Free Vibration Analysis of Functionally Graded Beam with Linearly Varying Thickness

2020 ◽  
pp. 891-904
Author(s):  
Rajat Jain ◽  
Mihir Chandra Manna
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Functionally Graded Beam ◽  
Varying Thickness

Exact solution for free vibration analysis of linearly varying thickness FGM plate using Galerkin-Vlasov’s method

2020 ◽  
pp. 146442072098049
Author(s):  
V Kumar ◽  
SJ Singh ◽  
VH Saran ◽  
SP Harsha
Keyword(s):  
Free Vibration ◽  
Variable Thickness ◽  
Vibration Analysis ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Constant Thickness ◽  
Engineering Structures ◽  
Property Variation ◽  
Varying Thickness

The present paper investigates the free vibration analysis for functionally graded material plates of linearly varying thickness. A non-polynomial higher order shear deformation theory is used, which is based on inverse hyperbolic shape function for the tapered FGM plate. Three different types of material gradation laws, specifically: a power (P-FGM), exponential (E-FGM), and sigmoid law (S-FGM) are used to calculate the property variation in the thickness direction of FGM plate. The variational principle has been applied to derive the governing differential equation for the plates. Non-dimensional frequencies have been evaluated by considering the semi-analytical approach viz. Galerkin-Vlasov’s method. The accuracy of the preceding formulation has been validated through numerical examples consisting of constant thickness and tapered (variable thickness) plates. The findings obtained by this method are found to be in close agreement with the published results. Parametric studies are then explored for different geometric parameters like taper ratio and boundary conditions. It is deduced that the frequency parameter is maximum for S-FGM tapered plate as compared to E- and P-FGM tapered plate. Consequently, it is concluded that the S-FGM tapered plate is suitable for those engineering structures that are subjected to huge excitations to avoid resonance conditions. In addition, it is found that the taper ratio is significantly affected by the type of constraints on the edges of the tapered FGM plate. Some novel results for FGM plate with variable thickness are also computed that can be used as benchmark results for future reference.

Free vibration analysis of a single edge cracked symmetric functionally graded stepped beams

2020 ◽  
Vol 23 (16) ◽  
pp. 3415-3428
Author(s):  
Yusuf Cunedioglu ◽  
Shkelzen Shabani
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Crack Depth ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Single Edge ◽  
Functionally Graded Beam ◽  
Cross Sectional

Free vibration analysis of a single edge cracked multi-layered symmetric sandwich stepped Timoshenko beams, made of functionally graded materials, is studied using finite element method and linear elastic fracture mechanic theory. The cantilever functionally graded beam consists of 50 layers, assumed that the second stage of the beam (step part) is created by machining. Thus, providing the material continuity between the two beam stages. It is assumed that material properties vary continuously, along the thickness direction according to the exponential and power laws. A developed MATLAB code is used to find the natural frequencies of three types of the stepped beam, concluding a good agreement with the known data from the literature, supported also by ANSYS software in data verification. In the study, the effects of the crack location, crack depth, power law gradient index, different material distributions, different stepped length, different cross-sectional geometries on natural frequencies and mode shapes are analysed in detail.

Free Vibration Analysis the Cracked FGM Beam under Bending-Torsion Loading, Using GDQ Method

2013 ◽  
Vol 330 ◽  
pp. 942-947 ◽  
Author(s):  
Alireza Daneshmehr ◽  
D.J. Inman ◽  
A. Mohammadi Fakhar
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Functionally Graded Beam ◽  
Fg Beam ◽  
Fgm Beam

This paper presents a theoretical investigation of free vibration analysis of a functionally graded beam (FGM) under the bending-torsion loading using a classical elasticity theory. The FG beam is assumed to have an open edge crack. It is assumed that the material properties of the simply-supported cracked beam, vary along the beam thickness following a polynomial distribution in the thickness direction. This analysis is based on the linear fracture mechanics. First of all, governing equations and boundary conditions of the FG beam are derived using Hamilton's principle. The governing equations are solved using generalized differential quadrature (GDQ) method. By applying GDQ method, the governing differential equations convert to a linear system of algebraic equations. Then solving the eigenvalue problem, natural frequencies of the FG beam can be found. The results indicate that natural frequencies in the presence of a crack are affected by the crack ratio and location.

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