The structural-acoustic coupled vibration problem of submerged cylindrical shells, which are widely used in submarines, torpedoes, and other underwater vehicles, has received a great deal of attention. A major current focus on those studies is how to ensure a quite low level of acoustic radiation from those structures. Up till now, many researchers have attempted to layer the viscoelastic or piezoelectric material sublayer on the internal surface of the underwater structures to resist the vibration and noise. Hasheminejad and Rajabi^[1] found that the acoustic scattering caused by a plane incident wave could be cancelled when a piezoelectric layer was adhered to the internal surface of an orthotropic cylindrical shell. However, quite a high level of the energy input was usually required to obtain a better damping effect due to the tiny piezoelectric coefficients. Moreover, the reliability of this kind of treatment was rather poor due to the control spillover or breakdown voltage of the piezoelectric layer. Ruzzene et al.^{[2, 3]} treated passive constrained layer damping (i.e., PCLD) on the longitudinal ribs of a cylindrical shell to restrain the acoustic radiation, and better damping effects were achieved. Meanwhile, Saravanan et al.^[4] found the damping factors of the whole structures were proved to be enhanced when the PCLD treatment was applied. Unfortunately, the flexibility of the PCLD treatment was insufficient since they could not be self-adaptive to the working condition. Recently, some researchers^{[5, 6]} declared that the active constrained layer damping (i.e. ACLD), which replaced the constrained layer of the PCLD structure with the piezoelectric material, could achieve a better damping effect than the traditional active control (i.e., treating the piezoelectric material on the structure) or PCLD treatment due to its low energy input and being highly self-adaptive. Ro and Baz^[5] located some ACLD blocks on an elastic plate to attenuate the noise of a rectangular acoustic cavity with five rigid walls, and good damping effect was acquired. Subsequently, Ray and Balaji^[6] employed some ACLD blocks on a cylindrical panel, in order to reject the noise of the same acoustic cavity as mentioned in Ref.  [5]. However, there remains a need for further work on the acoustic problem of submerged structures with ACLD treatment, since little work was done on the acoustic characteristic of them.

By convention, the methods adopted to deal with the acoustic-structural problems are numerical methods such as the finite element method (FEM), boundary element method (BEM) or finite/boundary element method. Soize^[7] pointed out that although FEM and BEM were proved to be mature and convenient enough to operate, they still tended to be inaccurate when handling issues in the medium high frequency range. Moreover, as expected by Amini andWilton, ^[8] applying those methods, the solving process would turn out to be much more difficult and complex, since the corresponding transformation should be done into the conventional Helmholtz boundary integral equation to guarantee the solution uniqueness of the Helmholtz external problem. From this point of view, those numerical methods were not always precise and convenient enough to demonstrate the common acoustic characteristic of structures in the whole frequency band. Hence, analytical representations of the acoustic pressure for some simple underwater structures are always pursued by many scholars. To date, analytical representations of the acoustic pressure of single layer cylindrical shells with ribs or deck-type internal plates have been achieved.^{[9– 14]} Among them, a finite length cylindrical shell model was mainly developed to obtain the sound pressure of an underwater cylindrical shell by introducing two infinite rigid baffles at the ends of the shell. However, as the infinite rigid baffles would change the real shape of the external liquid field, a little deviation might be brought in that model. Thus, hemi-spherical or flat plate rigid baffles gradually replaced the conventional infinite ones to obtain a more accurate solution.^{[13– 15]} As for the solution for the acoustic pressure of submerged laminated cylindrical shells, numerical methods such as FEM, BEM, and FEM/BEM are still the major choices. Recently in the last twenty years, a virtual source simulation technique, which was based on the superposition principle and the uniqueness theorem for the Helmholtz problem, has been proposed and proved to be effective for solving the acoustic boundary value problem. This technique to solve the acoustic radiation and scattering problems for structures was developed, and good approximate precision was achieved.^{[16– 19]} Up to now, the virtual source simulation technique has been developed to be a mature method of computing the acoustic boundary value problem. According to the number of the acting positions and the density function for the virtual sources, this technique could be mainly classified into four types: a one-point multipole method, a multi-point monopole method, a multi-point dipole method, and a multi-point multipole method. As for the one-point multipole method, the density function was usually expressed in the form of some spherical wave functions. Hence, the convergent velocity for this method is dependent on the geometric shape of the surface of the vibrating body. When the geometric shape of the vibrating boundary seriously deviates from a sphere, the convergent velocity of the one-point multipole method tends to be slow. In that case, the other three methods are preferred to be adopted.^{[17, 18]} Meanwhile, to avoid the singularity of the acoustic radiation, the virtual points are assumed to be located in the inside or in the outside of the real structure, and on a virtual boundary in a similar shape with the real boundary to guarantee the precision and efficiency.^[19]

