How to Solve a Mixed 2 DOF Vibration Problem with Incorrect Free Body Diagram?

[Matthias85]

I am struggling with the following question, it is a mixed (lever and linear) 2DOF vibration problem, something I never came across before. I am afraid I am missing something on the FBD, thus the differential equations of motions are wrong.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[AlephZero]

You did the right things to set up the equations of motion, but what are the horizontal displacements of the springs $k_1$ and $k_2$?

They are not attached at a distance $L$ from the pivot, so they are not $L\theta$.

Also, be careful with the signs. You have the amount of stretch in $k_1$ as $L\theta +x$ in one equation and $L\theta -x$ in the other. They can't both be right!

 

[Matthias85]

I see, distance L for each spring is given in question, is it k₂ (L/4)θ+x and k₁(L/2)θ ?

In which case the differential equation of motion become (now with correct signs)
Iθ + k₂ (L/4)θ+x - k₁(L/2)θ =0
mx - k₂ (L/4)θ+x =0

 

[AlephZero]

I see, distance L for each spring is given in question, is it k₂ (L/4)θ+x and k₁(L/2)θ ?

The L/2 and L/4 are right.

With θ and x defined as in the diagram in the question, (L/4)θ and x are both positive to the right. So, is the extension of the spring (L/4)θ+x or (L/4)θ-x ?