Vibration characteristics of axially loaded tapered Timoshenko beams made of functionally graded materials by the power series method

In the present article, a semi-analytical technique to investigate free bending vibration behavior of axially functionally graded non-prismatic Timoshenko beam subjected to a point force at both ends is developed based on the power series expansions. The beam is assumed to be made of linear elastic and isotropic material with constant Poisson ratio. Material properties including the elastic modulus and mass density vary continuously through the beam axis according to the volume fraction of the constituent materials based on exponential and power-law formulations. Based on Timoshenko beam assumption and using small displacements theory, the free vibration behavior is governed by a pair of second order differential equations coupled in terms of the transverse deflection and the angle of rotation due to bending. According to the power series method, the exact fundamental solutions are found by expressing the variable coefficients presented in motion equations including cross-sectional area, moment of inertia, material properties and the displacement components in a polynomial form. The free vibration frequencies are finally determined by solving the eigenvalue problem of the obtained algebraic system. Four comprehensive examples of axially non-homogeneous Timoshenko beams with variable cross-sections are presented to clarify and demonstrate the performance and convergence of the proposed procedure. Moreover, the effects of various parameters like crosssectional profile and material variations, taper ratios, end conditions and concentrated axial load are evaluated on free vibrational behavior of Timoshenko beam. The obtained outcomes are compared to the results of finite element analysis in terms of ABAQUS software and those of other available numerical and analytical solutions. The competency and efficiency of the method is then remarked.