Kinematic Analysis of "Wild Wild West"

Nin is the number of teeth on the smaller gear (attached to the handle) while Nout is the number of teeth on the larger gear (attached to the train tracks).

Nin = 6

Nout = 12

Angular Velocity Ratio: m_V = -Nin/Nout = -0.5

Torque Ratio / Mechanical Advantage: m_T = -Nout/Nin = -2

The negative sign is used for external gears. The train tracks turn at half the angular velocity of the handle but with twice the torque. This makes sense, considering that the tracks must support the weight of the train.

The above graph shows the absolute values of the angular and linear velocity of the train for a given absolute angular velocity of the handle. While the output angular velocity is half that of the input, the train itself moves very quickly. This makes sense considering that the train is at the edge of the tracks.

The images above depict the finished mechanism with a coordinate axis so that the origin is at the center of the train tracks, the tracks rotate around the z-axis, and the handle rotates around the y-axis. When the handle rotates around the positive y-axis, the tracks rotate around the negative z-axis. Similarly, the tracks rotate around the positive z-axis when the handle rotates around the negative y-axis. So, the signs of the input and output angular velocities are always opposites. This makes sense because the device uses external gears. 

Assuming power is conserved, the (absolute value) ratio of force on the train to force on the handle (Fout/Fin) is equal to the (absolute value) ratio of velocity on the handle to the velocity of the train (Vin/Vout). The radius of the handle is r1 and distance of the train from the center of the tracks is r2.

r1 = 3.3 cm

r2 = 7 cm

The absolute value of Vin/Vout equals the absolute value of m_T (torque ratio) times the ratio of the radii (r1/r2).

Vin/Vout = m_T * r1/r2 = .94 

Fout/Fin = .94.