18.3 - Kinematic Analysis

Introduction

Once we decided to use helical cam as a key mechanism, the most critical analysis necessary is the position analysis of the cam. During the proposal stage, we leveraged this website to design a position profile that approximates the bottom part of the longhorn. The website offers parameters p, a, b, c, and d which are used to change the prime radius and 4 different dwell heights. However, given that we need to design for a helical cam which contains profiles for more than one full rotation, and the fact that the actual longhorn has a more delicate shape than 4 dwell heights, we decide to write our own analysis tool using Python.

Disk cam design and analysis

To start, we first try to design a disk cam and follower system. To make our own life simpler, we choose the axis of translational motion by the follower to be aligned with the radial axis of the cam.

       Figure 18.3-1a  Skeleton diagram of a generic disk cam + follower            Figure 18.3-1b  Skeleton diagram of our disk cam + follower

With that, our fundamental guiding principle is that the displacement of the follower is determined by the radius of cam at the contact point with the follower. In other words,

Here, r_θ is the radius of the disk cam at rotation angle θ, d is the displacement of follower from its origin. By slightly rearranging the equation, we can then easily design the disk cam based on the displacement v.s. rotation angle graph that we want.

To "draw" the longhorn, we divide it into different sections and use natural cubic spline to plan out the path. We started out with the design where the bottom part of the long horn (blue path) will take one rotation of disk cam while the top part (orange path) will take one rotation of a presumably different cam. However, having the helical cam direction in mind, it is perfectly feasible to have more than one disk cam rotation to draw either the top or the bottom part. If we pair the cam rotation and the cylindrical paper rotation properly, where the paper rotates more slowly than the cam, then we effectively increase the resolution because we encode less information in one full rotation of a cam. Hence, we produce the following displacement v.s. rotation angle graph where top part and bottom part of the long horn each takes 8 rotations to complete.

Figure 18.3-2  Disk cam's radius profile v.s. rotational angle

Helical cam design

Now that we know how to design a disk cam given a certain curved path, we need to figure out how to follow multiple disk cams with different profiles/radii one after the other seamlessly. The solution is a helical cam, where we employ a z direction to have a continuously changing radius across multiple rotations. The design of z axis value is straightforward; we simply need to decide a pitch for the helix, then the z value can be calculated as

With that, we create the following helical cam profile that encodes the entire longhorn:

Figure 18.3-3  Various views of the helical cam profile

Now putting everything together, we have a working helical cam and follower system.

Gear train to connect cam rotation and paper cylinder rotation

Based on the analysis above, the helical cam will rotate 4 full circles to draw the bottom part, and then another 4 full rotations to draw the top part. In this process, we will need the paper cylinder to rotate 1 full circle for drawing the bottom part and another full circle for the top part. Therefore, we need to design a gear train with an overall gear ratio of 1:4, where the motor will direct drive the paper cylinder while the output shaft that spins the helical cam will rotate at 4 times the angular speed.  

Slider analysis

Since we use linear bearings extensively in our cam follower system, it's important that we get the ratios correctly to minimize binding. 

Thus, we have the following ratio constraints:

Reference

https://www.cs.cmu.edu/~rapidproto/mechanisms/chpt6.html