2. Kinematic Analysis- Position and Velocity

For the velocity analysis, I had to make the assumption that the bevel gears were cone-shaped with intersecting apices. The input and output angular velocity relation of the gears can be modeled as:

where N1 refers to the number of teeth of the pinion (6) and N2 is the number of teeth of the driven gear (12) making the above relationship simplify to:

This relation of N1/N2 is referred to as the gear ratio and is 0.5. The torque ratio is the reciprocal of the gear ratio and is 2. In addition, I measured the radius of the train path to be 6.2 cm. 

Positional Analysis:

For the input/output analysis regarding the position, I plotted the position with respect to time and then respect to the crank angle.

The first plot, below, shows the x and y position of the front of the train as a function of time. The train's start position is opposite to that of the tunnel. A new point is added to the graph every half second and this point corresponds to the position of the train at that time (i.e. when a point is added at the 2-second mark, the train would actually be located at that position on the mechanism). To do this, I set the input angular velocity to be a constant value of 1 rad/s allowing the train to move in a CW rotation. Then, I calculated the output angular velocity using the gear ratio (1/2). To find the new angular position, I multiplied a time vector by the angular velocity output (theta = t*omega). I then plotted the new x and y position.

In addition, the plot, below and on the left, shows the x and y positions of the front of the train as time progresses (same start position as before). The input velocity is a constant 1 rad/s as the time goes from 0 to 13 seconds (approximately the time needed for the train to make a full rotation since it truly takes 12.57 seconds). I also plotted the x and y position as a function of the crank angle, below and on the right. Based on the gear ratio, it should take approximately two full rotations of the crank angle to get 1 full rotation of the train. This prediction is supported by the graph which shows it takes around 720° to achieve a full rotation.

Velocity Analysis

I analyzed how the input angular velocity affects the train’s linear velocity. I varied the angular velocity from 1 to 10 (to simulate turning the crank CW). Then, using the velocity ratio and gear ratio, I determined the driven gear’s angular velocity, which was equal to the turntables angular velocity. I then converted this angular velocity into linear velocity using the relationship: 

The graph below (on the left) shows how the train’s linear velocity changes as the input angular velocity is changed. As the input angular velocity increases, the train’s linear velocity also increases in a linear fashion. This makes sense due to these relationships all being directly proportional. In addition, the graph on the right shows the x and y components of the train's velocity as the crank is rotated at 1 rad/sec (same start position as before).