Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

A novel approximate solution for nonlinear problems of vibratory systems

  • Edalati, Seyyed A. (Department of Civil and Environmental Engineering, Tarbiat Modares University) ;
  • Bayat, Mahmoud (Department of Civil Engineering, College of Engineering, Bandar Abbas Branch, Islamic Azad University) ;
  • Pakar, Iman (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University) ;
  • Bayat, Mahdi (Young Researchers and Elite Club, Mashhad Branch, Islamic Azad University)
  • Received : 2015.10.09
  • Accepted : 2016.01.11
  • Published : 2016.03.25
⟨ Previous Next ⟩

Abstract

In this research, an approximate analytical solution has been presented for nonlinear problems of vibratory systems in mechanical engineering. The new method is called Variational Approach (VA) which is applied in two different high nonlinear cases. It has been shown that the presented approach leads us to an accurate approximate analytical solution. The results of variational approach are compared with numerical solutions. The full procedure of the numerical solution is also presented. The results are shown that the variatioanl approach can be an efficient and practical mathematical tool in field of nonlinear vibration.

Keywords

References

  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. 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
  4. 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
  5. 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.
  6. 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. Theory., 77, 50-58. https://doi.org/10.1016/j.mechmachtheory.2014.02.009
  7. 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
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. Bararnia, H., Domairry, G., Gorji, M. and Rezania, A. (2010), "An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM", Numer. Meth. Part. Differ. Eq., 26(1), 1-13. https://doi.org/10.1002/num.20404
  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", Nonlin. Anal., 70(4), 1732-1737. https://doi.org/10.1016/j.na.2008.02.056
  20. Cai, X.C. and Liu, J.F. (2011), "Application of the modified frequency formulation to a nonlinear oscillator", Comput. Math. Appl., 61(8), 2237-2240. https://doi.org/10.1016/j.camwa.2010.09.025
  21. Chen, S.S. (2009), "Application of the differential transformation method to the free vibrations of strongly non-linear oscillators", Nonlin. Anal. Real World Appl., 10(2), 881-888. https://doi.org/10.1016/j.nonrwa.2005.06.010
  22. 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
  23. 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
  24. 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 Solit. Fract., 36(1), 157-166. https://doi.org/10.1016/j.chaos.2006.06.023
  25. Filobello-Nino, U., Vazquez-Leal, H., Benhammouda, B., Perez-Sesma, A., Jimenez-Fernandez, V., Cervantes-Perez, J., Sarmiento-Reyes, A., Huerta-Chua, J., Morales-Mendoza, L. and Gonzalez-Lee, M. (2015), "Analytical solutions for systems of singular partial differential-algebraic equations", Discrete Dyn. Nature Soc., Article ID 752523.
  26. Ganji, D., Nourollahi, M. and Rostamian, M. (2007), "A comparison of variational iteration method with Adomian's decomposition method in some highly nonlinear equations", Int. J. Sci. Tech., 2(2), 179-188.
  27. He, J.H. (2007), "Variational approach for nonlinear oscillators", Chaos Solit. Fract., 34(5), 1430-1439. https://doi.org/10.1016/j.chaos.2006.10.026
  28. 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
  29. He, J.H. (2008), "An improved amplitude-frequency formulation for nonlinear oscillators", Int. J. Nonlin. Sci. Numer. Simul., 9(2), 211-212.
  30. 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
  31. 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
  32. 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.
  33. 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
  34. 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.
  35. 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
  36. 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
  37. Radomirovic, D. and Kovacic, I. (2015), "An equivalent spring for nonlinear springs in series", Eur. J. Phys., 36(5), 055004. https://doi.org/10.1088/0143-0807/36/5/055004
  38. 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
  39. Sadighi, A. and Ganji, D. (2008), "Analytic treatment of linear and nonlinear Schrodinger equations: a study with homotopy-perturbation and Adomian decomposition methods", Phys. Lett. A, 372(4), 465-469. https://doi.org/10.1016/j.physleta.2007.07.065
  40. 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.
  41. 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
  42. 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
  43. Xu, L. (2010), "Application of Hamiltonian approach to an oscillation of a mass attached to a stretched elastic wire", Comput. Math. Appl., 15(5), 901-906.
  44. Xu, N. 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
  45. Xu, R., Li, D.X., Jiang, J.P. and Liu, W. (2015), "Nonlinear vibration analysis of membrane SAR antenna structure adopting a vector form intrinsic finite element", J. Mech., 31(3), 269-277. https://doi.org/10.1017/jmech.2014.97
  46. Zeng, D.Q. and Lee, Y.Y. (2009), "Analysis of strongly nonlinear oscillator using the max-min approach", Int. J. Nonlin. Sci. Numer. Simul., 10(10), 1361-1368.
  47. 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

Cited by

  1. Effect of granulated rubber on shear strength of fine-grained sand vol.9, pp.5, 2017, https://doi.org/10.1016/j.jrmge.2017.03.008
  2. Prediction of Liquefaction Potential of Sandy Soil around a Submarine Pipeline under Earthquake Loading vol.10, pp.2, 2019, https://doi.org/10.1061/(ASCE)PS.1949-1204.0000349
  3. Modeling of compressive strength of cemented sandy soil pp.1568-5616, 2019, https://doi.org/10.1080/01694243.2018.1548535
  4. Shear behavior of fiber-reinforced sand composite vol.12, pp.5, 2019, https://doi.org/10.1007/s12517-019-4326-z
  5. Accurate semi-analytical solution for nonlinear vibration of conservative mechanical problems vol.61, pp.5, 2016, https://doi.org/10.12989/sem.2017.61.5.657
  6. Numerical study of the hydraulic excavator overturning stability during performing lifting operations vol.11, pp.5, 2016, https://doi.org/10.1177/1687814019841779
  7. Investigation of the deformability properties of fiber reinforced cemented sand vol.33, pp.17, 2016, https://doi.org/10.1080/01694243.2019.1619224