On the vibrations of axially graded Rayleigh beams under a moving load

•Vibration of axially graded Rayleigh beams under a moving loads is investigated.•Closed-form expressions for the occurrence of various phenomena are presented.•Effects of geometrical and material characteristics on the vibration are examined. In the present investigation, the forced and free vibrations of axially functionally graded (AFG) Rayleigh and Euler-Bernoulli (EB) beams subjected to a moving load are studied and compared, aiming at performance enhancement of transportation systems. Also, for the first time, a precise mathematical modeling is obtained to analyze the influence of various key factors such as axial material gradation and rotary inertia factor on the critical speed, dynamic magnification factor, mechanisms of cancellation, and maximum free vibration of the system. Model verification is performed with the available results in the literature, and a good agreement is observed. Furthermore, the dynamical responses of the system acquired from the analytical and numerical approaches are in good agreement. It is demonstrated that for the gradient parameter, which is lower and higher than the critical value, the material properties variation has a reverse effect on the forced-free vibration amplitudes. Besides, it is concluded that, compared with the conventional isotropic EB beams, by selecting appropriate values of rotary inertia factor and gradient parameter in the AFG Rayleigh beams, the cancellation and maximum free vibration phenomena can be controlled. The results of this study can serve as a comprehensive benchmark to optimally design inhomogeneous structures under moving loads.