Analysis of Wild Wild West

The relationship of two interlocking gears can be described by the velocity ratio between them. The velocity ratio is equal to the ratio between the radius of the driving gear and the radius of the driven gear. Since the number of teeth on the gear is proportional to the radius of the gear, the ratio between the number of teeth of the driving gear and the number of teeth of the driven gear also equals the velocity ratio. If the driven gear is larger than the driving gear, then the angular velocity is reduced and the torque is increased. If the driven gear is smaller than the driving gear, then the angular velocity is increased and the torque is decreased.

As such, given the angular velocity of the driving gear and the number of teeth on each gear, the angular velocity of the driven gear can also be found. 

Time was then calculated in order to calculate the angular position of the driven gear. By putting the the angular velocity and angular position of the driving gear into the following equation, time was calculated. The equation was then used with the time and the angular velocity of the driven gear to calculate the angular position of the driven gear. 

Given a constant input angular velocity and a range of input positions, the following results were produced. 

By plotting the Y-position of a point on the platform and handle, using , the motion of the wheel and handle over time can be seen. In the time that it takes the handle to make a full rotation, the platform only makes half of a rotation.

I then decided to determine what the relationship between the velocities of the two gears would look like if velocity was not constant. Given a constant acceleration and a range of angular positions, the angular velocity of the driving gear was calculated: 

The angular velocity of the driven gear was then calculated using the gear teeth ratio, -N1/N2. Time and the angular position of the driven gear were then calculated in the same way that they were calculated for the constant velocity analysis. The angular acceleration of the driven gear was then calculated as well using the angular velocity of the driven gear and time. This produced the following results.

The angular position graph is much the same as the angular position graph for constant velocity; for each radian that the driving gear moves, the driven gear moves half a radian.

The angular velocity graph, however, is different. Due to the constant acceleration, the angular velocity of the gears increases gradually. The angular velocity of the driving gear increases faster than the angular velocity of the driven gear, indicating that the driven gear has a lower angular acceleration. The plots of the angular accelerations can be seen in the following graph. Both gears have constant angular accelerations, but the angular acceleration of the driven gear is half that of the driving gear.

As a result of the increasing angular velocities, the handle completes a full rotation in less time than with the constant angular velocity. However, the platform still does not complete a full rotation, although it does complete more than half of a rotation.

Through these results, it can be seen that when the driven gear is larger than the driving gear, the driven gear will take longer to complete a full rotation than the driving gear, whether velocity is constant or not.