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A solution of vibration and stability problem of an axially loaded cylindrical shell with a four lobed cross section of variable thickness
Abstract
Based on the thin-shell theory and using the computational transfer matrix approach and the Romberg integration method, the free vibration and elastic buckling behaviors of a cylindrical shell with a four lobed cross section of circumferential variable thickness subjected to axially compressive loads is presented. Modal displacements of the shell can be described by trigonometric functions and Fourier's approach is used to separate the variables. The governing equations of the shell are formulated in terms of eight firstorder differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations are written in a matrix differential equation. The transfer matrix is derived from the differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying a numerical integration in the circumferential direction. The natural frequencies and critical loads beside the mode shapes are calculated numerically in terms of the transfer matrix elements for the symmetric and antisymmetric vibration-modes. The influences of the thickness variation of cross-section and radius variation at lobed corners of the shell on the natural frequencies, mode shapes and critical loads are illustrated.
... Ritz method was used to analyse joined thick conical-cylindrical shells having variable thickness by Kang (2012). Khalifa (2012) used transfer matrix approach and Romberg intergration method for analyzing free Corresponding author, Associate Professor, E-mail: visu20@yahoo.com; viswanathan@utm.my ...
The paper proposes to characterize the free vibration behaviour of non-uniform cylindrical shells using spline approximation under first order shear deformation theory. The system of coupled differential equations in terms of displacement and rotational functions are obtained. These functions are approximated by cubic splines. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector which are spline coefficients. Four and two layered cylindrical shells consisting of two different lamination materials and plies comprising of same as well as different materials under two different boundary conditions are analyzed. The effect of length parameter, circumferential node number, material properties, ply orientation, number of lay ups, and coefficients of thickness variations on the frequency parameter is investigated.
The paper proposes to characterize the free vibration behaviour of non-uniform cylindrical shells using spline approximation under first order shear deformation theory. The system of coupled differential equations in terms of displacement and rotational functions are obtained. These functions are approximated by cubic splines. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector which are spline coefficients. Four and two layered cylindrical shells consisting of two different lamination materials and plies comprising of same as well as different materials under two different boundary conditions are analyzed. The effect of length parameter, circumferential node number, material properties, ply orientation, number of lay ups, and coefficients of thickness variations on the frequency parameter is investigated.
The transfer matrix approach is used and the Flügge’s shell theory is modeled to investigate the free vibration behaviour of a cylindrical shell with a four-lobed cross-section with reduced thickness over part of its circumference. Modal displacements of the shell can be described by trigonometric functions and Fourier’s approach is used to separate the variables. The vibration equations of the shell are reduced to eight first-order differential equations in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the nonlinear differential equations of the cylindrical shells by introducing the trigonometric functions in the longitudinal direction and applying a numerical integration in the circumferential direction. The proposed method is used to get the vibration frequencies and the corresponding mode shapes of symmetrical and antisymmetrical type-modes. Computed results indicate the sensitivity of the frequency parameters and the bending deformations to the geometry of the non-uniformity of the shell, and also to the radius of curvature at the lobed corners.
Buckling under the uniform external pressure and low-frequency vibrations of a thin circular closed cylindrical shell are considered. The shell has variable thickness and a curvilinear edge. Asymptotic expansions of the localized buckling and vibration modes are derived and analyzed. Approximate explicit formulae for the critical external pressure and the fundamental vibration frequency are obtained.
The stability of elastic cylindrical thin shells of variable thickness along the directrix and of elastic properties varying continuously depending on the thickness coordinates, subject to a uniform external pressure which is a power function of time, was investigated. Firstly, fundamental relations and the modified Donell-type stability equations of the non-homogeneous elastic cylindrical thin shells with variable thickness were derived. Applying Galerkin's method, a differential equation having a variable coefficient depending on time wass obtained and by applying the method of Sachenkov and Baktieva (1978) to these equations, general formulas for static and dynamic critical loads, corresponding wave numbers, and the dynamic factor were obtained. The appropriate formulas for elastic cylindrical thin shells made of homogeneous and non-homogeneous materials with uniform thickness were specifically obtained from these formulas. Finally, performing the computations, the effects of linear and sinusoidal variation of shell thickness, linear, parabolic, and exponential variation of elastic properties, and the variation of variation factor due to time on critical parameters were investigated.
