Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Employing Biot's theory of wave propagation in liquid saturated porous media, axially symmetric vibrations of fluid-filled and empty poroelastic circular cylindrical shells of infinite extent are investigated for different wall-thicknesses. Let the poroelastic cylindrical shells are homogeneous and isotropic. The frequency equation of axially symmetric vibrations each for a pervious and an impervious surface is derived. Particular cases when the fluid is absent are considered both for pervious and impervious surfaces. The frequency equation of axially symmetric vibrations propagating in a fluid-filled and an empty poroelastic bore, each for a pervious and an impervious surface is derived as a limiting case when ratio of thickness to inner radius tends to infinity as the outer radius tends to infinity. Cut-off frequencies when the wavenumber is zero are obtained for fluid-filled and empty poroelastic cylindrical shells both for pervious and impervious surfaces. When the wavenumber is zero, the frequency equation of axially symmetric shear vibrations is independent of nature of surface, i.e., pervious or impervious and also it is independent of presence of fluid in the poroelastic cylindrical shell. Non-dimensional phase velocity for propagating modes is computed as a function of ratio of thickness to wavelength in absence of dissipation. These results are presented graphically for two types of poroelastic materials and then discussed. In general, the phase velocity of an empty poroelastic cylindrical shell is higher than that of a fluid-filled poroelastic cylindrical shell.The phase velocity of a fluid-filled bore is higher than that of an empty poroelastic bore. Previous results are shown as a special case of present investigation. Results of purely elastic solid are obtained.     

Resonant response of orthotropic cylindrical shells

The resonant response of simply supported, thin and thick, orthotropic cylindrical shells is determined by using modal analysis, for three levels of hysteretic damping. This response is significantly changed if the larger of the elastic constants associated with normal stresses refers to the meridional, instead of the circumferential, direction. These changes may be related to changes in the relevant natural frequency. The causes of the latter are studied by considering the modal amplitudes and the contributions to the strain energy function associated with the various stress resultants.     

Damping properties of layered cylindrical shells,vibrating in axially symmetric modes

The damping of cylindrical shells coated with unconstrained layers of viscoelastic material either on one side of the shell (inside or outside) or on both sides is estimated. The basic equations of motion are derived which describe harmonic forced flexural damped vibrations in axisymmetric modes. For pure sinusoidal modes expressions for the overall loss factors are given. The damping properties of cylindrical shells of finite length, coated on the inside or outside, or on both sides (symmetrically or unsymmetrically) are compared. Classical thin shell theory is used for the analysis. It is shown how two-layered damped shells differ from two-layered damped beams. The extent of damping reduction in shells resulting from the fact that the shell cross-section is closed is discussed.     

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.     

Free interfacial vibrations in cylindrical shells

The 2D equations in the Kirchhoff-Love theory are subjected to asymptotic analysis in the case of free interfacial vibrations of a longitudinally inhomogeneous infinite cylindrical shell. Three types of interfacial vibrations, associated with bending, super low-frequency semi-membrane, and extensional motions, are investigated. It is remarkable that for extensional modes natural frequencies have asymptotically small imaginary parts caused by a weak coupling with propagating bending waves. Bending and extensional vibrations correspond to Stonely-type plate waves, while semi-membrane ones are strongly dependent on shell curvature and do not allow flat plate interpretation. The paper represents generalization of the recent authors' publication [Kaplunov et al., J. Acoust. Soc. Am. 107, 1383-1393 (2000)] dealing with edge vibrations of a semi-infinite cylindrical shell.     

Nonlinear vibrations of clamped-free circular cylindrical shells

Only experimental studies are available on large-amplitude vibrations of clamped-free shells. In the present study, large-amplitude nonlinear vibrations of clamped-free circular cylindrical shell are numerically investigated for the first time. Shells with perfect and imperfect shape are studied. The Sanders-Koiter nonlinear shell theory is used to calculate the elastic strain energy. Shell displacement fields (longitudinal, circumferential and radial) are expanded by means of a double mixed series, i.e. harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. All boundary conditions are satisfied. The system is discretized by using natural modes of the shell and Lagrange equations by an energy approach, retaining damping through Rayleigh's dissipation function. Different expansions involving from 18 to 52 generalized coordinates are used to study the convergence of the solution. The nonlinear equations of motion are numerically studied by using arclength continuation method and bifurcation analysis. Numerical responses to harmonic radial excitation in the spectral neighborhood of the lowest natural frequency are compared with experimental results available in literature. The effect of geometric imperfections and excitation amplitude are numerically investigated and fully explained.     

