Acoustic signature of a submarine hull under harmonic excitation

The structural and acoustic responses of a submarine under harmonic force excitation are presented. The submarine hull is modelled as a cylindrical shell with internal bulkheads and ring stiffeners. The cylindrical shell is closed by truncated conical shells, which in turn are closed at each end using circular plates. The entire structure is submerged in a heavy fluid medium. The structural responses of the submerged vessel are calculated by solving the cylindrical shell equations of motion using a wave approach and the conical shell equations with a power series solution. The far-field radiated sound pressure is then calculated by means of the Helmholtz integral. The contribution of the conical end closures on the radiated sound pressure for the lowest circumferential mode numbers is clearly observed. Results from the analytical model are compared with computational results from a fully coupled finite element/boundary element model.     

Three methods for analyzing forced vibration of a fluid-filled cylindrical shell

The forced vibration of an infinite elastic circular cylindrical shell filled with fluid is studied. Three methods are employed to analyze the forced vibration problem of this shell-fluid coupled system, that is, wave propagation approach (wave mode superposition), theorem of residues and a numerical integral method. In order to explain these methods more explicitly, before being used to investigate the vibration of an infinite fluid-filled elastic circular cylindrical shell, all these three methods are employed firstly to analyze the forced vibration problem of an infinite beam and an infinite elastic circular cylindrical shell in vacuo. Advantage and disadvantage of these three methods are discussed and their interesting relationship is revealed. That is, to any circumferential wavenumber and frequency of the external force, there is an unchangeable relationship between the general coordinates of various waves in the wave propagation approach and the residuals in the theorem of residues.     

The input mobility of an infinite circular cylindrical elastic shell filled with fluid

The force input mobility of an infinite elastic circular cylindrical shell filled with fluid is derived by using the spectral equations of motion. Mobilities are evaluated and their physical interpretations are discussed for a steel shell of thickness $\text{h}\text{a}\text{= 0·05}$ filled with water and vibrating in the n = 0, 1 and 2 circumferential modes. The results are subsequently used to analyze the related situations of wave transmission through a radial ring constraint and the far field vibrational energy distributions between the contained fluid and the shell wall for line and point driving forces.     

Acoustic radiation from the submerged circular cylindrical shell treated with active constrained layer damping

Based on the transfer matrix method of exploring the circular cylindrical shell treated with active constrained layer damping(i.e., ACLD), combined with the analytical solution of the Helmholtz equation for a point source, a multi-point multipole virtual source simulation method is for the first time proposed for solving the acoustic radiation problem of a submerged ACLD shell. This approach, wherein some virtual point sources are assumed to be evenly distributed on the axial line of the cylindrical shell, and the sound pressure could be written in the form of the sum of the wave functions series with the undetermined coefficients, is demonstrated to be accurate to achieve the radiation acoustic pressure of the pulsating and oscillating spheres respectively. Meanwhile, this approach is proved to be accurate to obtain the radiation acoustic pressure for a stiffened cylindrical shell. Then, the chosen number of the virtual distributed point sources and truncated number of the wave functions series are discussed to achieve the approximate radiation acoustic pressure of an ACLD cylindrical shell. Applying this method, different radiation acoustic pressures of a submerged ACLD cylindrical shell with different boundary conditions, different thickness values of viscoelastic and piezoelectric layer, different feedback gains for the piezoelectric layer and coverage of ACLD are discussed in detail. Results show that a thicker thickness and larger velocity 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, laying a thicker viscoelastic layer is not always a better treatment to achieve a better acoustic characteristic.     

Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections

The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of some of the lowest natural frequencies is investigated. Donnell's non-linear shallow-shell theory is used and the solution is obtained by the Galerkin method. Several expansions involving 16 or more natural modes of the shell are used. The boundary conditions on the radial displacement and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The non-linear equations of motion are studied by using a code based on the arclength continuation method. A series of accurate experiments on forced vibrations of an empty and water-filled stainless-steel shell have been performed. Several modes have been intensively investigated for different vibration amplitudes. A closed loop control of the force excitation has been used. The actual geometry of the test shell has been measured and the geometric imperfections have been introduced in the theoretical model. Several interesting non-linear phenomena have been experimentally observed and numerically reproduced, such as softening-type non-linearity, different types of travelling wave response in the proximity of resonances, interaction among modes with different numbers of circumferential waves and amplitude-modulated response. For all the modes investigated, the theoretical and experimental results are in strong agreement.     

