An Analytical Solution for the Vibration and Far-Field Sound Radiation Analysis of Finite, Semisubmerged Cylindrical Shells

Guo, Wenjie; Yang, Zhou; Han, Yueyang

doi:https://doi.org/10.1155/2021/9996960

Shock and Vibration

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 9996960 | https://doi.org/10.1155/2021/9996960

An Analytical Solution for the Vibration and Far-Field Sound Radiation Analysis of Finite, Semisubmerged Cylindrical Shells

Wenjie Guo,¹Zhou Yang,¹and Yueyang Han^2,3

Academic Editor: Abdullah Seçgin

Received29 Mar 2021

Revised15 Aug 2021

Accepted02 Sept 2021

Published21 Sept 2021

Abstract

The vibration response and far-field sound radiation of a semisubmerged, finite cylindrical shell with low-frequency excitation are studied. The solution to this problem can be divided into two steps. The first step is to apply the wave propagation approach to determine the vibration response of the cylindrical shell. In the cylindrical coordinate system, the Flügge shell equations and Laplace equation are used to describe the cylindrical shell and surrounding fluid so that the vibration responses of the shell can be addressed analytically. The fluid free surface effect is taken into account by applying the sine series to force the velocity potential on the free surface to be zero. Furthermore, compared with the FEM (the finite element method), the present method is not only reliable but also effective. In the second step, the far-field sound radiation is solved by the Fourier transform technique and the stationary phase method in accordance with the vibration responses of the shell from the previous step. The boundary element method is applied to validate the reliability of the acoustical radiation calculation. The circumferential directivity of far-field sound pressure is discussed, and it is found that the maximum value of the sound pressure always appears directly under the structure when the driving frequencies are relatively low. Besides, in consideration of simplicity and less computation effort, the present method can be used for the rapid prediction of the vibration and far-field sound pressure of a semisubmerged cylindrical shell with low-frequency excitation.

1. Introduction

Vibroacoustical characteristics of cylindrical shells are the subject studied extensively due to their widespread use in many industrial applications, such as underwater vehicles, aircrafts, and pipelines. There are many researchers [1, 2] who have studied the shell and plate vibration problems. And with respect to the coupling effect of acoustics and structure, previous works [3–11] focused mainly on shells that were in complete contact with a fluid. So, the influence of a free surface on the vibration and sound radiation of the cylindrical shell tended to be neglected. Obviously, as far as the structure located close to the water surface or afloat on the water surface is concerned, it is unreasonable to consider the fluid region around the cylindrical shell unbounded. Cylindrical shells partially coupled with the fluid (the shell centerline is parallel to the free surface) are frequently encountered in ocean engineering, such as ships or submarines floating on the sea.

The coupling dynamic behavior, in particular the free vibration behavior, has attracted a lot of attention from researchers. In order to study the vibration of a finite cylindrical shell in a partially liquid-filled state, Amabili [10] applied appropriate boundary conditions on the free surface. Afterward, using the above method, Amabili [11] further studied the vibration of a cylindrical shell in a partially immersed state. For the purpose of solving the problem about the vibration of a partially liquid-filled and submerged finite cylindrical shell, Ergin [12] combined the boundary integral method and the image method to study it for the first time. Lakis et al. [13, 14] made the element division in the circumferential direction of the finite cylindrical shell and found that the energy method is very suitable for the analysis of the vibration and sloshing of the finite cylindrical shell in a partially liquid-filled state. Li et al. [15] adopted the wave propagation method and Fourier transform technique to study the vibroacoustic characteristics of a semisubmerged infinite cylindrical shell. Guo et al. [16] investigated the free and forced vibration of a semisubmerged finite cylindrical shell by applying the Rayleigh–Ritz method.

