Non-Axisymmetric Vibrations of Stepped Cylindrical Shells Containing Cracks

A comparative analysis of the dynamic stability of cylindrical and conical shells with the same geometric and mechanical characteristics under periodic uniformly distributed axial compression was presented. The study of the stability of steady periodic vibrations of thin elastic shells was based on the joint use of the method of curvilinear grids, the projection method and the parameter continuation method combined with the Newton–Kantorovich method. Geometrically nonlinear relations of the thin elastic shells theory are formulated on the basis of the vector approximation of the displacements function in the general curvilinear coordinate system in tensor form and satisfy the Kirchhoff-Love hypothesis. The discretization of the differential equations of the steady forced vibrations in the direction of the generating shells using the method of curvilinear grids was carried out. The components of the elements displacement vectors of the shells middle surface in the circular directionare approximated by trigonometric series. Reduction of the number of generalized coordinates of the discrete dynamic model of shells steady forced vibrations was performed by the Bubnov-Galerkin basis reduction method. A transition from vector ordinary differential equations to a nonlinear system of algebraic equations was made. The construction of a mathematical model of the dynamic stability of steady forced nonlinear vibrations of thin elastic shells was performed according to Floquet's theory using the projection method. The criterion for the loss of stability was the equality to zero of the determinant of the matrix of linearized equations of steady forced nonlinear vibrations of shells according to the Lyapunov theorem. A comparative analysis of frequencies and modes of natural vibrations of cylindrical and conical shells with the same geometric and mechanical characteristics and boundary conditions was performed. Nonlinear steady vibrations of the shells due to periodic axial compression were studied. The critical values of the dynamic load and the corresponding forms of loss of shell stability in the range of lower frequencies of their natural vibrations were obtained.

Stable axisymmetric vibrations of a thick-walled circular cylindrical shell made of a visco-elastic material are examined under pressure distributed in cylindrical surfaces which changes in time according to the law of harmonic. A system of four partial differential equations of the second order for defining the constituents of radial and axial displacements is written without any preliminary suppositions about the character of changes in unknown values on thickness. A spline-collocation method is used for decreasing the system dimension under an arbitrary way of fixing of sections. A corresponding boundary problem is numerically solved using a discrete Godunov method of orthogonalization.As an example the vibrations of shells with the same radius of the surface under various values of length and thickness. A comparison of results according to a strict and classic theory based on Kirchhoff−Love hypotheses is performed. Some peculiarities of changes on the radius of the radial displacement constituents which can not be detected in the frames of any approximated theory, assuming constancy of bending on the shell thickness are noted. Key words: axisymmetric vibrational bending, thick-walled cylindrical shell, visco-elastic material.

Parametric resonance in cylindrical shells: a case study in the nonlinear vibration of structural shells

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this paper, the stability of a thin cylindrical sheel subjected to an axial leakage flow is discussed. Here, we focus on a lateral vibration of a cylindrical shell, that is the beam-like vibration of a shell. The coupled equations between a shell and a fluid are obtained by using the Donnell's shell theory and the Navier Stokes equation. The influence of the axial velocity on the unstable vibration phenomena is clarified concerning the beam-like vibration mode of a shell. The applicable range of Donnel shell theorry is confirmed for axial leakage flow-induced vibration of a thin cylindrical shell. The numerical results on the shell theory are compared with those on beam theory which have been already reported by the authors. And, the numerical parameter studies are done for several dimensions of shell and fluid.

