Abstract:
The problem of the flexural vibrations of structural members namely
uniform Rayleigh beams, non-uniform Rayleigh beams and isotropic
rectangular plates under the actions of concentrated loads traveling at varying
velocities is studied in this thesis. The governing equations of motions in all
the cases are fourth-order partial di flerential equations with variable and
singular coefficients. The main objective of this study is to obtain closed form
solutions to this class of problem for all pertinent boundary conditions.
In order to obtain a closed form solution to these dynamical problems the
property or the Dirac delta function as an even function was used to express it
in series form and an approach due to the generalized finite integral transform
was employed in the case or uni form Rayleigh beam to obtain a sequence of
coupled second order ordinary differential equations. Unlike in the case of
uniform Rayleigh beam, the non-uniform Rayleigh beam is resistant to the
generalized integral transform technique. Thus, a modification of the
generalized Galerkin's method was resorted to in order to reduce the fourth
order partial differential equation governing the motion of the non-uniform
beam to a sequence of coupled second order ordinary differential equations. In
the case of two-dimensional plate problems, the two-dimensional analogue of
the generalized finite integral transform mentioned above was used to transform
the governing equation of motion to a sequence of coupled second order
ordinary differential equations. The modified asymptotic method of Struble
was then used to simplify these equations, while variation of parameters
technique was employed in conjunction with Fresnel sine and Fresnel cosine