The sound radiation from structures partially coupled with fluid has received growing attention. Wu et al. [17, 18] studied the vibroacoustic characteristics of a fluid-filled cylindrical shell semisubmerged in a coaxial flow with the Fourier transform technique, and the effects of parameters of the cylindrical shell on the vibration were discussed. Zhou and Joseph [19] employed the finite element method (FEM) with the boundary element method (BEM) to analyze the vibration and sound radiation from an underwater ribbed cylindrical shell. Brunner et al. [20, 21] proposed a fast multipole BEM coupled with the FEM to investigate the vibroacoustic behavior of a partly immersed finite cylindrical shell and a ship model. Jouaillec and Jacquart [22] adopted the FEM and boundary integral method to study the sound radiation from a point-excited cylindrical shell partially immersed in the fluid. Salaün [23] applied the Fourier series expansion method to study the far-field sound radiation of a semisubmerged cylindrical shell at relatively high driving frequencies.

Generally speaking, there is a lack of an accurate and efficient method that can be used not only in the vibration response study of semisubmerged finite cylindrical shells but also in the far-field acoustic radiation analysis of ones, to the authors’ knowledge. Thus, this analytical method is used to deal with such problems in this paper. It is necessary to split the problem into two parts before applying the analytical method to calculation. In this way, the accuracy can be guaranteed, and the calculation efficiency can be improved. First of all, the vibrational response of the finite cylindrical shell must be obtained: it is a good idea to use the wave propagation method to deduce the fluid-structure coupling vibration equations, considering that the fluid is mainly distributed in the axial interval of the finite cylindrical shell. Secondly, the sound radiation of the finite cylindrical shell must be obtained. In this step, it would be well if there was a semi-infinite cylindrical acoustical baffle on each side of the finite cylindrical shell, combining the Fourier transform with the stationary phase method， the far-field sound pressure of the finite cylindrical shell could be further gained. In order to make the far-field acoustic radiation easier to get across, the circumferential directivity of far-field sound pressure is discussed.

2. Theoretical Analysis of the Shell Vibration

2.1. The Motion Equations of the Cylindrical Shell

There is a simply supported finite thin circular cylindrical shell in Figure 1, whose length, thickness, radius, Young’s modulus, Poisson’s ratio, and density are, respectively, L, h, R, E, μ, and ρ. Meanwhile, this cylindrical shell model submerges in such a fluid, whose density and sound velocity are, respectively, ρ_f and c_f.

The hypothesis of this paper is a semi-infinite fluid domain. That is to say, there is a free surface that divides the whole space into two parts: the upper part is a vacuum and the lower part is the fluid domain. Since the cylindrical coordinate system (r, x, φ) is very convenient to solve and calculate in this paper, this coordinate system is adopted, whose origin is point O. Furthermore, the analytical solution for the vibration of the cylindrical shell can be obtained by using this coordinate system. Besides, at the point (R, x₀, φ₀) of the cylindrical shell, there is a harmonic point force f_r, whose amplitude and driving frequency are separately F₀ and f₀.

The motion state of the cylindrical shell can be represented by the Flügge shell equations [24]:

The meaning of each character in the formula is as follows: U represents the axial displacement of the shell, V represents the tangential displacement, W represents the radial displacement, and [L] denotes the classical Flügge differential operator for the thin shell theory:

The meaning of f_p is the fluid pressure on the cylindrical shell, which can be defined aswhere p is the fluid pressure on the outer surface of the cylindrical shell.

Because the motion of the cylindrical shell is a kind of harmonic vibration, the axial displacement, the tangential displacement, the radial displacement, and the fluid pressure can be separately written as follows:

The meaning of each character in the aforementioned formulas is as follows: U_mn represents the axial displacement amplitude, V_mn represents the tangential displacement amplitude, and W_mn represents the radial displacement amplitude. F_mn refers to the amplitudes of harmonic point force f_r, and f_mn refers to the amplitudes of the fluid pressure f_p on a cylindrical shell. As for m and n, they separately represent the expansion coefficients in the axial and the circumferential directions. Besides, the circular frequency is expressed with ω, and the axial wavenumber is expressed with k_m, and k_m = mπ/L [25] and .

