Combined Transnational & Rotational System Transfer Function

1. The problem statement, all variables and given/known data

I’m given the following diagram

And asked to find the transfer function [itex]G(s) = \frac{X(s)}{T(s)}[/itex] and seem to be having some difficulty doing so.

2. Relevant equations

3. The attempt at a solution

Apparently this is free body diagram

I seem to be having difficulty understanding how this is the free body diagram. To me it would seem as if the the [itex]3 kg m^{2}[/itex] mass with [itex]T(t)[/itex] applied to would make the [itex]3 kg m^{2}[/itex] with teeth on it rotate clockwise. The free body diagram above seems to agree with this [itex]T(t)[/itex] is draw clockwise. I don’t understand why [itex]J_{eq}s^{2}Θ(s)[/itex] and [itex]D_{eq}sΘ(s)[/itex] are drawn counterclockwise. I don’t understand why the force [itex]F_{r}[/itex] acting on the [itex]3 kg m^{2}[/itex] mass with teeth as a result of the transnational system is equal to [itex]F_{2}[/itex] or why the force is drawn counterclockwise.

For the rectangular mass, I don’t really understand what [itex]F(s)[/itex] in the diagram. Apparently it would seem to be the net force on the mass. In which case according to the diagram, the mass is being displaced downwards. Apparently [itex]F(s) = (2s^{2}+2s+3)X(s)[/itex]. I’m confused as to why. I know that the force of gravity is pulling down the block by [itex]mg[/itex]. I assume that [itex]2s^{2}X(s)[/itex]. I understand that [itex]s^{2}X(s)[/itex] is the acceleration of the mass. I however don’t understand how [itex]mg = ma[/itex]. I understand that the damper is pulling the mass down by the force [itex]2sX(s)[/itex], so this makes since. I understand that the force of the spring acting on the mass is [itex]3X(s)[/itex]. I’m just confused by the following term [itex]s^{2}X(s)[/itex] and am unsure where it comes from.

Thanks for any help.

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