The manual transmission shown below has five forward ratios

The manual transmission shown below has five forward ratios and one reverse. The input gear (green) has 24 teeth. The meshing gear (red) on the lay shaft has 32 teeth. If the input gear starts at 500 rpm and accelerates at a constant 10 rad/s^2for 6 seconds, find the angular velocity and displacement of the output shaft (yellow) if the transmission is in 1^st gear (the blue gear has 40 teeth and the red has 16 for 1^st gear). When in 1^st, the yellow shaft and blue gear marked 1 are connected).

Solution

First we have to find the overall gear ratio in 1st gear.

Gear ratio from green to red = 24/32 = 3/4

Meshing gear (red) on the laying shaft rotates with same angular velocity as the red gear with 16 teeth.

Gear ratio from red to blue = 16/40 = 2/5

Overall gear ratio = 3/4 x 2/5 = 3/10

Input angular velocity = 500 rpm = 52.36 rad/s

Final angular velocity after acceleration = initial angular velocity + angular acceleration x time (Newton\'s equation of motion)

Final angular velocity = 52.36 + 10 x 6 = 112.36 rad/s

Output angular velocity = 112.36 x 3/10 = 33.708 rad/s

Displacement of the output shaft = 322 rotations per minute