superscript ‘þ’ denotes the Moore-Penrose inverse, SP and the size of TCP n and Tn is ðP þ 1Þ � 8P. According to the form of Eqs. (20) and (22), p_ Cn and p_ Sn are further extended to matrices KCFn and KSFn , and the expressions are 3 3 2 2

7 6 7 6 ΚC 6 Fn;1 7 7 6 7 6 0 7 6 7 6 7 6 ΚC Fn;2 7 6 7 6 7 6 0 7 6 7 6 C ΚFn ¼ 6 ΚCFn;3 7 7 6 7 6 ⋮ 7 6 7 6 0 7 6 7 6 7 6 ΚC 6 Fn;P 7 7 6 7 6 ΚC 6 Fn;ðPþ1Þ 7 5 4

; ΚSFn

7 6 7 6 ΚS 6 Fn;1 7 7 6 7 6 0 7 6 7 6 7 6 ΚS Fn;2 7 6 7 6 7 6 0 7 6 7 6 ¼ 6 ΚSFn;3 7 7 6 7 6 ⋮ 7 6 7 6 0 7 6 7 6 7 6 ΚS 6 Fn;P 7 7 6 7 6 ΚS 6 Fn;ðPþ1Þ 7 5 4

8P�8P

(48)

8P�8P

where 2

ΚCFn;i

3 CP ⋅cos α T i n;i 6 7 6 7 0 6 7 ¼6 7 6 TCP 7 ⋅sin α i n;i 4 5

2

ΚSFn;i

0

3 SP ⋅cos α T i n;i 6 7 6 7 0 6 7 ¼6 7 6 TSP 7 ⋅sin α i n;i 4 5

(49)

Fig. 12. Contributions from different circumferential mode numbers to radi­ ated sound pressure of the combined conical-cylindrical-spherical shell: (a) FeildA (b) FeildB.

0 4�8P

4�8P

In Eq. (49), it should be mentioned that two rows in ΚCFn;i and ΚSFn;i are nonzero vectors, which is attributed to that the sound pressure is perpendicular to the vibrating surface and it is decomposed into radial and axial parts in accordance with the global cylindrical coordinate. Introducing the extended matrices to the vibration governing equa­ tion (19), the final vibro-acoustic governing equation of submerged shells of revolution is " # " #! � � ( C) ΚCH Fe 0 ΚCF 0 XC þ ⋅ (50) ¼ XS 0 ΚSH FSe 0 ΚSF

spherical and conical-cylindrical-spherical shells are compared with ones in appropriate references or calculated by FEM/BEM to analyze accuracy of present method for complex shells, and influences of pa­ rameters on vibro-acoustic characteristics of combined conicalspherical-spherical shells are further discussed. 3.1. Free vibrations of independent shells Free vibrations of three independent shells submerged in water are discussed. For the independent shells, sound speed and mass density of external water are 1500m=s and 1000kg=m3 .

where matrices ΚCF and ΚSF denote external acoustic pressure acting on the shell and they are ( C � ΚF ¼ diag KCF0 ; KCF1 ; ⋯; KCFN � (51) ΚSF ¼ diag KSF0 ; KSF1 ; ⋯; KSFN

3.1.1. Cylindrical shells A clamped cylindrical shell studied by Zhang (2002) is firstly employed to test the convergence and validity of present method. The length, thickness and radius of the cylindrical shell are 20 m, 0.01 m and 1 m, respectively. Material parameters of the shell are Young’s modulus

By solving the vibro-acoustic governing equation (50), vibration and acoustic results of shells of revolution with arbitrary shapes can be easily obtained.

2:1 � 1011 N=m2 , Poisson’s ratio 0.3 and density 7850kg=m3 . Natural frequencies of the first eight modes are tabulated in Table 1. To discuss the convergence of present method, the cylindrical shell is uniformly divided into 60, 80 and 100 segments. From the table it is clearly found that natural frequencies of present method rapidly converge as the number of segments increases. Results of present method with 100 segments are in excellent agreement with the ones in literature. In addition, it can be concluded that present method is of higher accuracy after comparing frequencies of present method with the ones of two methods in the reference.

