Elastic Circular Cylindrical Shell Research Articles

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part I: Diffraction–radiation analysis

Abstract The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such an effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.

Energy trapping in power transmission through a circular cylindrical elastic shell by finite piezoelectric transducers

We study the transmission of electric energy through a circular cylindrical elastic shell by acoustic wave propagation and piezoelectric transducers. Our mechanics model consists of a circular cylindrical elastic shell with finite piezoelectric patches on both sides of the shell. A theoretical analysis using the equations of elasticity and piezoelectricity is performed. A trigonometric series solution is obtained. Output voltage and transmitted power are calculated. Confinement and localization of the vibration energy (energy trapping) is studied which can only be understood from analyzing finite transducers. It is shown that when thickness-twist mode is used the structure shows energy trapping with which the vibration can be confined to the transducer region. It is also shown that energy trapping is sensitive to the geometric and physical parameters of the structure.

Interaction between a submerged evacuated cylindrical shell and a shock wave—Part II: Numerical aspects of the solution

Abstract The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed.

Acoustic characterization and prediction of the cut-off dimensionless frequency of an elastic tube by neural networks

A neural network is developed to predict cut-off dimensionless frequencies of the antisymmetric circumferential waves (Ai) propagating around an elastic circular cylindrical shell of different radius ratio b/a (a, outer radius; b, inner radius). The useful data to train and test the performances of the model are determinated from calculated trajectories of natural modes of resonances or extracted from time-frequency representations of Wigner-Ville of the acoustic backscattered time signal obtained from a computation. In this work, the studied tubes are made of aluminum or stainless steel. The material density, the radius ratio b/a, the index i of the antisymmetric waves, and the propagation velocities in the tube, are selected like relevant entries of the model of neural network. During the development of the network, several configurations are evaluated. The optimal model selected is a network with two hidden layers. This model is able to predict the cut-off dimensionless frequencies with a mean relative error (MRE) of about 1%, a mean absolute error (MAE) of 3.10(-3) k1a, and a standard error (SE) of 10(-3) k1a (k1a is the dimensionless frequency, k is the wave number in water).

External shock loading on a submerged fluid-filled cylindrical shell

The interaction between a submerged fluid-filled elastic circular cylindrical shell and an external shock wave is considered. The study focuses on the internal acoustic field. A linear formulation of the problem is considered. A semi-analytical solution is obtained and used to simulate the interaction. A variety of phenomena are observed in the internal fluid, including the reflection and focusing of the internal acoustic wave as well as the radiation into the fluid of elastic waves propagating in the shell. Throughout the paper, the results of numerical simulations are compared with available experimental data, and a good agreement is observed. The solution developed appears to be suitable for use as a benchmark. Engineering relevance of the phenomena observed is discussed.

Stability loss of thin-walled cylindrical tubes, subjected to longitudinal compressive forces and external corrosion

The paper is devoted to investigation of stability loss of thin-walled elastic circular cylindrical pipes subjected to simultaneous action of longitudinal compressive forces and uniform corrosion from outside. The pipe is treated as a thin closed elastic circular cylindrical shell which can be described within Kirchhoff–Love approximation. The critical time of stability loss of the pipe is found, using the upper critical load value for static stability loss of the pipe, not subjected to corrosion, and the law of corrosion rate. Numerical results were obtained for pipes, made from steel and aluminum with different initial wall thickness.

Green’s Function for a Closed, Infinite, Circular Cylindrical Elastic Shell

An acceptable variant of the Koiter–Morley equations for an elastically isotropic circular cylindrical shell is replaced by a constant coefficient fourth-order partial differential equation for a complex-valued displacement-stress function. An approximate formal solution for the associated “free-space” Green’s function (i.e., the Green’s function for a closed, infinite shell) is derived using an inner and outer expansion. The point wise error in this solution is shown rigorously to be of relative order (h∕a)(1+h∕a∣x∣), where h is the constant thickness of the shell, a is the radius of the mid surface, and ax is distance along a generator of the mid surface.

