Dynamic Response to Moving Distributed Masses of Pre-stressed Uniform Rayleigh Beam Resting on Variable Elastic Pasternak Foundation

The dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation is examined. The equation governing this problem is a fourth order partial differential equation with variable and singular co-efficients. To solve this cumbersome equation, the method of Galerkin approach is adopted to reduce the governing differential equation to a sequence of coupled second order ordinary differential equation which is then simplified further with modified asymptotic method of Struble. The more simplified equation is solved using the Laplace transformation technique. The closed form solutions obtained are analyzed in order to show the conditions of resonance, and to show that resonance is attained earlier in moving mass system than in the moving force system. The results in plotted graphs show that as the axial force, the rotatory inertia, foundation modulus and shear modulus increase, the deflection of the elastically supported non-uniform Rayleigh beam decreases in each case. The transverse deflections of the beam on variable Pasternak elastic foundation are higher under the action of moving masses than those when only the force effects of the moving load are considered. This implies that resonance is reached faster in moving mass problem than in moving force problem.