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 ?

 

[HalfLostAndSearching]

As a scientist, it is important to approach problems with a systematic and analytical mindset. In this case, the first step would be to review and understand the relevant equations and principles related to 2 DOF vibrations. This could include equations for forces, moments, and free body diagrams.

Next, it may be helpful to break down the problem into smaller, more manageable components. This could involve separating the lever and linear components and analyzing them individually before combining them into a mixed system.

It is also important to carefully review the assumptions made in the problem and ensure they are valid for the given scenario. If any assumptions are incorrect, this could lead to incorrect equations and solutions.

If you are still struggling with the problem, it may be beneficial to seek assistance from a colleague or instructor who has experience with 2 DOF vibrations or mixed systems. They may be able to provide insights or alternative approaches that can help you solve the problem.

Remember to document your steps and calculations carefully, as this can help identify any errors or misunderstandings. With a thorough and methodical approach, you can overcome challenges and successfully solve the problem at hand.