scholarly journals Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams Based on Chebyshev Collocation Method

2020 ◽  
Vol 25 (3) ◽  
pp. 436-444
Author(s):  
Wei-Ren Chen
Keyword(s):  
Collocation Method ◽  
Volume Fraction ◽  
Axial Direction ◽  
Functionally Graded ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Bending Vibration ◽  
Cross Sectional ◽  
Chebyshev Collocation ◽  
Uniform Cross Section

The bending vibration behavior of a non-uniform axially functionally graded Euler-Bernoulli beam is investigated based on the Chebyshev collocation method. The cross-sectional and material properties of the beam are assumed to vary continuously across the axial direction. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem associated with the free vibration. Some calculated results are compared with numerical results in the published literature to validate the accuracy of the present model. A good agreement is observed. The effects of the taper ratio, volume fraction index, and restraint types on the natural frequency of axially functionally graded beams with non-uniform cross section are examined.

Vibration Analysis of Third-Order Shear Deformable FGM Beams with Elastic Support by Chebyshev Collocation Method

2018 ◽  
Vol 18 (05) ◽  
pp. 1850071 ◽  
Author(s):  
Nuttawit Wattanasakulpong ◽  
Tinh Quoc Bui
Keyword(s):  
Collocation Method ◽  
Volume Fraction ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Constant Factor ◽  
Functionally Graded ◽  
Spring Constant ◽  
Chebyshev Collocation Method ◽  
Third Order ◽  
Chebyshev Collocation

In this paper, we present new results of natural frequencies for the functionally graded beams based on Chebyshev collocation method and the third-order shear deformation theory (TSDT), without requiring any shear correction factors. The beams are assumed to be elastically supported by translational and rotational springs, or simply known as elastically restrained ends. The material compositions of the beams across the gradient direction are described by different mathematical models including the simple power law, exponential and Mori–Tanaka models, and their effects on the response of beams are analyzed. We first present the Chebyshev collocation formulation of the coupled differential equations of motion for free vibration of FGM beams considering different boundary conditions, and then verify the results obtained by the proposed approach against reference ones. A parametric study is also performed for parameters such as thickness, spring constant factor, material volume fraction index, etc. The present numerical results reveal that the proposed method can offer accurate frequency results for the FGM beams as compared with those available in the literature. The results also indicate that the spring constant factors have a significant effect on the frequencies of the beams.

Vibration Analysis of Bidirectional Functionally Graded Timoshenko Beams Using Chebyshev Collocation Method

2020 ◽  
pp. 2150009 ◽  
Author(s):  
Wei-Ren Chen ◽  
Heng Chang
Keyword(s):  
Collocation Method ◽  
Vibration Analysis ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Timoshenko Beams ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Proposed Model ◽  
Good Agreement

This paper studies the vibration behaviors of bidirectional functionally graded (BDFG) Timoshenko beams based on the Chebyshev collocation method. The material properties of the beam are assumed to vary simultaneously in the beam length and thickness directions. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem for free vibration analysis. To validate the accuracy of the proposed model, some calculated results are compared with those obtained by other investigators. Good agreement has been achieved. Then the effects of slenderness ratios, material distribution types, gradient indexes, and restraint types on the natural frequency of BDFG beams are examined. Through the parametric study, the influences of the various geometric and material parameters on the vibration characteristics of BDFG beams are evaluated.

Vibration characteristics of multiple functionally graded nonuniform beams

2020 ◽  
pp. 107754632095676
Author(s):  
Ma’en S Sari ◽  
Sameer Al-Dahidi
Keyword(s):  
Collocation Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Spectral Collocation Method ◽  
Transverse Motion ◽  
Algebraic Equations ◽  
Spectral Collocation ◽  
Engineering Structures ◽  
Chebyshev Spectral Collocation ◽  
Chebyshev Spectral Collocation Method

Based on the Euler–Bernoulli beam theory, the natural vibration behavior of functionally graded nonuniform multiple beams has been investigated. It is assumed that the beams are joined by elastic translational springs, and the properties of the beams vary along the axial direction. The Chebyshev spectral collocation method has been used to convert the governing differential equations of transverse motion into a system of algebraic equations that are put in the matrix–vector form. Then, the dimensionless transverse frequencies are obtained by solving the eigenvalue problem. The influence of several factors, such as the stiffness parameters of the coupling translational springs, the properties of the cross section of the beams, and the boundary conditions on the frequencies, has been carried out. The results generated from the Chebyshev spectral collocation method have been verified by comparing them with those reported in other studies and references from the literature. Several numerical examples have been presented and discussed to analyze the system under consideration. The authors hope that the findings of the current study are helpful in designing and characterizing multiple nonuniform thin engineering structures.

Chaos Analysis and Control in Fractional Order Systems Using Fractional Chebyshev Collocation Method

2016 ◽  
Author(s):  
Arman Dabiri ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Collocation Method ◽  
Fractional Order ◽  
Fractional Order System ◽  
Order System ◽  
Integer Number ◽  
Algebraic Equations ◽  
Chebyshev Collocation Method ◽  
Differentiation Matrix ◽  
Chebyshev Collocation ◽  
Collocation Points

In this paper, fractional Chebyshev collocation method is proposed to study Lyapunov exponents (LEs) and chaos in a fractional order system with nonlinearities. For this purpose, the solution of the fractional order system is discretized by N-degree Gauss-Lobatto-Chebyshev (GLC) polynomials where N is an integer number. Then, the discrete orthogonality relationship for the Chebyshev polynomials is used to obtain the fractional Chebyshev differentiation matrix. The differentiation matrix is then used to convert the nonlinear fractional differential equations to a system of nonlinear algebraic equations with the collocation points as the unknowns. The dominant LE (other than the zero LE) that corresponds to the time dimension is then computed by measuring the exponential rate of the trajectory deviations initiated slightly off the attractor point. The proposed technique is implemented to a damped driven pendulum with fractional order damping and the convergence of the dominant LE is studied versus the number of Chebyshev collocation points. The LE analysis is also verified by studying the system time and frequency responses for different values of the bifurcation parameter. Furthermore, the LE obtained by the proposed method for the analogous integer order system is compared with those obtained by the Jacobian technique and Grüwald-Letnikov approximation. Finally a fractional state feedback controller is designed to control the chaotic system to a desired equilibrium or periodic trajectory such that the error dynamics are time invariant or time periodic, respectively. The numerical example studied is the damped driven pendulum with fractional dampers.

An embedded formula of the Chebyshev collocation method for stiff problems

2017 ◽  
Vol 351 ◽  
pp. 376-391 ◽  
Author(s):  
Xiangfan Piao ◽  
Sunyoung Bu ◽  
Dojin Kim ◽  
Philsu Kim
Keyword(s):  
Collocation Method ◽  
Stiff Problems ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation
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