Non-linear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading

2008 ◽  
Vol 83 (2) ◽  
pp. 201-211 ◽  
Author(s):  
M. Darabi ◽  
M. Darvizeh ◽  
A. Darvizeh
Keyword(s):  
Dynamic Stability ◽  
Cylindrical Shells ◽  
Axial Loading ◽  
Linear Analysis ◽  
Functionally Graded ◽  
Non Linear

scholarly journals Nonlinear Vibrations of Functionally Graded Cylindrical Shells: Effect of the Geometry

2012 ◽  
Author(s):  
Matteo Strozzi ◽  
Francesco Pellicano ◽  
Antonio Zippo
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Cylindrical Shells ◽  
Nonlinear Partial Differential Equations ◽  
Linear Analysis ◽  
Finite Amplitude ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Displacement Fields

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies.

A tangent stiffness–based approach to study free vibration of shear-deformable functionally graded material rotating beam through a geometrically non-linear analysis

2017 ◽  
Vol 52 (5) ◽  
pp. 310-332 ◽  
Author(s):  
Suman Pal ◽  
Debabrata Das
Keyword(s):  
Free Vibration ◽  
Functionally Graded Material ◽  
Thermal Loading ◽  
Linear Analysis ◽  
Functionally Graded ◽  
Rotating Beam ◽  
Governing Equations ◽  
Tangent Stiffness ◽  
Non Linear ◽  
Graded Material

An improved mathematical model to study the free vibration behavior of rotating functionally graded material beam is presented, considering non-linearity up to second order for the normal and transverse shear strains. The study is carried out considering thermal loading due to uniform temperature rise and using temperature-dependent material properties. Power law variation is assumed for through-thickness symmetric functional gradation of ceramic–metal functionally graded beam. The effects of shear deformation and rotary inertia are considered in the frame-work of Timoshenko beam theory. First, the rotating beam configuration under time-invariant centrifugal loading and thermal loading is obtained through a geometrically non-linear analysis, employing minimum total potential energy principle. Then, the free vibration analysis of the deformed beam is performed using the tangent stiffness of the deformed beam configuration, and employing Hamilton’s principle. The Coriolis effect is considered in the free vibration problem, and the governing equations are transformed to the state-space to obtain the eigenvalue problem. The solution of the governing equations is obtained following Ritz method. The validation is performed with the available results, and also with finite element software ANSYS. The analysis is carried out for clamped-free beam and for clamped–clamped beam with immovably clamped ends. The results for the first two modes of chord-wise and flap-wise vibration in non-dimensional speed-frequency plane are presented for different functionally graded material compositions, material profile parameters, root offset parameters and operating temperatures.

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