Infinite Circular Cylindrical Shell Research Articles

Functionally graded viscoelastic core characteristics on vibroacoustic behavior of double-walled cylindrical shells in a subsonic external flow

The present study is concerned with an analytical solution for calculating sound transmission loss through an infinite double-walled circular cylindrical shell with two isotropic skins and a polymeric foam core. Accordingly, the two-walled cylindrical shell is stimulated applying an acoustic oblique plane wave. The equations of motion are derived according to Hamilton’s principle using the first-order shear deformation theory for every three layers of the construction. Additionally, by the aid of employing the Zener mathematical model for the core of polymeric foam, mechanical properties are determined. To authenticate the results of this study, the damping of the core layer goes to zero. Therefore, the numerical results in this special case are compared with those of isotropic shells. The results prove that the presented model has high accuracy. It is also designated that decreasing the power-law exponent of the core leads to improving the sound transmission loss through the thickness of the construction. Besides, in addition to probe some configurations versus alterations of frequencies and dimensions, the convergence algorithm is provided. Consequently, it is realized that by increasing the excitation frequency, the minimum number of modes to find the convergence conditions is enhanced. The results also contain a comparison between the sound transmission loss coefficient for four different models of a core of a sandwiched cylindrical shell. It is comprehended that the presented model has a transmission loss coefficient more than the other types of the core at high frequencies.

Vibration Power Flow of an Infinite Cylindrical Shell Submerged in Viscous Fluids

In the previous investigations of the vibroacoustic characteristics of a submerged cylindrical shell in a flow field, the fluid viscosity was usually ignored. In this paper, the effect of fluid viscosity on the characteristics of vibration power flow in an infinite circular cylindrical shell immersed in a viscous acoustic medium is studied. Flügge’s thin shell theory for an isotropic, elastic, and thin cylindrical shell is employed to obtain the motion equations of the structure under circumferential-distributed line force. Together with the wave equations for the viscous flow field as well as continuity conditions at the interface, the vibroacoustic equation of motion in the coupled system is derived. Numerical analysis based on the additional-damping numerical integral method and ten-point Gaussian integral method is conducted to solve the vibroacoustic coupling equation with varying levels of viscosity. Then, the variation of the input power flow against the nondimensional axial wave number in the coupled system with different circumferential mode numbers is discussed in detail. It is found that the influence of fluid viscosity on the vibroacoustic coupled system is mainly concentrated in the low-frequency band, which is shown as the increase of the crest number and amplitude of the input power flow curves.

A study on buckling of piezoelectric circular cylindrical shell

The infinite length piezoelectric circular cylindrical shell is considered in the present study, which is subjected to axisymmetric external pressure and applied electrical voltage in radial direction. The mathematical formulation of the system is adopted simultaneous linear homogeneous ordinary differential equations as governing equations. These governing equations are further solved by using finite difference method. Numerical results of the analysis provides comparative study of conventional method with FDM method and show that piezoelectric effect improves stability of shell.

Solving Equations of Free Vibration for a Cylindrical Shell Rotating on Rollers by the Fourier Method

The small free vibrations of an infinite circular cylindrical shell rotating about its axis at a constant angular velocity are considered. The shell is supported on n absolutely rigid cylindrical rollers equispaced on its circle. The roller-supported shell is a model of an ore benefication centrifugal concentrator with a floating bed. The set of linear differential equations of vibrations is sought in the form of a truncated Fourier series containing N terms along the circumferential coordinate. A system of 2N–n linear homogeneous algebraic equations with 2N–n unknowns is derived for the approximate estimation of vibration frequencies and mode shapes. The frequencies ω k , k = 1, 2, …, 2N–n, are positive roots of the (2N–n)th-order algebraic equation D(ω2) = 0, where D is the determinant of this set. It is shown that the system of 2N–n equations is equivalent to several independent systems with a smaller number of unknowns. As a consequence, the (2N–n)th-order determinant D can be written as a product of lower-order determinants. In particular, the frequencies at N = n are the roots of algebraic equations of an order is lower than 2 and can be found in an explicit form. Some frequency estimation algorithms have been developed for the case of N > n. When N increases, the number of found frequencies also grows, and the frequencies determined at N = n are refined. However, in most cases, the vibration frequencies can not be found for N > n in an explicit form.

