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


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 Du_ng  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. Baltan_as, J.P., Trueba, J.L., and Sanju_an, M.A.F.  Energy dissipation in a nonlinearly damped Du_ng  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 amplitude", 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 _nd  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 Du_ng oscillator with damping effect  using the modi_ed di_erential 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. Nanouid ow  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 di_erential quadrature element  (DQE) methods", Physica E: Low-dimensional  Systems and Nanostructures, 105, pp. 68{82 (2018).  214 M. Mohammadian and M. Shariati/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 203{214  25. Sheikholeslami, M. and Ganji, D.D. Magnetohydrodynamic  ow in a permeable channel _lled with  nanouid", 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 di_erential equations of Vanderpol,  Rayleigh and Du_ng 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 Du_ng-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. Signi_cant 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 di_erential 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 di_erential  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 di_erent  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).