Application of AG method and its improvement to nonlinear damped oscillators

Document Type : Article

Authors

1 Department of Mechanical Engineering, Kordkuy Center, Gorgan Branch, Islamic Azad University, Kordkuy, P.O. Box 488164-4479, Iran.

2 Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran.

Abstract

In this paper, a new and innovative semi analytical technique, namely Akbari-Ganji’s method (AGM), is employed for solving three nonlinear damped oscillatory systems. Applying this method to nonlinear problems is very simple because in solving process only a trial solution, the main differential equation and its derivatives are required. The analytical solutions obtained by the AGM are utilized to study the impact of amplitude on nonlinear frequency and damping ratio. It is found that the AGM leads to acceptable results for the problems considered in this paper. Also, in order to obtain a more accurate solution, instead of using a trial solution with higher-order terms which may result in complicated and time consuming mathematical calculations, the solution obtained by AGM is improved via variational iteration method (VIM). The usefulness and effectiveness of the approach is demonstrated through comparison of the obtained results with those achieved by the numerical method. Hence, the AGM can be applied to nonlinear problems consisting of significant nonlinear damping terms and, if necessary, can be easily improved. 

Keywords

Main Subjects


References:
1. Afsharfard, A. and Farshidianfar, A. Design of nonlinear impact dampers based on acoustic and damping behavior",
International Journal of Mechanical Sciences,65(1), pp. 125{133 (2012).2. Leung, A.Y.T., Guo, Z., and Yang, H.X. Residue
harmonic balance analysis for the damped Dung resonator driven by a van der Pol oscillator", International Journal of Mechanical Sciences, 63(1), pp. 59{65 (2012). 3. Mansoori Kermani, M. and Dehestani, M. Solving the nonlinear equations for one-dimensional nanosized model including Rydberg and Varshni potentials and Casimir force using the decomposition method", Applied Mathematical Modelling, 37(5), pp. 3399{3406 (2013). 4. Zhang, W., Wang, F.-X., and Zu, J.W. Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation", Chaos, Solitons & Fractals, 24(4), pp. 977{998 (2005). 5. Mohammadian, M. and Shariati, M. Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method", Chinese Journal of Physics, 55(1), pp. 47{58 (2017).
6. Mohammadian, M. and Akbarzade, M. Higher-order approximate analytical solutions to nonlinear oscillatory systems arising in engineering problems", Archive of Applied Mechanics, 87(8), pp. 1317{1332 (2017). 7. Mohammadian, M. Application of the global residue harmonic balance method for obtaining higher-order approximate solutions of a conservative system", International Journal of Applied and Computational Mathematics, 3(3), pp. 2519{2532 (2017). 8. Mohammadian, M., Pourmehran, O., and Ju, P. An iterative approach to obtaining the nonlinear frequency
of a conservative oscillator with strong nonlinearities", International Applied Mechanics, 54(4), pp. 470{479 (2018).
9. Ganji, S.S., Ganji, D.D., Davodi, A.G., and Karimpour, S. Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach", Applied Mathematical Modelling, 34(9), pp. 2676{2684 (2010).
10. Yazdi, M.K., Ahmadian, H., Mirzabeigy, A., and Yildirim, A. Dynamic analysis of vibrating systems with nonlinearities", Communications in Theoretical Physics, 57(2), pp. 183{187 (2012). 11. De Rosa, M.A. and Lippiello, M. Nonlocal Timoshenko
frequency analysis of single-walled carbon nanotube with attached mass: An alternative Hamiltonian approach", Composites Part B: Engineering, 111, pp. 409{418 (2017). 12. Yazdi, M.K., Mirzabeigy, A., and Abdollahi, H. Nonlinear oscillators with non-polynomial and discontinuous elastic restoring forces", Nonlinear Science Letters A, 3(1), pp. 48{53 (2012). 13. Baltanas, J.P., Trueba, J.L., and Sanjuan, M.A.F. Energy dissipation in a nonlinearly damped Dung oscillator", Physica D: Nonlinear Phenomena, 159(1{2), pp. 22{34 (2001). 14. Liao, S.-J. An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying mplitude", International Journal of Non-Linear Mechanics, 38(8), pp. 1173{1183 (2003). 