Distorted cylindrical shell response to internal acoustic excitation below the cut-off frequency

The mechanism of coupling between a plane wave acoustic mode and a non-axisymmetric structural mode of a thin walled, circular, cylindrical shell, via the geometrical distortion of the shell, is considered. A theoretical model of response below the acoustic cut-off frequency is used to estimate the vibration level induced in a tube having small circumferential variations of wall thickness, radius and modulus of elasticity. The results have been confirmed experimentally.     

Analysis of axisymmetric shell structures under axisymmetric loading by the flexibility method

A method is developed for the static stress and deformation analysis of axisymmetric shells under axisymmetric loading by reduction of the shell to ring sections. In particular, the wall thickness of the shell may vary and the method is applicable to the analysis of shells with irregular meridional geometry. Explicit expressions for the influence coefficients for each ring element are derived. In the development of these expressions, exact evaluation of stresses in the circumferential direction of the ring is used. The distribution of stresses in the meridional direction of the ring element is assumed to be linear with each element. By using the derived influence coefficients, the unknown forces at the junctures of the ring elements are found by the standard flexibility method of indeterminate structural analysis. Subsequently, the displacements and internal stresses are determined. Example solutions for a flat circular plate under transverse loading and for a cylindrical shell under a boundary edge loading show excellent agreement with solutions found by solving the governing differential equations.     

充水圆柱壳声散射的环绕波分析

研究了内部充水圆柱壳散射声场的构成。采用弹性薄壳理论和Sommerfeld-Watson变换(SWT)方法导出散射声场的解析解。数值搜索环绕波的复波数极点,比较圆柱壳体内部真空和内部充水两种情况下表面环绕波的相速度、衰减等性质。用镜反射波和环绕波的叠加合成反向散射形态函数,与简正级数解符合良好。计算流体附加环绕波的回波时序并与实验数据对照。结果表明,流体附加环绕波的再辐射是内部充水圆柱壳的形态函数出现更加丰富的共振峰和回波结构形成的重要原因。      

Double-walled carbon nanotube with surrounding elastic medium under axial pressure

In this paper, the buckling behavior and critical axial pressure of double-walled carbon nanotubes (DWCNTs) with surrounding elastic medium are investigated. A double-shell (circular cylindrical shell) model is presented and the effects of surrounding elastic medium on the outer tube and the van der Waals forces between two adjacent tubes are taken into account. The analysis and the numerical solution method are based on the classical theory of plates and shells and the Galerkin method. Equations are derived for the critical axial forces and pressures of DWCNTs; the critical axial forces and pressures are calculated for different axial half sine wavenumbers and circumferential sine wavenumbers and compared with those for single-walled carbon nanotubes (SWCNTs).Results indicate that the critical axial force of a DWCNT is higher than that of an SWCNT, but the critical axial pressure of a DWCNT is lower than the critical axial pressure of a SWCNT. Although the critical axial force of a DWCNT decreases as the axial half sine wavenumbers increase, it rises as the circumferential sine wavenumbers increase.     

LINE FORCE RECEPTANCE OF AN ELASTIC CYLINDRICAL SHELL WITH HEAVY EXTERIOR FLUID LOADING

A time-harmonic line force is applied to an infinite elastic cylindrical shell immersed in compressible fluid. The force may also have axial harmonic dependence. The formal solution for the shell displacement is obtained as the sum of circumferential harmonics and evaluated in the asymptotic limit of heavy exterior fluid-loading. The resulting asymptotic expressions for the elements of the receptance matrix, either at the line of application of the force, or elsewhere on the shell surface, are simple trigonometric functions of the shell and fluid parameters, and show excellent agreement with numerical evaluation of the circumferential harmonic series over a wide frequency range.     

