Levai's Gear Configuration 12

Gear Configuration 12

flickr:4998463511

Here,

a = 1
f = 2
l = 6

from which, we get

(1)
\begin{align} \frac {\omega_{6/1}}{\omega_{2/1}}& =\frac {\omega_6-\omega_1}{\omega_2-\omega_1}\\ \end{align}
(2)
\begin{align} \frac {\omega_{6/1}}{\omega_{2/1}}& =\left(\frac {-N_2}{N_3}\right)\left(\frac {-N_3}{N_4}\right)\left(\frac {N_4}{N_5}\right)\left(\frac {N_5}{N_6}\right)\\ \end{align}

and,

(3)
\begin{align} &\left(\frac {N_4}{N_5}\right)=1 \end{align}

so,

(4)
\begin{align} &\frac {\omega_{6/1}}{\omega_{2/1}}& =\left(\frac {N_2N_5}{N_4N_6}\right)\\ \end{split} \end{align}

Therefore,

(5)
\begin{align} \frac {\omega_6-\omega_1}{\omega_2-\omega_1}&=\left(\frac {N_2N_5}{N_4N_6}\right)\\ \end{align}

from (1) and the expression above, we get

(6)
\begin{align} \omega_6 & =\left(\frac {N_2N_5}{N_4N_6}\right)\left(\omega_2 - \omega_1\right) + \omega_1\\ \end{align}

This is the angular velocity of the Ring gear in terms of the angular velocity of the Arm and the angular velocity of the Sun gear for the configuration shown.