2. Kinematic Analysis of Wild Wild West

This mechanism has two objectives – to change the output plane and to increase the magnitude of the linear velocity at the output.

To change the plane of rotation, the mechanism uses a simplified version of bevel gears as depicted in Figure 2. Instead of having a conical shape, the gears are two dimensional and the spacing between the teeth of one gear is wide enough such the tooth of another gear can fit through it and push on its teeth (Figure 3). Thus, the input motion is in the xz plane while the output is in the xy plane.

Figure 2. Gear Mechanism

Figure 3.  Close-up of Gear Mechanism

2.1 Input-Output Relationships

Figure 4. Mechanism Schematic

 is the input angular velocity, is the angular velocity of the driving gear,  is the angular velocity of the driven gear and  is the output angular velocity.

Since the driving gear (gear 1) is connected to the input through the shaft, they share the same angular velocity. Or, 

Since at the point of contact of the two gears, the velocities are equal,  

Therefore, 

Rearranging and substituting the radii for the number of teeth because the teeth dimensions are the same,

If gear 1 rotates clockwise in the xz plane, gear 2 also rotates clockwise but in the xy plane. The driven gear and the output share the same angular velocity because they are connected by the same shaft.

So, 

Therefore, the output velocity, 

Since  , the output velocity, 

Also, since , the output angular acceleration 

Finally, the Mechanical Advantage:

For this mechanism,

r_in= 2.3 cm, r_out= 6.9 cm, N₁= 6, N₂= 12

Since the ratio of the output radius to input radius (3) is more than the gear ratio (2) the mechanical advantage is less than one and the velocity of the output is more than the input velocity. Also, the output rotates once for two complete rotations at the input.  This mechanism is interesting because the train at the output moves at a faster linear velocity than the input but at a smaller angular velocity.

Since the gear ratio is 2, it reduces the angular velocity but doubles the torque. This makes it easier for the user to rotate the larger disk at the output. 

Figure 5 plots the output angular velocity versus the input angular velocity and Figure 6 plots the magnitude of the output linear velocity versus the magnitude of the input linear velocity. 

Figure 5. Input-Output relationship of angular velocity

Figure 6. Input-Output relationship of magnitude of linear velocity

2.2. Input-Output relationships versus time. 

To analyze the input-output relationships over time, it was assumed that the mechanism starts from rest and the input is under a constant angular acceleration of 0.1 rad/s². Figures 7 and 8 plot the distance travelled and the velocity magnitudes of the train (output) and the user (input). These plots also include the distance travelled and velocity of the train if the train was placed at a radius that was equal to the input radius (in the model, the train is at three times the input radius). 

Figure 7. Distance moved by input, output and output at a smaller radius versus time.

Figure 8. Magnitude of velocity of input, output and output at a smaller radius versus time.

Figures 7 and 8 show that the train moves by a much larger distance than the input and it also moves much faster than the input. If the train was placed at a smaller radius, the output velocity would have been smaller than the input. Such a mechanism would have a mechanical advantage more than one and would be useful for a mechanism that would need force amplification.

Therefore, the mechanism successfully makes it easier for the user to move the train at a fast speed.