Find the eigenfrequencies for systems

[roeb]

Homework Statement

$$(M + m) \ddot x + m l \ddot\theta - ml \dot\theta ^2\theta = 0$$

$$\ddot\theta + \frac{\ddot x}{l} + \frac{g}{l}\theta = 0$$

I am not quite sure how to get a x(t) and theta(t) that actually fit for these equations...
For the second equation, I was thinking something like
x(t) = At^2 + Bt+C
theta(t) = Dt^2 + Et + C, but of course that doesn't work.

I know how to find the eigenfrequencies for systems that say have a Ae^(iwt) term in them, but for something like this, I have no idea... If I could get the correct forms of x(t) and theta(t) I think I could probably find them, but I am a bit lost in how to get the forms for something like this.

 

[Dr.D]

I would think in terms of trying to cast this as a matrix eigenproblem, but that last term in the first equation somewhat messes things up with the theta-dot^2 factor. Are you sure you have this written correctly?

 

[roeb]

Thanks for your reply.
It is written correctly; however, I *may* (not positive but upon thinking about it..) be able to say that the theta-dot^2 term is small so that it is effectively zero...
$$(M + m) \ddot x + m l \ddot\theta = 0$$
$$\ddot\theta + \frac{\ddot x}{l} + \frac{g}{l}\theta = 0$$

I'm afraid I'm not quite sure how to proceed even if this were the case. I am not familiar exactly with matrix eigenproblems.

If this were a pure math problem I suppose I would do something like
(A-I*lambda)x = 0 but since I don't have the system of equations in terms of x(t) and theta(t) I'm not quite sure what to do.

 

[Dr.D]

"...since I don't have the system of equations in terms of x(t) and theta(t) I'm not quite sure what to do." But if you were to substitute x = X*sin(omega*t) and theta = Theta*sin(omega*t), then you would have such a system.

But dropping that theta-dot^2 term seems shaky unless you have some physical argument for it.