scholarly journals UNCOUPLED VIBRATIONS IN FUNCTIONALLY GRADED TIMOSHENKO BEAM

2016 ◽  
Vol 54 (6) ◽  
pp. 785 ◽  
Author(s):  
Nguyen Tien Khiem ◽  
Nguyen Ngoc Huyen
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Flexural Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Material Parameters ◽  
Flexural Vibrations ◽  
Fgm Beam

Free vibration of FGM Timoshenko beam is investigated on the base of the power law distribution of FGM. Taking into account the actual position of neutral plane enables to obtain general condition for uncoupling of axial and flexural vibrations in FGM beam. This condition defines a class of functionally graded beams for which axial and flexural vibrations are completely uncoupled likely to the homogeneous beams. Natural frequencies and mode shapes of uncoupled flexural vibration of beams from the class are examined in dependence on material parameters and slendernes

In-Plane Free Vibration of Uniform Circular Arches Made of Axially Functionally Graded Materials

2019 ◽  
Vol 19 (08) ◽  
pp. 1950084 ◽  
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Free Vibration ◽  
Young’S Modulus ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Young's Modulus ◽  
Mass Density ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Governing Equations ◽  
Direct Integral

This study focused on the in-plane free vibration of uniform circular arches made of axially functionally graded (AFG) materials. Based on the dynamic equilibrium of an arch element, the governing equations for the free vibration of an AFG arch are derived in this study, where arbitrary functions for the Young’s modulus and mass density are acceptable. For the purpose of numerical analysis, quadratic polynomials for the Young’s modulus and mass density are considered. To calculate the natural frequencies and corresponding mode shapes, the governing equations are solved using the direct integral method enhanced by the trial eigenvalue method. For verification purposes, the predicted frequencies are compared to those obtained by the general purpose software ADINA. A parametric study of the end constraint, rotatory inertia, modular ratio, radius parameter, and subtended angle for the natural frequencies is conducted and the corresponding mode shapes are reported.

scholarly journals Mode shape analysis of multiple cracked functionally graded beam-like structures by using dynamic stiffness method

2017 ◽  
Vol 39 (3) ◽  
pp. 215-228 ◽  
Author(s):  
Tran Van Lien ◽  
Ngo Trong Duc ◽  
Nguyen Tien Khiem
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Stiffness ◽  
Beam Theory ◽  
Functionally Graded ◽  
Theoretical Development ◽  
Mode Shapes ◽  
Functionally Graded Beam ◽  
Stiffness Method ◽  
Dynamic Stiffness Method ◽  
Fgm Beam

Mode shapes of multiple cracked beam-like structures made of Functionally Graded Material (FGM) are analyzed by using the dynamic stiffness method. Governing equations in vibration theory of multiple cracked FGM beam are derived on the base of Timoshenko beam theory; power law variation of material; coupled spring model of crack and taking into account the actual position of neutral axis. A general solution of vibration in frequency domain is obtained and used for constructing dynamic stiffness matrix of the multiple cracked FGM Timoshenko beam element that provides an efficient method for modal analysis of multiple cracked FGM frame structures. The theoretical development is illustrated by numerical analysis of crack-induced change in mode shapes of multi-span continuous FGM beam.

Free Vibration of a Spinning Stepped Timoshenko Beam

2000 ◽  
Vol 67 (4) ◽  
pp. 839-841 ◽  
Author(s):  
S. D. Yu ◽  
W. L. Cleghorn
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Linear Transformations ◽  
Stepped Beam ◽  
Uniform Cross Section ◽  
Vibration Problems

The finite element method is employed in this paper to investigate free-vibration problems of a spinning stepped Timoshenko beam consisting of a series of uniform segments. Each uniform segment is considered a substructure which may be modeled using beam finite elements of uniform cross section. Assembly of global equation of motion of the entire beam is achieved using Lagrange’s multiplier method. The natural frequencies and mode shapes are subsequently reduced with the help of linear transformations to a standard eigenvalue problem for which a set of natural frequencies and mode shapes may be easily obtained. Numerical results for an overhung stepped beam consisting of three uniform segments are obtained and presented as an illustrative example. [S00021-8936(01)00101-5]

An exact solution for free vibration of thin functionally graded rectangular plates

2010 ◽  
Vol 225 (3) ◽  
pp. 526-536 ◽  
Author(s):  
A Hasani Baferani ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Different Boundary Conditions

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

Differential Transformation Approach for Free Vibration Analysis of a Centrifugally Stiffened Timoshenko Beam

2005 ◽  
Vol 128 (2) ◽  
pp. 170-175 ◽  
Author(s):  
C. Mei
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Variable Coefficients ◽  
Mode Shapes ◽  
Governing Equations ◽  
Transformation Technique ◽  
Differential Transformation ◽  
Transformation Approach

In this paper, the differential transformation approach is applied to analyze the free vibration of centrifugally stiffened Timoshenko beam structures. Such structures involve variable coefficients in the governing equations, which in general cannot be solved analytically in closed form. Both the natural frequencies and the mode shapes are obtained using the differential transformation technique. Numerical examples are presented and results are compared with available results in the literature.

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.

