Abstract
In order to obtain the equations of motion of vibratory systems, we will need a mathematical description of the forces and moments involved, as function of displacement or velocity, solution of vibration models to predict system behavior requires solution of differential equations, the differential equations based on linear model of the forces and moments are much easier to solve than the ones based on nonlinear models, but sometimes a nonlinear model is unavoidable, this is the case when a system is designed with nonlinear spring and nonlinear damping. Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained; homotopy perturbation method (HPM) is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic; the enhanced HPM is used to solve the nonlinear shock absorber and spring equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Palm, W.J.: Mechanical vibration, vol. 1. ISBN 0-471-34555-5, 9
Vidyasagar, M.: Nonlinear systems analysis, 2nd edn. Prentice-Hall, Englewood Cliffs, ISBN 0-13-623513-1 (1993)
Khalil, H.: Nonlinear systems, vol. 2. ISBN 964-6213-22-7, 8
He J.H.: The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput. 151(1), 287–292 (2004)
Ganji D.D.: The application of He’s homotopy perturbation method to nonlinear equation arising in heat transfer. Phys. Lett. A 355, 337–341 (2006)
Ganji D.D., Rajabi A.: Assessment of homotopy perturbation and perturbation method in heat radiation equations. Int. Commun. Heat Mass Transf. 33(3), 391–400 (2006)
He J.H.: Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 350(1–2), 87–88 (2006)
He J.H.: Application of homotopy perturbation method to nonlinear wave equations. Chaos Solut. Fractals 26(3), 695–700 (2005)
He J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6(2), 207–208 (2005)
Ganji D.D.: The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Phys. Lett. A 355(4–5), 337–341 (2006)
Ganji D.D.: The application of He’s homotopy perturbation method to nonlinear equation arising in heat transfer. Phys. Lett. A 355, 337–341 (2006)
Ganji D.D., Rajabi A.: Assessment of homotopy perturbation and perturbation method in heat radiation equations. Int. Commun. Heat Mass Transf. 33(3), 391–400 (2006)
Adomian G.: A review of the decomposition method in applied mathematics. Math. Anal. Appl. 135, 501–544 (1988)
Hosein Nia S.H., Soltani H., Ghasemi J., Ranjbar A.N., Ganji D.D.: Maintaining the stability of nonlinear differential equations by the enhancement of HPM. Phys. Lett. A 372(16), 2855–2861 (2008)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Fereidoon, A., Rostamiyan, Y., Akbarzade, M. et al. Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Arch Appl Mech 80, 641–649 (2010). https://doi.org/10.1007/s00419-009-0334-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-009-0334-x