3. Results and discussion In this part, vibration and acoustic results of some typical shells of revolution are studied. Free vibrations of three independent shells, including submerged cylindrical, conical and spherical shells, are firstly discussed to test convergence and validity of the proposed method. Then, vibro-acoustic responses of combined spherical-cylindrical10

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Ocean Engineering 189 (2019) 106345

As the artificial spring technology is employed to uniformly analyze elastic and classic boundary conditions, assigning appropriate stiffness constants is important. Effects of stiffness constants on frequencies of the first two modes are shown in Fig. 4. For each curve in the figure, only one stiffness constant at the left edge increases from 10 to 1016 and the other 7 ones are assigned as 1016 . It is found that frequencies dramati­ cally increase as the stiffness constant Ku increases, whereas frequencies corresponding to other constants almost keep unchanged. Furthermore, frequencies are not increased or decreased when the constants are too large or too small, which means that free, fixed and elastic boundary conditions can be dealt through adopting appropriate constants.

thickness 0.01 m are discussed in this subsection. Material parameters

are Young’s modulus 2:1 � 1011 N=m2 , Poisson’s ratio 0.3 and density

7850kg=m3 . Differing from divisions of previous two shells, a new parameter Δφ is introduced, as shown in Fig. 5, and the axial length of different segments may be different. On one hand, to maintain the ac­ curacy of present method, segments around two poles should be narrow enough while segments at other parts may be longer. On the other hand, the efficiency of present method will be dramatically reduced as the length of all segments is consistent. Consequently, different Δφ is adopted for different regions to guarantee both accuracy and efficiency. Δφ2 is adopted at the interval ½φ1 ; φ2 � whereas Δφ1 is adopted at the � � intervals ½0 ; φ1 � and ½φ2 ; 180 �. To test the convergence of present method, three different cases, defined as Cas1, Cas2 and Cas3, are � analyzed. For three cases, φ1 and φ2 are uniform and they are 3 and � � 177 . Δφ1 and Δφ2 are different in different cases, and they are 1 and 3� for Cas1, 0.5� and 1.5� for Cas2 and 0.2� and 1� for Cas3. Correspond­ ingly, the total number of segments is 64, 128 and 204 for three cases. Natural frequencies of a complete spherical shell submerged in water are tabulated in Table 3. It is found that the convergence of present method is rapid. Although discrepancies between Cas2 and Cas3 still exist, they are smaller than the ones between Cas1 and Cas2. In the following analysis, the division of Cas3 will be adopted as spherical shells involved in. What’s more, the discrepancy of frequencies of pre­ sent method and the reference (Berot and Peseux, 1998) is negligible, which demonstrates the high accuracy of present method.

3.1.2. Conical shells The second independent shell of revolution is a conical shell, which was studied by Caresta and Kessissoglou (2008). The radii of two ends of the conical shell are 0.5 m and 3.25 m. The length in meridional direc­ tion and thickness of shell are 8.90 m and 0.014 m. Young’s modulus,

Poisson’s ratio and density are 2:1 � 1011 N=m2 , 0.3 and 7800kg=m3 . In Table 2, convergence and comparison of frequencies of the conical shell with different classic boundary conditions are listed. In the table, the number of segments adopted to discuss convergence of present method is P ¼ 40, 60 and 80. For 3 kinds of boundaries, the convergence of present method is rapid. More importantly, differences between results of present method and the literature are negligible, except for a few modes. 3.1.3. Spherical shells Free vibrations of a complete spherical shell with radius 2.5 m and

Fig. 14. Sound pressure contributed from different parts of the combined conical-cylindrical-spherical shell subjected to radial force: (a) FieldA (b) FieldB.

Fig. 13. Effects of external excitations on radiated sound pressure of the combined conical-cylindrical-spherical shell: (a) FeildA (b) FeildB. 11

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3.2. Vibro-acoustic responses of combined shells

The thickness of conical, cylindrical and hemispherical shells is 0.008 m.

Vibro-acoustic responses of two combined shells, combined spherical-cylindrical-spherical and combined conical-cylindricalspherical shells, will be studied hereafter.

son’s ratio υ ¼ 0:3 and density ρ ¼ 7800Kg=m3 . To consider structure damping, complex Young’s modulus E ¼ Eð1 jηs Þ is introduced and ηs is 0.01. Sound speed and mass density of external fluid are 1500m=s and