Substituting equation (4) into equation (1), the following equations can be further gained after the orthogonalization:

Matrix [T] contains multiple elements, whose expressions are as follows:

From the perspective of dimension, Ω belongs to a nondimensional frequency，which is only related to the ring frequency of the cylindrical shell, and

It is not difficult to obtain the vibration equation about radial displacement by solving equation (5).where .

The expression for f_r is as follows:

The meaning of each character in the aforementioned formula is as follows: F₀ is the maximum value of the harmonic point force f_r, δ( ) is the delta function, and (x_0,φ₀) is the position of f_r.

Substituting equation (8) into equation (4), F_mn can be further gained after the orthogonalization:

2.2. The Motion Equations of the Fluid

Combined with the incompressibility of the fluid and velocity potential function , a function meeting the fluid Laplace equation, the characteristics of the fluid field can be reflected.

Consider that the liquid at infinity is still; that is,

At the free surface, both the gravity effect and the surface tension of the fluid could be neglected [26]. Therefore, the fluid boundary condition at the free surface can be expressed as

The separation of variables method can be employed to solve the Laplace equation [10], and the velocity potential function which meets all the conditions could be expressed as follows:where denotes the amplitude of the velocity potential function and K_a( ) is the ath-order modified Bessel function of the second kind. Besides, sound pressure should be equal to 0 for an arbitrary point on the free surface. Thus, the velocity potential would be equivalent to zero on the free surface. That is why we choose the sine series.

2.3. The Velocity Continuity Condition

Since the radial velocity on the outer surface of the cylindrical shell should be equal to that of the fluid,

By conducting orthogonal processing and integrating about equation (14), one can getwhere is a weight function, and equation (15) is a typical Galerkin integral equation.

Furthermore, the amplitude of the velocity potential function can be deduced from equation (15):where

The load p which the fluid imposed on the cylindrical shell can be calculated by Bernoulli’s equation:

By substituting equations (13) and (17) into equation (18), the fluid pressure p is expressed as

By applying orthogonal processing about equation (3), one can get

Thus, the amplitude of the load on the shell’s surface imposed by the fluid pressure can be obtained after solving equation (20).where .

By substituting equations (9) and (21) into (5), the coupling equation of the system can be expressed aswhere .

Since there exist infinite series in equation (22), a truncation process is required to make the solution possible. The truncated numbers of a, m, and n are K, M, and N, where N represents the truncated numbers of N and B, and n = b.

Then, equation (22) can be rewritten aswhere .

For the free vibration analysis, there is no exciting force, and the above problem changes into the solution of eigenvalues.

3. Theoretical Analysis of the Far-Field Sound Radiation

In this section, the far-field sound radiation of the semisubmerged finite shell will be derived. In Figure 2, point P is a far-field observation point in the acoustic field. In the figure, the length of line segment OP is defined as R₀, the angle formed by rotating counterclockwise from the vector OP to the positive half of the x-axis is defined as θ， and the vector OP is written as . It is obvious that the coordinate system (R₀, θ, φ) is a typical spherical coordinate system.

In the x-direction, the fluid is free and unbounded, which would bring about such a problem that the domain of the sound field is incompatible with that of the shell’s surface velocity. In order to avoid this problem, we have processed the cylindrical shell properly, that is, setting up a semi-infinite rigid column barrier [26] at each end of the shell, respectively. Therefore, the radial displacement of the shell is rewritten as

When dealing with the acoustic radiation problem, the fluid is considered as an acoustic medium, and it is compressible. Thus, the sound pressure in the fluid field should satisfy the Helmholtz equation:where is the compression wavenumber, and c_f is the sound speed in the fluid.

Since the free surface exists, the sound pressure release boundary condition on the free surface should be met:

Therefore, the sound pressure expression, which satisfies the sound pressure release boundary, can be expressed as

In order to solve the sound-structural coupling model, the Fourier transform and Fourier inverse transform are introduced as follows:

The meaning of each character in the aforementioned formulas is as follows: refers to the function of f(x), k_x refers to the wavenumber , and they are all in the wavenumber domain.

