DOI QR코드

DOI QR Code

Analytical study of nonlinear vibration of oscillators with damping

  • Bayat, Mahmoud (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahdi (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Pakar, Iman (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University)
  • 투고 : 2014.07.01
  • 심사 : 2014.11.06
  • 발행 : 2015.07.25

초록

In this study, Homotopy Perturbation Method (HPM) is used to solve the nonlinear oscillators with damping. We have considered two strong nonlinear equations to show the application of the method. The Runge-Kutta's algorithm is used to obtain the numerical solution for the problems. The method works very well for the whole range of initial amplitudes and does not demand small perturbation and also sufficiently accurate to both linear and nonlinear physics and engineering problems. Finally to show the accuracy of the HPM, the results have been shown graphically and compared with the numerical solution.

키워드

참고문헌

  1. 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
  2. Atmane, H.A., Tounsi, A., Ziane, N. and Mechab, I. (2011), "Mathematical solution for free vibration of sigmoid functionally graded beams with varying cross-section", Steel Compos. Struct., 11(6), 489-504. https://doi.org/10.12989/scs.2011.11.6.489
  3. Cordero, A., Hueso, J.L., Martinez, E. and Torregros, J.R. (2010), "Iterative methods for use with nonlinear discrete algebraic models", Math. Comput. Model., 52(7-8), 1251-1257. https://doi.org/10.1016/j.mcm.2010.02.028
  4. Bayat, M., Pakar, I. and Emadi, A. (2013a), "Vibration of electrostatically actuated microbeam by means of homotopy perturbation method", Struct. Eng. Mech., 48(6), 823-831. https://doi.org/10.12989/sem.2013.48.6.823
  5. Bayat, M., Bayat, M. and Pakar, I. (2014a), "The analytic solution for parametrically excited oscillators of complex variable in nonlinear dynamic systems under harmonic loading", Steel Compos. Struct., 17(1), 123-131. https://doi.org/10.12989/scs.2014.17.1.123
  6. Bayat, M., Pakar, I. and Bayat, M. (2014b), "An accurate novel method for solving nonlinear mechanical systems", Struct. Eng. Mech., 51(3), 519-530. https://doi.org/10.12989/sem.2014.51.3.519
  7. Bayat, M., Bayat, M. and Pakar, I. (2014c), "Forced nonlinear vibration by means of two approximate analytical solutions", Struct. Eng. Mech., 50(6), 853-862. https://doi.org/10.12989/sem.2014.50.6.853
  8. Bayat, M., Pakar, I. and Cveticanin, L. (2014d), "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities : An analytical approach", Mech. Mach. Theo., 77, 50-58. https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  9. Bayat, M., Pakar, I. and Cveticanin, L. (2014e), "Nonlinear vibration of stringer shell by means of extended Hamiltonian approach", Arch. Appl. Mech., 84(1), 43-50. https://doi.org/10.1007/s00419-013-0781-2
  10. Bayat, M., Bayat, M. and Pakar, I. (2014f), "Nonlinear vibration of an electrostatically actuated microbeam", Latin Am. J. Solid. Struct., 11(3), 534-544. https://doi.org/10.1590/S1679-78252014000300009
  11. Bayat, M., Pakar, I. and Bayat, M. (2013b), "On the large amplitude free vibrations of axially loaded Euler-Bernoulli beams", Steel Compos. Struct., 14(1), 73-83. https://doi.org/10.12989/scs.2013.14.1.073
  12. Bayat, M. and Pakar, I. (2013c), "Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses", Earthq. Eng. Eng. Vib., 12(3), 411-420. https://doi.org/10.1007/s11803-013-0182-0
  13. Bayat, M., Bayat, M. and Pakar, I. (2014g), "Accurate analytical solutions for nonlinear oscillators with discontinuous", Struct. Eng. Mech., 51(2), 349-360. https://doi.org/10.12989/sem.2014.51.2.349
  14. Bayat, M. and Abdollahzadeh, G. (2011), "On the effect of the near field records on the steel braced frames equipped with energy dissipating devices", Latin Am. J. Solid. Struct., 8(4), 429-443. https://doi.org/10.1590/S1679-78252011000400004
  15. Bayat, M. and Pakar, I. (2012a), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347. https://doi.org/10.12989/sem.2012.43.3.337
  16. 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.