Analysis of Dispersion Characteristics of an Infinite Cylindrical Shell Submerged in Viscous Fluids Considering Hydrostatic Pressure

In previous investigations of the vibro-acoustic characteristics of a submerged cylindrical shell in a flow field, the fluid viscosity was usually ignored. In this paper, the effect of fluid viscosity on the vibrational dispersion characteristics of an infinite circular cylindrical shell immersed in a viscous acoustic medium and subject to hydrostatic pressure is studied. The Flügge's thin shell theory for an isotropic, elastic, and thin cylindrical shell is employed to obtain the equations of motion of the structure. Together with the wave equations for the viscous flow field as well as continuity conditions at the interface, the dispersion equation of the coupled system is derived. Numerical analysis based on a winding-number integral method is conducted to solve the dispersion equation for the shell loaded with viscous fluids with varying levels of viscosity. Then the variations of the dispersion characteristic, the amplitude ratio of complex waves and the relative difference parameter against the nondimensional axial wave number in the coupled system with different circumferential mode numbers are discussed in detail. It is found that the influence of fluid viscosity on dispersion characteristics of the propagating waves is more significant in the low-frequency range than at high frequencies. As for the complex waves, the amount of the waves in the coupled system and the cut-off frequency is dependant on the fluid viscosity coefficients.

Dynamic cylindrical shell equations by power series expansions

The dynamics of an infinite circular cylindrical shell is considered. The derivation process is based on power series expansions of the displacement components in the radial direction. By using these expansions in the three-dimensional equations of motions, a set of recursion relations is identified expressing higher displacement coefficients in terms of lower order ones. The new approximate shell equations are hereby obtained from the boundary conditions, resulting in a set of six partial differential equations. These equations are believed to be asymptotically correct and it is, in principle, possible to go to any order. Equations of order up to and including the square of the thickness h 2 are presented explicitly. This set of six equations are reduced to five as well as three equations and compared to classical theories. Dispersion curves, together with the eigenfrequencies for a 2D case, are calculated using exact, classical and expansion theories. It is shown that the approximate equations containing order h 2 are in general as good as or better than the established theory of the same order.

Parity violating cylindrical shell in the framework of QED

We present calculations of Casimir energy (CE) in a system of quantized electromagnetic (EM) field interacting with an infinite circular cylindrical shell (which we call ‘the defect’). Interaction is described in the only QFT-consistent way by Chern–Simon action concentrated on the defect, with a single coupling constant a. For the regularization of UV divergencies of the theory, we use the Pauli–Villars regularization of the free-EM action. The divergencies are extracted as a polynomial in the regularization mass M, and they renormalize the classical part of the surface action. We reveal the dependence of CE on the coupling constant a. Corresponding Casimir force is attractive for all values of a. For a → ∞, we reproduce the known results for CE for perfectly conducting cylindrical shell first obtained by DeRaad and Milton. As a future task for solving existing arguments on observational status of (rigid) self-pressure of a single object, we propose for investigation a system which we call ‘Casimir drum’.

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.

Casimir energy of a semi-circular infinite cylinder

The Casimir energy of a semi-circular cylindrical shell is calculated by making use of the zeta function technique. This shell is obtained by crossing an infinite circular cylindrical shell by a plane passing through the symmetry axes of the cylinder and by considering only half of this configuration. All the surfaces, including the cutting plane, are assumed to be perfectly conducting. The zeta functions for scalar massless fields obeying the Dirichlet and Neumann boundary conditions on the semi-circular cylinder are constructed exactly. The sum of these zeta functions gives the zeta function for the electromagnetic field in question. The relevant plane problem is considered also. In all the cases the final expressions for the corresponding Casimir energies contain the pole contributions which are the consequence of the edges or corners in the boundaries. This implies that further renormalization is needed in order for the finite physical values for vacuum energy to be obtained for given boundary conditions.

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.

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.

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.

Approximate elasticity solution for a long and thick laminated circular cylindrical shell of revolution

An approximate three-dimensional elasticity solution is presented for an infinite, thick, orthotropic as well as laminated circular cylindrical shell of revolution subjected to distributed pinch load. The validity and the accuracy of the results of approximate solution has been established by comparing it with the results of an exact three-dimensional elasticity solution for single and multilayered (hybrid) shell of revolution. Numerical results have been presented for cross-ply laminated (0/90/0 and 90/0/90) infinite circular cylindrical shell of revolution subjected to axisymmetric band and distributed pinch loads. These results have been compared with the classical and first-order shear deformation theories of Flugge and Donnell to assess the accuracy and limitations of the twodimensional shell theories.