Material parameters are Young’s modulus E ¼ 2:1 � 1011 N=m2 , Pois­

1000kg=m3 . The left end of conical shell is clamped. The start and end frequencies are 1 Hz and 100 Hz, and the frequency step is 0.5 Hz. The global coordinate of driving point ðr; θ; zÞ is ð0:6; 0; 3:7428Þ, and it is defined as PointA. Excepting the driving point, responses of PointB, which is located at the junction of cylindrical and spherical shells, are also discussed. The coordinate of PointB is ð0:6; 0; 5:9928Þ. Radiated sound pressure of one near-field point (FeildA) and one far-field point (FeildB) is considered, and the global coordinates of two filed points are ð1; 0; 3:7428Þ and ð100; 0; 3:7428Þ. For present method, the conical, cylindrical and spherical shells are divided into 30, 90, and 102 seg­ ments, respectively, and the truncated circumferential mode number is 10. Since results in the reference are incomplete, FEM/BEM is addi­ tionally employed to discuss the validity of present method. The finite element model is developed in Ansys16.0 through SHELL181 elements. The conical, cylindrical and spherical shells are meshed to 160 elements in circumferential direction. In axial direction, the conical and cylin­ drical shells are divided into 60 and 240 elements. The total number of elements is 52800. The boundary element model is developed in Virtual. Lab 13.6, and the total number of elements is 6912. To ensure the reliability of results of FEM/BEM, the first 800 modes of the finite element model are calculated and the maximum frequency is larger than 1000 Hz. Displacement and sound pressure responses are expressed as

3.2.1. Combined spherical-cylindrical-spherical shells Vibro-acoustic responses of the combined spherical-cylindricalspherical shell studied by Peters et al. (2014) and Qu et al. (2015) are discussed in current sub-section. The radius and length of middle cy­ lindrical shell is 3.25 m and 45 m. The thickness of cylindrical shell and two hemispherical shells is equal and it is 0.04 m. The combined shell is made of steel with Young’s modulus E ¼ 2:1 � 1011 N=m2 , Poisson’s

ratio υ ¼ 0:3 and density ρ ¼ 7860kg=m3 . To consider structure damping (ηs ¼ 0:01), Young’s modulus E is replaced by a complex one E ¼ Eð1 jηs Þ. Axial ring force, transverse ring force and axial point force are forced at the intersection between left hemispherical shell and cylindrical shell, respectively. The schematic diagram of three kinds of excitations is shown in Fig. 6. For ring and point excitations, the magnitude is 1N. The frequency range is 1–100 Hz, and frequency step is 0.5 Hz. The combined spherical-cylindrical-spherical shell is submerged in heavy or light acoustic medium, respectively. Hemispherical and cylindrical shells are divided into 102 and 90 segments, and the total number of segments is 294. The density and sound speed of heavy acoustic medium (water) are 1000kg=m3 and 1482m=s, and the ones of

light acoustic medium (air) are 1:204kg=m3 and 340m=s. The sound power lever 10 log10 ðPS =10 12 Þ is adopted to discuss the validity of present method, and PS denotes the amplitude of radiated sound power. Fig. 7 presents radiated sound power of the combined sphericalcylindrical-spherical shell subjected to axial and transverse ring forces as the external fluid is air. Only breathing modes (n ¼ 0) are excited by axial ring force, whereas only beam modes (bending modes, n ¼ 1) are considered for transverse ring force. For two forces, curves of three methods are almost overlapped and some slight discrepancies appear at resonant peaks. As it is known, amplitudes of resonant peaks are mainly controlled by structure damping and the difference between natural and exciting frequencies. On one hand, different methods lead to small dis­ crepancies of natural frequencies. On the other hand, structure damping is not considered by Peters et al. (2014). These reasons lead to negligible discrepancies of amplitudes of resonant peaks. As the combined spherical-cylindrical-spherical shell is submerged in water, the radiated sound power of the combined shell subjected to transverse ring force and axial point force is shown in Fig. 8. For transverse ring force, the curve of present method highly coincides with the one of the reference (Peters et al., 2014). For axial point force, the truncated circumferential mode number of present method is 10, which satisfies the requirement of convergence. At low frequencies, three curves are consistent. At high frequencies, although apparent differ­ ences occur at some peaks, especially two peaks of the reference in 90–100 Hz, the tendency of three curves is similar. The reasons are similar with the ones of light acoustic medium. In addition, the heavy acoustic medium decreases natural frequencies of the combined shell and more modes appear in the same frequency range (1–100 Hz). Since some modes may not be excited by using different methods, the discrepancy of three curves will occur at some frequencies. On the whole, results of present method agree well with the reference and FEM/BEM, and it is concluded that present method can predict accurate vibro-acoustic responses of combined spherical-cylindrical-spherical shells. 3.2.2. Combined conical-cylindrical-spherical shells Vibro-acoustic responses of a combined conical-cylindrical-spherical shell, which was studied by Jin et al. (2018), are discussed. The radius and length of cylindrical shell are 0.6 m and 4.5 m. For the conical shell, radii at two ends are 0.2 m and 0.6 m, and the semi-vertex angle is 15� .

