According to the framework of the Flügge's shell theory, the Winkler and Pasternak foundations model, the transfer matrix approach and the Romberg integration method, the vibration behavior of an isotropic and orthotropic cylindrical shell with variable thickness is investigated. The governing equations of the shell based on the Pasternak foundation model are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling which comes from the variable curvature and thickness of the shell. The vibration equations of the shell are reduced to eight first order differential equations in the circumferential coordinate. Using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the Winkler and Pasternak foundations moduli, the thickness ratio, and the orthotropic parameters are demonstrated.
A new vibration behavior is presented for an elastic oval cylindrical shell having circumferentially variable thickness with complex radius of curvature of an isotropic and orthotropic material. Based on the framework of the Flügge’s shell theory, the transfer matrix approach and the Romberg integration method, the governing equations of motion that have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling which comes from variable curvature and thickness of shell. 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 proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for symmetric and anti-symmetric modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different types of vibration modes of shells 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.
Based on the framework of Flügge's shell theory, transfer matrix approach and Romberg integration method, we investigated how the thermal gradient affects the vibration behavior of rotating isotropic and orthotropic oval cylindrical shells. The governing equations of orthotropic oval cylindrical shells, under parabolically varying thermal gradient around its circumference, with consideration of the effects of initial hoop tension and centrifugal forces due to the rotation are derived, and they are put in a matrix differential equation as a boundary-value problem. As a semianalytic solution, the trigonometric functions are used with Fourier's approach to approximate the solution in the longitudinal direction, and also to reduce the two-dimensional problem in to an one-dimensional one. Using the transfer matrix approach, the equations can be written in a matrix differential equation of first-order and solved numerically as an initial-value problem. The proposed model is applied to get the natural frequencies and vibratory displacement of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration behavior to the rotational speed, the thermal gradient, the ovality and orthotropy of the shell is studied for different type-modes of vibration. The present method is found to be accurate when compared with the results available in the literature.
In this paper, based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, the vibration behavior of an elastic oval cylindrical shell with parabolically varying thickness along of its circumference resting on the Winkler‐Pasternak foundations is investigated. The theoretical analysis of the governing equations of the shell is formulated to overcome the mathematical difficulties of mode coupling of variable curvature and thickness of shell. Using the transfer matrix of the shell, the vibration equations based on the Winkler‐Pasternak foundations are written in a matrix differential equation of first order in the circumferential coordinate and solved numerically. The proposed model is applied to get the vibration frequencies and the corresponding mode shapes of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration characteristics and bending deformations to the Winkler‐Pasternak foundations moduli, thickness variation, ovality and orthotropy of the shell is studied for different type‐modes of vibration.
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