Free vibration study of layered cylindrical shells by collocation with splines

The free vibration of circular cylindrical thin shells, made up of uniform layers of isotropic or specially orthotropic materials, is studied using point collocation method and employing spline function approximations. The equations of motion for the shell are derived by extending Love's first approximation theory. Assuming the solution in a separable form a system of coupled differential equations, in the longitudinal, circumferential and transverse displacement functions, is obtained. These functions are approximated by Bickley-type splines of suitable orders. The process of point collocation with suitable boundary conditions results in a generalized eigenvalue problem from which the values of a frequency parameter and the corresponding mode shapes of vibration, for specified values of the other parameters, are obtained. Two types of boundary conditions and four types of layers are considered. The effect of neglecting the coupling between the flexural and extensional displacements is analysed. The influences of the relative layer thickness, a length parameter and a total thickness parameter on the frequencies are studied. Both axisymmetric and asymmetric vibrations are investigated. The effect of the circumferential node number on the vibrational behaviour of the shell is also analysed.     

Analysis of circumferential waves on a water-filled cylindrical shell

The formation of scattering field from a water-filled cylindrical shell was studied. The analytic solutions of scattering field are derived using elastic thin shell theory and Sommerfeld-Watson Transformation(SWT) method.Complex wave-number poles of circumferential waves are found numerically,the phase speed and attenuation of circumferential waves between the situation of a hollow cylindrical shell and a water-filled cylindrical shell are compared. The synthesis of backscattering form functions which are sum of specular reflection component and circumferential waves is consistent with normal mode result.The calculated echo sequences of additional fluid circumferential waves are compared with experimental results. The results show that richer resonance peaks appeared in the backscattering form functions of a water-filled cylindrical shell and the formation of echo’s structure are due to re-radiation effects of additional fluid circumferential waves.     

Active control of low-frequency sound radiation by cylindrical shell with piezoelectric stack force actuators

A novel active control method of sound radiation from a cylindrical shell under axial excitations is proposed and theoretically analyzed. This control method is based on a pair of piezoelectric stack force actuators which are installed on the shell and parallel to the axial direction. The actuators are driven in phase and generate the same forces to control the vibration and the sound radiation of the cylindrical shell. The model considered is a fluid-loaded finite stiffened cylindrical shell with rigid end-caps and only low-frequency axial vibration modes are involved. Numerical simulations are performed to explore the required control forces and the optimal mounting positions of actuators under different cost functions. The results show that the proposed force actuators can reduce the radiated sound pressure of low-frequency axial modes in all directions.     

Vibration analysis of long cylindrical shells using acoustical excitation

Acoustical excitation along with two other methods was used to excite a long circular cylindrical shell, with simply supported boundary conditions. By comparing different types of excitation, the acoustical method was discovered to have many advantages over other methods of excitation used by previous researchers. Five different analytical methods based on the Love and Flugge theories, were also studied. The objective of this study was to identify the accuracy of each theory, in predicting the natural frequencies and mode shapes of a long circular cylindrical shell. A study was made to compare the predictions of the five analytical methods with experimental measurements. Interesting theoretical and experimental observations were observed for the long shell. Finally, a simple method is proposed to reduce the errors found in some of the analytical methods.     

Free vibrations of circular cylindrical shells

The finite element method is used to predict the dynamic behaviour of circular cylindrical shells in free vibrations. A suitable shape function for the circumferential displacement distribution has been proposed. This reduces the three-dimensional character of the problem to a two-dimensional one. The simultaneous iteration method to determine the eigen-frequencies and eigenvectors is utilised for solving the eigenvalue problem. The accuracy of the method has been checked by verifying the results of known cases. Finally an experimental shell structure containing elastic rings welded at the ends has also been analysed and the experimental results compared with the theoretical ones.     

The receptance method applied to ring-stiffened cylindrical shells: Analysis of modal characteristics

The receptance method is applied to determine the natural frequencies and mode shapes of circular cylindrical shells stiffened by rings. The receptances of cylindrical shell and of a ring to forces in the radial and circumferential directions are derived in terms of the modal characteristics of each. A matrix equation of free vibration, which must be solved by an iterative technique, results by eliminating the angular variable. An iterative solution is practical, since the size of the matrices remains at two times the number of stiffening rings, regardless of the number of modes of the unstiffened cylinder and rings included in the solution. The validity of the method is demonstrated by comparing results for specific cases with the results obtained theoretically and experimentally by others. When various stiffener configurations are being considered for a given cylindrical shell, the modal characteristics of the shell without stiffeners may be calculated once and used repeatedly to calculate the frequencies of the stiffened shell configurations. The form of the results offers potential for simplifications which are presented in a companion paper.     