Fig. 15. Sound pressure contributed from different compartments of the combined conical-cylindrical-spherical shell subjected to axial force: (a) FieldA (b) FieldB. 12

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Ocean Engineering 189 (2019) 106345

Fig. 16. Surface sound pressure of the combined conical-cylindrical-spherical shell.

resonant peaks occur for radial excitation, far-field sound pressure radiated from the combined shell excited by axial force is closely equal to the one of radial force at some frequencies, such as the resonant peak around 60 Hz. As a result, far-field sound pressure generated by axial force should be taken into account. Fig. 14 and Fig. 15 compare sound pressure contributed by different compartments as the combined conical-cylindrical-spherical shell sub­ jected to radial and axial forces, respectively. For both forces, contri­ butions of the cylindrical shell are dominant whereas the ones of the conical shell are insignificant. On the other hand, contributions of the spherical shell are non-ignorable, especially for the far-field sound pressure as the combined shell subjected to axial force at high fre­ quencies. Additionally, amplitudes of cylindrical and spherical shells are larger than the total ones at some frequencies, which is attributed to the phase difference of different compartments of the combined shell. To more visually illustrate contributions of different compartments, surface sound pressure of the combined shell is shown in Fig. 16. At 4.5 Hz and 54.5 Hz, no matter what the excitation is, surface sound pressure of the overall shell is similar and the one of the spherical shell is obvious. At 78.5 Hz, surface sound pressure of the combined shell under two exci­ tations is apparently different. The surface sound pressure of spherical shell is obvious as the combined shell excited by axial force, whereas the surface sound pressure of spherical shell is negligible for radial force, which further states that the spherical shell significantly contributes to radiated sound pressure of the combined conical-cylindrical-spherical shell.

20 log10 ðD0 =10 12 Þ and 20 log10 ðPa =10 6 Þ in the following analysis, and D0 and Pa denote amplitudes of displacement and sound pressure. Fig. 9 compares radial displacements and radiated sound pressure of different methods. For the radial displacement of driving point (PointA), results in the reference are also shown. It is found that the curve of FEM/ BEM agrees well with the one of Jin et al. (2018) in 1–70 Hz, and some small discrepancies occur in 70–100 Hz. The discrepancies mainly result from differences of natural frequencies of two methods. Correspond­ ingly, it can be concluded that results of FEM/BEM are reliable. Comparing displacements and sound pressure of present method and FEM/BEM, it is clearly observed that results of present method are in excellent agreement with the ones of FEM/BEM excluding some shifts of resonant peaks at high frequencies. In Fig. 10, surface displacements of the combined shell are shown for some resonant peaks, which are labeled in Fig. 9(b). Although frequencies of resonant peaks are different, surface displacements of two methods are almost identical. To further discuss the validity of present method, directivity patterns about radiated sound pressure of some frequencies are plotted in Fig. 11. Field points of the directivity pattern are located at the plane of mid cross-section of cylindrical shell, and the distance between the field points and the axis of the combined shell is 1 m. Four sub-figures correspond to the first four resonant peaks in Fig. 9(c). In the figure, curves of present method agree well with the ones of FEM/BEM, and the discrepancy results from small differences of amplitudes of resonant peaks, as shown in Fig. 9(c). Besides, the directivity pattern of four subfigures almost corresponds to the circumferential mode number 1, 2, 3 and 4, respectively. To further discuss the phenomenon, contributions of different circumferential modes to the radiated sound pressure are shown in Fig. 12. For FieldA, the first four circumferential mode numbers (1, 2, 3 and 4) are mainly contributed to the first four resonant peaks. For sound pressure of far-field point (FieldB), contributions of n ¼ 1 are dominant and other modes are negligible, which illustrates the reason why the resonant peaks of FieldB are obviously less. Further­ more, it can be deduced that beam modes (n ¼ 1) may be more efficient to generate far-field radiated sound pressure. Fig. 13 compares radiated sound pressure of the combined conicalcylindrical-spherical shell subjected to radial and axial forces. Although sound pressure radiated from the combined shell excited by radial force is obviously greater than the one of axial force and more

