A Is the Approach for Verifying Lagrangian Acceptable?

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The discussion centers on the verification of a Lagrangian approach for a complex structure supported by a slew bearing. The system is described as stable despite a single pinned connection, with gravity being irrelevant to its dynamics. A spring's interaction with a wall is highlighted, assuming they are in contact, and the damping term is noted as part of the system's dynamics. The poster seeks validation of their approach, particularly regarding the spring's behavior and energy calculations. The inquiry emphasizes the need for clarity on the spring's arc shape and its implications on the system's stability and energy.
Mishal0488
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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?
 

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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.
 
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