Maxi-element analysis for free vibrations of orthotropic shells of revolution

The theory of free vibration for orthotropic shells of revolution with arbitrary homogeneous boundary conditions is developed. The essence of the method is to decompose the overall shell into a number of so-called cylindrical, conical, and plate “maxi-elements”. Since the eigenfunctions of each of the individual maxi-elements are analytically determined directly from the solution of the governing differential equations, the procedure has the advantage of requiring significantly fewer elements compared with the usual finite element or finite difference procedures. For the conical shell and the plate, the method of solution is novel, while the solution for the cylindrical shell has been published by the first author. The versatility and accuracy of the method is shown through the inclusion of a number of examples which present the excellent correlation with test results and other numerical schemes.     

Free vibrations of cylindrical shells with elastic-support boundary conditions

This paper concerns the free vibrations of cylindrical shells with elastic boundary conditions. Based on the Flügge classical thin shell theory, the equations of motion for the cylindrical shells are solved by the method of wave propagations. The wave numbers are obtained by directly solving an eighth order equation. The elastic-support boundary conditions can be arbitrarily specified in terms of 8 independent sets of distributed springs. All the classical homogeneous boundary conditions can be considered as the special cases when the stiffness for each set of springs is equal to either infinity or zero. The present solutions are validated by the results previously given by other researchers and/or obtained using finite element models. The effects on the frequency parameters of elastic restraints are investigated for shells of different geometrical characteristics.     

On radially symmetric vibrations of orthotropic non-uniform disks including shear deformation

This paper deals with the free axisymmetric vibrations of orthotropic circular plates with linear variation in thickness. The analysis is based on a set of two differential equations derived by an extension of Mindlin's shear theory for plates. On simplification and algebraic manipulation, one of the dependent variables is eliminated from the governing equations of motion, giving rise to a fourth order linear differential equation with variable coefficients. The resulting differential equation is solved numerically by the Chebyshev collocation technique. Frequencies and mode shapes for the first five modes of vibration are computed for different plates.     

Effect of heterogeneity on the axisymmetric vibrations of cylindrical shells

The influence of large heterogeneity on the axisymmetric vibration characteristics of thin, composite cylindrical shells is studied, both analytically and numerically. In the neighborhood of the axisymmetric breathing mode, frequency spectra for shells of infinite and finite length are shown to be influenced qualitatively as well as quantitatively by large deviations from material and geometric symmetry in layer arrangement. A study of mode coupling in a semi-infinite shell is made for both end modes and modes with stationary frequency with real finite wave number, the latter being uniquely generated by a special class of heterogeneity. In each case, analytical estimates are given for frequencies, wave numbers, and modal amplitudes as functions of material and geometric properties of the shell.     

Simplified equations and solutions for the vibration of orthotropic cylindrical shells

Starting with Love type equations of motion for orthotropic circular cylindrical shells, the theory is simplified by assumptions similar to those in the Donnell-Mushtari-Vlasov development for isotropic shells. Closed form solutions for simply supported cases are then obtained. Results of two example cases are compared with finite element results and are shown to agree well. It is argued that this simplified approach allows easy assessment of the influence of design parameter changes.     

Non-linear vibrations of laminated cylindrical shallow shells under thermomechanical loading

The geometrically non-linear vibrations of linear elastic composite laminated shallow shells under the simultaneous action of thermal fields and mechanical excitations are analysed. For this purpose, a model based on a very efficient p-version first-order shear deformation finite element, with hierarchical basis functions, is employed. The equations of motion are solved in the time domain by a Newmark implicit time integration method. The model and code developed are partially validated by comparison with published data. Parametric studies are carried out in order to study the influence of temperature change, initial curvature, panel thickness and fibre orientation on the shells’ dynamics.     

Sound transmission through layered cylindrical shells with applied damping treatment

Sound transmission through closed circular cylindrical shells with unconstrained damping treatment and sandwich shells with constrained damping treatment is considered in an acoustoelastic formulation in which the full interaction between the structural vibration and the internal cavity pressure field are taken into account. Only the axisymmetric modes of vibration of the shell are considered in an initial formulation. The structural response and the sound transmission characteristics for an external random pressure field are computed through an efficient matrix inversion procedure. Results indicate insensitivity of noise reduction and structural response to variations in core parameters of the sandwich shell. The sandwich shell has better noise transmission characteristics in the higher frequency ranges as compared to an equivalent layered shell with unconstrained damping treatment.     

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.     

Free vibrations of laminated composite cylindrical shells with an interior rectangular plate

The free vibration analysis of a laminated composite cylindrical shell with an interior rectangular plate is performed by the analytical and experimental methods. The frequency equations of vibration of the shell including the plate are formulated by using the receptance method. To obtain the free vibration characteristics before the combination of two structures, the energy principle based on the classical plate theory and Love's thin shell theory is adopted. The numerical results are compared with the results from an experiment, as well as a finite element analysis, to validate the current formulation. The influences of the length-to-radius ratio (L_S/a) and radius-to-thickness ratio (a/h_S) of the shell and fiber orientation angles (Θ) of symmetric cross- and angle-ply composite materials on the natural frequencies of a cylindrical laminated combined shell are also discussed in details.