International Journal of Multidisciplinary Research and Publications ISSN (Online): 2581-6187 257 Ogunyebi Segun Nathaniel, Adedowole Alimi, and Famuagun Kayode Samuel, “Response of Non-uniform Damped Beam to Mobile Distributed Loads on Variable Elastic Foundation,” International Journal of Multidisciplinary Research and Publications (IJMRAP), Volume 6, Issue 2, pp. 257-260, 2023. Response of Non-uniform Damped Beam to Mobile Distributed Loads on Variable Elastic Foundation Ogunyebi Segun Nathaniel 1 , Adedowole Alimi 2 , Famuagun Kayode Samuel 3 1 Department of Mathematics, Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria-+234 2 Department of Mathematical Sciences, Adekunle Ajasin University, Akungba-Akoko, Ondo State, Nigeria-+234 3 Department of Mathematics, Adeyemi College of Education, Ondo, Ondo State, Nigeria-+234 Email address: segun.ogunyebi@eksu.edu.ng, alimi.adedowole@aaua.edu.ng, famuagunks@ace.edu.ng Abstract—An analytical procedure for the flexural motion of non- uniform structure on variable subgrade with moving weight is studied. The motion equation is solved by an assumed mode technique to obtain second order differential equation which is then solved by Laplace method and convolution theory. The effects of the variable elastic foundation, as well as damping intensity and torsional rigidity of the prismatic beam having moving distributed weight are assessed and the results displayed in plotted curves. Keywords— Damping, Moving load, Non-uniform beam, Torsional rigidity, Variable foundation. I. INTRODUCTION Owing to its significant relevance in structural and construction engineering, extensive works have been conducted to predict the behavioral pattern of beam-like on foundation subgrade traversed by the presence of load moving at uniform speed [1-4]. The appearances of damping in vibration of elastic structures have great effects and also useful for design engineering. Many authors have worked in this subject area for both beam and plate structural element. Crandall [5] studied the behavior of damping in role and special area where small amount of damping has an exaggerated importance in determining the dynamic behavior of a system are examined. Mousa and Reza [6] gave novel approach for free vibrational synthesis of the cracked cantilever beam having a breaking crack by taking into account the effect of the distributed structural damping. Robin and Rana [7] analyzed the vibrations of isotropic/orthotropic damped plate whose thickness vary and lying on foundation. Practical problems in the structural dynamics especially under moving loads considers beam parameter to be vary. Here, the distribution of the non-uniform characteristics may be assumed as power function [8]. Also, when the structure has variable cross-section that is the beam parameters, mas and moment of inertia are considered as varying along the length of the structure [9, 10]. Gutierrez and Laura [11] presented dynamical analysis of a non-uniform cross-sectional structure traversed by concentrated load. In a later development, the vibrational behavior of a beam with cross- section beam having concentrated mass and force by FEM was considered by Ahmadian et al [12]. Taha [13] obtained a closed form solution for damped free vibration of a non- uniform shear beam resting on an elastic foundation. Recent studies on distributed moving weight lying on elastic subgrade were addressed by authors [14-17]. Zhong et al [18] presented dynamic instability of a simply supported rectangular plate attached with the arbitrary concentrated masses owing to parametric resonance excited by an in-plane uniformly distributed periodic load along two opposite edges. Ogunyebi et al [19] examined vibration of non-prismatic beam-like lying on variable foundation subgrade by mobile concentrated forces. In the paper, the dynamic effect of rotatory inertia is neglected. Very recently, Ogunyebi [20] developed an analytical procedure for the solution of plate type structural members due to the influence of torsional rigidity and other vital parameters. The fourth order governing differential equation is addressed by the versatile method of Shadnam et al. In the work, a rise in the values of these structural parameters produces a noticeable effect on the critical velocity of the plate – type member. In this present study, the effects of damping and torsional rigidity on variable non-uniform elastic foundation under moving distributed load is extensively studied to determine the dynamic role of the vital input characteristics in the motion equation. II. THEORY AND FORMULATION Fig. 1: Schematic diagram of moving distributed loads on non-uniform beam Let us consider non-uniform thin beam moving with constant velocity lying on variable foundation subgrade and subjected to MDM system. Thus, vibration equation is 2 2 2 * 2 2 2 (,) (,) () () W t W t EJ m t            +       