JOURNAL BROWSE
Search
Advanced SearchSearch Tips
A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
A GENERAL MULTIPLE-TIME-SCALE METHOD FOR SOLVING AN n-TH ORDER WEAKLY NONLINEAR DIFFERENTIAL EQUATION WITH DAMPING
Azad, M. Abul Kalam; Alam, M. Shamsul; Rahman, M. Saifur; Sarker, Bimolendu Shekhar;
  PDF(new window)
 Abstract
Based on the multiple-time-scale (MTS) method, a general formula has been presented for solving an n-th, n = 2, 3, , order ordinary differential equation with strong linear damping forces. Like the solution of the unified Krylov-Bogoliubov-Mitropolskii (KBM) method or the general Struble's method, the new solution covers the un-damped, under-damped and over-damped cases. The solutions are identical to those obtained by the unified KBM method and the general Struble's method. The technique is a new form of the classical MTS method. The formulation as well as the determination of the solution from the derived formula is very simple. The method is illustrated by several examples. The general MTS solution reduces to its classical form when the real parts of eigen-values of the unperturbed equation vanish.
 Keywords
oscillation;non-oscillation;asymptotic method;
 Language
English
 Cited by
 References
1.
G. N. Bojadziev, Damped forced nonlinear vibrations of systems with delay, J. Sound Vibration 46 (1976), 113-120. crossref(new window)

2.
G. N. Bojadziev, Two variables expansion method applied to the study of damped non-linear oscillations, Nonlinear Vibration Problems 21 (1981), 11-18.

3.
G. N. Bojadziev, Damped nonlinear oscillations modeled by a 3-dimensional differential system, Acta Mech. 48 (1983), no. 3-4, 193-201. crossref(new window)

4.
N. N. Bogoliubov and Yu. A. Mitropolskii, Asymptotic Methods in the Theory of Non- linear Oscillations, Gordan and Breach, New York, 1961.

5.
A. Hassan, The KBM derivative expansion method is equivalent to the multiple-time- scales method, J. Sound Vibration 200 (1997), no. 4, 433-440. crossref(new window)

6.
B. Z. Kaplan, Use of complex variables for the solution of certain nonlinear systems, J. Computer Methods in Applied Mechanics and Engineering 13 (1978), 281-291. crossref(new window)

7.
N. N. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, New Jersey, 1947.

8.
I. S. N. Murty, A unified Krylov-Bogoliubov method for solving second order nonlinear systems, Int. J. Nonlinear Mech. 6 (1971), 45-53. crossref(new window)

9.
I. S. N. Murty, B. L. Deekshatulu, and G. Krisna, On asymptotic method of Krylov- Bogoliubov for overdamped nonlinear systems, J. Frank Inst. 288 (1969), 49-64. crossref(new window)

10.
A. H. Nayfeh, Perturbation Methods, John Wiley & Sons, New York-London-Sydney, 1973.

11.
A. H. Nayfeh, Introduction to Perturbation Techniques, Wiley-Interscience [John Wiley & Sons], New York, 1981.

12.
I. P. Popov, A generalization of the asymptotic method of N. N. Bogolyubov in the theory of non-linear oscillations, Dokl. Akad. Nauk SSSR (N.S.) 111 (1956), 308-311.

13.
R. A. Rink, A procedure to obtain the initial amplitude and phase for the Krylov- Bogoliubov method, J. Franklin Inst. 303 (1977), 59-65. crossref(new window)

14.
M. Shamsul Alam, A unified Krylov-Bogoliubov-Mitropolskii method for solving n-order nonlinear systems, J. Franklin Inst. 339 (2002), 239-248. crossref(new window)

15.
M. Shamsul Alam, A unified Krylov-Bogoliubov-Mitropolskii method for solving n-th order nonlin- ear systems with varying coefficients, J. Sound and Vibration 265 (2003), 987-1002. crossref(new window)

16.
M. Shamsul Alam, A modified and compact form of Krylov-Bogoliubov-Mitropolskii unified KBM method for solving an n-th order nonlinear differential equation, Int. J. Nonlinear Mech. 39 (2004), 1343-1357. crossref(new window)

17.
M. Shamsul Alam, M. Abul Kalam Azad, and M. A. Hoque. A general Struble's technique for solving an n-th order weakly nonlinear differential system with damping, Int. J. Nonlinear Mech. 41 (2006), 905-918. crossref(new window)