Keywords:-
Article Content:-
Abstract
This research investigates the effect of flexural rigidity on dynamic response to moving load of damped shear beam resting on an Vlasov foundation when subjected to moving load traveling at a constant velocity. The governing equations are coupled second-order partial differential equations. The finite Fourier series method was employed to transform the coupled second-order partial differential equations into a set of coupled second-order ordinary differential equations. The resulting simplified equations that characterize the motion of the beam-load system were then solved using Laplace transformation alongside the convolution theory to derive the solutions. Extensive analyses were performed to assess the impact of flexural rigidity on the transverse displacement and rotation of damped shear beams of different lengths when subjected to the moving load traversing at a constant velocity. Furthermore, the research investigates how flexural rigidity affects the critical velocities of the vibrating system. The results indicate that both the transverse displacement and rotation of the beam significantly decrease as flexural rigidity increases. Additionally, it was observed that an increase in flexural rigidity correlates with a rise in critical velocity, suggesting a more stable dynamic system. From a practical standpoint, these findings clearly demonstrate that flexural rigidity plays a crucial role in enhancing the dynamic stability of the beam under the influence of the moving load.
References:-
References
Fryba, L. 1972. Vibration of solids and structures under moving loads, Noordhoof International Publishing Groningen, The Netherland.
Civalek, O. and Kiracioglu, O. 2010. Free vibration analysis of Timoshenko bean by DSC method, International Journal of Numerical Methods in Biomedical Engineering, 26(12), 1890-1898.
Ogunbamike, O.K., 2012. Response of Timoshenko beams on Winkler foundation subjected to dynamic load, International Journal of Scientific and Technology Research, 1(8), 48-52.
Nguyen, P.T., Pham, D.T. and Phuong-Hoa, H. 2016. A new foundation model for dynamic analysis of beams on nonlinear foundation subjected to a moving mass, Procedia Engineering, 142(2), 165-172.
Santos, H.A.F.A. 2024. A new finite element formulation for dynamic analysis of beams moving loads, Computers and Structures, 298, 23-35.
Steele, C.R. 1971. Beams and sells with moving loads, Internal Journal of Solids and Structures, 7, 1171-1198.
Andi, E.A., Oni, S.T. and Ogunbamike, O.K. 2014. Dynamic Analysis of a finite Simply Supported Uniform Rayleigh beam under travelling distributed loads, Journal of Nigerian Association of Mathematical Physics, 26, 125-136.
Ogunbamike, O.K. and Owolanke, O.A. 2022. Convergence of analytical solution of the Initial-Boundary value moving mass problem of beams resting on Winkler foundation, Electronic Journal of Mathematical Analysis and Applications 10(1), 129-136.
Ma, J., Wang, J., Wang, C., Li, D. and Guo, Y. 2024. Vibration response of beam supported by finite-thickness elastic foundation under a moving concentrated force, Journal of Mechanical Science and Technology, 38, 595-604.
Ajijola, O. O. 2025. Analysis of Transverse Displacement and Rotation Under Moving Load of Prestressed Damped Shear Beam Resting on Vlasov Foundation. African Journal of Mathematics and Statistics Studies, Vol. 8 No.1.
Fryba, L. 1976. Non-stationary response of a beam to a moving random force. Journal of Sound and Vibration, Vol. 46.
Oni, S. T. and Awodola, T. O. 2010. Dynamic response under a moving load of an elastically supported non-prismatic Bernoulli-Euler beam on variable elastic foundation, Latin American Journal of Solids and Structures, Vol 3.
Thambiratnam, D. and Zhuge, Y. 1996. Dynamic analysis of beams on an elastic foundation subjected to moving loads. Journal of sound and vibration, 198(2), 149-169.
Mallik, A. K., Chandra, S. and Singh, A. B. 2006. Steady-state response of an elastically supported infinite beam to a moving load. Journal of sound and vibration, 291(3), 1148-1169.
Adams, G. G. 1995. Critical speeds and the response of a tensioned beam on an elastic foundation to repetitive moving loads, Int. J. Mech. Sci., 37, 773 781.
Awodola, T. O. 2007. Variable velocity influence on the vibration of simply supported bernoulli-Euler beam under exponentially varying magnitude moving load, J. Math. Stat., 3, 228–232. 1
Rao, G. V. 2000. Linear dynamics of an elastic beam under moving loads, J. Vib. Acoust., 122, 281–289.
Sigueira, L.O., Cortez, R.L. and Hoefel, S.S. 2019. Vibration analysis of an axial-loaded Euler-Bernoulli beam on two-parameter foundation, 25th ABCM International Congress of Mechanical Engineering, Uberlandia, Brazil, 1-4.
Crandall, S. H. 1970. The role of damping in vibration theory. Journal of Sound and Vibration, Vol. 21 No.1.
Rezaee, M., Hassannejad, R. 2010. Damped free vibration analysis of a beam with a fatigue crack using energy balance method. International Journal of the Physical Sciences, Vol. 35 No.6.
