ANALYTICAL-NUMERICAL METHOD FOR SOLVING NONLINEAR DYNAMICAL SYSTEMS
DOI:
https://doi.org/10.46881/ajsn.v5i0.121Keywords:
Deflection, Dynamical Systems, Finite difference Method, Load, Nonlinear Equations, SystemsAbstract
This paper investigates the analytical-numerical method for solving nonlinear dynamical systems. The governing partial differential equation of order four was transformed to Ordinary differential equation using analytical method. The finite difference method was used to transform the approximate governing equation. It was shown from the graph of deflection against distance that the deflection increases as the value of distance increases and also from the graph of deflection against time, the deflection increases with increase in time. The result is in agreement with the existing resultsReferences
LeissaA.W. (1989); Closed form exact solutions for the steady state vibrations of continuous systems subjected to distributed exciting forces, Journal of sound and vibration 134 pg. 435- 453.
Achenbach, J. D. & Sun, C. T. (1965): Moving load on a flexibly supported Timoshenko beam, International Journal of Solids and Structures 1,353-370.
Adel A. Al-Azzawi (2011): Analysis of Timoshenko Beam resting on a nonlinear compressional and frictional Winkler foundation: ARPN, Journal of Engineering and Applied Sciences. Asian Publishing Network (ARPN). Vol 6. No 11. Pp: 1 – 14.
Arturo O. Cifuentes (1989): Dynamic response of a beam excited by a moving mass.Beam to uniform Partially Distributed moving loads, Finite Elements in Analysis and Design 5,237-246.
Dehestani, M. Mofid, A. Vafai (2009): Investigation of critical influential speed for moving mass problems on beams,Applied Mathematical Modeling 33,3885-3895.
Esmalzadeh E. and Ghorashi 1997: Vibration analysis of Timoshenko Beams traversed by uniform partially distributed moving masses. Proceeding of the international conference in Engineering Application of Mechanics Tehran 2, 238 - 323.
Foda, M.A. &Abduljabbar, Z. (1997): Journal of Sound and Vibration 230,493-506. A dynamic Green function formulation for the response of a beam structure to a moving mass.
Fryba, L. (1972): "Vibration of Solids and S t r u c t u r e s u n d e r M o v i n g Loads",Noordhoff International, Groningen.
Gbadeyan J.A. and Oni S.T. (1992): Dynamic Response of Moving Concentrated Masses of Elastic Plates on a non-Winkler Elastic Foundation. Journal of Sound and Vibration 154(2) : 343 - 358.
Gbadeyan J.A. &Oni S.T. (1995): Dynamics behaviour of Beams and rectangular plates under moving loads. Journal of Sound and Vibration. 182 (5) pp. 677 – 695.
Lueschen, G.G.G, Bergman&Macfarland, D.M. (1996):Green functions for uniform Timoshenko Beams, Journal of sound and vibration 194, 93-102.
Michaltsos, G.T. &Kounadis, A.N. (2001): The Effects of Centripetal and Coriolis Forces on the Dynamic Response of Light Bridges Under Moving Loads,Journal of Sound and Vibration 247,261-277.
Gbolagade, A.W ,Gbadeyan, J.A., S.O.S. Olabamiji, O. Otolorin (2003): Response of Elastically Connected Beams subjected to moving forces .Nigeria Journal of Mathematics and Applications, 16: 67-79.
Hankun, Wang and Goong Chen (1991): Asymptotic locations of Eigen frequencies of Euler-Bernoulli Beam with nonhomogeneous structural and viscous damping coefficients. Journal of control and optimization.Vol. 29.No. 2. Pp. 347 – 367.
J.W. Nicholson, L.A Bergman (1986); Free vibration of combined dynamical systems , American Society of Civil Engineers Journal of Engineering Mechanics 112 pg. 1- 13.
Jia-Jang Wu, A.R. Whittaker, M.P. Cartmell (2000): The use of finite element techniques for calculating the dynamic response of structures to moving loads,Computers& Structures 78, 789-799.
Kargarnovin M. H. &Younesian, D. (2005): Response of beams on nonlinear viscoelastic foundations to harmonic moving loads, Computers & Structures, 83 1865-1877.
Kargarnovin, M. H. &Younesian, D. (2004) : Dynamics of Timoshenko beams on Pasternak foundation under moving load, Mechanics Research Communications, 31, 713-723.
L. A Bergman, J.W. Nicholson (1985); Forced vibration of a damped combined linear system, American Society of Mechanical Engineers , Journal of vibration, Acoustics, Stress, and Realiability in Design 107 pg. 275 - 281.
L. Fryba (1972); Vibration of Solids and Structures under Moving loads, Noordhoff International, Groningen.
L.A. Bergman, D.M McFarland (1988); On the vibration of a point – supported linear distributed systems, American Society of Mechanical Engineers, Journal of Vibration, Acoustics, Stress, Realiability in design 110 pg. 458 – 592.
Lee, H. P. (1998) : Dynamic response of a Timoshenko beam on a Winkler foundation subjected to a moving mass, Applied Acoustics, 55 (3) 203-215.
M. Gurgoze, H. Erol (2001); Determination of the frequency response function of a cantilevered beam simply supported inspan, Journal of sound and vibration 247, pg 372 – 378.
M.A Foda, Z. Abduljabbar (1998); A dynamic Green function formulation for the response of a beam structure to a moving mass, Journal of sound and vibration 210 pg. 295 - 306
Mehri, B. A Davar,&Rahmani O. (2009): Dynamic Green Function Solution of Beams Under a Moving Load with Different Boundary Conditions. ScientiaIranica 16,273-279.
Michaltos G., Sopianopoulos D., &Kounadis A.N (1996) :The effect of moving mass and other parameters on the dynamic response of a simply supported beam, Journal of sound and Vibration 191, 357-362.
Michaltsos, G.T. (2001): Journal of Sound and Vibration 258,359-372. Dynamic behavior of a single-span beam Subjected to loads moving with variable speeds.
Mohan CharanSethi(2012) ; Dynamic response under moving mass, a thesis submitted in partial fulfillment of the requirement for the degree of bachelor of technology in mechanical Engineering, Department of Mechanical engineering, National institute of technology, Rourkelas.
MuhammedUncuer, BozidarMarinkovic and HurKoser (2007): Simulation of damped – free and damped-damped microbeams dynamics for nonlinear mechanical switch application: Except from the proceeding of the COMSOL, Conference 2007.
Ojih, P.B., Ibiejugba M.A. and Adejo B.O. (2014): Dynamic Response under moving concentrated loads of uniform Rayleigh beam resting on Pasternak foundation. International Journal of Scientific and Engineering Research: Volume 5, Issue 6: 1 – 21.
PiotriKoziol and ZdzislawHryniewicz (2012): Dynamic Response of a beam resting on a nonlinear foundation to a moving load: Coiflet - based solution shock and vibration 19:995 – 1007.
