An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

• Published in: Journal of the Franklin Institute Vol. 348; no. 5; pp. 899 - 916
• Main Authors: Shamsul Alam, M., Roy, Kamalesh Chandra, Rahman, M. Saifur, Mossaraf Hossain, Md
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.06.2011; Elsevier
• Subjects: Approximation; Damping; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Harmonic balance method; Harmonics; Mathematical analysis; Mathematical models; Nonlinearity; Perturbation methods; Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Harmonic; Mean field approximation; Perturbation method; Vibration damping; Non linear control; Vibration control; Wave damping; Non linear effect; Non linear system; Modeling; Harmonic balance
• ISSN: 0016-0032; 1879-2693
• DOI: 10.1016/j.jfranklin.2011.03.001

Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.