The dynamic response to moving masses of rectangular plates with general classical boundary conditions and resting on variable Winkler elastic foundation is investigated in this work. The governing fourth order partial differential equation is solved using a technique based on separation of variables, the modified method of Struble and the integral transformations. Numerical results in plotted curves are then presented. The results show that as the value of the rotatory inertia correction factor Ro increases, the response amplitudes of the plate decrease and that, for fixed value of Ro, the displacements of the plate decrease as the foundation modulus Fo increases for the variants of the classical boundary conditions considered. The results also show that for fixed Ro and Fo, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. For the rectangular plate, for the same natural frequency, the critical speed for moving mass problem is smaller than that of the moving force problem for all variants of classical boundary conditions, that is, resonance is reached earlier in moving mass problem than in moving force problem. When Fo and Ro increase, the critical speed increases, hence, risk is reduced.
This work investigates the problem of dynamic response to variable-magnitude moving distributed masses of Bernoulli-Euler beam resting on bi-parametric elastic foundation. The governing equation is a fourth order partial differential equation with variable and singular co-efficients. This equation is reduced to a set of coupled second order ordinary differential equation by the method of Garlerkin. For the solutions of these equations, two cases are considered; (1) the moving force case -when the inertia is neglected and (2) the moving mass case -when the inertia term is retained. To solve the moving force problem, the Laplace transformation and convolution theory are used to obtain the transversedisplacement response to a moving variable-magnitude distributed force of the Bernoulli-Euler beam resting on a bi-parametric elastic foundation. For the solution of the moving mass problem, the celebrated struble's technique could not simplify the coupled second order ordinary differential equation with singular and variable co-efficient because of the variability of the load magnitude; hence use is made of a numerical technique, precisely the Runge-Kutta of fourth order is used to solve the moving mass problem of the response to variable-magnitude moving distributed masses of Bernoulli-Euler beam resting on Pasternak elastic foundation. The analytical and the numerical solutions of the moving force problem are
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