15. Cveticanin, L. Oscillators with nonlinear elastic and damping forces", Computers & Mathematics with  applications, 62(4), pp. 1745{1757 (2011). 16. Shamsul Alam, M., Roy, K.C., Rahman, M.S., and Mossaraf Hossain, M. An analytical technique to find approximate solutions of nonlinear damped oscillatory systems", Journal of the Franklin Institute, 348(5), pp. 899{916 (2011). 17. Wu, B.S. and Sun, W.P. Construction of approximate analytical solutions to strongly nonlinear damped oscillators", Archive of Applied Mechanics, 81(8), pp. 1017{1030 (2011). 18. Nourazar, S. and Mirzabeigy, A. Approximate solution for nonlinear Dung oscillator with damping effect using the modified differential transform method", Scientia Iranica, 20(2), pp. 364{368 (2013). 19. Cveticanin, L. On the truly nonlinear oscillator with positive and negative damping", Applied Mathematics and Computation, 243, pp. 433{445 (2014). 20. Ene, R.D., Marinca, V., and Marinca, B. Free oscillations of a nonlinear oscillator with an exponential nonviscous damping", Applied Mechanics and Materials, 801, pp. 38{42 (2015). 21. Herisanu, N. and Marinca, V. An optimal homotopy asymptotic approach to a damped dynamical system of a rotating electrical machine", Applied Mechanics and Materials, 801, pp. 202{206 (2015). 22. Razzak, M.A. and Molla, M.H.U. A new analytical technique for strongly nonlinear damped forced systems", Ain Shams Engineering Journal, 6(4), pp.1225{1232 (2015).23. Sheikholeslami, M. and Ganji, D.D. Nano fluid  flow and heat transfer between parallel plates considering Brownian motion using DTM", Computer Methods in Applied Mechanics and Engineering, 283, pp. 651{663 (2015). 24. Mohammadian, M., Hosseini, S.M., and Abolbashari, M.H. Lateral vibrations of embedded hetero-junction carbon nanotubes based on the nonlocal strain gradient theory: Analytical and differential quadrature element (DQE) methods", Physica E: Low-dimensional Systems and Nanostructures, 105, pp. 68{82 (2018). 214 M. Mohammadian and M. ShariatiScientia Iranica, Transactions B: Mechanical  ngineering 27 (2020) 203{214. 25. Sheikholeslami, M. and Ganji, D.D. Magnetohydrodynamic  flow in a permeable channel filled with nano fluid", Scientia Iranica, B, 21(1), pp. 203{212 (2014).
26. Ebaid, A., Rach, R., and El-Zahar, E. A new analytical solution of the hyperbolic Kepler equation using the Adomian decomposition method", Acta Astronautica, 138, pp. 1{9 (2017). 27. Pakdemirli, M. Perturbation-iteration method for strongly nonlinear vibrations", Journal of Vibration and Control, 23(6), pp. 959{969 (2017). 28. Akbari, M.R., Ganji, D.D., Majidian, A., and Ahmadi, A.R. Solving nonlinear differential equations of Vanderpol,Rayleigh and Dung by AGM", Frontiers of Mechanical Engineering, 9(2), pp. 177{190 (2014). 29. Mirgolbabaee, H., Ledari, S.T., and Ganji, D.D. New approach method for solving Dung-type nonlinear oscillator", Alexandria Engineering Journal, 55(2), pp.1695{1702 (2016).30. Akbari, M.R., Ganji, D.D., Nimafar, M., and Ahmadi, A.R. Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach", Frontiers of Mechanical Engineering, 9(4), pp. 390{401 (2014).31. Akbari, M.R., Ganji, D.D., Rostami, A.K., and Nimafar, M. Solving nonlinear differential equation governing on the rigid beams on viscoelastic foundation by AGM", Journal of Marine Science and Application, 14(1), pp. 30{38 (2015). 32. Mirgolbabaee, H., Ledari, S.T., and Ganji, D.D. An assessment of a semi analytical AG method for solving nonlinear oscillators", New Trends in Mathematical Sciences, 4(1), pp. 283{299 (2016). 33. He, J. A new approach to nonlinear partial differential equations", communications in Nonlinear Science and Numerical Simulation, 2(4), pp. 230{235 (1997). 34. Daeichi, M. and Ahmadian, M. Application of variational iteration method to large vibration analysis of slenderness beams considering mid-plane stretching", Scientia Iranica. Transactions B, Mechanical Engineering, 22(5), pp. 1911{1917 (2015). 35. Mohammadian, M. Application of the variational iteration method to nonlinear vibrations of nanobeams induced by the van der Waals force under dierent boundary conditions", The European Physical Journal Plus, 132(4), p. 169 (2017). 36. Witten, M. and Siegel, D. A kinetics model of abrin binding in a virus transformed lymphocyte cell culture", Bulletin of Mathematical Biology, 44(4), pp. 453{476 (1982).