LOCALIZED FAMILIES OF BENDING WAVES IN A THIN MEDIUM-LENGTH CYLINDRICAL SHELL UNDER PRESSURE

The initial-boundary-value problem for the equations describing motion of a thin, medium-length, non-circular cylindrical shell is examined. The shell edges are not necessarily plane curves, with the conditions of a joint support, a rigid clamp or a free edge being considered as the boundary conditions. The shell is supposed to experience normal internal (or external) dynamic pressure which may be non-uniform in the circumferential direction. It is assumed that the initial displacements and velocities of the points at the shell middle surface are functions decreasing rapidly away from some generatrix. Using the complex WKB method the asymptotic solution of the governing equations is constructed by superimposing localized families (wave packets) of bending waves running in the circumferential direction. The dependence of frequencies, group velocities, amplitudes and other dynamic characteristics upon variable pressure and geometrical parameters of the shell are studied. As an example, the wave forms of motion of a circular cylindrical shell with sloping edges under growing dynamic pressure are considered. The effect of localization of bending vibrations near the longest generator as well as the effects of reflection, focusing and increasing amplitude in the running wave packets are revealed.     

Space harmonic analysis of sound radiation from a submerged periodic ring-stiffened cylindrical shell

An analytical method is developed to study radiated sound power characteristics from an infinite submerged periodically stiffened cylindrical shell excited by a radial cosine harmonic line force. The harmonic motion of the shell and the pressure field in the fluid are described by Flügge shell equations and Helmholtz equation, respectively. By using periodic theory of space harmonic analysis, the response of the periodic structure to harmonic excitations has been obtained by expanding it in terms of a series of space harmonics. Radiated sound power on the shell wall along the axial direction and the influence of different parameters on the results are studied, respectively. A conclusion is drawn that the stiffeners have a great influence at low and high frequencies while have a slight influence at intermediate frequencies for low circumferential mode orders. The work will give some guidelines for noise reduction of this kind of shell.     

Analysis of wave propagation in a thin composite cylinder with periodic axial and ring stiffeners using periodic structure theory

Wave propagation characteristics of a thin composite cylinder stiffened by periodically spaced ring frames and axial stringers are investigated by an analytical method using periodic structure theory. It is used for calculating propagation constants in axial and circumferential directions of the cylindrical shell subject to a given circumferential mode or axial half-wave number. The propagation constants corresponding to several different circumferential modes and/or half-wave numbers are combined to determine the vibrational energy ratios between adjacent basic structural elements of the two-dimensional periodic structure. Vibration analyses to validate the theoretical development have been carried out on sufficiently detailed finite element model of the same dimension and configuration as the stiffened cylinder and very good agreement is obtained between the analytical and the dense finite element results. The effects of shell material properties and the length of each periodic element on the wave propagation characteristics are also examined based on the current analytical approach.     

Parametric study on sound radiation from an infinite fluid-filled/semi-submerged cylindrical shell

This work presents the parametric study on the far-field sound pressure radiated from an infinite fluid-filled/semi-submerged cylindrical shell excited by a radial point load. Here, the exterior fluid is non-viscous, isotropic and irrotational coaxial flow. The formula of the radial velocity of the shell in wave-number domain is developed by using the wave-number domain approach (WDA). Then, the analytic expressions are derived for the far-field sound pressure radiating from the shell by using the same method presented in Salaün [Journal of the Acoustical Society of America 90 (1991) 2173]. The influences of parameters such as fluid velocity, structural damping, position of the force, and structural thickness on the far-field sound pressure are investigated. The sound pressure is shown to be very different from the one in the case of a fluid-filled/full-submerged cylindrical shell. Furthermore, it is shown that the pressure and the resonance frequency would increase with the fluid velocity increasing for downstream propagation. The reverse is true for upstream propagation. Moreover, the far-field sound pressure is related to the position and frequency of the excited force. In addition, the influences of structural damping and thickness are shown to be very important.