Lagrangian, 2 DOF (rotation with torsion, springs)

[Kolodny]

Homework Statement

[PLAIN]http://mityaka.com/users/kolodny/img/lagprob.png

Homework Equations

L = T - V

T = $$\frac{1}{2}$$*m*U²

V_s = $$\frac{1}{2}$$*k*x²

The Attempt at a Solution

I worked out the equations of motion as:

F_L = m*$$\ddot{y}$$+k*y-k*b*$$\theta$$

F_L*e = I_G*$$\ddot{\theta}$$+(k*b+k_T)*$$\theta$$+k*b*yAnd from here I'm not sure where to proceed. I understand in general that I would have to find the $$\dot{y}$$ and $$\dot{\theta}$$, and then the displacement for the potential energy, but given the equations of motion I'm not sure how to go about doing that.

If someone could direct me to an example of a 2 DOF Lagrangian problem, that would be awesome. My textbook (Palm's System Dynamics) doesn't cover Lagrangians at all for some odd reason, and I can't find anything other than general explanations of Lagrangians, as well as some 1 DOF problems, through Google.

Edit: Nevermind, I was making it a lot harder than it should be. For some reason I thought I had to derive the y-dot and theta-dot in terms of position and use that, but I've got it now.

 

[BigJDubs]

The Lagrangian is L = T - V, with T = \frac{1}{2}*m*\dot{y}^2 + \frac{1}{2}*IG*\dot{\theta}^2V = \frac{1}{2}*k*(y-b*\theta)^2 + k*b*y*\thetaThe equations of motion are found by taking the time derivative of the Lagrangian:m*\ddot{y}+k*y-k*b*\theta = 0IG*\ddot{\theta}+(k*b+kT)*\theta+k*b*y = 0