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An Algorithm for Autonomous Non-linear Dynamical Equations

Published online by Cambridge University Press:  07 June 2016

A Simpson
A Simpson*
Affiliation:
University of Bristol
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Summary

The method of Beecham and Titchener is extended to systems with n degrees of freedom and is shown to be a combination of the averaging principle and the method of variation of parameters. In this extended form, the method provides a powerful solution algorithm for non-linear problems such as those which arise in aircraft structural dynamics and aeroelasticity. The method is exemplified in application to a two-degree-of-freedom damped non-linear oscillator and to a binary (flexure-aileron) non-linear flutter system. The method is finally extended to non-linear differential equations in first-order form such as those which arise commonly in flight mechanics.

Information

Type
Research Article
Information
Aeronautical Quarterly , Volume 28 , Issue 3 , August 1977 , pp. 211 - 234
Copyright
Copyright © Royal Aeronautical Society. 1977

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References

1 Beecham, L J Titchener, I M Some notes on an approximate solution for the free oscillation characteristics of non-linear systems typified by ẍ + F(x, ẋ) = 0. RAE TR 69172, August 1969.Google Scholar
2 Bogoliubov, N N Mitropolski, Y A Asymptotic Methods in the Theory of Nonlinear Oscillations. Blaisdell, New York, 1965.Google Scholar
3 Krylov, N Bogoliubov, N N An Introduction to Non-Linear Mechanics. Princeton University Press, 1947.Google Scholar
4 Simpson, A An extension of Beecham’s method to nonlinear systems having n degrees of freedom. Paper A4, University of Loughborough Symposium on Non-Linear Mechanics, 1972.Google Scholar
5 SirJeffreys, H LadyJeffreys, B S Methods of Mathematical Physics. Cambridge University Press, 1956.Google Scholar
6 Titchener, I M Development of a technique for the analysis of non-linear dynamic characteristics of a flight vehicle. Paper A1, University of Loughborough Symposium on Non-Linear Mechanics, 1972.Google Scholar
7 Woodcock, D L Structural nonlinearities. In Vol IV, AGARD Manual on Aeroelasticity.Google Scholar
8 Birdsall, D L The effects of structural non-linearities on flutter. PhD thesis, Department of Aeronautical Engineering, University of Bristol, July 1965.Google Scholar
9 Shen, S F An approximate analysis of certain non-linear flutter problems. Journal of the Aerospace Sciences, Vol 26, pp 25–32, 45, 1959.CrossRefGoogle Scholar
10 Simpson, A A generalisation of Kron’s eigenvalue procedure. Journal of Sound and Vibration, Vol 26, pp 129–139, 1973.Google Scholar