1st Gear Solution

Find: Functional Relationship between Input and Output
Given: Clutch 1 engaged, Brake 1 engaged

Choose First, Last, and Arm

First - Gear 5
Last - Gear 3
Arm - Arm 4

(1)
\begin{align} {\omega_{out}}= fn({\omega_{in1}},{\omega_{in_2}}) \end{align}
(2)
\begin{align} {\omega_{first}}={\omega_5}={\omega_{in_1}} \end{align}
(3)
\begin{align} {\omega_{arm}}={\omega_4}={\omega_{out}} \end{align}
(4)
\begin{align} {\omega_{last}={\omega_3}={\omega_{in_2}}=0 \end{align}
(5)
\begin{align} \frac{\omega_{\frac{l}{a}}}{\omega_{\frac{f}{a}}} = \frac{\omega_l-\omega_a}{\omega_f-\omega_a} = \frac{\omega_{\frac{3}{4}}}{\omega_{\frac{5}{4}}} = \frac{\omega_3-\omega_4}{\omega_5-\omega_4} \end{align}
(6)
\begin{align} -\frac{N_5}{N_1}*\frac{N_1}{N_3}=-\frac{N_5}{N_3}=\frac{\omega_3-\omega_4}{\omega_5-\omega_4} \end{align}
(7)
\begin{align} \omega_4=\frac{\frac{N_5}{N_3}}{(\frac{N_5}{N_3}+1)}*\omega_5 \end{align}

camera