As mentioned above, ACLD is generally considered to be a better damping treatment on vibration and acoustic suppression. Thus, up to now, considerable investigations, including modeling, control strategy and analysis method etc., have been placed on ACLD cylindrical shells. Focusing on the geometric characteristic of an ACLD cylindrical shell, Saravanan et al.^[4] developed a semi-analytical finite element model and used the FEM to demonstrate its dynamic characteristic. Lee and Kim^[20] proposed a spectral finite element for ACLD beams. The corresponding spectral finite element method (SFEM) mentioned in the paper was more efficient than conventional FEM. However, it turned out to be difficult to establish the spectral finite element for complex structures such as plates and shells. Shen^[21] developed the dynamic differential ordinary equations for the PCLD or ACLD shell. Baz and Chen^[22] derived the axis-symmetric control equations of an ACLD cylindrical shell and explored the boundary control strategy for the shell. To date, by virtue of the precise integral method with high accuracy, a transfer matrix method has been explored for solving the dynamic problem of ACLD cylindrical shells.^{[23– 25]} According to this method, when the external excitation can be expressed in the form of the Fourier series expansion in the circumferential direction, the radial displacement and its derivation of the point at the surface of the cylindrical shell can be resolved, which makes it possible and convenient to settle the acoustic radiation pressure for submerged ACLD cylindrical shells.

In this paper, the transfer matrix method is further developed for the acoustic radiation from a submerged cylindrical shell with ACLD. Firstly, the control equations of the shell are expressed in the frequency domain. Then, an analytical solution is given for the radiation acoustic pressure by a multi-point multipoles method with undetermined coefficients. Next, applying the coupling condition, these coefficients are determined. After that, a sound pressure level at an arbitrary point in the external liquid field can be provided. Finally, damping effects of different parameters of the ACLD treatment on the acoustic radiation of the sandwich cylindrical shell are discussed.

As shown in Figs.  1 and 2, a cylindrical shell fully treated with ACLD is immerged in the liquid. The radius, thickness and length of the host shell are expressed by R₁, h₁, L respectively. The thickness and radius for the viscoelastic layer are h₂ and R₂. As for the piezoelectric layer they are denoted by h₃ and R₃. In this paper, only the inverse piezoelectric effect is considered for the piezoelectric layer, which is transversely isotropic and polarized in the radial direction. Meanwhile, the piezoelectric layer is assumed to be thin enough. Hence, the radial component of the electric field intensity E_z could be taken to be a constant. The electric field intensities and the piezoelectric effect in the other two directions could be neglected also.

Fig.  1.

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	Fig.  1. Submerged ACLD cylindrical shell with hemi-spherical end caps.

Fig.  2.

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	Fig.  2. Cross section for the ACLD cylindrical shell.

Like Refs.  [13] and [14], two semi-spherical rigid baffles are linked at the ends of the cylindrical shell in Fig.  1. Meanwhile, a uniform feedback control voltage is applied to the piezoelectric layer, which means that the distribution function of the control voltage in the circumferential direction, i.e., R(θ ), is equal to unit one. In this case, the electric field intensity in the piezoelectric layer is^[21]

in which, k_p and k_d denote the feedback gain coefficients of the radial displacement and velocity respectively, w(x₀, θ ₀) is the amplitude of the radial displacement at the feedback referring point with the coordinate (x₀, θ ₀), e^{iω t} the time factor, and ω the angular frequency for the external harmonic excitation.

