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http://dx.doi.org/10.12989/sss.2015.15.5.1311

Mathematical solution for nonlinear vibration equations using variational approach  

Bayat, M. (Department of Civil Engineering, Mashhad Branch,Islamic Azad University)
Pakar, I. (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University)
Publication Information
Smart Structures and Systems / v.15, no.5, 2015 , pp. 1311-1327 More about this Journal
Abstract
In this paper, we have applied a new class of approximate analytical methods called Variational Approach (VA) for high nonlinear vibration equations. Three examples have been introduced and discussed. The effects of important parameters on the response of the problems have been considered. Runge-Kutta's algorithm has been used to prepare numerical solutions. The results of variational approach are compared with energy balance method and numerical and exact solutions. It has been established that the method is an easy mathematical tool for solving conservative nonlinear problems. The method doesn't need small perturbation and with only one iteration achieve us to a high accurate solution.
Keywords
Variational Approach (VA); nonlinear vibrations; energy balance method; numerical method;
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Times Cited By KSCI : 5  (Citation Analysis)
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1 Bayat, M. and Pakar, I. (2012a), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng.Mech., 43(3), 337-347.   DOI   ScienceOn
2 Bayat, M., Pakar, I. and Domaiirry, G, (2012b), "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review", Latin American J. Solids Struct., 9(2), 145- 234 .
3 Bayat, M., Pakar, I. and Bayat, M. (2013a), "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell", Steel Compos. Struct., 14(5), 511-521.   DOI   ScienceOn
4 Bayat, M. and Pakar, I. (2013b), "Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses", Earthq. Eng. Eng. Vib., 12(3), 411-420 .   DOI
5 Bayat, M. and Pakar, I. (2013c), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52.   DOI
6 Bayat, M., Pakar, I. and Cveticanin, L. (2014a), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: an analytical approach", Mech. Machine Theory, 77, 50-58.   DOI   ScienceOn
7 Bayat, M., Pakar, I. and Cveticanin, L. (2014b), "Nonlinear vibration of stringer shell by means of extended Hamiltonian approach", Arch. Appl. Mech., 84(1), 43-50.   DOI   ScienceOn
8 Bayat, M., Bayat, M. and Pakar, I. (2014c), "Nonlinear vibration of an electrostatically actuated microbeam", Latin American J. Solids Struct., 11(3), 534- 544.   DOI
9 Cordero, A., Hueso, J.L., Martinezand, E. and Torregros, J.R. (2010), "Iterative methods for use with nonlinear discrete algebraic models", Math. Comput. Model., 52(7-8), 1251-1257.   DOI   ScienceOn
10 Dehghan, M. and Tatari, M. (2008), "Identifying an unknown function in a parabolic equation with over specified data via He's variational iteration method", Chaos, Solitons Fractals, 36(1), 157-166.   DOI   ScienceOn
11 He, J.H (2007), "Variational approach for nonlinear oscillators", Chaos, Solitons Fractals, 34(5), 1430-1439.   DOI
12 He J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212.   DOI
13 He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillators", Mech. Res. Commun., 29(2), 107-111.   DOI
14 Kuo, B.L. and Lo, C.Y. (2009), "Application of the differential transformation method to the solution of a damped system with high nonlinearity", Nonlinear Anal., 70(4), 1732-1737.   DOI
15 Mehdipour, I., Ganji, D.D. and Mozaffari, M. (2010), "Application of the energy balance method to nonlinear vibrating equations", Current Appl. Phys., 10(1), 104-112.   DOI
16 Nayfeh, A.H. and Mook, D.T. (1973), Nonlinear Oscillations, Wiley, New York.
17 Odibat, Z., Momani, S, and Suat Erturk, V. (2008), "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 197(2), 467-477.   DOI
18 Pakar, I. and Bayat, M. (2011a), "Analytical solution for strongly nonlinear oscillation systems using energy balance method", Int. J. Phy. Sci., 6(22), 5166- 5170.
19 Pakar, I. and Bayat, M. (2012a), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroengineering, 14(1), 423-429.
20 Bayat, M. and Pakar, I. (2011a), "Nonlinear free vibration analysis of tapered beams by hamiltonian approach", J. Vibroengineering, 13(4), 654-661.
21 Bayat, M. and Pakar, I. (2011b), "Application of he's energy balance method for nonlinear vibration of thin circular sector cylinder", Int. J. Phy. Sci., 6(23), 5564-5570.
22 Bayat, M., Pakar, I. and Shahidi, M. (2011c), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
23 Shaban, M., Ganji, D.D. and Alipour, A.A. (2010), "Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems", Current Appl. Phys., 10(5), 1267-1285.   DOI
24 Pakar, I. and Bayat, M. (2012b), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroengineering, 14(1), 216-224.
25 Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler_Bernoulli beams", Acta Phys. Polonica A, 123(1), 48-52.   DOI
26 Pakar, I. and Bayat, M. (2013b), "Vibration analysis of high nonlinear oscillators using accurate approximate methods", Struct. Eng. Mech., 46(1), 137-151.   DOI   ScienceOn
27 Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133.   DOI   ScienceOn
28 Wu, G. (2011), "Adomian decomposition method for non-smooth initial value problems", Math. Comput. Model., 54(9-10), 2104-2108.   DOI   ScienceOn
29 Xu, N. and Zhang, A. (2009), "Variational approachnext term to analyzing catalytic reactions in short monoliths", Comput. Math. Appl., 58(11-12), 2460-2463.   DOI   ScienceOn
30 Xu, L. (2008), "Variational approach to solution of nonlinear dispersive K(m, n) equation", Chaos, Solitons Fractals, 37(1), 137-143.   DOI
31 Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlinear Sci. Numer. Simul., 10 (10), 1361-1368.