4. Conclusions A semi-analytic method is developed to predict vibro-acoustic re­ sponses of arbitrary shells of revolution. For the shell, it is firstly divided to some narrow shell segments in axial direction and all segments are treated as conical shells. By utilizing Flügge shell theory and expanding displacements as power series, four displacements and four forces at any cross-section of conical shells are expressed in terms of 8 unknown co­ efficients. For acoustic pressure of external fluid, the surface Helmholtz integral equation is reduced to the line integral along the generator line through expanding pressure and velocity as Fourier series. After mesh­ ing the generator line into several 3-node isoparametric elements and 13

K. Xie et al.

Ocean Engineering 189 (2019) 106345

considering relationships between displacements and velocities, the acoustic pressure can be expressed as unknown coefficients of dis­ placements of all segments. To obtain the structure-acoustic coupling system, continuity conditions between adjacent segments, which include modifications introduced by acoustic pressure, and elastic boundary conditions are established, and they can readily be assembled to the finial vibro-acoustic governing equation of submerged shells of revolution. To test the accuracy and convergence of present method, natural frequencies of submerged cylindrical, conical and spherical shells and vibro-acoustic responses of combined spherical-cylindricalspherical and conical-cylindrical-spherical shells are listed and suffi­ ciently compared with the ones in appropriate references or calculated by FEM/BEM, which demonstrates rapid convergence and high accuracy of the semi-analytic model. For combined conical-cylindrical-spherical shells, beam modes are more efficient to generate far-field radiated sound pressure. In addition, the spherical shell significantly contributes to far-field sound pressure as the combined shell subjected to axial forces.

As compared with theoretical models in references, present model is of wider application for vibro-acoustic analysis of shells of revolution with arbitrary shapes, and both independent and combined shells of revolution are included. In contrast to traditional finite elementboundary element method, the efficiency of present method is much higher and it will be more obvious for shells with straight generator line. Although ring stiffeners and bulkheads are not considered, the model can be easily extended to include those stiffening members only if modeling bulkheads and rings with rectangular, L and T cross-sections as combinations of annular plates with different semi-vertex angles and radii. Acknowledgements This work is supported by Fundamental Research Funds for the Central Universities of China (Grant No. 2662018QD032) and National Natural Science Foundation of China (Grand No. 51779098).

Appendix C. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.oceaneng.2019.106345. Appendix A Differential operators in Eq. (1) are s2 s ∂ ∂ 1 υ ∂2 1 υ h2 c2 ∂2 þ þ þ 2þ 2 2 2 R ∂x ∂x R 2R ∂θ 2R2 12 R2 ∂θ2

L11 ¼

L12 ¼

1 þ υ ∂2 2R ∂x∂θ

3

L13 ¼

υc ∂ R ∂x

h2 sc3 12 R4

L21 ¼

1 þ υ ∂2 3 υ s ∂ þ 2R ∂θ∂x 2 R2 ∂θ

L22 ¼

L23 ¼

sc R2

1 ∂2 R2 ∂θ2

sc R2

υÞs2 2

h2 c ∂3 þ 12 R ∂x3 L32 ¼

c ∂ R2 ∂ θ

L33 ¼

c2 R2

υ2 Þ ∂2 E ∂t2

h2 s2 c ∂ 12R3 ∂x

þ

h2 c ∂3 1 υ h2 c ∂3 þ 3 12 R ∂x 2R3 12 ∂x∂θ2

3

υ h2 sc ∂2 2 12 R4 ∂θ2

ð1

h2 sc 1 þ υ ∂2 12 R4 2 ∂θ2

υÞ

∂2 ∂x∂θ

υÞ

(A.5)

h2 c 3 υ ∂3 12 R2 2 ∂x2 ∂θ

(A.6)

h2 sc3 12 R4

(A.7)

h2 c ð1 υÞ ∂3 2 12 R3 ∂x∂θ2

h2 sc ð3 þ υÞ ∂2 h2 s2 c 3 þ υ ∂ h2 c 3 υ ∂3 þ þ 2 12 R3 ∂x∂θ 12 R4 2 ∂θ 12 R2 2 ∂x2 ∂θ c2 þ 2