Robin, Rana, U. S. 2013. Numerical study of damped vibration of orthotropic rectangular plates of variable thickness. Journal of Orissa Mathematical Society, 32(2).
Famuagun, K.S. 2023. Influence of Damping Coefficients on the Response to Moving Distributed Masses of Rayleigh Beams Resting on Bi-Parametric Subgrade. Journal of the Nigerian Association of Mathematical Physics, 71 – 86.
Rayleigh, J.W.S. 1877. The theory of sound. Macmillan London.
Caughey, T.K. and O’ Kelly, M.E.J. 1965. Classical normal modes in damped linear dynamic systems. ASME, Journal of Applied Mechanics, 32, 583-588.
Liu, Q., Wang, Y., Sun, P. and Wang, D. 2022. Comparative analysis of viscous damping model and hysteretic damping model, Journal of Applied Science, 12(23), 1-13.
Ogunbamike, O.K. 2020. Seismic analysis of simply supported damped Rayleigh beams on elastic foundation, Asian Research Journal of Mathematics, 16(11), 31-47.
Ogunbamike, O.K. 2021. Damping effects on the transverse motions of axially loaded beams carrying uniform distributed load, Applications of Modelling and Simulation 5, 88-101.
Esen, I. 2011. Dynamic response of a beam due to accelerating moving mass using finite element approximation, Mathematical and Computational Application, 16(1), 171 182.
Lei, Y., Murmu, T., Adhikari, S. and Friswell, M.I. 2013. Dynamic characteristics of damped viscoelastic nonlocal Euler-Bernoulli beam, European Journal of A/Solids, 42, 125-136.
Ogunbamike, O.K. and Oni, S.T. 2019. Flexural Vibration to partially distributed masses of non-uniform Rayleigh beams resting on Vlasov foundation with general boundary conditions. Journal of the Nigerian Mathematical Society, Vol. 38 No.1.
Rajib U.l., Alam Uzzal, Rama B. Bhat, Waiz Ahmed 2012. Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation. Shock and Vibration, Vol. 19.
Jimoh S.A. 2017. Analysis of non-uniformly prestressed tapered beams with exponentially varying thickness resting on Vlasov foundation under variable harmonic load moving with constant velocity. International Journal of Advanced Research and Publications, 1(5), 135-142.
Rajib U.l., Alam Uzzal, Rama B. Bhat and Waiz Ahmed 2012. Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation. Shock and Vibration, 19, 205–220.
Oni, S. T. and Ogunbamike, O. K. 2014. Dynamic Behaviour Of non-prismatic Rayleigh beam on Pasternak foundation and under partially distributed masses moving at varying velocities. Journal of Nigerian Mathematical Society, 33 (1-3).
Jimoh Sule Adekunle and Awelewa Oluwatundun Folakemi 2017. Dynamic Response of Non-Uniform Elastic Structure Resting on Exponentially Decaying Vlasov Foundation under Repeated Rolling Concentrated Loads. Asian Research Journal of Mathematics 6(4), 1-11.
Ogunbamike, O. K. and Oni, S. T. 2014. Flexural vibration to partially distributed masses of non-uniform Rayleigh beam resting on Vlasov foundation with general boundary conditions. Journal of Nigerian Mathematical Society, 38(1), 55-88.
Timoshenko, S. 1921. On the correction for shear of the differential equation for transverse vibration of prismatic bars, Philosophy Mag. c 774-776.
Ajijola, O. O. 2024. Dynamic Response to Moving Load of Prestressed Damped Shear Beam Resting on Bi-parametric Elastic Foundation. African Journal of Mathematics and Statistics Studies 7(4), 328-342.
DOI: 10.52589/AJMSS0JZLWMLB.
Ajijola Olawale Olaonipekun 2025. Analysis of Transverse Displacement and Rotation Under Moving Load of Prestressed Damped Shear Beam Resting on Vlasov Foundation. African Journal of Mathematics and Statistics Studies 8(1), 31-46. DOI: 10.52589/AJMS.
Ajijola O. O, Ogunbamike, O. K. and Adedowole 2025. c, Journal of Nigerian Association of Mathematical Physics, 70, 33-44.
Ajijola, O. O. 2025. Axial Force Influence on Transverse Displacement and Rotation under Moving Load of Elastically Supported Damped Shear Beam. International Journal of Mathematics and Computer Research, 13(4), 5051-5059.
Ajijola, O. O. 2025. Effect of Damping Coefficient on Transverse Displacement and Rotation Under Moving Load Of Elastically Supported Prestressed Shear Beam. International Journal of Engineering Technology Research & Management, 9(6), 497-515.
Okafor, F.O and O Oguaghamba O 2008. Effects of flexural rigidity of reinforcement bars on the fundamental natural frequency of reinforced concrete slabs, Nigerian Journal of Technology, 28(2), 48-57