Free vibration analysis of a type of tapered beams by using Adomian decomposition method

Free Transverse Vibration Analysis of Axially Functionally Graded Tapered Euler-Bernoulli Beams through Spline Finite Point Method

“…And so far, few analytical solutions are found for the free vibration of the axially FG tapered beams due to the difficulty of mathematical treatment of the problem; for example, Elishakoff et al [12,13] proposed the close-form solutions for axially FG tapered beam with material gradation that follows the polynomial law using the semi-inverse method; however, the proposed semi-inverse method cannot be applied for graded beams of any axial nonhomogeneity. Therefore, a large number of numerical methods have been proposed in the reported literatures by other researchers, according to which lots of numerical models, based on different beam theories, namely, the Euler-Bernoulli beam theory (EBT) [2,3,[14][15][16][17][18][19][20] and the Timoshenko beam theory (TBT) [21][22][23][24][25], have been constructed to investigate the free vibration of the tapered beams considering the material property homogeneity or nonhomogeneity. However, it is well understood that the effects of the shear deformation and rotary inertia of the cross section for the long slender beams can be neglected; thus the EBT models are capable of analyzing the vibration problems of the long slender beams and the reliable results with high accuracy can be achieved.…”

A rigid multibody method for free vibration analysis of beams with variable axial parameters

“…The other cases use numerical methods for solving the mode shape equation obtained by applying the method of separation of variables. Thus, the Frobenius method was used by Naguleswaran (1992Naguleswaran ( , 1994aNaguleswaran ( , 1994b, the Rayleigh-Ritz method was used by Kim and Dickinson (1988), and the Adomian decomposition method was applied by Hsu et al (2008) and Mao and Pietrzko (2012). In Tong et al (1995), the analytical solution for vibrations of stepped Timoshenko and Euler-Bernoulli beams was given.…”

Free Vibrations of Spinning Beams Under Nonclassical Boundary Conditions Using Adomian Modified Decomposition Method

“…The AMDM has been applied to the free vibration problems for beam structures, and the method has furnished reliable results in providing analytical approximation that converges rapidly. [11][12][13][14][15] By using the AMDM, the governing di®erential equations for the spinning beam become a recursive algebraic equation system. The boundary conditions become simple algebraic frequency equations that are suitable for symbolic computations.…”