Dynamic Characteristics of Functionally Graded Timoshenko Beams by Improved Differential Quadrature Method

This study proposes an effective method to enhance the accuracy of the Differential Quadrature Method (DQM) for calculating the dynamic characteristics of functionally graded beams by improving the form of discrete node distribution. Firstly, based on the first-order shear deformation theory, the governing equation of free vibration of a functionally graded beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam axial displacement, transverse displacement, and cross-sectional rotation angle by considering the effects of shear deformation and rotational inertia of the beam cross-section. Then, ignoring the shear deformation of the beam section and only considering the effect of the rotational inertia of the section, the governing equation of the beam is transformed into the eigenvalue problem of ordinary differential equations with respect to beam transverse displacement. Based on the differential quadrature method theory, the eigenvalue problem of ordinary differential equations is transformed into the eigenvalue problem of standard generalized algebraic equations. Finally, the first several natural frequencies of the beam can be calculated. The feasibility and accuracy of the improved DQM are verified using the finite element method (FEM) and combined with the results of relevant literature.