Magneto-electro-elastic semi-analytical models for free vibration and transient dynamic responses of composite cylindrical shell structures

A new closed form formulation of three-dimensional (3-D) refined higher-order shell theory (RHOST) to analyze the free vibration of composite circular cylindrical shells has been presented in this article. The shell is considered to be laminated with orthotropic layers and simply supported boundary conditions. The proposed theory is used to investigate the effects of the in-plane and rotary inertias as well as transverse normal and shear strains on the dynamic response of thick composite cylindrical shells. The trapezoidal shape factor of the shell element is incorporated to obtain accurate stress-resultants. Using Hamilton’s principle, the equations of motion are obtained and solved in terms of the Galerkin method. Numerical results for the natural frequencies are verified by making comparison with the 3-D exact elasticity iterative solutions in the literature. In addition, the validity of the results is further verified by ABAQUS. According to the results, for thick composite cylinders with large length-to-radius and orthotropic ratios, through thickness exact integration yields accurate stress-resultants for proper prediction of the natural frequencies.

A linear free vibration analysis is presented for computing the natural frequencies of a grid-stiffened carbon-nanotube-reinforced composite cylindrical shell. The carbon nanotubes (CNTs) are supposed to be oriented unidirectionally across the shell thickness, and the elastic moduli of the CNT-reinforced polymer composite are calculated using a modified rule of mixtures. In order to obtain the equivalent stiffness parameters of the grid-stiffened cylindrical shell, the smeared stiffness method is employed. The stiffeners are assumed to support the transverse shear loads, bending moments, and axial loads. An analytical method based on the first-order shear deformation theory is used. To validate the results obtained, a 3-D finite-element model is employed using the ABAQUS CAE software.

An analytical study is presented for vibration and acoustic radiation of a finite thin orthotropic composite cylindrical shell excited by a harmonic concentrated force in a hygroscopic environment. The modal analysis method is used to solve the governing equations. Theoretical results are presented in natural vibration, radial quadratic velocity, sound power and radiation efficiency, with uniform incremental moisture content. Furthermore, different stiffness, length and thickness are set respectively to research the effects of the material and structure parameters variation of the orthotropic cylindrical shell on the vibration and acoustic radiation characteristics. It is found that the natural frequencies decrease with an increase of moisture content. The modal indices associated with the lowest frequency mode reaches the modal indices corresponding to the lowest buckling mode near the critical buckling moisture content with moisture content. The radial quadratic velocity and sound radiation power decrease with the incremental moisture content in the lower frequency band. The vibration and acoustic response decrease with the enhanced stiffness. The increasing length has little impact on the sound radiation and the thickened cylindrical shell weakens the sound radiation response.

In this paper, we analyze the axisymmetric vibrations of a cylindrical shell of which the thickness varies in the axial direction by using the thin cylindrical shell theory and an improved thick cylindrical shell theory. We solve the equations of vibration exactly by using a series solution. As numerical examples, we obtain the natural frequencies and the mode shapes of cylindrical shells with both ends clamped, simply supported and free and examine the variations of the natural frequencies and the mode shapes for the variations of the thickness. And also, we make clear the influences of the shear deformation and the rotatory inertia upon the natural frequencies and the mode shapes.

The effect of rotary inertia of added masses on vibrations of empty and fluid-filled circular cylindrical shells

Vibration and damping analysis of thick sandwich cylindrical shells with a viscoelastic core under arbitrary boundary conditions

Free vibration of nonlinear deformation ring-and stringer-stiffened nonuniform cylindrical shell

Axially symmetric vibrations of fluid-filled poroelastic circular cylindrical shells

Free vibrations of partial fluid-filled orthotropic circular cylindrical shells are investigated using the Dynamic Stiffness Method (DSM) or Continuous Element Method (CEM) based on theFirst Order Shear Deformation Theory (FSDT) and non-viscous incompressible fluid equations. Numerical examples are given for analyzing natural frequencies and harmonic responses of cylindrical shells partially and completely filled with fluid under various boundary conditions. The vibration frequencies for different filling ratios of cylindrical shells are obtained and compared with existing experimental and theoretical results which indicate that the fluid filling can reduce significantly the natural frequencies of studiedcylindrical shells. Detailed parametric analysis is carried out to show the effects of some geometrical and material parameters on the natural frequencies of orthotropic cylindrical shells. The advantages of this current solution consist in fast convergence, low computational cost and high precision validating for all frequency ranges.