Elastic buckling of an axially compressed open circular cylindrical shell

AbstractThe paper is devoted to stability problem of an axially compressed open elastic circular cylindrical shell. Two curvilinear edges of the open cylindrical shell are simply supported while two straight edges are free. Two load cases of the shell are assumed. The first load case ‐ was the invariable axially normal force intensity, and the second load case ‐ the linearly varying axially normal force intensity. Critical loads of the shell for both load cases are determined. These critical loads are compared with a classical load for closed circular cylindrical shell. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interaction of a spherical shockwave and an elastic circular cylindrical shell

The paper studies the interaction of a spherical shock wave with an elastic circular cylindrical shell immersed in an infinite acoustic medium. The shell is assumed infinitely long. The wave source is quite close to the shell, causing deformation of just a small portion of the shell, which makes it possible to represent the solution by a double Fourier series. The method allows the exact determination of the hydrodynamic forces acting on the shell and analysis of its stress state. Some characteristic features of the stress state are described for different distances to the wave source. Formulas are proposed for establishing the safety conditions of the shell.

Acoustic insulation provided by circular and infinite plane walls

The transmission of sound energy is a complex phenomenon on account of the number and the interdependence of variables involved. In this context the mathematical modelling of this phenomenon becomes difficult. Of the models that have been developed, some of them only take a limited number of variables into account, leading to simplified models such as the Mass Law. Among the simplified methods for evaluating the sound insulation provided by a single panel are those described by Sewell [1], Sharp [2], Cremer [3] and Callister et al. [4]. Novikov [5] used the mass law with the addition of a correction coefficient to study the sound insulation of finite plates, at low frequencies. Osipov et al. [6] studied the sound transmission through a single panel using an infinite plate model and a room–plate–room model. They noticed that at low frequencies the results are strongly influenced by the geometry and the dimensions of the room–wall–room system, in addition to the properties of the panel material. Most of the articles published that take the full coupling between the fluid and the solid structure into account are restricted to low frequencies, because of the computational cost entailed, particularly when modelling closed spaces. This kind of analysis is simpler in the case of circular enclosures, for which analytical solutions can be derived. Other authors in a different context have tackled the study of circular cylindrical shells. Veksler et al. [7] studied the modal resonances of submerged elastic circular cylindrical shells filled with air, when struck by plane harmonic acoustic waves. Their work used a standard resonance scattering theory, and paid special attention to the generation of bending waves. Bao et al. [8] analyzed the various circumferential waves generated when a circular cylindrical shell submerged in a fluid and filled with another fluid is disturbed by a normally incident plane wave. Maze et al. [9], in a related work, analyzed the existence of various guided acoustic circumferential modes in a water-filled tube. ARTICLE IN PRESS

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.

INTERACTION OF A SPHERICAL SHOCK WAVE AND A SUBMERGED FLUID-FILLED CIRCULAR CYLINDRICAL SHELL

The complete three-dimensional interaction between a spherical shock wave and a submerged fluid-filled elastic circular cylindrical shell is considered. A hybrid analytical–numerical solving procedure is established. An exact analytical solution in the form of double Fourier series with time-depending coefficients is obtained for the hydrodynamic pressure. Displacements of a shell are approached analytically to reduce the problem to a set of systems of ordinary differential equations, which are treated numerically. Detailed analysis of the interaction is performed with emphasis given to the stress–strain state. A few important features of the interaction process have been found. In particular, it has been shown that the interior fluid not only substantially affects the magnitude of displacements and stresses, but also dramatically changes the nature of the interaction. It has been found that the absolute maximum of stresses can neither be caused by a direct action of a shock wave nor by a constructive superpose of elastic waves in the shell, but by the pressure wave propagating in the interior fluid. This fact seems to be of essential importance for engineering applications, especially when safety is a primary design concern. Another important result is that the maximum stresses are attained at large times, which makes use of early time asymptotics leading to incorrect results. The proposed semi-analytical approach seems to be computationally attractive and suitable for extensive numerical simulations.

VIBRATIONAL POWER FLOW INPUT AND TRANSMISSION IN A CIRCULAR CYLINDRICAL SHELL FILLED WITH FLUID

In this paper, the characteristics of vibrational power flow in an infinite elastic circular cylindrical shell filled with fluid are investigated. The Flügge thin shell theory and Helmholtz equation are employed respectively in analyzing the wave motion in the shell wall and the sound field in the fluid. The coupling condition on the inner surface of the shell wall is introduced to obtain the vibrational equation of this coupled system. By using Fourier transform and its inverse transform, the input power into the coupled system excited by a driving force is studied. Along the shell axial direction, the transmission of the power flow carried by different internal forces of the shell wall and by the fluid is also studied. The results show that the input power flow and the power flow transmission depend mainly upon the characteristics of the free propagating waves in this coupled system.

