Mathematical solution for nonlinear vibration equations using variational approach

다국어 초록 (Multilingual Abstract)

In this paper, we have applied a new class of approximate analytical methods called Variational Approach (VA) for high nonlinear vibration equations. Three examples have been introduced and discussed.

The effects of important parameters on the response of the problems have been considered. Runge-Kutta’s algorithm has been used to prepare numerical solutions. The results of variational approach are compared with energy balance method and numerical and exact solutions. It has been established that the method is an easy mathematical tool for solving conservative nonlinear problems. The method doesn’t need small perturbation and with only one iteration achieve us to a high accurate solution.

참고문헌 (Reference)

1 I. Pakar, "Vibration analysis of high nonlinear oscillators using accurate approximate methods" 국제구조공학회 46 (46): 137-151, 2013

2 Xu, N., "Variational approachnext term to analyzing catalytic reactions in short monoliths" 58 (58): 2460-2463, 2009

3 Xu, L., "Variational approach to solution of nonlinear dispersive K(m, n) equation" 37 (37): 137-143, 2008

4 He, J. H, "Variational approach for nonlinear oscillators" 34 (34): 1430-1439, 2007

5 Shen, Y.Y., "The max–min approach to a relativistic equation" 58 (58): 2131-2133, 2009

6 Bayat, M., "Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems : a review" 9 (9): 145-234, 2012

7 He, J. H., "Preliminary report on the energy balance for nonlinear oscillators" 29 (29): 107-111, 2002

8 Bayat, M., "On the approximate analytical solution to non-linear oscillation systems" 20 (20): 43-52, 2013

9 Pakar, I., "On the approximate analytical solution for parametrically excited nonlinear oscillators" 14 (14): 423-429, 2012

10 Bayat, M., "Nonlinear vibration of stringer shell by means of extended Hamiltonian approach" 84 (84): 43-50, 2014

11 Bayat, M., "Nonlinear vibration of an electrostatically actuated microbeam" 11 (11): 534-544, 2014

12 Bayat, M., "Nonlinear free vibration of systems with inertia and static type cubic nonlinearities : an analytical approach" 77 : 50-58, 2014

13 Bayat, M, "Nonlinear free vibration analysis of tapered beams by hamiltonian approach" 13 (13): 654-661, 2011

14 M. Shaban, "Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems" 한국물리학회 10 (10): 1267-1285, 2010

15 Bayat, M., "Nonlinear dynamics of two degree of freedom systems with linear and nonlinear stiffnesses" 12 (12): 411-420, 2013

16 Nayfeh, A.H., "Nonlinear Oscillations" Wiley 1973

17 Cordero, A., "Iterative methods for use with nonlinear discrete algebraic models" 52 (52): 1251-1257, 2010

18 Dehghan, M., "Identifying an unknown function in a parabolic equation with over specified data via He’s variational iteration method" 36 (36): 157-166, 2008

19 Odibat, Z., "Generalized differential transform method : application to differential equations of fractional order" 197 (197): 467-477, 2008

20 I. Mehdipour, "Application of the energy balance method to nonlinear vibrating equations" 한국물리학회 10 (10): 104-112, 2010

21 Kuo, B.L., "Application of the differential transformation method to the solution of a damped system with high nonlinearity" 70 (70): 1732-1737, 2009

22 Bayat, M., "Application of he’s energy balance method for nonlinear vibration of thin circular sector cylinder" 6 (6): 5564-5570, 2011

23 Pakar, I., "Analytical study on the non-linear vibration of Euler-Bernoulli beams" 14 (14): 216-224, 2012

24 Pakar, I, "Analytical solution for strongly nonlinear oscillation systems using energy balance method" 6 (6): 5166-5170, 2011

25 Mahmoud Bayat, "Analytical solution for nonlinear vibration of an eccentrically reinforced cylindrical shell" 국제구조공학회 14 (14): 511-521, 2013

26 Zeng, D.Q., "Analysis of strongly nonlinear oscillator using the max–min approach" 10 (10): 1361-1368, 2009

27 Bayat, M., "Analysis of nonlinear vibration of coupled systems with cubic nonlinearity" 17 (17): 620-629, 2011

28 He J. H., "An improved amplitude-frequency formulation for nonlinear oscillators" 9 (9): 211-212, 2008

29 Pakar, I., "An analytical study of nonlinear vibrations of buckled Euler_Bernoulli beams" 123 (123): 48-52, 2013

30 Wu, G., "Adomian decomposition method for non-smooth initial value problems" 54 (54): 2104-2108, 2011

31 M. Bayat, "Accurate analytical solution for nonlinear free vibration of beams" 국제구조공학회 43 (43): 337-347, 2012