DOI QR코드

DOI QR Code

Vibration analysis of high nonlinear oscillators using accurate approximate methods

  • Pakar, I. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University) ;
  • Bayat, M. (Department of Civil Engineering, Mashhad Branch, Islamic Azad University)
  • 투고 : 2012.09.12
  • 심사 : 2013.03.31
  • 발행 : 2013.04.10

초록

In this paper, two new methods called Improved Amplitude-Frequency Formulation (IAFF) and Energy Balance Method (EBM) are applied to solve high nonlinear oscillators. Two cases are given to illustrate the effectiveness and the convenience of these methods. The results of Improved Amplitude-Frequency Formulation are compared with those of EBM. The comparison of the results obtained using these methods reveal that IAFF and EBM are very accurate and can therefore be found widely applicable in engineering and other science. Finally, to demonstrate the validity of the proposed methods, the response of the oscillators, which were obtained from analytical solutions, have been shown graphically and compared with each other.

키워드

참고문헌

  1. Andrianov, I.V., Awrejcewicz, J. and Manevitch, L.I. (2004), Asymptotical Mechanics of Thin-Walled Structures, Springer, Verlag Berlin Heidelberg, Germany.
  2. Awrejcewicz, J., Andrianov, I.V. and Manevitch, L.I. (1998), Asymptotic Approaches in Nonlinear Dynamics, Springer, Verlag Berlin Heidelberg, Germany.
  3. Amiro, I.Y. and Zarutsky, V.A. (1981), "Studies of the dynamics of ribbed shells", Appl. Mech., 17(11), 949-962.
  4. Ba datl, S.M., Ozkaya, E., Ozyiit, H.A. and Tekin, A. (2009), "Nonlinear vibrations of stepped beam systems using artificial neural networks", Struct. Eng. Mech., 33(1), 15-30. https://doi.org/10.12989/sem.2009.33.1.015
  5. Bayat, M., Pakar, I. and Domaiirry, G. (2012a), "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review", Latin American Journal of Solids and Structures, 9(2), 145-234.
  6. Bayat, M. and Pakar, I. (2013a), "On the approximate analytical solution to non-linear oscillation systems", Shock and vibration, 20(1), 43-52. https://doi.org/10.1155/2013/549213
  7. Bayat, M., Pakar, I., Shahidi, M., (2011), "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity", Mechanika, 17(6), 620-629.
  8. Bayat, M., Pakar, I. and Bayat, M. (2013b), "On the large amplitude free vibrations of axially loaded Euler- Bernoulli beams", Steel and Composite Structures, 14(1), 73-83. https://doi.org/10.12989/scs.2013.14.1.073
  9. Bayat, M. and Pakar, I. (2012b), "Accurate analytical solution for nonlinear free vibration of beams", Structural Engineering and Mechanics, 43(3), 337-347. https://doi.org/10.12989/sem.2012.43.3.337
  10. Fu, Y.M., Zhang, J. and Wan, L.J. (2011), "Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS)", Curr Appl Phys., 11(3), 482-485. https://doi.org/10.1016/j.cap.2010.08.037
  11. Ganji, D.D., Bararnia, H., Soleimani, S. and Ghasemi, E, (2009), "Analytical solution of Magneto- Hydrodinamic flow over a nonlinear stretching sheet", Modern Physics Letters B, 23, 2541-2556. https://doi.org/10.1142/S0217984909020692
  12. Geng, L. and Cai, X.C. (2007), "He's frequency formulation for nonlinear oscillators", European Journal of Physics, 28(5), 923-93. https://doi.org/10.1088/0143-0807/28/5/016
  13. He, J.H. (2010a), "Hamiltonian approach to nonlinear oscillators", Physics Letters A., 374(23), 2312-2314. https://doi.org/10.1016/j.physleta.2010.03.064
  14. He, J.H., Zhong, T. and Tang, L. (2010b), "Hamiltonian approach to duffing-harmonic equation", Int. J. Nonlin. Sci. Num., 11(S1) 43-46.
  15. He, J.H. (2006), "Some asymptotic methods for strongly nonlinear equations", Int. J. Mod. Phys. B., 20(10), 1141-1199. https://doi.org/10.1142/S0217979206033796