The perturbed stress state of a closed cylindrical shell with longitudinal cracks on its inner surface

Using the Rice-Levy line spring model we find an approximate solution of the problem of elastic balance of a closed infinite circular cylindrical shell weakened by a periodic system of longitudinal interior cracks of identical length. The stress intensity factors are found at the center of a crack for various numbers of cracks with various parameters.

A study of the perturbed stress state of a closed cylindrical shell with a system of part-through transverse cracks

We study the elastic equilibrium of a closed infinite circular cylindrical shell with a system of surface cracks of identical length and depth. We use the method of singular integral equations together with the modeling of solid matter in the plane of a part-through crack by irregularly distributed “line springs”. We conduct a numerical analysis of the variation of the relative stress intensity factor at the center of a crack as a function of the parameters of a crack and the number of cracks. We study cracks located on both the interior and exterior surface of the shell.

Increased accuracy in the application of the Sommerfeld–Watson transformation to acoustic scattering from cylindrical shells

The Sommerfeld–Watson transformation (SWT), when applied to problems of sound scattering from elastic bodies with separable geometries, allows the scattered field to be displayed in terms of waves rather than in terms of modes, and so gives a more meaningful picture of the scattering mechanisms at moderate and high frequencies. As typically applied, however, the SWT has left a discontinuity in the calculated field as a function of field point. This discontinuity is not inherent in the method, but is a result of not accounting for the effects of nearby poles in the evaluation of the line integral which represents the geometric part of the scattered field (essentially the specular term). This paper considers this effect in the particular case of broadside scattering from an infinite circular cylindrical elastic shell in an acoustic medium. The scattered field in the backward half-space, as computed from the corrected wave solution, is shown to be in excellent agreement with the field calculated from the normal mode (partial wave) series for frequencies corresponding to ka=1.5 and above, for a steel shell whose wall thickness is one percent of its radius. Shell theory, rather than the theory of elasticity, is used.

The generalized Stoneley and Rayleigh waves on cylindrical shells.

This paper dealt with acoustical scattering resulting from a plane harmonic wave normally incident upon an infinite elastic circular cylindrical shell immersed in water and filled with air. Various circumferential surface modes have been experimentally and numerically observed. The so called l=0 and l=1 circumferential waves are considered. A numerical study of their behavior at high frequency, when the relative thickness of the shell remains constant, is achieved in the complex wave-number plane. The results are presented in the form of trajectories in the complex wave-number plane and of curves of phase velocity and attenuation by reradiation in water plotted as function of reduced frequency. At low frequency, the trajectory of the circumferential waves on shells depends strongly on the value of the relative thickness of the shell. Nevertheless, it is shown that, at high frequency, the effect of the internal interface becomes negligible and, for all values of the relative thickness considered, the l=0 wave (respectively, l=1 wave) on cylindrical shells tends toward the Stoneley wave (respectively, Rayleigh wave) on solid cylinder and, in the limit of very high frequency, when the curvature becomes negligible, their common limit is their flat-surface counterparts.

Acoustic radiation from rib-reinforced infinite cylindrical shell

The acoustic radiation from rib-reinforced cylindrical shells is effected by the interaction between the shell and reinforcing ribs. This interaction includes normal, shear, and moment reactions of the rib to shell motion. An analytic model for the farfield acoustic radiation from a fluid-loaded infinite circular cylindrical shell with periodic rib supports is developed. The ribs are modeled as circular rings where the reactive forces, moments, and shears are applied by the shell at the outer edge of the ring. Approximate solutions for the force, moment, and shear impedances of the ribs are developed and incorporated into the acoustic radiation model to explore the dependence of the acoustic radiation from ribbed cylindrical shells on rib/shell interaction.

Axisymmetrical impulsive responses of an infinite circular cylindrical shell filled with liquid.

Axisymmetrical impulsive responses of an infinite circular cylindrical shell filled with liquid.

Axisymmetrical impulsive responses of an infinite circular cylindrical shell filled with liquid.

In this paper, dynamic interaction phenomena on solid and liquid interfaces are discussed. Axisymmetrical responses of an infinite circular cylindrical shell perfectly filled with liquid are analyzed, based on Flugge's theory for a circular cylindrical shell and the potential theory for the ideal fluid under conditions of the impulsive external band pressure given on the outer surface of the shell. The deflection and the moment of the shell and the pressure in the fluid are evaluated by using the numerical inversion of the Laplace transformation method. The approximated solution for the shell with an equivalent mass on it is analyzed and is evaluated, abased on the solution for the solid and liquid interaction.