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Application of Mathieu's Equation to Stability of Non-Linear Oscillator

Published online by Cambridge University Press:  03 November 2016

N. W. McLachlan
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Extract

The differential equation for y, the lateral displacement of an electrically-driven tuning fork, is

where a>0, c, κ small>0. (cẏ2-2κ), the coefficient in the damping term, is positive or negative according as cẏ2>2κ or cẏ2<2κ. During periodic motion, the coefficient changes sign, such that the inherent loss per period is compensated exactly by the energy supplied from the driving agent. In the language of “electronics”, loss corresponds to a “positive” resistance and energy supply to a “negative” resistance, both of which vary periodically.

Type
Research Article
Information
The Mathematical Gazette , Volume 35 , Issue 312 , May 1951 , pp. 105 - 107
Copyright
Copyright © The Mathematical Association 1951

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References

* Rayleigh, , Phil. Mag., 15, 229, 1883.CrossRefGoogle Scholar

† McLachlan, , Ordinary Non-linear Differential Equations (Oxford, 1950),Google Scholar which will be designated by N.D.E.

* McLachlan, , Mathieu Functions (Oxford, 1947),Google Scholar designated by M.F.