Mathieu Equation

[netjaws]

Trying to investigate parametric roll, and don't actually need to SOLVE the Mathieu equation - just need to know whether the solution will be bounded or unbounded.

Any helps?


Joshua N Straume

[Guillermo]

May this help?

http://www.eagle.org/news/TECH/Marin..._JRP_Param.pdf

http://webphysics.davidson.edu/Proje...er/node14.html

[netjaws]

Interestingly, that first article is what posed my question!

That is - how do we find whether the solution is bounded or unbounded, without actuallly solving it?

[Guillermo]

Originally Posted by netjaws

Interestingly, that first article is what posed my question!

That is - how do we find whether the solution is bounded or unbounded, without actuallly solving it?

Good question. I don't know.

[tspeer]

Originally Posted by netjaws

Trying to investigate parametric roll, and don't actually need to SOLVE the Mathieu equation - just need to know whether the solution will be bounded or unbounded...

I think Guillermo has put you on the right road - the Christian paper says the imaginary part of the solution's exponent has to be zero for a bounded solution, and the boundary curves shown on the Ince-Strutt diagram _are_ the conditions that separate the bounded from the unbounded solutions.

You might try looking for a Lyapunov function for the problem and finding the bounds on that. Zounes and Rand use the function L(phi) = p*phi and present a number of ways of finding the stability contours. Zhang and Li use the same Lyapunov function, too, so I suspect it's the way to go.

It looks like a really hairy problem!