The flexural vibrations of the walls of thin cylinders are considered. In this type of vibration many forms of nodal pattern may exist owing to the combination of circumferential and axial nodes. Theoretical expressions are developed for the natural frequencies of cylinders with freely-supported and fixed ends and a comparison is made with the frequencies obtained experimentally.
In practice, the ends of cylinders are subjected to a certain degree of fixing by end-plates, flanges, etc., and the natural frequencies thus lie between the corresponding values for freely-supported and fixed ends. To make possible the estimation of such frequencies, a method is devised in which an equivalent wavelength factor is used. This factor represents the wavelength of the freely-supported cylinder that would have the same frequency as the cylinder under consideration when vibrating in the same mode. The results of experimental investigations with various end thicknesses and flange dimensions are recorded, and from these the equivalent factors are derived.
Sets of curves calculated for cylinders with freely-supported ends and covering a range of cylinder thicknesses are given. From these it is possible to obtain close approximation to the frequencies of cylinders under other end conditions by the use of an appropriate factor. An example is given of frequency calculations for a large air-receiver for which two frequencies were identified by experiment.
Starting from Flügge's three equations of motion for a uniform thin cylindrical shell, the paper gives a general solution, from which the dependence of natural frequencies on shell dimensions and mode number can be investigated for any end conditions. This solution requires the assumption of a natural frequency and the determination of the corresponding shell length for the prescribed end conditions. Numerical results are given for shells with clamped ends and for shells with free ends; the variation of frequency factor and of mode shape with dimensional and mode parameters is shown and the accuracy of approximate theories assessed.
An approximate analysis for studying the free vibration of a noncircular cylindrical shell having circumferentially varying thickness is presented. The analysis is formulated to overcome the mathematical difficulties associated with mode coupling caused by variable shell wall curvature and thickness. Accurate results have been obtained for symmetric and antisymmetric vibration modes of shells with oval cross sections. This analysis can be applied to any noncircular cylinder if the curvature along the circumference can be expressed as a function of its perimetrical length. Similarly, any general thickness variation that can be described by the perimetrical length (or the angle θ) can also be analyzed. Thus, the analysis can be applied to nonsymmetric cylindrical shells with both closed and open profiles.
This article presents a review of the research work related to the mechanical behavior of non-circular cylindrical shells and shell segments. To this end, after a brief reference to the basic nomenclature that is mainly used, it initially provides quite a general framework for most of the relevant governing equations employed in the relevant literature. It proceeds with a review of the corresponding dynamic analyses, which are primarily grouped according to the geometrical configuration of the noncircular shell considered and secondarily according to the type of the mathematical model employed. These deal with the dynamics of closed cylindrical shells and open cylindrical panels based on classical (CST) or transverse shear deformable shell theories (SDST). The static analyses reviewed next are divided according to the nature of the physical problem considered and deal with small as well as with large deflections of statically loaded non-circular cylindrical shells. These include both linearized and geometrically nonlinear elastic stability analyses as well as the very few relevant studies that assumed an elastic-plastic response of the shell material constitution. This review article contains 196 references.
An analysis is presented for the vibration and stability of an orthotropic circular cylindrical shell subjected to an axial static load by use of the transfer matrix approach. The applicability of the thin-shell theory is assumed, and the governing equations of free vibration of the shell are written in a matrix differential equation by using the transfer matrix of the shell. Once the matrix has been determined by a solution to the equation, the natural frequencies and the critical loads are calculated numerically in terms of the elements of the matrix for a given set of boundary conditions at the edges. This method is applied to orthotropic circular cylindrical shells simply supported at the edges, and the effects of the length ratio, orthotropy, and axial load on the vibration and stability are studied.
This paper is concerned with the free vibrations of circular cylindrical shells with rigid intermediate supports. The simplest form of thin shell theory according to Donnell and the reputed best first order approximation theory of Sanders are considered. An automated Rayleigh-Ritz method is adopted to evaluate the natural frequencies and the mode shapes of the structures. A proposed unified set of Ritz functions is used to span the displacement fields of various types and combinations of end boundary conditions. Comparison and convergence studies are carried out to verify and establish the appropriate number of Ritz functions to produce results with an acceptable order of accuracy. Shells with any number of intermediate supports and any combination of end boundary conditions can be analyzed by the proposed method. Effects of intermediate supports and their locations on natural frequencies of the shells are studied and the results presented for easy access by design engineers.