  16. He, J.H. (2002), "Preliminary report on the energy balance for nonlinear oscillations", Mechanics Research Communications, 29(2), 107-111. https://doi.org/10.1016/S0093-6413(02)00237-9
  17. He, J.H. (2004), "Solution of nonlinear equations by an ancient chinese algorithm", Appl. Math. Compute. 151(1), 293-297. https://doi.org/10.1016/S0096-3003(03)00348-5
  18. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlin. SCi. Numer. Simul., 9(2), 211-212.
  19. Jalaal, M., Ganji, D.D. and Ahmadi, G. (2010), "Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media", Advanced Powder Technology, 21(3), 298-304. https://doi.org/10.1016/j.apt.2009.12.010
  20. Koiter, W.T. (1966), "On the nonlinear theory of thin elastic shells", Proc. Kon. Ned. Ak. Wet, ser B, 69(1), 1-54.
  21. Konuralp, A. (2009), "The steady temperature distributions with different types of nonlinearities", Computers and Mathematics with Applications, 58(11-12), 2152-2159. https://doi.org/10.1016/j.camwa.2009.03.007
  22. Kural, S. and Ozkaya, E. (2012), "Vibrations of an axially accelerating, multiple supported flexible beam", Structural Engineering and Mechanics, 44(4), 521-538. https://doi.org/10.12989/sem.2012.44.4.521
  23. Liu, J.F. (2009), "He's variational approach for nonlinear oscillators with high nonlinearity", Computers and Mathematics with Applications, 58(11-12), 2423-2426. https://doi.org/10.1016/j.camwa.2009.03.074
  24. Manevitch, A.I. (1972), "Stability and optimal design of reinforced shells", Visha Shkola, Kiev-Donetzk. (in Russian)
  25. Oztur, B. and Coskun, S.B. (2011), "The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation'', Structural Engineering and Mechanics, 37(4), 415-425. https://doi.org/10.12989/sem.2011.37.4.415
  26. Pakar, I., Bayat, M. and Bayat, M. (2011), "Analytical evaluation of the nonlinear vibration of a solid circular sector object", Int. J. Phy. Sci. 6(30), 6861-6866.
  27. Pakar, I. and Bayat, M. (2012a), "Analytical study on the non-linear vibration of Euler-Bernoulli beams", Journal of vibroengineering, 14(1), 216-224.
  28. Pakar, I., Bayat, M. and Bayat, M. (2012b), "On the approximate analytical solution for parametrically excited nonlinear oscillators", Journal of Vibroengineering, 14(1), 423-429.
  29. Pakar, I. and Bayat, M. (2013), "An analytical study of nonlinear vibrations of buckled Euler_Bernoulli beams", Acta Physica Polonica A, 123(1), 48-52. https://doi.org/10.12693/APhysPolA.123.48
  30. Piccardo,G. and Tubino, F. (2012), "Dynamic response of Euler-Bernoulli beams to resonant harmonic moving loads", Structural Engineering and Mechanics, 44(5), 681-704. https://doi.org/10.12989/sem.2012.44.5.681
  31. Shen, Y.Y. and Mo, L.F. (2009), "The max-min approach to a relativistic equation", Comput. Math. Appl. 58, 2131-2133. https://doi.org/10.1016/j.camwa.2009.03.056
  32. Thongmoon, M. and Pusjuso, S. (2010), "The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations", Nonlinear Analysis: Hybrid Systems, 4(3), 425-431 https://doi.org/10.1016/j.nahs.2009.10.006
  33. Xu, L. and He, J.H. (2010), "Determination of limit cycle by Hamiltonian approach for strongly nonlinear oscillators", Int. J. Nonlin. Sci. 11(12), 1097-1101.
  34. Zhang, H.L., Xu, Y.G. and Chang, J.R. (2009), "Application of He's energy balance method to a nonlinear oscillator with discontinuity", International Journal of Nonlinear Sciences and Numerical Simulation, 10 (2), 207-214.
  35. Zhang, H.L. (2008), "Application of He's frequency-amplitude formulation to an force nonlinear oscillator," Int. J. Nonlin. Sci. Numer. Simul., 9(3), 297-300. https://doi.org/10.1515/IJNSNS.2008.9.3.297
  36. Zarutsky, V.A. (1993), "Oscillations of ribbed shells", Inter. Appl. J. Mech., 29(10), 837-841. https://doi.org/10.1007/BF00855264

