5 - Penguin Motion Profile

The penguin motion profile was plotted based on the displacements I measured in the previous section. I defined zero degrees to be approximately the point where the straight edge of the cam falls away from contact with the follower, and the follower returns to contact with the cam's base circle. No displacement was measured for the first 50 degrees and so I set the displacement, velocity, and acceleration of the penguin to zero for the first 50 degrees. Using Excel, I made a trend line for the rest of the displacement data points. The first and second derivative of this trend line define the velocity and acceleration of the penguin, respectively. The penguin's motion profile plots are given in Figures 13 - 15. The velocity plot is given in both in/s and in/rad, and the acceleration plot is given in both in/s^2 and in/rad^2. For the plots given in units with time, the time derivatives were taken of the trend line. These equations assume that θ = ωt, where ω is a constant angular velocity. For the plots given in units with radians, the trend line was multiplied by d/dθ for velocity and multiplied by d^2/dθ^2 for acceleration.

Figure 13. Ready To Fly Penguin Displacement And Trend Line.    

Figure 14. Ready To Fly Penguin Velocity and Comparison to Point of Contact Plot. 

Figure 15. Ready To Fly Penguin Velocity and Comparison to Point of Contact Plot. 

Recall the fundamental law of cam design, which states that the cam function must be continuous through the first and second derivatives of displacement across the entire interval (360 degrees). While this is true for most of the interval, because I set the first 50 degrees to zero for displacement, velocity, and acceleration before using the trend line, there is a small jump in the velocity and acceleration plots at 50 degrees. The jump in the velocity plot is almost negligible, but the jump in the acceleration plot is noticeable. Had I taken measurements in smaller increments around the 50 degree mark, the acceleration curve may be smoother.

Notice that the point of contact distance plot is also overlaid on the velocity (in/rad) plot. 

The radius of the cam can also be defined by the following vector loop:

Where c is the distance from the cam's axis of rotation to the center of curvature and rho is the radius of curvature.

Recall Equations 1 and 2:

Equations 1 and 10 can be set equal to each other and separated into real and imaginary components:

The velocity in in/rad is obtained by taking the derivative of Equation 12 with respect to angle theta. This derivative is given in Equation 13.

As you can see, the point of contact distance is equal to the velocity in units of in/rad. Figure 14 compares my measurements for point of contact distance to the first derivative (d/dθ) of the displacement trend line. The point of contact distance plot does follow the velocity plot well, but it is not a perfect match. As I stated previously, the point of contact distance was difficult to measure and there is some uncertainty in how accurate my measurements were since they were taken with my caliper from outside the model base rather than inside and against the cam.

I also noticed that the displacement of the penguin could be described by trigonometry equations after reaching the maximum displacement value. From 296 degrees to 350 degrees, the maximum radius of the cam is in contact with the follower and forms the triangle:

Figure 16. Triangle defining displacement (296 deg to 350 deg).

Where displacement, s, is calculated with the following equation:

And β = θ - 296 deg. From 350 degrees to 360 degrees, the maximum radius of the cam loses contact with the follower and the follower slides down the straight edge of the cam and forms the triangle:

Figure 17. Triangle defining displacement (350 deg to 360 deg).                                 Figure 18. Center to edge length, b.

Where b is the follower's center to edge length (not the radius of the follower since the cams do not line up with the follower's center) and displacement, s, is calculated with the following equation:

Equations 14 and 15 are plotted with my own displacement measurements and the displacement trend line over the angle range of 296 degrees to 360 degrees in Figure 19. As you can see, these equations can also describe the displacement of the penguin after 296 deg. 

Figure 19. Ready To Fly Penguin Velocity and Comparison to Point of Contact Plot.