In this paper, the buckling behaviors of axially functionally graded and non-uniform Timoshenko beams were investigated. Based on the auxiliary function and power series, the coupled governing equations were converted into a system of linear algebraic equations.
The present work utilizes the straight forward implementation utilizing the Chebyshev polynomial-to-cosine change of variable together with a proper selection of internal collocation points based on the boundary conditions. The differential equation that governs the free longitudinal vibration of a axially graded bar reads as :. Here l denotes the axial coordinate, u l , t denotes longitudinal displacement at any position l and time t.
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The classical boundary conditions considered are:. We now outline the solution methodology for Eq. Applying the transformation, the solution function W l is now a function of x and is assumed to be of the form:. It is to be noticed that the trigonometric form of the Chebyshev polynomial which is a change of coordinate is used for the representation of the approximate solution.
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This is coupled with analytic differentiation and collocation at internal points followed by enforcement of boundary conditions at end points. If we collocate Eq. Then we get a system of equations that is expressed as a matrix eigen value equation. The resulting equation is then solved using a standard eigensolver. Finally, we apply the method to obtain the non-dimensional frequencies of vibration of a functionally graded axial bar. In all the examples considered, the rods are of unit length. The details of the work carried out in this section are presented in Table 1.
A diagrammatic view of typical rods considered herein is given in Fig. Table 1. A layout of the numerical work carried out in Section 4. To study the convergence behaviour of the PS method in obtaining the frequency values of the vibration of non-uniform rods, some examples have been examined. Abrate  has presented exact analytical solution for longitudinal free vibration of bars with polynomial area variations. The exact non-dimensional frequency parameters of rods given  are set as basis values and the relative error of the PS method with N values varying from 10 to 16 is analyzed.
It is observed that the proposed method requires less number of grid points to obtain a relative error of the order of 10 -5 in the computation of the first six non-dimensional frequency values of a non-uniform rod with polynomial variation of cross section area. As a second example we examine the convergence of the PS method by considering the free axial vibration of a tapered fixed-free bar.
The area of cross-section A x and mass density are assumed to vary linearly i. The computations are carried out for a unit length rod.
Differential and Integral Equations
It is observed that the present method achieves a relative error of order 10 -6 with fewer grid points. A Schematic view of typical inhomogeneous rods. Table 2. Table 3. We examine our results with the non-dimensional frequency values presented in  as basis values. Table 4. Table 5. Comparison of non-dimensional frequencies of fixed-fixed non-uniform rods.
The GFEM method is an iterative approach whose main goal is to increase the accuracy of eigenvalues related to a chosen vibration mode. In the method  each tangent frequency was obtained by different iterative analysis. The results presented in  brought out the narrow precision of GFEM than the c-version of composite element method and the h -version of FEM in free longitudinal vibration analysis of uniform and non-uniform straight bars for the same degree of freedom.
Eigenvalues of Inhomogenous Structures by Laurie Kelly (2004, Hardcover)
The results show that the PS method which is easier and simple to implement achieves the same accuracy as that of the adaptive GFEM with relatively few grid points. The results obtained are given in Tables 6 and 7 and are found to be in good agreement with the exact results obtained by . Table 6. Table 7. Non-dimensional natural frequencies of clamped-free fixed-free rods with. A comparison of the values obtained is carried out with the corresponding the exact values in . The results bring out the accuracy of the method.
These values are presented along with the corresponding exact values given in . Though exact values are available for 3 significant digits, the approximate values are given for 6 significant digits for future comparison. Table 8. Table 9.
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Both the methods are known to considerably improve the convergence rate of conventional Differential Transform Method DTM. It is observed that the frequencies are significantly dependent on the cross-section area which is in turn dependent on both the height and breadth taper ratios. The ease of implementation is the added advantage. Table Non-dimensional frequencies of a fixed-free axially FG bar. It is observed from the table that the results obtained using PSM is slightly lower than that of . Moreover, it is observed from literature , that the stiffness method generally provides results that are higher than the exact ones.
We therefore, feel that the results obtained here are of high precision with fewer collocation points. Non-dimensional frequencies of a fixed-fixed axially FG bar. Non-dimensional frequencies of a FG tapered axial bar. The main contribution of the present study consists in applying a pseudospectral formulation for solving the governing differential equation of motion for tapered functionally graded axial bars.
The method utilizes a Chebyshev polynomial-to-cosine change of variable that simplifies the computer implementation and achieves good precision. Free longitudinal vibrations of functionally graded tapered axial bars by pseudospectral method Sri Harikrishna Pillutla 1 , Sudheer Gopinathan 2 , Vasudeva Rao Yerikalapudy 3. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Views Reads Downloads Introduction Functionally graded FG material structures find extensive use in modern engineering as their material properties can be tailored to meet the requirements of different applications . Pseudospectral method The spectral methods arise from the fundamental problem of approximation of a function by interpolation on an interval and are very much successful for the numerical solution of differential equations . Mathematical formulation and solution methodology The differential equation that governs the free longitudinal vibration of a axially graded bar reads as : 5.