ROTATORY INERTIA INFLUENCE ON THE RESPONSE TO VARIABLE-MAGNITUDE MOVING LOADS OF RAYLEIGH BEAM RESTING ON PASTERNAK FOUNDATION

Abstract

The problem of evaluating the Rotatory inertia influence on the response to variable- magnitude moving loads of Rayleigh Beam resting on Pasternak foundation is investigated in this research. The problem is governed by fourth order partial differential equation. The Generalized Finite Integral Technique with the series representation of Heaviside function is used to reduce the equation to a set of second order ordinary differential equations. The Laplace transform technique in conjunction with convolution theory is used to obtain the transverse displacement response to a moving force of the dynamical problem. The moving mass case of the dynamical problem is analytically unsolvable due to the variability of the loads magnitude. Hence, the Finite Element Method in conjunction with Newmark method is first used to solve the moving force case of the problem and the solution compared with the analytical solution of the moving force in order to validate the accuracy of the Finite Element Method in solving this kind of dynamical problem. The Finite element method is then used to solve the moving mass case of the problem. The numerical solution is shown to compare favorably with the analytical solution of the moving force problem. The result in plotted curves show that the response amplitudes of the beam decreases as the value of the Rotatory inertia(R) increases for both cases of moving force and moving mass problems. Also, it is shown that as the Pre-stress(N), foundation modulus(K) and shear modulus(G) increases, the deflection of the uniform Rayleigh beam decreases for both cases of moving force and moving mass. The results also show that the deflection amplitude of moving mass are higher than those of moving force and that at the same natural frequency, the critical speed for the uniform Rayleigh beam traversed by moving force is greater than that under the influence of a moving mass. Hence, resonance is reached earlier in the moving mass problem.Therefore, the ivmoving force solution is misleading as it is not a safe approximation to the moving mass problem, this indicate that the inertia terms must be considered for accurate evaluation of the response to moving loads of dynamical problems.