By substituting equation (27) into equation (25) and conducting the Fourier transform technique, equation (25) is further changed to

Then, the expression of can be acquired by applying the separation of variables method:where is the sound pressure amplitude in the wavenumber domain, is the ath-order Hankel function of the first kind, and is the radial wavenumber.

Afterward, the sound pressure in the wavenumber domain can be expressed as

Similarly, combined with the Fourier transform technique, the radial displacement can be converted into the wavenumber domain:

Accordingly, the expression of the radial displacement is as follows:where = is the displacement amplitude in the wavenumber domain.

At the interface of a cylindrical shell and external fluid, they can be regarded as continuous, so it is not difficult to gain the following equation:

Substituting equations (31) and (33) into equation (34), the amplitude of sound pressure can be further gained after the orthogonalization:where

Based on the principle of the inverse Fourier transform, the expression of sound pressure can be obtained as follows:

According to the theory of far-field acoustics, the Hankel function can be approximately written as [22]where x = k_rr, and when r goes to infinity, x also goes to infinity.

Substituting equation (38) into equation (37), the sound pressure can be further written as

Converting equation (39) into the expression under the spherical coordinate system (R₀, r, x), where the conversion relationship is r = R₀sinθ and x = R₀cosθ, the sound pressure could be rewritten as follows:

The far-field sound pressure is examined in this study, and in the acoustic field [27] and electromagnetic field [28], it is often necessary to calculate the infinite integral, which could lead to a large amount of calculation and inconvenience. In order to solve this problem well, a common approach is to introduce the stationary phase method, which can transform the infinite integral problem into a linear operation one. Therefore, the solution of equation (40) is relatively easier.

Two related functions are given here to explain the method—the stationary phase method:

According to the aforementioned formulas, the infinite integral in equation (40) is further changed into

Then, the stationary phase method [27] is used to process equation (40), and the result iswhere k_s refers to the stationary phase point.

Actually, k_s is the solution of equation (43):

Therefore, the stationary phase point k_s = k_fcosθ, and equation (43) is further changed into

Thus, the expression of the far-field sound pressure in the spherical coordinate system is obtained aswhere = .

It should be mentioned that when the stationary phase point k_s = k_fcosθ is close to the axial wavenumber k_m, the expression of is also convergent; the reason is as follows:

To be clear, the calculation formula of the sound pressure level is SPL = 20log10(|p|/p₀), and the value of P₀ is 10⁻⁶ Pa.

4. Numerical Examples of the Shell Vibration

The relevant calculation parameters used in this paper are as follows: for the cylindrical shell, its elastic modulus E is 206 GPa, Poisson’s ratio μ is 0.3, density ρ is 7850 kg/m³, length L is 1.284 m, radius R is 0.18 m, and thickness h is 0.003 m; for the harmonic point force, its amplitude F₀ is 1 N; for the fluid, its density ρ_f is 1025 kg/m³, and sound velocity c_f is 1500 m/s. In this paper, the complex elastic modulus E_c is used instead of the elastic modulus E for calculation. The relationship between them is as follows: E_c = E (1 + iη), where η is the damping coefficient of the structure, and its value is 0.01.

4.1. The Convergence Analysis of the Radial Velocity

In Section 2, in order to calculate the infinite series in equation (23), we introduced the fixed-phase method, which is an approximate algorithm, so it is necessary to prove its convergence. The main proof process is as follows: if M, K, and N are equal, then the quadratic velocity V_m of the cylindrical shell can be further calculated, and its expression is , where the meaning of s is the surface area of the cylindrical shell. The calculation results of V_m are shown in Figure 3. The quadratic velocity level is defined as QVL = 10log10(V_m/V₀) (unit: dB), where V₀ = 10⁻¹² m²/s².

(a)

(b)

(c)

It is not hard to see in the figure that when N is close to 12, V_m begins to converge, so it is reasonable to take N = 14 in this paper.