According to the thoughts and operations described in Ref.  [25], by expanding the physical variables in the control equations of the host shell and the piezoelectric layer in the form of Fourier series expansion in the circumferential direction, and replacing the components of those variables by the dimensionless ones, after derivation and arrangement, the first order ordinary differential matrix control equation for the two layers can be derived respectively. Then, based on the method developed in Ref.  [23], by considering the interaction between the viscoelstic layer and the host shell or the piezoelectric layer, by reducing some intermediate variables and through processing, the control equation of the ACLD cylindrical shell in the frequency domain can be yielded as follows:^[25]

where ξ = x/L,

represents the integrated state vector of the ACLD cylindrical shell without considering the piezoelectric effect;

is the dimensionless vector of the external harmonic excitation vector; F_V denotes the generalized active control force vector acting on the piezoelectric layer, which has been given in Ref.  [25]; Ā , B̄ are called the coefficient matrices, and their components, together with , , , , , are described in Ref.  [23].

It is worth pointing out that the dynamic problem of an arbitrary ACLD cylindrical shell can be described by a set of matrix equations in the form of Eq.  (2) corresponding to the different values of circumferential wave number n (n = 0, 1, … , ∞ ). Meanwhile, the state variable in each of Z, F, and F_V is the dimensionless n-th component of the Fourier series expansion in the circumferential direction for the real physical variable.

In addition, the inhomogeneous terms appearing on the right-hand side of Eq.  (2) consist of vectors F and F_V · w(x₀, θ ₀). As for the shell shown in Fig.  1, F is composed of two parts: one is the determined mechanical external excitation vector F_er, and the other is the undetermined acoustic pressure vector P (P = L {0, 0, 0, 0, 0, 0, p, 0, 0, 0, 0, 0}^T/K⁽¹⁾, p is the acoustic pressure). Because P together with the term w(x₀, θ ₀) is dependent on the solution of Eq.  (2), the solution of Eq.  (2) is difficult to obtain. When F merely comprises F_er, w(x₀, θ ₀) can be obtained by an extended homogeneous capacity precision integration method and superposition principles, and then Eq.  (2) could be resolved successfully.^[25] Based on Ref.  [25], suppose that the acoustic pressure could be expressed in an analytical form as a sum of the Fourier series with wave numbers in the circumferential direction, the acoustic radiation problem for cylindrical shells shown in Fig.  1 should be solved as well.

Under the harmonic excitation, the acoustic pressure p(r) at any point Q(r) in the liquid field shown in Fig.  1 satisfies the following Helmholtz equation:

where, k = ω /c₀ is the wave number, c₀ is the sound propagation velocity in the liquid medium.

To ensure the solution uniqueness for Eq.  (3), p(r) should meet the Summerfield radiation condition in the far liquid field as well, that is,

Meanwhile, on the liquid-solid interaction surface, Neumann boundary condition is also given as

where r_s is the position vector of the point on the interaction surface S, v_r (r_s) is its normal velocity, while ρ ₀ represents the density of the liquid medium.

When the acoustic source is a point sound source located at P, the solution of Eq.  (3), which meets Eq.  (4), could be yielded in the spherical coordinate system as^[26]

in which ρ _PQ is the radius vector from the spherical center point Q to the source point P; ψ _PQ and ϕ are the polar angle and circumferential angle in the spherical coordinate system respectively; denotes the 2nd kind of spherical Hankel function with order represents the l-th order associated Legendre function with degree n. Function appearing in Eq.  (6) makes up a series of wave functions with order l and degree n. They are usually truncated with the finite integers M and N respectively. B_ln are the coefficients for these wave functions.