Penetration of an elastic circular cylindrical shell into an incompressible liquid

The plane unsteady problem of impact of a thin elastic cylindrical shell on the surface of an ideal incompressible liquid is considered. The initial stage of interaction between the body and the liquid when the stresses in the shell attain peak values is studied. The problem is treated in a linearized formulation and is solved numerically by the normal modes method within the framework of the Wagner approach. The numerical results agree with experimental data for various types of circular cylindrical shells made from mild steel.

Acoustic scattering by a closed semi-infinite fluid-loaded elastic cylindrical shell

A plane sound wave is obliquely incident upon a semi–infinite elastic circular cylindrical shell, closed at one end by a rigid but movable disk, with exterior fluid loading. Expanding the unknown pressure on the end plate as a Dini series and applying half–range Fourier transforms allows the problem to be formulated as a Wiener–Hopf equation with unknown coefficients which satisfy a system of simultaneous linear equations. The solution is presented for the axisymmetric, n = 0 harmonic, component of the scattered sound, which in the far–field consists of the cylindrically spreading waves, in appropriate regions, which would have been present if the shell had been infinitely long, together with a spherically spreading component which is due to the semi–infinite length of the shell. Some numerical results are presented for the spherically spreading component of the scattered field.

Scattering by cylindrical shells: The Scholte–Stoneley wave nature

A systematic study of the A wave revolving around circular cylindrical elastic shells immersed in water is performed. The resonance frequencies and halfwidths have been computed for a wide range of the b/a ratio (b is the inner radius and a the outer one). The A-wave evolution is studied from the case of the quasisolid cylinder to the case of the very thin shell (b/a equal to 0.99). The resonances are computed from the complex frequency plane; it is then possible to get the dispersion curves of the A wave. The phase velocities dispersion curves, plotted against the adimensional variable ktd (kt is the shear wave number and d the thickness of the shell), practically coincide in the low-frequency range. The resonance halfwidths are small in this frequency range. The A wave is a bending type wave. At higher frequencies, the A wave becomes analogous to a diffracted Franz wave.

The Plane Symmetrical Problem of Impact of a Slender Elastic Circular Cylindrical Shell on the Surface of a Liquid with Allowance for Separation

The Plane Symmetrical Problem of Impact of a Slender Elastic Circular Cylindrical Shell on the Surface of a Liquid with Allowance for Separation

VIBRATION POWER FLOW IN A FLUID-FILLED CYLINDRICAL SHELL

The vibrational power flow from a line circumferential cosine harmonic force into an infinite elastic circular cylindrical shell filled with fluid is studied. To analyze the response of the shell, an integrated numerical method along the pure imaginary axis of the complex wavenumber domain is used. The results are discussed for a steel shell filled with fluid and vibrating in then=0, 1 and 2 circumferential modes. In order to evaluate the effect of the fluid, the results of a shell filled with fluid are compared with those of a shellin vacuo.

FREE VIBRATION OF THICK CIRCULAR CYLINDRICAL SHELLS SUBJECTED TO AXIAL STRESSES

A two-dimensional higher-order shell theory is applied to the free vibration problems of a simply supported cylindrical shell subjected to axial stresses. The effects of higher-order deformations such as shear deformations with thickness changes and rotatory inertia on natural frequencies of thick elastic circular cylindrical shells are studied. Based on the power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order shell theory is derived through Hamilton's principle. Several sets of truncated approximate theories which can take into account the complete effects of higher-order deformations are applied to solve the vibration problems of a simply supported thick circular cylindrical shell. In order to assure the accuracy of the present theory, the convergence of the natural frequencies is examined and the results are compared with those obtained in existing theories.

Bending A wave in the scattering by a circular cylindrical shell: Its relation with the bending free modes

The steady-state problem of a plane acoustic wave scattered by an elastic circular cylindrical shell is considered. The standard resonance scattering theory (RST) procedure is used for the analysis of modal resonances. The bending waves, namely A and A0 waves, are investigated in detail. At a fixed order of modal resonance n, it is possible to obtain either the A wave resonance, or the A0 wave resonance, but never both of them together. The relation between the modal resonances of the peripheral waves A and A0 and the free modes of vibration of the shell is studied by means of the variation of the density ρ of the ambient liquid. When the density of the ambient liquid tends toward zero (at unchanged sound velocity in it), the resonance frequency approaches the one of the free vibration in two different ways. If the wave associated with the free mode has a velocity greater than the sound velocity in the fluid, the A0 wave resonance frequency tends to that of the free mode. If the wave associated with the free mode has a velocity lower than the sound velocity in the fluid, the A wave resonance frequency tends to that of the free mode. The A wave, which was up to now considered as an external wave, is rather internal at a low frequency and becomes more external when the frequency increases.