4.2. The Verification of the Present Method for Vibration Analysis

In this section, we will continue to demonstrate the reliability of the present method. In finite element software MSC Patran/Nastran, a cylindrical shell model with the external fluid is established (see Figure 4), and the model is simplified. For example, in MSC Patran/Nastran, the cylindrical shell is regarded as the shell element, and the virtual added mass matrix [29] in the virtual mass method is used to simulate the influence of the fluid on the free vibration and forced vibration of the cylindrical shell. In addition, the model is divided into 40 equal parts in both axial and tangential directions, with a total of 1600 elements, all of which are quadrilateral.

4.2.1. Free Vibration

In order to illustrate the calculation accuracy of the present method, the natural frequencies and mode shapes obtained by the present method are compared with those obtained by the FEM, respectively (see Table 1 and Figure 5), where f_a and f_b represent the natural frequencies obtained by the present method and the FEM, respectively. The relative error of the natural frequencies between both methods is defined as Err1 = |f_a − f_b|/f_b.

It is easy to see that the method is reliable and accurate from the comparison results of Figure 5 and Table 1. For the natural frequencies of the first ten orders, the relative errors are less than 1%, where mode number (i, j) indicates the axial mode number and circumferential mode number in Table 1, respectively.

In order to further demonstrate the accuracy and reliability of the present method, the natural frequencies calculated with the present method are compared with those in [11]. The material parameters in [11] are listed as follows: E = 206 GPa, μ = 0.3, ρ = 7680 kg/m³, and the dimensions of the shell are L = 0.664 m, R = 0.175 m, and thickness h = 0.001 m. The density of the fluid is ρ_f = 1000 kg/m³. The relative error of the natural frequencies obtained by the two methods is defined as Err2 = |f_pre–f_Ref|/f_Ref, where f_pre refers to the natural frequencies obtained by the present method and f_Ref represents the natural frequencies in [11].

In Table 2, the differences between results of the present method and [11] are relatively small (less than 3%), proving that the present method for the free vibration is accurate and reliable.

4.2.2. Forced Vibration

By using the same model parameters as those in Table 1, the point excitation is exerted at (R, L/2, −π/2), and then the dynamic responses at (R, L/4, −π/2) and (R, L/4, −π/2) can be obtained. They are evaluated by the radial displacement level, whose calculation formula is RDL = 20log10, where the value of is 10⁻¹²m.

The comparison between the finite element simulation results about RDL and the theoretical calculation ones is shown in Figure 6.

(a)

(b)

As shown in Figure 6, it is obvious that this method is reliable, and its calculation results are in good agreement with the finite element ones. More importantly, the computational efficiency of the present method is much higher than that of the finite element method. The former is about 20 times faster than the latter.

It is noteworthy that this method can solve the problem of low frequency very well. When the frequency of simple harmonic point force is relatively high (more than 500 Hz), it is not so suitable. The reason for this phenomenon lies in the following: in software Nastran, the fluid is regarded as virtual mass and infinite in the axial direction of the cylindrical shell; however, when the problem is solved analytically, the fluid is limited in the axial direction of the cylindrical shell; besides, the calculation accuracy of the analytical method depends on the frequency of the harmonic point force.

5. Numerical Analysis of the Far-Field Sound Radiation

5.1. The Verification of the Accuracy of the Stationary Phase Method

It can be seen from the first section that the solution content of this paper is mainly vibration response and acoustic radiation. Since the vibration response has been calculated in the previous sections, the task of this section is to solve the sound radiation and compare this result with that of the half-space boundary element method (BEM) [26].

The excitation point is located at (R, L/2, π/2), and its maximum value F₀ is 1 N. When R₀ = 1000 m, θ = π/4, and φ = 0∼π, the sound pressure level calculated by the present method and that calculated by the half-space boundary element method are calculated, respectively, and the differences between them are compared (see Figure 7).

(a)

(b)

(c)

(d)

As shown in the figure, the results gained by the two methods are very consistent, and the calculation process of the present method (the stationary phase method) is actually a linear calculation process, so this method is not only reliable but also efficient. Under the same conditions, the computational speed of the present method is 40 times faster than that of the half-space BEM.

