# 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 $G(s) = \frac{X(s)}{T(s)}$ 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 $3 kg m^{2}$ mass with $T(t)$ applied to would make the $3 kg m^{2}$ with teeth on it rotate clockwise. The free body diagram above seems to agree with this $T(t)$ is draw clockwise. I don’t understand why $J_{eq}s^{2}Θ(s)$ and $D_{eq}sΘ(s)$ are drawn counterclockwise. I don’t understand why the force $F_{r}$ acting on the $3 kg m^{2}$ mass with teeth as a result of the transnational system is equal to $F_{2}$ or why the force is drawn counterclockwise.

For the rectangular mass, I don’t really understand what $F(s)$ 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 $F(s) = (2s^{2}+2s+3)X(s)$. I’m confused as to why. I know that the force of gravity is pulling down the block by $mg$. I assume that $2s^{2}X(s)$. I understand that $s^{2}X(s)$ is the acceleration of the mass. I however don’t understand how $mg = ma$. I understand that the damper is pulling the mass down by the force $2sX(s)$, so this makes since. I understand that the force of the spring acting on the mass is $3X(s)$. I’m just confused by the following term $s^{2}X(s)$ and am unsure where it comes from.

Thanks for any help.

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