This paper investigates the flexural vibration of a finite nonuniform Rayleigh beam resting on an elastic foundation and under travelling distributed loads. For the solution of this problem, in the first instance, the generalized Galerkin method was used. The resulting Galerkin’s equations were then simplified using the modified asymptotic method of Struble. The simplified second-order ordinary differential equation was then solved using the method of integral transformation. The closed form solution obtained was analyzed and results show that, an increase in the values of foundation moduli K and rotatory inertia correction factor R0 reduces the response amplitudes of both the clamped-clamped nonuniform Rayleigh beam and the clamped-free nonuniform Rayleigh beam. Also for the same natural frequency, the critical speed for the moving distributed mass problem is smaller than that for the moving distributed force problem. Hence resonance is reached earlier in the former. Furthermore, resonance conditions for the dynamical system are attained significantly by both R0 and K for the illustrative end conditions considered.

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The problem of the flexural vibrations of a rectangular plate having arbitrary supports at both ends is investigated. The solution technique which is suitable for all variants of classical boundary conditions involves using the generalized two-dimensional integral transform to reduce the fourth order partial differential equation governing the vibration of the plate to a second order ordinary differential equation which is then treated with the modified asymptotic method of Struble. The closed form solutions are obtained and numerical analyses in plotted curves are presented. It is also deduced that for the same natural frequency, the critical speed for the system traversed by uniformly distributed moving forces at constant speed is greater than that of the uniformly distributed moving mass problem for both clamped-clamped and simple-clamped end conditions. Hence resonance is reached earlier in the uniformly distributed moving mass system. Furthermore, for both structural parameters considered, the response amplitude of the moving distributed mass system is higher than that of the moving distributed force system. Thus, it is established that the moving distributed force solution is not an upper bound for an accurate solution of the moving distributed mass problem.

This paper presents a mini review on the moving distributed force problem of a non-uniform prestressed simply supported Rayleigh beam resting on an elastic foundation. The problem, which is reduced to a nonhomogeneous second order ordinary differential equation, is tackled by means of Laplace transforms and Convolution theory. Analytical solutions are presented for the boundary condition under consideration. Conditions under which resonance occur are established. The theoretical considerations find application in calculations relating to dynamic stresses in railways, bridges and foundation for all types of highways.

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