A novel approximate solution for nonlinear problems of vibratory systems

- Journal title : Structural Engineering and Mechanics
- Volume 57, Issue 6, 2016, pp.1039-1049
- Publisher : Techno-Press
- DOI : 10.12989/sem.2016.57.6.1039

Title & Authors

A novel approximate solution for nonlinear problems of vibratory systems

Edalati, Seyyed A.; Bayat, Mahmoud; Pakar, Iman; Bayat, Mahdi;

Edalati, Seyyed A.; Bayat, Mahmoud; Pakar, Iman; Bayat, Mahdi;

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

variational approach method;nonlinear vibration;numerical method;

Language

English

Cited by

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.

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.

3.

Bayat, M. and Pakar, I. (2013a), "On the approximate analytical solution to non-linear oscillation systems", Shock Vib., 20(1), 43-52.

4.

Bayat, M. and Pakar, I. (2012a), "Accurate analytical solution for nonlinear free vibration of beams", Struct. Eng. Mech., 43(3), 337-347.

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.

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.

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.

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.

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.

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.

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.

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

14.

Bayat, M., Bayat, M. and Pakar, I. (2014g), "Accurate analytical solutions for nonlinear oscillators with discontinuous", Struct. Eng. Mech., 51(2), 349-360

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

28.

He, J.H. (2010), "Hamiltonian approach to nonlinear oscillators", Phys. Lett. A, 374(23), 2312-2314.

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.

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.

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.

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.

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

37.

Radomirovic, D. and Kovacic, I. (2015), "An equivalent spring for nonlinear springs in series", Eur. J. Phys., 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.

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.

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.

42.

Wu, G. (2011), "Adomian decomposition method for non-smooth initial value problems", Math. Comput. Model., 54(9-10), 2104-2108.

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.

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.

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.