Besides, since the driven frequencies are relatively low, the directivity of far-field sound pressure in the circumferential direction is monotonous. Moreover, it can be seen that the maximum value of the sound pressure in the circumferential direction is at the angle of the excitation position (φ₀ = π/2) in Figure 7.

5.2. Analysis of the Circumferential Directivity of the Far-Field Sound Radiation

In order to learn more about the relationship between the circumferential directivity and the excitation position, we discussed the circumferential directivity at different excitation position angles φ₀. In particular, φ₀ is set to be π/2, π/4, 0, −π/4, and −π/2; when R₀ = 1000 m, θ = π/3, and φ = 0∼π, the circumferential directivity of the sound pressure is studied at a frequency of 350 Hz.

It is found in Figure 8 that the maximum value of the sound pressure on the circumferential direction is located at the bottom, and it is independent of the excitation position. The physical explanation for the directivity is given below.

The sound pressure mentioned in this paper is that at infinity, and the distance from any point of the cylindrical shell to the sound pressure point studied is the same, so the cylindrical shell can be replaced by a point source.

According to the image method [30, 31], another equidistant (the distance between the point source and free surface) and opposite phase source is applied to force the sound pressure on the free surface to be zero. In fact, the above physical model is called a dipole model.

Furthermore, it is not difficult to deduce the sound pressure of the dipole model [27]:where the meaning of A is the maximum value of the sound pressure of the point source, r is the distance between the point source and the sound pressure point studied, and D represents the distance between the source and the image source (D/R < 2).

In fact, the exact value of D is unknown, but it does not affect our analysis of sound pressure directivity. This is because the driving frequency is relatively low, and the product of k_f and D is less than π/2 (k_fD < π/2); thus, when sinφsinθ takes a maximum value, the sound pressure also does.

Therefore, this is a good explanation for why the extreme value of sound pressure always appears at φ = π/2 and why the circumferential directivity diagram has good symmetry, although the velocity distribution is asymmetrical.

Besides, from equation (48), the sound pressure will also reach a maximum value obviously when θ = π/2. For the purpose of illustrating the viewpoint, we calculated the sound pressure at different angles θ when the excitation position angle φ₀ is equal to π/4. In particular, θ is set to be π/8, π/4, and π/2; when R₀ = 1000 m and φ = 0∼π, the circumferential directivity of the sound pressure is studied at the same driven frequency (350 Hz). Figure 9 presents the circumferential directivity of the sound pressure with different angles θ. From Figure 9, it is seen that the sound pressures increase with the increase of the angle θ. This further proves that the dipole model is suitable to explain the law of directivity.

6. Conclusion

An approximate analytical method is proposed to study the vibration and far-field acoustic radiation of a semisubmerged, finite cylindrical shell with low-frequency excitation. In order to improve the efficiency of the solving process, the solution to this problem can be divided into two parts. After comparing the results of the present method with those of the FEM and BEM, this method is very reliable and can be used to calculate far-field sound pressure with high efficiency.

Besides, the results show the following sound pressure characteristics. The circumferential directivity diagram has good symmetry, although the velocity distribution is asymmetrical. Furthermore, the circumferential directivity of far-field sound pressure is independent of the excitation position, and the maximum value of the sound pressure always appears at the angle φ equal to π/2 or the angle θ equal to π/2 (directly under the structure). In addition, the sound pressures increase with the increase of the angle θ.

Data Availability

All the data included in this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors wish to express their gratitude to the Youth Science Foundation of Jiangxi Province (Contract no. 20202BABL214049) and the Youth Science Foundation of Jiangxi Provincial Education Department (Contract no. GJJ190339).

