Is the Approach for Verifying Lagrangian Acceptable?

[Mishal0488]

TL;DR Summary

Looking for an opinion on a derivation of a Lagrangian mechanics problem I am solving.

Hi Guys

Please refer to the attached document for my derivation.
The image presents the system in plan view, I know one my think that it is unstable structure based on a single pinned connection however this is a simplification of a complex structure sitting on a slew bearing.

Gravity does not play a part in this system. The rod essentially rotates with it's center of mass at l2, there is also a spring connected to the system which will impact into a wall. The system assumes that the spring and wall are already in contact.

Note that the D term is the damping of the system.

Is the approach I have used acceptable?

 

[anuttarasammyak]

In the last line you posutulate that the spring shrinks keeping the shape of arc .　Further,
$$U=\frac{1}{2}kl_1^2(\theta-\theta_0)^2$$
for
$$\theta-\theta_0 <0$$
otherwise 0 where $l_1\theta_0$ is natural length of arc shape spring.