FLEXURAL MOTIONS UNDER MOVING DISTRIBUTED MASSES OF BEAM-TYPE STRUCTURES ON VLASOV FOUNDATION AND HAVING TIME-DEPENDENT BOUNDARY CONDITIONS

Abstract

This thesis focuses on dynamic response to moving distributed masses of beam-type of elastic structural members resting on Vlasov foundation having time-dependent boundary conditions. Beam-type structures considered is uniform and non-uniform elastic beams. The response of structural members under moving loads generally involves classical boundary condition where the end conditions are homogenous. However, it happens frequently, that vibration problems have to deal with continuous system of which one or more boundaries are constrained to undergo displacements or traction which varies with time. In this case, the dynamical problem involves a boundary-initial-value problem consisting of a non-homogenous partial differential equation of motion having non-homogenous boundary and initial condition. In cases like this, the boundary conditions are not stationary and on this account, solutions are not in general, obtainable by the classical methods. The main purpose of this study is to obtain closed form solutions to this class of dynamical problems for all variants of classical boundary conditions. In the first instance, the Mindlin and Goodman’s technique is used to transform the governing non-homogeneous fourth order partial differential equations with non-homogeneous boundary conditions into non-homogeneous fourth order partial differential equations with homogeneous boundary conditions. The resultant simplified equations are then treated using the Generalized Finite Integral transform or Generalized Galerkin’s method with series representation of Heaviside function, a modification of Struble’s asymptotic methods and the integral transformation techniques in conjunction with convolution theory. An important feature of this technique is that it is capable of tackling this class of problems for all variants of commonly encountered time-dependent boundary conditions. From the closed form solutions obtained in both cases of uniform and non- uniform beams resting on Vlasov foundation for the same natural frequency, the critical speed for the moving distributed mass problem is smaller than that of the moving distributed force problem for all variants of classical boundary conditions considered. Thus, resonance is reached earlier in the moving distributed mass problem. Furthermore, the transverse displacements response for the moving distributed force and moving distributed mass were calculated for various time t and the various results obtained were presented in plotted curves. Also, it is found that the moving distributed force solution is not an upper bound for the accurate solution for the moving distributed mass problem in the cases of all the structural members considered. Thus, the inertia term of the moving distributed load often neglected must be considered for accurate computation of the vehicle-track interaction. Finally analyses further show that an increase in the values of the structural parameters namely, axial force N, foundation stiffness K, shear modulus G and rotatory inertia correction factor R0 reduce the response amplitudes of the uniform and non-uniform elastic beam resting on Vlasov foundation for all variants of classical boundary conditions, and under moving distributed loads with time-dependent boundary conditions.