Problem with the harmonic oscillator equation for small oscillations

[PhillipLammsoose]

TL;DR Summary

I have an extra term in my harmonic oscillator equatiion, does it destroy the solution for small frequencies?

Hey, I solved a problem about a double pendulum and got 2 euler-lagrange equations:

1) x''+y''+g/r*x=0
2) x''+y'' +g/r*y=0 (where x is actually a tetha and y=phi)

the '' stand for the 2nd derivation after t, so you can see the basic harmonic oscillator equation with a term x'' or y'' that bother me. How do these terms effect the endresult for the frequency of small oscillations? I know w^2 (omega) would be w^2=g/r for a classic form of the equation, but what about the x'' or y''?

thanks for helping, I hope this was readable.

 

[vanhees71]

I'm a bit puzzled by your equations. Maybe you give some details about how you parametrized the problem?

In any case, for such a problem of linear coupled ODE's with constant coefficients the usual ansatz is
$$x=A \exp(\mathrm{i} \omega t), \quad y=B \exp(\mathrm{i} \omega t).$$
This leads to a eigenvalue problem, which determines $\omega$ (the frequencies of the fundamental modes of the system) and, for each $\omega$, $A$ and $B$ (the corresponding eigenvectors).

 

[PhillipLammsoose]

I solved this by seeing that x has to equal y in the 2 equations, so I got a new equation with $$2\ddot{x}+\frac{g}{r}x=0$$ which was the generic harmonic equation I was looking for.

 

[vanhees71]

Well, this is one of the two normal modes. You need to find another one. For me it's still not clear that these are the right equations for the double pendulum...