Nonlinear vibration of mechanical systems by means of Homotopy perturbation method
Keywords:
Energy balance method, Homotopy perturbation method, nonlinear oscillators, Runge-Kutta algorithm.Abstract
In this study, it has been tried to present a new approximate method by usingHomotopy Perturbation Method (HPM) for high nonlinear problems. Three differentexamples are considered and the application of the Homotopy perturbation method isstudied. Runge-Kutta algorithm is used to obtain numerical results. Another analyticalmethod called Energy Balance Method (EBM) is applied to compare the results ofHPM and Runge-Kutta algorithm. It has been shown that only one iteration of themethod prepares high accurate solution for whole domain. It has been established thatHomotopy perturbation method does not need any linearization and overcome thelimitations of the perturbation methods.
References
Bayat, M. & Pakar, I. 2011a. Application of He’s Energy Balance Method for Nonlinear vibration of thin
circular sector cylinder, International Journal of Physical Sciences, 6(23):5564-5570.
Bayat, M. & Pakar, I. 2011b. Nonlinear Free Vibration Analysis of Tapered Beams by Hamiltonian
Approach, Journal of vibroengineering, 13(4): 654-661.
Bayat, M. & Pakar, I. 2012. Accurate analytical solution for nonlinear free vibration of beams, Structural
Engineering and Mechanics, 43(3): 337-347.
Bayat, M. & Pakar, I. 2013a. On the approximate analytical solution to non-linear oscillation systems,
Shock and vibration, 20(1), 43-52.
Bayat, M. & Pakar, I. 2013b. Nonlinear dynamics of two degree of freedom systems with linear and
nonlinear stiffnesses, Earthquake Engineering and Engineering Vibration, 12 (3): 411-420.
Bayat, M. Pakar, I. & Bayat, M. 2013. Analytical solution for nonlinear vibration of an eccentrically
reinforced cylindrical shell, Steel and Composite Structures, 14(5): 511-521.
Bayat, M., Bayat, M. & Pakar, I. 2014. “Nonlinear vibration of an electrostatically actuated microbeam”,
Latin American Journal of Solids and Structures, 11(3), 534 – 544.
Bayat, M., Pakar, I. & Cveticanin, L. 2014a. Nonlinear vibration of stringer shell by means of extended Hamiltonian approach, Archive of Applied Mechanics, 84(1): 43–50.
Bayat, M., Pakar, I. & Cveticanin L. 2014b. Nonlinear free vibration of systems with inertia and static
type cubic nonlinearities: an analytical approach, Mechanism and Machine Theory, 77(7): 50–58.
Bayat, M., Pakar, I. & Domaiirry, G. 2012. Recent developments of Some asymptotic methods and their
applications for nonlinear vibration equations in engineering problems: A review, Latin American
Journal of Solids and Structures, 9(2):145 – 234 .
Bayat, M., Pakar, I. & Shahidi, M. 2011. Analysis of nonlinear vibration of coupled systems with cubic
nonlinearity, Mechanika, 17(6): 620-629.
Bor-Lih, K. & Cheng-Ying, L. 2009. Application of the differential transformation method to the solution
of a damped system with high nonlinearity, Nonlinear Analysis: Theory, Methods & Applications.
(4):1732–1737.
Cordero, A, Hueso, J. L., Martinez, E. & Torregrosa, J, R. 2010. Iterative methods for use with
nonlinear discrete algebraic models, Mathematical and Computer Modelling, 52(7-8):1251-1257.
Dehghan, M. & 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.
He, J. H. 1999. Homotopy perturbation technique, Computer methods in applied mechanics and
engineering, 178(3-4): 257-262.
He, J. H. 2007. Variational approach for nonlinear oscillators, Chaos, solitons & Fractals , 34(5):1430-
He, J. H. 2008. An improved amplitude-frequency formulation for nonlinear oscillators, International
Journal of Nonlinear Sciences and Numerical Simulation, 9(2): 211-212.
Mehdipour, I., Ganji, D. D. & Mozaffari, M. 2010. Application of the energy balance method to
nonlinear vibrating equations, Current Applied Physics, 10(1): 104-112.
Nayfeh, A. H. & Mook, D. T. 1973. Nonlinear Oscillations, Wiley, New York.
Odibat, Z., Momani, S. & Suat Erturk, V. 2008. Generalized differential transform method: application
to differential equations of fractional order, Applied Mathematics and Computation. 197(1): 467–
Pakar, I. & Bayat, M. 2012. “Analytical study on the non-linear vibration of Euler-Bernoulli beams,
Journal of vibroengineering, 14(1): 216-224.
Pakar, I. & Bayat, M. 2013a. An analytical study of nonlinear vibrations of buckled Euler_Bernoulli
beams, Acta Physica Polonica A, 123(1): 48-52.
Pakar, I. & Bayat, M. 2013b. Vibration analysis of high nonlinear oscillators using accurate approximate
methods, Structural Engineering and Mechanics, 46(1):137-151.
Pakar, I., Bayat, M. & Bayat, M. 2012. On the approximate analytical solution for parametrically excited
nonlinear oscillators, Journal of vibroengineering, 14(1): 423-429.
Shen, Y. Y. & Mo, L. F. 2009. The max–min approach to a relativistic equation, Computers & Mathematics
with Applications. 58(11): 2131–2133.
Wu G, 2011. Adomian decomposition method for non-smooth initial value problems”, Mathematical and
Computer Modelling, 54(9-10): 2104-2108.
Xu, Nan, & Zhang, A. 2009. Variational approach next term to analyzing catalytic reactions in short
monoliths, Computers & Mathematics with Applications, 58(11-12): 2460-2463.