References

Ö. Civalek and A. K. Baltacıoglu, “Free vibration analysis of laminated and FGM composite annular sector plates,” Composites Part B: Engineering, vol. 157, pp. 182–194, 2019.
View at: Publisher Site | Google Scholar
O. Civalek and M. Ülker, “Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates,” Structural Engineering & Mechanics, vol. 17, no. 1, pp. 1–14, 2004.
View at: Publisher Site | Google Scholar
P. R. Stepanishen, “Modal coupling in the vibration of fluid‐loaded cylindrical shells,” Journal of the Acoustical Society of America, vol. 71, no. 4, pp. 813–823, 1982.
View at: Publisher Site | Google Scholar
W. Williams, N. G. Parke, D. A. Moran, and C. H. Sherman, “Acoustic radiation from a finite cylinder,” Journal of the Acoustical Society of America, vol. 36, no. 12, pp. 2316–2322, 1964.
View at: Publisher Site | Google Scholar
H. Darvish Gohari, M. Zarastvand, and R. Talebitooti, “Acoustic performance prediction of a multilayered finite cylinder equipped with porous foam media,” Journal of Vibration and Control, vol. 26, no. 11-12, pp. 899–912, 2020.
View at: Publisher Site | Google Scholar
X. M. Zhang, “Frequency analysis of submerged cylindrical shells with the wave propagation approach,” International Journal of Mechanical Sciences, vol. 44, no. 7, pp. 1259–1273, 2002.
View at: Publisher Site | Google Scholar
R. Talebitooti, M. Zarastvand, and H. Gohari, “Investigation of power transmission across laminated composite doubly curved shell in the presence of external flow considering shear deformation shallow shell theory,” Journal of Vibration and Control, vol. 24, no. 19, pp. 4492–4504, 2018.
View at: Publisher Site | Google Scholar
J. F. M. Scott, “The free modes of propagation of an infinite fluid-loaded thin cylindrical shell,” Journal of Sound and Vibration, vol. 125, no. 2, pp. 241–280, 1988.
View at: Publisher Site | Google Scholar
M. R. Zarastvand, M. H. Asadijafari, and R. Talebitooti, “Improvement of the low-frequency sound insulation of the poroelastic aerospace constructions considering Pasternak elastic foundation,” Aerospace Science and Technology, vol. 112, Article ID 106620, 2021.
View at: Publisher Site | Google Scholar
M. Amabili, “Free vibration of partially filled, horizontal cylindrical shells,” Journal of Sound and Vibration, vol. 191, no. 5, pp. 757–780, 1996.
View at: Publisher Site | Google Scholar
M. Amabili, “Flexural vibration of cylindrical shells partially coupled with external and internal fluids,” Journal of Vibration and Acoustics, vol. 119, no. 3, pp. 476–484, 1997.
View at: Publisher Site | Google Scholar
A. Ergin, “An approximate method for the free vibration analysis of partially filled and submerged, horizontal cylindrical shells,” Journal of Sound and Vibration, vol. 207, no. 5, pp. 761–767, 1997.
View at: Publisher Site | Google Scholar
A. Selmane and A. A. Lakis, “Vibration analysis of anisotropic open cylindrical shells subjected to a flowing fluid,” Journal of Fluids and Structures, vol. 11, no. 1, pp. 111–134, 1997.
View at: Publisher Site | Google Scholar
A. A. Lakis, G. Bursuc, and M. H. Toorani, “Sloshing effect on the dynamic behavior of horizontal cylindrical shells,” Nuclear Engineering and Design, vol. 239, no. 7, pp. 1193–1206, 2009.
View at: Publisher Site | Google Scholar
T. Y. Li, P. Wang, and X. Zhu, “Prediction of far-field sound pressure of a semi-submerged cylindrical shell with low-frequency excitation,” J. Vib. Acoust., ASME, vol. 139, no. 4, Article ID 041002, 2017.
View at: Publisher Site | Google Scholar
W. J. Guo, T. Y. Li, Y. Y. Miao, and X. Zhu, “Vibration characteristics research of finite cylindrical shells semi-submerged,” Applied Mechanics and Materials, vol. 863, pp. 163–169, 2017.
