Free Vibration Analysis of a Rotating Mori–Tanaka-Based Functionally Graded Beam via Differential Transformation Method

2015 ◽  
Vol 41 (2) ◽  
pp. 577-590 ◽  
Author(s):  
Farzad Ebrahimi ◽  
Mohadese Mokhtari
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Transformation Method ◽  
Functionally Graded ◽  
Differential Transformation Method ◽  
Functionally Graded Beam ◽  
Differential Transformation

Free Vibration Analysis of Rotating Tapered Bresse-Rayleigh Beams Using the Differential Transformation Method

2009 ◽  
Author(s):  
Dominic R. Jackson ◽  
S. Olutunde Oyadiji
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Free Vibration Analysis ◽  
Transformation Method ◽  
Differential Transformation Method ◽  
Algebraic Equations ◽  
Differential Transformation ◽  
Rayleigh Beam

The free vibration characteristics of a rotating tapered Rayleigh beam is analysed in this study. First, the strain-displacement relationship for the rotating beam is formulated and used to derive the kinetic and strain energies in explicit analytical form. Second, Hamilton’s variational principle is used to derive the governing differential equation of motion and the associated boundary conditions. Third, the Differential Transformation Method (DTM) is applied to reduce the governing differential equations of motion and the boundary conditions to a set of algebraic equations from which the frequency equation is derived. Next, a numerical algorithm implemented in the software package Mathematica is used to compute the natural frequencies of vibration for a few paired combinations of clamped, pinned and free end conditions of the beam. Also, the variation of the natural frequencies of vibration with respect to variations in the rotational speed, hub radius, taper ratio and the slenderness ratio is studied. The results obtained from the Bresse-Rayleigh theory are compared with results obtained from the Bernoulli-Euler and Timoshenko theories to demonstrate the accuracy and relevance of their application.

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.

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