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

Abstract

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 Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

“…where Cb and Ch denote the width and height taper ratios, respectively, and E 0 , I 0 ,  0 , and A 0 denote the elastic modulus, the moment of inertia, the beam's mass per unit volume, and the cross-sectional area, respectively, at x = 0. The results of this study are in good agreement with those of Chen [2].…”

“…Various studies have focused on the vibration analysis of AFG Euler−Bernoulli beams. Chen [2] investigated the bending behavior of a non-uniform AFG Euler−Bernoulli beam based on the Chebyshev collocation method; the Chebyshev differentiation matrices were used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem associated with the free vibration. Soltani et al [3] applied the FDM to evaluate natural frequencies of non-prismatic beams with different boundary conditions and resting on variable one-or two-parameter elastic foundations.…”

Vibration Analysis of Axially Functionally Graded Timoshenko Beams with Non-uniform Cross-section

“…Thus, the dynamic problems of engineering structures constructed from FGMs have received considerable attention, especially for the beam members commonly used in bridges, buildings and machine components. In the past decades, most studies numerically dealt with the dynamics of FGM beams with material properties graded in the thickness direction based on various beam theories (Simsek (2010); Thai and Vo (2012); Nguyen et al (2013); Pradhan and Chakraverty (2014); Su and Banerjee (2015); Wattanasakulpong and Mao (2015); Chen andChang (2017, 2018); Ding et al (2018); Esen (2019)). It is noted that the strength and weight of beam structures, which affect its vibration behavior, can be optimized by changing the crosssectional and material properties along the beam length direction.…”

Vibration Analysis of Axially Functionally Graded Non-Prismatic Euler-Bernoulli Beams Using the Finite Difference Method

“…Various studies have focused on the vibration analysis of AFG Euler−Bernoulli beams, most of which using numerical methods. Chen [2] investigated the bending behavior of a non-uniform AFG Euler−Bernoulli beam based on the Chebyshev collocation method; the Chebyshev differentiation matrices were used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem associated with the free vibration. Soltani et al [3] applied the FDM to evaluate natural frequencies of non-prismatic beams with different VIBRATION ANALYSIS OF AXIALLY FUNCTIONALLY GRADED EULER−BERNOULLI BEAMS boundary conditions and resting on variable one-or two-parameter elastic foundations.…”