Natural frequencies and mode shapes of variable thickness elastic cylindrical shells resting on a Pasternak foundation

Abstract

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.

Parametric study of three-dimensional vibration of viscoelastic cylindrical shells on different boundary conditions

“…(2014) employed the Kelvin–Voigt model to study forced vibration of viscoelastic FGM annular structures under external dynamic loading. Ahmed (2016) computed natural frequencies and corresponding mode shapes of a CS resting on a Pasternak foundation using Flügge's shell theory. Based on the theory of elasticity, Aliyari Parand and Alibeigloo (2017) studied static and free vibration behavior of a cylindrical sandwich shell with FG core layer and viscoelastic interfaces using DQM.…”

Dynamic analysis of stepped composite cylindrical shells surrounded by Pasternak elastic foundations based on the continuous element method

“…Recently, scientists employed different methods and theories to study the dynamic behavior of those shells. Ahmed [1] studied the vibration of an elastic oval cylindrical shell with parabolically varying thickness and along of its circumference surrounded by Pasternak foundations based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method. Qu et al used a Domain Decomposition Method (DDM) to evaluate the vibration of stepped circular cylindrical and conical shells [2] and laminated orthotropic conical shells on Pasternak foundation [3].…”

Analysis of the Natural Frequencies of Oscillations of the “Payload–locating Block” System for Orbital Injection

“…Наибольшее распространение в силу своей простоты получили однопараметрические и двухпараметрические модели Винклера и Пастернака. Несмотря на невысокую точность этих моделей поведения упругого основания, они широко распространены, поскольку позволяют быстро изучить качественную картину поведения колебательной системы [1][2][3][4][5][6][7][8][9]. Развитие указанных моделей оснований позволяет ввести в рассмотрение отдельные особенности их поведения [10], в частности возможность упругопластических эффектов [11], работу только на сжатие [12], отличие инерционного и безынерционного поведения основания [2].…”