DYNAMIC ANALYSIS OF ELASTIC STRUCTURES RESTING ON BI-PARAMETRIC VLASOV FOUNDATIONS AND UNDER DISTRIBUTED MASSES MOVING AT VARYING VELOCITIES

Abstract

In this thesis, the problem of the dynamic analysis of elastic structures resting on Bi-parametric Vlasov foundation namely; uniform Rayleigh beams, non-uniform Rayleigh beams and isotropic rectangular plates under the actions of partially distributed masses travelling at varying velocities is studied. Most of the works available in literature are those involving concentrated loads and the foundation on which the structures lie have been the widely criticized Winkler foundation. The governing equations of motion of the dynamical systems in all the cases are fourth order non-homogeneous partial differential equation with variable and singular coefficients. As most of the authors in open literature concentrate on numerical simulation, 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. The reason for this is simple. Solutions so obtained often shed light on vital information about the vibrating system. Subsequently, the closed form solutions are analysed. In order to achieve the above purpose, the Heaviside function which describes the partially distributed load was first, expressed in series form so as to simplify the transformation of the partial differential equation. An approach due to the generalized finite integral transform is then employed in the case of uniform Rayleigh beam to obtain a sequence of coupled second order Ordinary Differential Equations. In case of non-uniform Rayleigh beam, the former technique fails and the generalized Galerkin’s method was then resorted to in order to simplify and reduce the fourth order partial differential equation with variable and singular coefficients to a sequence of coupled second order Ordinary Differential Equations. For the two-dimensional isotropic rectangular plate problem, the twodimensional analogue of the generalized finite integral transform mentioned above was employed to transform the governing equation of motion 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, a modified asymptotic method of Struble was then used to simplify the equations, while variation of parameters technique was employed in conjunction with Fresnel sine and Fresnel cosine identities to obtain analytical solutions to the dynamical problems. From the closed form solutions obtained in the case of uniform Rayleigh 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. The same results obtained in the cases of non-prismatic Rayleigh beams and isotropic rectangular plate resting on Vlasov foundations and under the actions of distributed loads travelling at varying velocities. Furthermore, the transverse displacements response for the partially distributed moving force and partially distributed moving mass were calculated for various time t and the various results obtained were presented in plotted curves. From all illustrated examples, it was 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 three structural members considered. Thus, the inertial term of the moving distributed load often neglected must be considered for accurate computation of the vehicle-track interaction. Thus, important result has also been reported for the cases of structures resting on Winkler foundation and carrying concentrated moving loads. 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 correction factor R0 reduce the response amplitudes of the uniform Rayleigh beam resting on Vlasov foundation and under distributed loads moving with varying velocities. The same behaviour characterizes the non-prismatic Rayleigh beam and isotropic rectangular plate 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.