Reverse Gear Solution
To solve 3rd gear, we will need to solve 3 equations.
First equation,
f = 6
a = 7
l = 3
from which, we get
(1)\begin{align} \frac {\omega_{3/7}}{\omega_{6/7}}& =\frac {\omega_3-\omega_7}{\omega_6-\omega_7}\\ \end{align}
(2)
\begin{split} \frac {\omega_{3/7}}{\omega_{6/7}}& =\left(\frac {-N_6}{N_3}\right)\\ \end{split}
Since $\omega_3$ = 0, (2) Becomes
(3)\begin{split} \frac {\omega_7}{\omega_{6/7}}& =\left(\frac {N_6}{N_3}\right)\\ \end{split}
simplifying, we get
(4)\begin{align} \omega_7 & =\left(\frac {N_6\omega_6}{N_3+N_6}\right)\\ \end{align}
Second equation,
f = 7
a = 3
l = 5
Because $\omega_3$ = 0, we get
(5)\begin{align} \frac {\omega_{5/3}}{\omega_{7/3}}& =\frac {\omega_5}{\omega_7}\\ \end{align}
(6)
\begin{split} \frac {\omega_5}{\omega_{7}}& =\left(\frac {-N_7}{N_5}\right)\\ \end{split}
therefore from (6) we get
(7)\begin{align} \omega_5 & =\left(\frac {-N_7\omega_7}{N_5}\right) \end{align}
Plugging in $\omega_7$ we get:
(8)\begin{align} \omega_5 & =\left(\frac{-N_7N_6\omega_6}{N_5\left(N_3+N_6\right) }\right)\\ \end{align}
Lastly, equation three,
f = 5
a = 4
l = 3
Because $\omega_3$ = 0, we get
(9)\begin{align} \frac {-\omega_4}{\omega_{5/4}}& =\frac {-\omega_4}{\omega_5-\omega_4}\\ \end{align}
(10)
\begin{split} \frac {-\omega_4}{\omega_{5/4}}& =\left(\frac {-N_5}{N_3}\right)\\ \end{split}
therefore from (9) and (10) and the expression above, we get
(11)\begin{align} \omega_4 & =\left(\frac {N_5}{N_3}\right)\left(\omega_5 - \omega_4\right) \\ \end{align}
simplifying, we get
(12)\begin{align} \omega_4 & =\left(\frac{N_5}{\left(N_3+N_4\right) }\right)\omega_5\\ \end{align}
Substituting $\omega_5$ = 0, we get:
(13)\begin{align} \omega_4 & =\left(\frac{N_5}{\left(N_3+N_4\right) }\right)\left(\frac{-N_7N_6\omega_6}{N_5\left(N_3+N_6\right) }\right)\\ \end{align}
Therefore, our solution is:
(14)\begin{align} \omega_4 & =\left(\frac{-N_7N_6}{\left(N_3+N_4\right)\left(N_3+N_6\right) }\right)\omega_6\\ \end{align}
With $\omega_6$ as our input.
page revision: 4, last edited: 17 Sep 2010 20:14