Mathieu oscillator: parametric resonance

[samreen]

hey, i need help in solving the equation of a mathieu oscillator (ignoring damping) and showing how the condition for max parametric resonance is doubling of the natural frequency . ( got viva 2morro. I am so going to suck)

D^2x + K(t)x =0
(Ko is the constant natural frequency when no perturbation present. D^2 is the second order time derivative of displacement x) . and, um, i kno precious little maths. for differential equations with variable coeffs, jst Frobenius method to seek series solns.

 

[masqau]

if ur
K(t)=a-2q cos(2t)
than ur differential equation is Matthieu differential equation.

 

[samreen]

yes it is. but i need to develop that solution for the viva

 

[masqau]

In mathieu equation , q=Mathieu Characteristic parameter, and its solution is available.
If ur system is quantum than the wave function is Floquet solution.
http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf

for solution
http://www.physics.drexel.edu/~tim/open/mat/node2.html

for ur information, Mathieu equation is special case of Hill's equation.

 

[samreen]

hey, brilliant! thanx

 

[masqau]

U welcome

 

[babamarysol]

Hi masqau,
can you explain me the procedure that i found in your first link

http://eqworld.ipmnet.ru/en/solutions/ode/ode0234.pdf

"Selecting a sufficiently large m and omitting the term with the maximum number in the recurrence relations we can obtain approximate relations for the eigenvalues a (b) with respect to parameter q. Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients A (B) to zero, we obtain an algebraic equation for finding a(q) (or b(q))"

I try to put into practice but without success :-(

 

[masqau]

dear
i am sending the link of a book written by Abramowitz and Stegun.
hope this will solve ur problem,
Good luck

http://calc.entisoft.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=722

 

[babamarysol]

Thanks