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.

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.

Effects of Damaged Boundaries on the Free Vibration of Kirchhoff Plates: Comparison of Perturbation and Spectral Collocation Solutions

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
Ma’en Sari ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Free Vibration ◽  
Collocation Method ◽  
Separation Of Variables ◽  
Free Vibration Analysis ◽  
Perturbation Parameter ◽  
Damage Parameter ◽  
Chebyshev Collocation Method ◽  
Chebyshev Collocation ◽  
Kirchhoff Plates ◽  
Grid Points

In order to compare numerical and analytical results for the free vibration analysis of Kirchhoff plates with both partially and completely damaged boundaries, the Chebyshev collocation and perturbation methods are utilized in this paper, where the damaged boundaries are represented by distributed translational and torsional springs. In the Chebyshev collocation method, the convergence studies are performed to determine the sufficient number of the grid points used. In the perturbation method, the small perturbation parameter is defined in terms of the damage parameter of the plate, and a sequence of recurrent linear boundary value problems is obtained which is further solved by the separation of variables technique. The results of the two methods are in good agreement for small values of the damage parameter as well as with the results in the literature for the undamaged case. The case of mixed damaged boundary conditions is also treated by the Chebyshev collocation method.

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.

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