Longhorn Cam Analysis

Cam Analysis

Plausibility Analysis

In order to establish that a cam (and therefore an image) was viable for the mechanism, both followers had to be able to traverse the entire outer edge of their cams and the motor must be able to rotate through the forces introduced by the springs of each follower. The two immediate failure cases for an image are as follows:

• 1.) If the change in cam radius from one degree to the next is greater than the radius of the follower, the follower will not be able to climb up to the next degree, the mechanism will bind up, and the image will be incomplete.

• 2.) If the motor cannot produce an adequate reaction force to counter the force of each follower’s spring, the follower will pinch the cam in place, the mechanism will jam, and the image will be incomplete.

In the above image, the vertical Reaction Force of the Cam must meet the Force of the Spring in order to keep the cam rotating. This force is found with the following formula: F = (T/r)*cos(θ) where F is the Cam Reaction Force, T is the motor torque, r is the current radius, and theta is the angle from the current radius to the next radius. As a note, the horizontal Reaction Force of the Follower is what introduced the flexure to the follower's rod that bound up the system in early designs. In later designs, the double pillow block and linear ball bearings set up on each of the rods would reduce this flexure to tolerable levels.

Resolution Analysis

Once the cams were deemed viable, we found it of interest to know how well the cam's information would be transmitted to the output drawing. The metric we used for resolution is the percentage of radii that the follower would be able to read off of the cam. We reasoned that the only time an outside follower on a cam would not be able to keep contact with the cam is if there was a radius that was smaller than its neighbors so that the follower would simply glide over the notch. In the case that the follower loses contact with the cam, we know that the x or y value (depending on which cam had the failure) for that radius will be mistranslated to the resulting image.

Both analyses (plausibility and resolution) are handled by a Matlab function we created that takes in the stall torque of the motor, diameter of the follower, spring length, spring k value, and arrays of x and y values from the cam synthesis code. The function evaluates each point from the arrays using the input parameters and return a 1 or 0 to indicate if the cam is possible or not, a percentage of the cam's accuracy, and two arrays - one array indicating if a point will cause the cam to fail and one array indicating if the point will have a resolution error. The points from these arrays are then marked on the previously generated images of the cams and the source image to show the user where any errors would occur and what type of error is present (blue circles represent accuracy errors, red x's represent points that cause the cam to fail).

Analysis Results

To perform the below analysis, we used a Matlab script, camPlausibility.m, which can also be accessed in our Github repository.

The first set of images are the Matlab results for an imagined setup where the motor has a low max torque, meaning that it will fail to react the spring force in some points along the sketch, resulting in not only resolution errors (blue circles), but points where the cam will fail completely (red x's.) It stands to reason that the points further to the right (where the x-follower spring experiences max compression) and the points further down the image (where the y-follower spring experiences max compression) would be prone to failure as those are the regions where the spring forces are strongest and most difficult for the motor to react. The setup prescribes a follower diameter of .2 inches, a motor with 50 oz-in stall torque, and a spring k value of 22.4 ounce-inches.

This next set of images show a setup where the diameter of the follower is very small, meaning that there are fewer resolution errors, but in practical implementations the smaller diameters seemed to bind up easier. The setup prescribes a follower diameter of .05 inches, a motor with 110 oz-in stall torque, and a spring k value of 22.4 ounce-inches.

The final set of images are the Matlab results for the setup of our final design where there are no points of failure, but some points where accuracy will be lost from the original image (marked with the blue circles.) The setup prescribes a follower diameter of .2 inches, a motor with 110 oz-in stall torque, and a spring k value of 22.4 ounce-inches.

The final product image is placed below for comparison to the prediction of our analysis. From visual inspection, our prediction was correct as the drawn image exhibits resolution errors at the top of the head, ends of the horns, the notches around the ears, and the sides of the nose.