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http://dx.doi.org/10.12989/sem.2014.51.3.519

An accurate novel method for solving nonlinear mechanical systems  

Bayat, Mahdi (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University)
Pakar, Iman (Young Researchers and Elites Club, Mashhad Branch, Islamic Azad University)
Bayat, Mahmoud (Department of Civil Engineering, College of Engineering, Mashhad Branch, Islamic Azad University)
Publication Information
Structural Engineering and Mechanics / v.51, no.3, 2014 , pp. 519-530 More about this Journal
Abstract
This paper attempts to investigate the nonlinear dynamic analysis of strong nonlinear problems by proposing a new analytical method called Hamiltonian Approach (HA). Two different cases are studied to show the accuracy and efficiency of the method. This approach prepares us to obtain the nonlinear frequency of the nonlinear systems with the first order of the solution with a high accuracy. Finally, to verify the results we present some comparisons between the results of Hamiltonian approach and numerical solutions using Runge-Kutta's [RK] algorithm. This approach has a powerful concept and the high accuracy, so it can be apply to any conservative nonlinear problems without any limitations.
Keywords
approximate frequency; Hamiltonian approach; analytical investigations;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 Nayfeh, A.H. (1973), Perturbation Methods, Wiley Online Library.
2 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.
3 Pakar, I., Bayat, M. and Bayat, M. (2012a), "On the approximate analytical solution for parametrically excited nonlinear oscillators", J. Vibroeng., 14(1), 423-429.
4 Bayat, M. and Pakar, I. (2011a), "Nonlinear free vibration analysis of tapered beams by Hamiltonian approach", J. Vibroeng., 13(4), 654-661.
5 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.
6 Bayat, M., Pakar, I. and Shahidi, M. (2011c), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
7 Bayat, M., Pakar, I. and Domaiirry, G. (2012a), "Recent developments of Some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review", Lat. Am. J. Solid. Struct., 9(2),145-234.
8 Bayat, M. and Pakar, I. (2012b), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347.   DOI   ScienceOn
9 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
10 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
11 Pakar, I. and Bayat, M. (2012b), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", J. Vibroeng., 14(1), 216-224.
12 Pakar, I. and Bayat, M. (2013a), "An analytical study of nonlinear vibrations of buckled Euler Bernoulli beams", Acta Physica Polonica A, 123(1), 48-52.   DOI
13 Pirbodaghi, T. and Hoseini, S. (2010), "Nonlinear free vibration of a symmetrically conservative two-mass system with cubic nonlinearity", J. Comput. Nonlin. Dyn., 5(1), 011006.
14 Ren, Z.F. and Gui, W.K. (2011), "He's frequency formulation for nonlinear oscillators using a golden mean location", Comput. Math. Appl., 61(8), 1987-1990.   DOI   ScienceOn
15 Wazwaz, A.M. (2007), "The variational iteration method: a powerful scheme for handling linear and nonlinear diffusion equations", Comput. Math. Appl., 54(7-8), 933-939.   DOI   ScienceOn
16 Bayat, M. and Pakar, I. (2013c), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52.   DOI
17 Bayat, M., Pakar, I. and Cveticanin, L. (2014a), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities: an analytical approach", Mech. Mach. Theor., 77, 50-58.   DOI   ScienceOn
18 Bayat, M., Bayat, M. and Pakar, I. (2014c), "Nonlinear vibration of an electrostatically actuated microbeam", Lat. Am. J. Solid. Struct., 11(3), 534-544.   DOI
19 Zeng, D.Q. (2009), "Nonlinear oscillator with discontinuity by the max-min approach", Chaos, Soliton. Fract., 42(5), 2885-2889.   DOI   ScienceOn
20 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
21 Belendez, A., Hernandez, A., Belendez, T., Neipp, C. and Marquez, A. (2008), "Higher accuracy analytical approximations to a nonlinear oscillator with discontinuity by He's homotopy perturbation method", Phys. Let. A, 372(12), 2010-2016.   DOI   ScienceOn
22 Fu, Y., Zhang, J. and Wan, L. (2011), "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)", Cur. Appl. Phys., 11(3), 482-485.   DOI   ScienceOn
23 Ganji, D.D., Gorji, M., Soleimani, S. and Esmaeilpour, M. (2009), "Solution of nonlinear cubic-quintic Duffing oscillators using He's Energy Balance Method", J. Zhejiang Univ.-Sci. A, 10(9), 1263-1268.   DOI
24 He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mech. Res. Commun., 29(2-3), 107-111.   DOI   ScienceOn
25 He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Phys. Let. A, 374(23), 2312-2314.   DOI   ScienceOn
26 Kaya, M. and Demirbag, S.A. (2009), "Application of parameter expansion method to the generalized nonlinear discontinuity equation", Chaos, Soliton. Fract., 42(4), 967-197.
27 Shou, D.H. (2009), "The homotopy perturbation method for nonlinear oscillators", Comput. Math. Appl., 58(11-12), 2456-2459.   DOI   ScienceOn
28 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