Parametric investigation of the size-dependent axially graded Rayleigh beams subjected to a moving load on Pasternak substrate

The vibration of axially graded Rayleigh and Euler-Bernoulli micro-beams under a moving load on Pasternak foundation is studied numerically and analytically. Accurate mathematical modeling is acquired to analyze the effect of various parameters such as longitudinal gradient parameter of material, whirling inertia factor, the stiffness of Pasternak foundation, and strain gradient parameter on the critical velocity, and cancellation mechanism and the maximum amplitude of vibrations. Natural frequencies are obtained and compared with available results in the technical literature. Closed-form expressions are extracted for dynamic magnification coefficient and maximum amplitude of free vibration. The changes in material characteristics of the system have inverse influences on the amplitude of free and forced vibrations for lower and higher values of the critical gradient parameters. It is concluded that in comparison with homogenous Euler-Bernoulli beams, in the axially graded Rayleigh micro-beams surrounded by shear Pasternak foundation. It can be controlled the cancellation and maximum free vibration phenomenon, by choosing the accurate values of gradient parameter, whirling inertia coefficient, the stiffness of foundation, and strain gradient parameter. Also, the results of the present study can be used as a criterion for the optimal design of heterogeneous structures under the moving loads.