Buckling and Free Vibration of Nonuniformly Heated Functionally Graded Carbon Nanotube Reinforced Polymer Composite Plate

2017 ◽  
Vol 17 (06) ◽  
pp. 1750064 ◽  
Author(s):  
Nivish George ◽  
P. Jeyaraj ◽  
S. M. Murigendrappa
Keyword(s):  
Temperature Field ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Plate ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Functionally Graded ◽  
Temperature Fields ◽  
Mode Shapes ◽  
Reinforced Polymer

Buckling and free vibration behavior of functionally graded carbon nanotube reinforced polymer composite plate subjected to nonuniform temperature fields have been investigated using finite element approach. The effective material constants of the plate are obtained using the extended rule of mixture along with efficiency parameters of the carbon nanotube (to include geometry-dependent material properties). Influence of boundary conditions, aspect ratio, functional grading of the carbon nanotube, nonuniform thermal loading on thermal buckling and free vibration behavior of the heated plate are analyzed. It is observed that temperature fields and functional grading are influenced on the critical buckling temperature of the plates. Further, nature of functional grading showed significant change in buckling mode shapes irrespective of the boundary conditions. The first few natural frequencies of the plate under thermal load decreases as the temperature increases and they are influenced significantly by the nature of temperature field. Variations in free vibration mode shapes of the square plates found with not significant change as temperature increases. However, free vibration modes of the rectangular plates are sensitive to the nature of temperature field whenever there is a free edge associated with the boundary condition. Influence of functional grading on the free vibration mode shapes is not significant in contrast with the free vibration natural frequencies. The magnitude of free vibration natural frequencies of functional grade-X type carbon nanotube reinforcement showed higher in comparison with other two types of reinforcements considered here.

Geometrically Non-Linear Free Vibration of Fully Clamped FGM Skew Plates Using Homogenization Technique

2015 ◽  
Vol 1105 ◽  
pp. 370-380
Author(s):  
Moulay Abdelali Hanane ◽  
Khalid El Bikri ◽  
Benamar Rhali
Keyword(s):  
Free Vibration ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Power Law Distribution ◽  
Skew Angle ◽  
Linear Mode ◽  
Homogenization Technique ◽  
Skew Plate ◽  
Non Linear ◽  
Good Agreement

The present work concerns the geometrically non-linear free vibration of fully clamped functionally graded skew plates (FGSP). The theoretical model based on Hamilton’s principle and spectral analysis is used. A homogenization technique has been developed to reduce the FGSP problem under consideration to that of an isotropic homogeneous skew plate. The material properties of the skew plate examined herein are assumed to be graded in the thickness direction of the plate according to the power-law distribution in terms of volume fractions of the constituents. Results are given for the linear and non-linear fundamental frequency considering different parameters. The non-linear mode shapes exhibit a maximum value in the bending stress at the centre of the plate. It is found also that the non-linear frequencies increase with increasing the amplitude of vibration and increasing the skew angle, which corresponds to the hardening type effect. A good agreement is found with published results.

scholarly journals Free vibrations of axially functionally graded horseshoe arch

2021 ◽  
Vol 9 (3) ◽  
pp. 251-262
Author(s):  
Gweon Sik Kim ◽  
Sang Jin Oh ◽  
Tae Eun Lee ◽  
Byoung Koo Lee
Keyword(s):  
Mechanical Properties ◽  
Finite Element ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Mass Density ◽  
Quadratic Function ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Parametric Studies

This paper deals with free vibrations of the axially functionally graded (AFG) horseshoe arch. The modulus of elasticity and the mass density of AFG material of arch are chosen as a univariate quadratic function. The differential equations with the boundary conditions that govern the free vibration of such arch are derived and numerically solved to calculate natural frequencies and mode shapes. Natural frequencies of this study agree well with those of the finite element ADINA. Parametric studies of the geometrical and mechanical properties of the arch on frequencies and mode shapes are performed and extensively discussed.

scholarly journals Free Vibration Analysis for Cracked FGM Beams by Means of a Continuous Beam Model

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
E Chuan Yang ◽  
Xiang Zhao ◽  
Ying Hui Li
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Crack Depth ◽  
Free Vibration Analysis ◽  
Bending Stiffness ◽  
Functionally Graded ◽  
Beam Model ◽  
3D Fem ◽  
Fgm Beam

Based on Euler-Bernoulli beam theory and a continuous stiffness beam model, the free vibration of rectangular-section beams made of functionally graded materials (FGMs) containing open edge cracks is studied. Assuming the material gradients follow exponential distribution along beam thickness direction, the conversion relation between the vibration governing equations of a FGM beam and that of an isotropic homogenous beam is deduced. A continuous function is used to characterize the bending stiffness of an edge cracked FGM beam. Thus, the cracked FGM beam is treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The characteristic equations of beams with different boundary conditions are obtained by transfer matrix method. To verify the validity of the proposed method, natural frequencies for intact and cracked FGM beams are calculated and compared with those obtained by three-dimensional finite element method (3D FEM) and available data in the literature. After that, further discussions are carried out to analyze the influences of crack depth, crack location, material property, and slenderness ratio on the natural frequencies of the cracked FGM beams.

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