Unfortunately, the concrete acoustic source is not always composed of some real point sources. According to the internal source density simulation technique^{[16– 19, 27]} for the acoustic problem of the revolution cavity and by employing superposition principles, a multi-point multipole method is adopted to deal with the radiation acoustic pressure. Generally speaking, these virtual point sources should be assumed to be located in the inside or outside of the real structure to avoid singularity. Hence, in the present paper, the virtual point source is assumed to be located on the axial line of the cylindrical shell. Meanwhile, this kind of location strategy could be a convenient one for programming. In that case, the acoustic pressure caused by each virtual sound source could be expressed in the form of the sum of some wave function series as shown in Eq.  (6). Then, by adjusting the density of these wave functions, i.e., the coefficient B_ln for each virtual sound source, the real acoustic source could be simulated and the approximate results of the acoustic pressure could be obtained. The detailed steps are given as follows.

Applying a convenient way, supposing that some virtual point sources marked as P_k (x_k, 0) are located eventually along the axial line of the cylindrical shell shown in Fig.  3, referring to Eq.  (6) and combining with superposition principles, the acoustic pressure at a field point Q could be written as

where NP denotes the total number of the virtual point sources, is one wave function component of the acoustic pressure caused by the k-th virtual point sound source, which could be expressed in detail as

and are the corresponding coefficients for these wave functions.

Fig.  3.

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	Fig.  3. Schematic diagram of the meridian plane at the fluid-solid interface.

According to Eqs.  (7) and (8), the acoustic pressure of Q is represented by the unknown coefficients , which need to be determined by the liquid-solid coupling condition. As for the cylindrical shell shown in Fig.  1, it could be simplified from Eq.  (5) into

where S_L, S_R, S_C represent the interaction surfaces between the liquid field and the left semi-spherical rigid baffle, the right one, and the cylindrical shell respectively. Meanwhile, for the shell of revolution, normal partial derivative of one of the components of acoustic pressure could be deduced as

where, α is the angle between the normal vector n_pkQ and the vector ρ _PkQ. Equation  (10) can be further developed for the shell shown in Fig.  3. For the limitation of space, the detailed formulations about it at any point on S_L, S_R, or S_C are derived in Appendix  A, which are expressed by Eq.  (A1), Eq.  (A2), and Eq.  (A3) respectively.

It can be seen from Eq.  (9) that can be determined by the point collocation method when the radial displacements of the cylindrical shell w are determined, which will be demonstrated in the following section.

Applying the extended homogeneous capacity precision integration method, for a given inhomogeneous term appearing in Eq.  (2), the radial displacement caused by it can be solved. Taking the generalized active force F_V (known) for example, the detailed process of solving Eq.  (2) is described in the following.

Firstly, F_V is approximated by a quadratic polynomial vector function

Then, by extended homogeneous capacity processing, equation  (2) could be written as

As demonstrated in Ref.  [23], equation  (12) can be solved by applying precision integration. Hence the radial displacement at any point of the base shell can be obtained.

Assume the radial displacement amplitudes caused by F_V, F_er, and , , and , respectively. The superscript n for these radial displacement amplitudes means that they were the n-th component of the Fourier series expansion of the real dimensionless one. Next, based on the superposition principles, the amplitude of the radial displacement at any point (ξ , θ ) on the surface of the cylindrical shell can be achieved as

According to Ref.  [24], substituting x = x₀ (ξ = ξ ₀ = x₀/L), θ = θ ₀ into Eq.  (13), one obtains

Substituting Eq.  (14) into Eq.  (13), w(x₀, θ ₀) could be eliminated from Eq.  (13). After that, w̄ (ξ , θ ) only depends on the coefficients . Then, the collocation algebraic equation for could be further described based on Eq.  (9). The detailed manipulation is illustrated in Appendix  B, in which can be solved from Eqs.  (B1)– (B6). Finally, the acoustic pressure at any point in the liquid field can be obtained from Eq.  (B7).