피인용 문헌

  1. On choice of initial guess in the variational iteration method and its applications to nonlinear oscillator vol.230, pp.6, 2016, https://doi.org/10.1177/0954408915569331
  2. Nonlinear Vibration Analysis Of Prebuckling And Postbuckling In Laminated Composite Beams vol.61, pp.2, 2015, https://doi.org/10.1515/ace-2015-0020
  3. Approximate analytical solution of nonlinear systems using homotopy perturbation method vol.230, pp.1, 2016, https://doi.org/10.1177/0954408914533104
  4. Accurate periodic solution for nonlinear vibration of thick circular sector slab vol.16, pp.5, 2014, https://doi.org/10.12989/scs.2014.16.5.521
  5. Accurate periodic solution for non-linear vibration of dynamical equations vol.7, pp.1, 2014, https://doi.org/10.12989/eas.2014.7.1.001
  6. A novel approximate solution for nonlinear problems of vibratory systems vol.57, pp.6, 2016, https://doi.org/10.12989/sem.2016.57.6.1039
  7. Nonlinear vibration of stringer shell: An analytical approach vol.229, pp.1, 2015, https://doi.org/10.1177/0954408913509090
  8. The analytic solution for parametrically excited oscillators of complex variable in nonlinear dynamic systems under harmonic loading vol.17, pp.1, 2014, https://doi.org/10.12989/scs.2014.17.1.123
  9. Nonlinear vibration of conservative oscillator's using analytical approaches vol.59, pp.4, 2016, https://doi.org/10.12989/sem.2016.59.4.671
  10. Nonlinear vibration of thin circular sector cylinder: An analytical approach vol.17, pp.1, 2014, https://doi.org/10.12989/scs.2014.17.1.133
  11. Vibration of electrostatically actuated microbeam by means of homotopy perturbation method vol.48, pp.6, 2013, https://doi.org/10.12989/sem.2013.48.6.823
  12. Study of complex nonlinear vibrations by means of accurate analytical approach vol.17, pp.5, 2014, https://doi.org/10.12989/scs.2014.17.5.721
  13. Nonlinear vibration of stringer shell by means of extended Hamiltonian approach vol.84, pp.1, 2014, https://doi.org/10.1007/s00419-013-0781-2
  14. Forced nonlinear vibration by means of two approximate analytical solutions vol.50, pp.6, 2014, https://doi.org/10.12989/sem.2014.50.6.853
  15. High conservative nonlinear vibration equations by means of energy balance method vol.11, pp.1, 2016, https://doi.org/10.12989/eas.2016.11.1.129
  16. Mathematical solution for nonlinear vibration equations using variational approach vol.15, pp.5, 2015, https://doi.org/10.12989/sss.2015.15.5.1311
  17. Analytical study of nonlinear vibration of oscillators with damping vol.9, pp.1, 2015, https://doi.org/10.12989/eas.2015.9.1.221
  18. Accurate analytical solutions for nonlinear oscillators with discontinuous vol.51, pp.2, 2014, https://doi.org/10.12989/sem.2014.51.2.349
  19. An accurate novel method for solving nonlinear mechanical systems vol.51, pp.3, 2014, https://doi.org/10.12989/sem.2014.51.3.519
  20. Nonlinear vibration of an electrostatically actuated microbeam vol.11, pp.3, 2014, https://doi.org/10.1590/S1679-78252014000300009
  21. Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell vol.14, pp.5, 2013, https://doi.org/10.12989/scs.2013.14.5.511
  22. Nonlinear vibration of Euler-Bernoulli beams resting on linear elastic foundation vol.15, pp.4, 2013, https://doi.org/10.12989/scs.2013.15.4.439
  23. Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems vol.61, pp.5, 2013, https://doi.org/10.12989/sem.2017.61.5.657
  24. Nonlinear Dynamic and Stability Analysis of an Edge Cracked Rotating Flexible Structure vol.21, pp.7, 2013, https://doi.org/10.1142/s0219455421500917
  25. Nonlinear Vibration of Axially Loaded Railway Track Systems Using Analytical Approach vol.40, pp.4, 2013, https://doi.org/10.1177/14613484211004190