DAMPING EFFECTS ON THE DYNAMIC RESPONSE TO MOVING DISTRIBUTED MASSES OF BEAM-TYPE STRUCTURES RESTING ON VLASOV FOUNDATION

Abstract

This thesis concerns the dynamic response to moving distributed masses of damped elastic beam-type structural members resting on Vlasov foundation. The damped beam-type structural members considered are uniform and non-uniform elastic beam structures. Two types of damping are incorporated into the beam structure formulation. The governing equations of motion of the dynamical systems in both cases are fifth order non-homogenous partial differential equations with variable and singular coefficients. The main objective of this study is to obtain closed form solutions to this class of dynamical problems for all variants of classical boundary conditions. Subsequently, the closed form solutions are analyzed. In order to obtain these closed form solutions to these dynamical problems, an approach due to the generalized finite integral transform is employed in the case of uniform damped beam-type problem to obtain a sequence of coupled second order Ordinary Differential Equations. In case of non-uniform damped beam-type problem, the generalized Galerkin’s method was used to simplify and reduce the fifth order partial differential equation with variable and singular coefficients to a sequence of coupled second order Ordinary Differential Equations. Since the resulting coupled second order Ordinary Differential Equations do not yield readily to classical methods in the two cases, a modified asymptotic method of Struble was then used to simplify the equations, while Laplace transformation in conjunction with the initial condition and convolution theory are used to obtain analytical solutions to the dynamical problems. From the closed form solutions obtained, in the case of uniform damped beam-type problem resting on Vlasov foundation the results show that the response amplitudes of the uniform damped beam under the action of distributed forces and masses decrease as the values of damping due to resistance to strain velocity are increased for all variants of classical boundary conditions considered. The same behavior was observed for the damping due to resistance to transverse displacement. However, damping due to resistance to transverse displacement has a more pronounced effect in reducing the response amplitude of the beam-type structure than the damping due to resistance to strain velocity. From the closed form solutions obtained in the case of uniform damped beam-type structure resting on Vlasov foundation for the same natural frequency, 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 than in the moving distributed force problem. The same results obtains in the cases of non-uniform damped beams. The transverse displacement response for the partially distributed moving force and partially distributed moving mass were calculated for various time and the various results obtained were presented in plotted curves. From all illustrative examples, it found that the moving distributed force solution is not an upper bound for the accurate solution for the moving distributed mass problem in the two cases of structural members considered. Thus, the inertia term of the moving distributed load often neglected must be considered for accurate computation and assessment of the vehicle-track interaction. Analyses further show that for all variants of classical boundary conditions, an increase in the values of the structural parameters namely, axial force N, foundation stiffness K, shear modulus G and rotatory inertia R0 reduces the response amplitudes of the uniform damped beam resting on Vlasov foundation under distributed loads moving. The same behaviour characterizes the non-uniform damped beam-type structure resting on Vlasov foundations. It was however noted that higher values of the structural parameters are required for a more noticeable effect on the structural response in the case of other boundary conditions than the case of simply supported boundary conditions for both moving distributed force and moving distributed mass problems.