Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators

Livija Cveticanin

doi:10.4028/www.scientific.net/AMM.801.3

Paper Titles

Preface and Committees

Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators
p.3

On the Hexapod Leg Control with Nonlinear Stick-Slip Vibrations
p.12

Analytical Study of the Oblique Impact of a Rod with a Flat Using an Elasto-Plastic Contact Model
p.25

An Application of the Optimal Homotopy Asymptotic Method to Generalized Van der Pol Oscillator
p.33

Free Oscillations of a Nonlinear Oscillator with an Exponential Non-Viscous Damping
p.38

On the Open Problem: An Optimal Analytical Solution for Duffing Oscillator
p.43

Assessment of Environmental Noise by Harmonica Index – Case Study: The City of Niš
p.51

Calculation of Noise Field in an Urban Area close to a Traffic Overpass-Case Study
p.60

HomeApplied Mechanics and MaterialsApplied Mechanics and Materials Vol. 801Generalization of the Krylov-Bogoliubov Method for...

Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators

Abstract:

In this paper the Krylov-Bogoliubov method for solving nonlinear oscillators is considered. Based on the original method, developed for the oscillator with small nonlinearity, a generalization is made to oscillators with strong nonlinearity. After rewriting the equation into two first order differential equations, the averaging procedure is introduced. Truly nonlinear differential equations are investigated where the linear term does not exist nor the linearization of the equation is possible. Solution is assumed in the form of the Ateb-function. After averaging the approximate solution for the oscillator is obtained. A numerical example is tested. It is shown that the difference between the analytical approximate solution and the exact numerical one is negligible.

You might also be interested in these eBooks

Acoustics & Vibration of Mechanical Structures II

View Preview

Info:

Periodical:

Applied Mechanics and Materials (Volume 801)

Pages:

3-11

DOI:

https://doi.org/10.4028/www.scientific.net/AMM.801.3

Citation:

Cite this paper

Online since:

October 2015

Authors:

Livija Cveticanin

Keywords:

Asymptotic Solution, Ateb-Function, Averaging Method, Nonlinear Vibrations

Export:

RIS, BibTeX

Price:

Permissions CCC:

Request Permissions

Permissions PLS:

Request Permissions

Сopyright:

Citation:

References

[1] N.M. Krylov, N.N. Bogoliubov, On Various Formal Expansions of Non-linear Mecahnics, Izdat. Zagalnoukr. Akad, Nauk, Kiev, 1934. (Ukranian).

Google Scholar

[2] N.M. Krylov, N.N. Bogolyubov, New Metods in Non-linear Mechanics, ONTI GTTI, Moscow-Leningrad, (1934).

Google Scholar

[3] N.M. Krylov, N.N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, (1947).

Google Scholar

[4] N.N. Bogoliubov, Y.A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon and Breach, New York, (1961).

Google Scholar

[5] N.N. Bogolubov, N.N. Bogolubov, Jnr., Introduction to Quantum Statistical Mechanics, Gordon and Breach, ISBN 2-88124-879-9, (1992).

Google Scholar

[6] L. Cveticanin, M. Kalami-Yazdi, Z. Saadatnia, H. Askari, Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power, International Journal of Nonlinear Sciences and Numerical Simulation, 11, 997-1001, (2010).

DOI: 10.1515/ijnsns.2010.11.12.997

Google Scholar

[7] L. Cveticanin, T. Pogany, Oscillator with a sum of non-integer order non-linearities, Journal of Applied Mathematics, doi: 10. 1155/2012/649050. 2012, Article ID 649050, 20 pages, (2012).

Google Scholar

[8] L. Cveticanin, Strongly Nonlinear Oscillators, Analytical Solutions, Springer, New York, (2014).

Google Scholar