Vibrations of Cylindrical Shells with Varying Thickness : Continued

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.

In this paper, the axisymmetric free vibrations of truncated conical shells are analyzed by means of an improved shell theory. The equations of vibration and the boundary conditions are obtained from the stational condition of Lagrangian of the shells. The equations of vibration are solved exactly by a solution in series for a conical shell with linearly varied thickness along its axis and the effects of boundary conditions, thickness and semi-vertex angle on the natural frequencies are investigated. As a special case of the conical shell, we compare this theory with Mindlin's circular plate theory and with Mirsky's cylindrical shell theory.

Free vibration characteristics of piezoelectric cylindrical shells with stepped thickness using an analytical symplectic approach

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

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.

Based on the Donnell’s approximations of the thin shell theory, this paper presents solutions for the problem of free non-axisymmetric vibration of stepped circular cylindrical shells with cracks. The shell under consideration is sub-divided into multiple segments separated by the locations of thickness variations. It is assumed that at thejth step there exists a circumferential surface crack with uniform depthcj. The influence of circular cracks with constant depth on the vibration of the shell is prescribed with the aid of a matrix of local flexibility. The latter is related to the coefficient of the stress intensity known in the linear fracture mechanics. Numerical results are obtained for cylindrical shells of stepped thickness containing cracks at re-entrant corners of steps. Shells with various combinations of boundary conditions can be analyzed by the proposed method. Furthermore, the influences of the shell thicknesses, locations of step-wise variations of the thickness and other parameters on the natural frequencies are examined. The results can be used for the approximate evaluation of dynamic parameters of cylindrical shells with cracks and flaws.

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

Asymptotic solutions are obtained for the eigenvalue problems of the inextensional free vibrations of circular cylindrical shells, considering all 45 possible combinations of the boundary conditions, characterizing the simply supported, the clamped, and the free ends. In addition to the well-known Rayleigh and Love types of inextensional vibrations for shells with the free-free ends, a type represented by a linear combination of those classical ones is found in cases where one end is free and the other is supported hi such a manner that it can move freely in the axial direction. The existence of the three types of inextensional mode is proved by an experiment, and the mode shapes are visualized by holographic interferometry.

A study of free vibration of orthotropic circular cylindrical shells is presented. The vibration control equations of shells are based on Flugge classical thin shell theory. Wave approach is used in the analysis, in which the boundary conditions of shells can be simplified according to the associated beam. The free vibration frequencies of shells can be obtained from a frequency polynomial equation of order 6. The parametric analysis of the free vibration of orthotropic cylindrical shells is investigated using a statistical method. The effects of geometrical parameters and material characteristics upon frequencies are investigated here. Multivariate analysis (MVA) can be a useful tool for this parametric study. Some statistical characteristics, including correlation analysis and ANOVA are applied. ANOVA has been conducted to predict the statistical significance of the various factors. Calculations are performed in the Minitab statistical software. The results show that the L/R, h/R and m have larger effects on the lowest frequency. The importance of input parameters is ranked according to their contributions to the total variance. A knowledge and data visualization approach, Self-organizing mapping (SOM) is also adopted here for mining some intrinsic characteristics of 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.

Free vibration analysis of cylindrical shells using the Haar wavelet method

Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions

The present study aims at the investigation of free vibration analysis of thin cylindrical shells surrounded by an elastic medium, which is distributed over a particular length of the shell. The Love’s shell theory is utilized along with the Winkler foundation to obtain the governing equations of motion. An exact series expansion method of solution is employed for any arbitrary boundary conditions. The excellent accuracy of the obtained results is validated by comparing to the available literature. Moreover, several case studies are performed to provide an insight into the influence of some practically important parameters on free vibrations of circular cylindrical shells, including stiffness and length of the partial foundation. Finally, it is concluded that the proposed method can accurately determine the exact solution to the free vibration of shells partially surrounded by elastic foundations with any boundary conditions.

This paper is concerned with a theoretical investigation of the free vibration of arbitrary shells of revolution by means of the classical bending theory of shells. A method is developed that is applicable to rotationally symmetric shells with meridional variations (including discontinuities) in Young's modulus, Poisson's ratio, radii of curvature, and thickness. By means of the method of this paper, the natural frequencies and the corresponding mode shapes of axisymmetric or nonsymmetric free vibration of rotationally symmetric shells can be obtained without a limitation on the length of the meridian of the shell. To illustrate the application of the method given in this paper to particular shells, stone results of free vibration of spherical and conical shells obtained earlier by means of the bending theory are reproduced by the general method of this paper, and a detailed comparison is made. In addition, paraboloidal shells and a sphere cone shell combination are considered, which have been previously analyzed by means of the inextensional theory of shells, and natural frequencies and mode shapes predicted by the bending theory are given.

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