(A.3) (A.4)

h2 2s3 c h2 s2 c ∂ þ 12 R4 12 R3 ∂x

� h2 c2 2 12 R4

(A.1) (A.2)

h2 ð1 υÞs2 c ∂ h2 sc 3 ð1 þ 4 ∂θ 12 R3 2 R 8

υc ∂ R ∂x

ρð1

υ s ∂ R2 ∂θ

υÞs ∂ 1 υ ∂2 h2 s2 c2 3 ð1 þ þ 2R ∂x 2 ∂x2 12 R4 2 2R � h2 ð1 υÞsc2 ∂ h2 c2 3 ∂2 ρð1 υ2 ∂2 ð1 υÞ 2 þ 3 2 ∂x 12 R 2 E 8 R ∂x ∂t2 ð1

c ∂ R2 ∂θ

L31 ¼

2

h 2 s2 c 2 12 R4

∂2 ∂θ 2

�

h2 4 r 12

ρð1 E

(A.8)

υ2 Þ ∂2 ∂t 2

(A.9)

where s ¼ sin α, c ¼ cos α, r4 ¼ r2 r2 ; r2 ¼ ∂2 =∂x2 þ ðs=RÞ∂=∂x þ ð1=R2 Þ∂2 =∂θ2 . E, ρ, h and υ denote the Young’s modulus, density, thickness and Poisson’s ratio. R represents the radius at x. Expressions of forces of conical shells are (Leissa, 1993) � � � � Eh ∂u 1 ∂v s c h2 c ∂2 w N¼ (A.10) þυ þ uþ w 2 2 1 υ ∂x R ∂θ R R 12 R ∂x

14

K. Xie et al.

M¼

Ocean Engineering 189 (2019) 106345

Eh3 12ð1 υ2 Þ

�

�

∂2 w c ∂v þυ 2 R ∂θ ∂x2

1 ∂2 w R2 ∂θ2

� � s ∂w c ∂u þ R ∂x R ∂x

(A.11)

� � Mxθ Nxθ þ R

T¼

S ¼ Qx þ

(A.12)

1 ∂Mxθ R ∂θ

(A.13)

where Nx θ ¼

� Eh 1 ∂u ∂v þ 2ð1 þ υÞ R ∂θ ∂x

Mθ ¼

Eh3 12ð1 υ2 Þ

Mxθ ¼

Eh3 12ð1 þ υÞ

Mθx ¼

Eh3 24ð1 þ υÞ

Qx ¼

�

�

�

s h2 vþ R 12

1 ∂2 w R2 ∂θ2

�

c ∂2 w c2 ∂v sc ∂w þ þ R2 ∂θ∂x R2 ∂x R3 ∂θ

∂2 w ∂x2

sc u R2

1 ∂2 w c ∂v s ∂w þ þ R ∂x∂θ R ∂x R2 ∂θ

sc v R2

2 ∂2 w 2s ∂w c ∂v þ þ R ∂x∂θ R2 ∂θ R ∂x

c ∂u R2 ∂θ

1 ∂ðRMÞ 1 ∂ðMθx Þ þ R ∂x R ∂θ

s ∂w R ∂x

υ

c2 w R2

sc2 v R3

�� (A.14)

� (A.15)

� (A.16) � sc v R2

(A.17)

s Mθ R

(A.18)

Appendix B Recurrence formulas about unknown coefficients of symmetric and antisymmetric modes are uniformly given as 6 X

an;mþ2 ¼

4 X

An;ai an;m

5þi

i¼1

3þi

4 X

Cn;ai cn;m

þ

i¼1

bn;mþ2 ¼ �

3þi

5þi

4þi

i¼1

i¼1

(B.1)

m�0

9 X

Bn;ci bn;m

�

3þi

i¼1

6 X

An;ci an;m

Cn;bi cn;m

�

i¼1

7 X

m�0

5 X

Bn;bi bn;m

þ

i¼1

3þi

i¼1

6 X

An;bi an;m

cn;mþ4 ¼

6 X

Bn;ai bn;m

�

4þi

Cn;ci cn;m

þ

6þi

m�0

i¼1

where the superscript ‘C’ or ‘S’ indicating different modes is omitted. The minus sign is appropriate for the antisymmetric modes. Detailed expressions of coefficients An;ai , Bn;ai , Cn;ai , An;bi , Bn;bi , Cn;bi , An;ci , Bn;ci and Cn;ci can be referred in the previous paper of authors (Xie et al., 2015). From above recurrence formulas, relationships of matrices DCn , DSn , FCn and FSn are

(B.2)

DSn ¼ TSC ∘DCn ; FSn ¼ TSC ∘FCn where ‘∘’ represents Hadamard product, and matrix TSC is 2 3 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 17 7 TSC ¼ 6 4 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1

(B.3)

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