FLEXURAL VIBRATIONS OF PRESTRESSED BERNOULLI-EULER BEAM RESTING ON ELASTIC FOUNDATION AND TRAVERSED BY MASSES TRAVELING AT VARIABLE SPEEDS

Abstract

The problem of flexural vibrations of prestressed Bernoulli-Euler beam resting on elastic foundation and traversed by concentrated masses traveling at variable speeds is studied in this thesis. Both cases of uniform and non-uniform Bernoulli-Euler beams involving fourth order partial differential equations having variable and singular coefficients are considered. Foremost, closed form solutions are obtained for both the problems of uniform and non-uniform Bernoulli-Euler beams. The solution technique is based on the generalized integral transforms, the generalized Galerkin' s method, the expansion of the Dirac Delta function in series form, a modification of the Struble's asymptotic method and the use of the generating functions of the Bessel functions. An important features of this robust technique is that it is applicable for all variants of classical boundary conditions for this class of problems. The closed form solutions are analyzed and numerical analysis in plotted curves are presented. The results show that for the same natural frequency, the critical speed for the uniform Bernoulli-Euler beams traversed by moving force is greater than that under the influence of a moving mass for both uniform and non-uniform Bernoulli-Euler beams. Hence resonance is reached earlier in the moving mass problem. The same results are obtained for the non-uniform Bernoulli-Euler beams. Furthermore, for fixed values of axial force N and foundation modulus K in all the illustrative examples considered, the moving force solution is not an upper bound for the accurate solution of the moving mass solution. It is also found that as the axial force N and the foundation modulus K increases, the amplitudes of both uniform and non-uniform Bernoulli-Euler beams under the action of moving loads traveling with variable velocities decrease. However, higher values of N and K are required for more noticeable effects in the case of other boundary conditions than those of simply supported end conditions. Finally, it is observed that relying on the moving force problem as a good approximation to a moving mass problem is not only misleading, but it is tragic.