View at: Publisher Site | Google Scholar
H. L. Li, C. J. Wu, and X. Q. Huang, “Parametric study on sound radiation from an infinite fluid-filled/semi-submerged cylindrical shell,” Applied Acoustics, vol. 64, no. 5, pp. 495–509, 2003.
View at: Publisher Site | Google Scholar
C. J. Wu, H. L. Chen, and X. Q. Huang, “Sound radiation from a finite fluid-filled/submerged cylindrical shell with porous material sandwich,” Journal of Sound and Vibration, vol. 238, no. 3, pp. 425–441, 2000.
View at: Publisher Site | Google Scholar
Q. Zhou and P. F. Joseph, “A numerical method for the calculation of dynamic response and acoustic radiation from an underwater structure,” Journal of Sound and Vibration, vol. 283, no. 3, pp. 853–873, 2005.
View at: Publisher Site | Google Scholar
D. Brunner, M. Junge, C. Cabos, and L. Gaul, “Vibroacoustic simulation of partly immersed bodies by a coupled fast BE‐FE approach,” Journal of the Acoustical Society of America, vol. 123, no. 5, p. 3418, 2008.
View at: Publisher Site | Google Scholar
D. Brunner, G. Of, M. Junge, O. Steinbach, and L. Gaul, “A fast BE-FE coupling scheme for partly immersed bodies,” International Journal for Numerical Methods in Engineering, vol. 81, no. 1, pp. 28–47, 2010.
View at: Publisher Site | Google Scholar
F. Jouaillec and G. Jacquart, “The acoustic radiation from a semisubmerged elastic cylinder,” Journal of the Acoustical Society of America, vol. 84, no. S1, p. S58, 1988.
View at: Publisher Site | Google Scholar
P. Salaün, “Effect of a free surface on the far‐field pressure radiated by a point‐excited cylindrical shell,” Journal of the Acoustical Society of America, vol. 90, no. 4, pp. 2173–2181, 1991.
View at: Publisher Site | Google Scholar
W. Flügge, Stress In Shells, Springer, Berlin Germany, 1973.
L. Xuebin, “Study on free vibration analysis of circular cylindrical shells using wave propagation,” Journal of Sound and Vibration, vol. 311, no. 3, pp. 667–682, 2008.
View at: Publisher Site | Google Scholar
A. Harari and B. E. Sandman, “Vibratory response of laminated cylindrical shells embedded in an acoustic fluid,” Journal of the Acoustical Society of America, vol. 60, no. 1, pp. 117–128, 1976.
View at: Publisher Site | Google Scholar
M. C. Junger, Sound, Structures and Their Interaction, M. I. T Press, Cambridge, UK, 1979.
Y. D. Li Yi-Ding, P. F. Zhang Hui, and H. Yu Miao, “Stationary phase method of calculating Čerenkov radiation spectrum of a charged particle moving in curved path,” Acta Physica Sinica, vol. 62, no. 10, Article ID 104103, 2013.
View at: Publisher Site | Google Scholar
H. S. Park, S. H. Choi, and N. S. Kim, “Identification of propeller singing phenomenon through vibration analysis of propeller blade,” in Proceedings of the Fifteenth International Offshore and Polar Engineering Conference, pp. 591–595, Seoul, Republic of Korea, June 2005.
View at: Google Scholar
W. Guo, T. Li, X. Zhu, Y. Miao, and G. Zhang, “Vibration and acoustic radiation of a finite cylindrical shell submerged at finite depth from the free surface,” Journal of Sound and Vibration, vol. 393, pp. 338–352, 2017.
View at: Publisher Site | Google Scholar
P. Wang, T.-Y. Li, X. Zhu, W.-J. Guo, and R. Nie, “An analytical solution for free flexural vibration of a thin cylindrical shell submerged in acoustic half-space bounded by a free surface,” International Journal of Structural Stability and Dynamics, vol. 18, no. 3, Article ID 1850042, 2018.
View at: Publisher Site | Google Scholar

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Copyright © 2021 Wenjie Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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