In order to discuss conveniently the acoustic radiation characteristic of submerged cylindrical shells with ACLD, the point Q₀ in the external liquid field with the coordinate (3, 0, L/2) is chosen as the observing point. Moreover, the sound pressure level (SPL) of that observing point, which can represent the performance index of the acoustic stealth for the structure, is defined as^[28]

where, P is the amplitude of the sound pressure, referring to sound pressure p_ref = 1.0× 10^{− 6}  Pa.

Based on the previous research on the ACLD cylindrical shell without liquid, a series of first order integrated ordinary differential matrix equations in the frequency domain is presented in this paper. These matrix equations can be smoothly decoupled when the external excitation acting on the shell is harmonic and known.

By the internal source density simulation method, some virtual point sound sources are assumed to be distributed evenly along the axial line of the cylindrical shell to approach to the real external excitation. A superposition solution of the radiation acoustic pressure is explored with unknown coefficients by the method called the multi-point multipole method. Then, by the extended homogenous capacity integration method and superposition principles, the acoustic radiation of the cylindrical shell can be demonstrated in the paper by this method, which is proved to be available when discussing the acoustic radiation problems of a revolution cavity.

Numerical results of the SPL of the acoustic radiation from the shell with different parameters display that thicker thickness and larger velocity feedback gain for the piezoelectric layer, and larger coverage of the ACLD layer can obtain a better damping effect for the whole structure in general. Whereas, a thinner thickness for the viscoelastic layer is a better choice when lower frequency takes a major part in the working condition for the structure, and vice versa. Meanwhile, optimization should be carried out to achieve suitable values of these parameters at a minimum cost. And the presented method could be further developed for designing the real underwater structures combined with optimization methods.

Denote the coordinate of the source point P_k as (l_Pk, 0) (0 ≤ l_Pk ≤ L), the field point on the surface of cylindrical shell Q_c as (x_Qc, R₁) (0 ≤ x_Qc ≤ L)), and let x_a = ω R_PkQc/c₀, x_b = (x_Qc− l_Pk)/R_PkQc, and , then the normal derivative for the component of the acoustic pressure at Q_c will be written as

Denote the coordinate of the point on the surface of the right rigid baffle Q_R as (L+ R₁ sinβ , R₁ cosβ ), let x_aR = ω R_PkQR/c₀, x_bR = (L+ R₁ sinβ − l_Pk)/R_PkQR, and , then the normal derivative for the component of the acoustic pressure at Q_R will yield

Denote the coordinate of the point on the surface of the left rigid baffle Q_L as (− R₁ sinγ , R₁ cosγ ), and let x_aL = ω R_PkQL/c₀, x_bL = (− R₁ sin γ − l_Pk)/R_PkQL, and , then the normal derivative for the component of the acoustic pressure at Q_L is

Substituting Eq.  (14) into Eq.  (13), after arrangement, one obtains

in which

Let

and choose N_c discrete field points along the meridian line of the cylindrical shell, denote the coordinates as ξ _i = i/(N_c − 1), (i = 0, … , N_c − 1), then from Eqs.  (9) and (A1), collocation equations for the points on the surface of cylindrical shell will be obtained as

where the sign sum(• ) represents the sum of vectors.

Similarly, choose N_R discrete field points on the surface of the right semi-spherical rigid baffle, for which coordinate could be deduced from the discrete angle β _i = iπ /2N_R, (i = 1, … , N_R) respectively, then from Eqs.  (9) and (A2), collocation equations for the points on the surface of the right semi-spherical rigid baffle will be yielded as

Next, choose N_L discrete field points on the surface of the left semi-spherical rigid baffle, for which the coordinate could be obtained from the discrete angle γ _i = iπ /2N_L, (i = 1, … , N_L) respectively, then from Eqs.  (9) and (A3), collocation equations for the points on the surface of the left semi-spherical rigid baffle will be obtained as

Till now, the whole collocation equations can be tidied up from Eqs.  (B3)– (B5) as

The coefficients can be achieved from