A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads

A mixed method is presented to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads. The theoretical formulations are based on Euler–Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations. The Rayleigh–Ritz method is employed to discretize the spatial partial derivatives and a step-by-step differential quadrature method (DQM) is used for the discretization of temporal derivatives. It is shown that the proposed mixed method is very efficient and reliable. Also, compared to the single-step methods such as the Newmark and Wilson methods, the DQM gives better accuracy using larger time step sizes for the cases considered. Moreover, effects of material properties of the FG beam and inertia of the moving load on the dynamic behavior of the system are investigated and analyzed.