# DAMPING EFFECT ON THE DYNAMIC RESPONSE OF SIMPLY SUPPORTED DAMPED BEAM UNDER THE ACTION OF DISTRIBUTED FORCES RESTING ON BI-PARAMETRIC FOUNDATION

FAMUAGUN K. S

Department of Mathematics,

Adeyemi College of Education Ondo

ABSTRACT

This paper concerns the dynamic response of simply supported damped beam resting on bi-parametric foundation. The damped beam-type structural members considered is uniform elastic beam structures. Two types of damping parameter are incorporated in the beam structure formulation. In most of the works available in literature, only damping due to viscous resistance to transverse displacement of the beam has been considered. The governing equations of motion of the dynamical systems is a fifth order non-homogenous partial differential equation with variable and singular coefficients. The main objective of this study is to obtain closed form solutions to this class of dynamical problems for simply supported boundary conditions. Subsequently, the closed form solutions are analyzed.

In order to obtain this closed form solutions to this dynamical problem, the generalized finite integral transform is employed to obtain 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 Laplace transformation in conjunction with the initial condition and convolution theory are used to obtain analytical solutions to the dynamical problem. From the closed form solutions obtained, theresults show that the response amplitudes ofthe uniform damped beam under the action of distributed forces decreases as the values of damping due to resistance to strain velocity are increases for the simply supported 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. The transverse displacement response for the partially distributed moving force were calculated for various time and the various results obtained were presented in plotted curves. Analyses further show that the 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 reduce the response amplitudes of the uniform damped beam resting on bi-parametric foundation under distributed loads.