- Volume 13 Issue 2
This paper concerns two new analytical approaches for solving high nonlinear vibration equations. Energy Balance method and Hamiltonian Approach are presented and successfully applied for nonlinear vibration equations. In these approaches, there is no need to use small parameters to solve and only with one iteration, high accurate results are reached. Numerical procedures are also presented to compare the results of analytical and numerical ones. It has been established that, the proposed approaches are in good agreement with numerical solutions.
conservative systems;nonlinear vibration;Energy Balance method;Hamiltonian Approach
- Akgoz, B. and Civalek, O. (2011), "Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations", Steel Compos. Struct., 11(5), 403-421. https://doi.org/10.12989/scs.2011.11.5.403
- Bayat, M. and Pakar, I. (2015a), "Mathematical solution for nonlinear vibration equations using variational approach", Smart Struct. Syst., 15(5), 1311-1327. https://doi.org/10.12989/sss.2015.15.5.1311
- Bayat, M., Bayat, M. and Pakar, I. (2015b), "Analytical study of nonlinear vibration of oscillators with damping", Earthq. Struct., 9(1), 221-232. https://doi.org/10.12989/eas.2015.9.1.221
- 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 Am. J. Solid. Struct., 9(2), 145-234.
- Cai, X.C. and Liu, J.F. (2011), "Application of the modified frequency formulation to a nonlinear oscillator", Comput. Math. Appl., 61(8), 2237-2240. https://doi.org/10.1016/j.camwa.2010.09.025
- Civalek, O. (2006), "Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation", J. Sound Vib., 294(4), 966-980. https://doi.org/10.1016/j.jsv.2005.12.041
- Civalek, O. (2013), "Nonlinear dynamic response of laminated plates resting on nonlinear elastic foundations by the discrete singular convolution-differential quadrature coupled approaches", Compos. Part B: Eng., 50, 171-179. https://doi.org/10.1016/j.compositesb.2013.01.027
- Cunedioglu, Y. and Beylergil, B. (2014), "Free vibration analysis of laminated composite beam under room and high temperatures", Struct. Eng. Mech., 51(1), 111-130. https://doi.org/10.12989/sem.2014.51.1.111
- He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillators", Mech. Res. Commun., 29(2), 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
- He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Phys. Lett. A, 374(23), 2312-2314. https://doi.org/10.1016/j.physleta.2010.03.064
- Huseyin, K. and Lin, R. (1991), "An Intrinsic multiple- time-scale harmonic balance method for nonlinear vibration and bifurcation problems", Int. J. Nonlinear Mech., 26(5), 727-740. https://doi.org/10.1016/0020-7462(91)90023-M
- Jamshidi, N. and Ganji, D.D. (2010), "Application of energy balance method and variational iteration method to an oscillation of a mass attached to a stretched elastic wire", Curr. Appl. Phys., 10(2), 484-486. https://doi.org/10.1016/j.cap.2009.07.004
- Lau, S.L., Cheung, Y.K. and Wu, S.Y. (1983), "Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems", J. Appl. Mech., ASME, 50(4), 871-876. https://doi.org/10.1115/1.3167160
- Mehdipour, I., Ganji, D.D. and Mozaffari, M. (2010), "Application of the energy balance method to nonlinear vibrating equations", Curr. Appl. Phys., 10(1), 104-112. https://doi.org/10.1016/j.cap.2009.05.016
- Pakar, I. and Bayat, M. (2015), "Nonlinear vibration of stringer shell: An analytical approach", Proc. Inst. Mech. Engineers, Part E: J. Process Mech. Eng., 229(1), 44-51.
- Sedighi, H.M. and Bozorgmehri, A. (2016), "Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory", Acta Mechanica, 227(6), 1575-1591. https://doi.org/10.1007/s00707-016-1562-0
- Sedighi, H.M., Koochi, A., Daneshmand, F. and Abadyan, M. (2015), "Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow", Int. J. Non-Linear Mech., 77, 96-106. https://doi.org/10.1016/j.ijnonlinmec.2015.08.002
- Shaban, M., Ganji, D.D. and Alipour, A.A. (2010), "Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems", Curr. Appl. Phys., 10(5), 1267-1285. https://doi.org/10.1016/j.cap.2010.03.005
- Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl., 58(11), 2131-2133. https://doi.org/10.1016/j.camwa.2009.03.056
- Xu, L. (2010), "Application of Hamiltonian approach to an oscillation of a mass attached to a stretched elastic wire", Comput. Math. Appl., 15(5), 901-906.
- Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlinear Sci. Numer. Simulat., 10(10), 1361-1368.
- Wu, G. (2011), "Adomian decomposition method for non-smooth initial value problems", Math. Comput. Model., 54(9-10), 2104-2108. https://doi.org/10.1016/j.mcm.2011.05.018