  17. Bayat, M. and Pakar, I. (2013a), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52. https://doi.org/10.1155/2013/549213
  18. Bayat, M., Pakar, I. and Bayat, M. (2013b), "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell", Steel Compos. Struct., 14(5), 511-521. https://doi.org/10.12989/scs.2013.14.5.511
  19. Bor-Lih, K. and Cheng-Ying, L. (2009), "Application of the differential transformation method to the solution of a damped system with high nonlinearity", Nonlinear Anal., 70(4), 1732-1737. https://doi.org/10.1016/j.na.2008.02.056
  20. 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
  21. Dehghan, M. and 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. https://doi.org/10.1016/j.chaos.2006.06.023
  22. 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, 484-486. https://doi.org/10.1016/j.cap.2009.07.004
  23. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos, Solitons Fractals, 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026
  24. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212. https://doi.org/10.1515/IJNSNS.2008.9.2.211
  25. 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
  26. 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
  27. Odibat, Z., Momani, S. and Suat Erturk, V. (2008), "Generalized differential transform method: application to differential equations of fractional order", Appl. Math. Comput., 197(2), 467-477. https://doi.org/10.1016/j.amc.2007.07.068
  28. Pakar, I., Bayat, M. and Bayat, M. (2014a), "Nonlinear vibration of thin circular sector cylinder: An analytical approach", Steel Compos. Struct., 17(1), 133-143. https://doi.org/10.12989/scs.2014.17.1.133
  29. Pakar, I., Bayat, M. and Bayat, M. (2014b), "Accurate periodic solution for nonlinear vibration of thick circular sector slab", Steel Compos. Struct., 16(5), 521-531. https://doi.org/10.12989/scs.2014.16.5.521
  30. Pakar, I., Bayat, M. and Bayat, M. (2011), "Analytical evaluation of the nonlinear vibration of a solid circular sector object", Int. J. Phys. Sci., 6(30), 6861-6866.
  31. Pakar, I. and Bayat, M. (2013), "Vibration analysis of high nonlinear oscillators using accurate approximate methods", Struct. Eng. Mech., 46(1), 137-151. https://doi.org/10.12989/sem.2013.46.1.137
  32. Rajasekaran, S. (2013), "Free vibration of tapered arches made of axially functionally graded materials", Struct. Eng. Mech., 45(4), 569-594. https://doi.org/10.12989/sem.2013.45.4.569
  33. Shahidi, M., Bayat, M., Pakar, I. and Abdollahzadeh, G.R. (2011), "Solution of free non-linear vibration of beams", Int. J. Phys. Sci., 6(7), 1628-1634.
  34. 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
  35. 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
  36. Xu, Nan and Zhang, A. (2009), "Variational approachnext term to analyzing catalytic reactions in short monoliths", Comput. Math. Appl., 58(11-12), 2460-2463. https://doi.org/10.1016/j.camwa.2009.03.035
  37. Xu, L. (2010), "Application of Hamiltonian approach to an oscillation of a mass attached to a stretched elastic wire", Math. Comput. Appl., 15(5), 901-906.
  38. Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlinear Sci. Numer. Simul., 10(10), 1361-1368.
  39. Zhifeng, L., Yunyao, Y., Feng, W., Yongsheng, Z. and Ligang, C. (2013), "Study on modified differential transform method for free vibration analysis of uniform Euler-Bernoulli beam", Struct. Eng. Mech., 48(5), 697-709. https://doi.org/10.12989/sem.2013.48.5.697

피인용 문헌

  1. Nonlinear vibration of conservative oscillator's using analytical approaches vol.59, pp.4, 2016, https://doi.org/10.12989/sem.2016.59.4.671
  2. Energy based approach for solving conservative nonlinear systems vol.13, pp.2, 2017, https://doi.org/10.12989/eas.2017.13.2.131
  3. Nonlinear vibration of multi-body systems with linear and nonlinear springs vol.25, pp.4, 2015, https://doi.org/10.12989/scs.2017.25.4.497
  4. APPROXIMATE ANALYTICAL SOLUTIONS TO NONLINEAR DAMPED OSCILLATORY SYSTEMS USING A MODIFIED ALGEBRAIC METHOD vol.62, pp.1, 2021, https://doi.org/10.1134/s0021894421010090