A Smooth Rise Function for Cam-Follower Systems

Consider a cam-follower mechanism, as outlined in [2]. Cam-follower mechanisms have an irregularly shaped circular object (topologically equivalent to a disk) called a cam that rotates at constant angular velocity with a follower mechanism that goes up and down (for an oscillating follower) in accord with the irregular circular shape of the cam.

Figure 1: A cam-follower mechanism (From https://www.researchgate.net/publication/279290006_Nonlinear_Passive_Cam-Based_Springs_for_Powered_Ankle_Prostheses)

Cams typically have a low dwell, rise, high dwell, fall, and again low dwell parts to their cycle. The radius of the low dwell region is constant; similarly, the radius of the high dwell region is constant.

Figure 2: Low Dwell, Rise Function, High Dwell, Fall Function, and Low Dwell (Repeating) for the Position Graph of the Follower in a Cam-Follower Mechanism (From https://youtu.be/NmQlarb7pzQ?si=p3qlk2EAlyWSOAGi&t=576)

The “name of the game,” or The Fundamental Law of Cam Design, in designing the cam in a cam-follower mechanism is to make the velocity and acceleration of the position function of the follower continuous and the jerk (third derivative of the position function) of the follower piecewise continuous (bounded everywhere and continuous apart from a finite number of jump discontinuities). Each jump discontinuity in the jerk introduces a Dirac delta function input into the acceleration response of the follower, creating vibrations and “noise” in the acceleration function. Minimizing the noise in the position, velocity, and acceleration responses of the follower is the “optimum” criterion in the optimal control engineering problem of designing cam-follower mechanism [2].

Figure 3: Cam-follower mechanism acceleration curve (a) with noise (above) (b) with noise still present in design but filtered out for display (below). (From https://youtu.be/NmQlarb7pzQ?si=Mr5TSVzQj-R6HBmf&t=732)

As outlined in [1], let \(h: \mathbb{R} \to \mathbb{R}\) be given by \[h(x) = \begin{cases}0 & x \le 0 \\ e^{-1/x}& x > 0 \end{cases}\] and let \(y_0: \mathbb{R} \to \mathbb{R}\) be given by \[\displaystyle y_0(x) = \frac{h(x)}{h(x) + h(1-x)}\]

Then \[\displaystyle y_0(0) = \frac{h(0)}{h(0)+h(1)} = \frac{0}{0+e^{-1}} = 0\] and \[\displaystyle y_0(1) = \frac{h(1)}{h(1)+h(0)} = \frac{e^{-1}}{e^{-1}+0} = 1\] so \(y_0(x)\) is a "unit rise function" from the low-dwell radius of 0 at 0 to the high-dwell radius of 1 at 1.

Also, all order derivatives exist and are continuous on all of \(\mathbb{R}\), and \(y_0^{(n)}(0) = 0\) and \(y_0^{(n)}(1) = 0\) for all order derivatives as all exponentials approach 0 as \(s\) approaches \(-\infty\) via \(x\) approaching \(0^+\) through \(\displaystyle s = -\frac{1}{x}\) faster than all polynomials in \(\displaystyle \frac{1}{x}\) approach \(\infty\) via \(x\) approaching \(0^+\), as is proven in [1].

[NB: The symmetric fall function is \[\displaystyle \left.\tilde{y}_0(x) = \frac{h(1-x)}{h(x) + h(1-x)}\right]\]

If we let \[y_{(R_1, R_2, \theta_1, \theta_2)}(x) = (R_2-R_1) \frac{h(x-\theta_1)}{h(x-\theta_1) + h(\theta_2 - x)} + R_1\] for \(0 < R_1 < R_2\) and \(\theta_1 < \theta_2\), then \[\displaystyle y_{(R_1, R_2, \theta_1, \theta_2)}(\theta_1) = (R_2-R_1)\frac{h(0)}{h(0)+h(\theta_2-\theta_1)} + R_1 = (R_2-R_1)\frac{0}{0+e^{\frac{-1}{\theta_2-\theta_1}}} + R_1 = (R_2-R_1)0+R_1 = R_1\] and \[\displaystyle y_{(R_1, R_2, \theta_1, \theta_2)}(\theta_2) = (R_2-R_1)\frac{h(\theta_2-\theta_1)}{h(\theta_2-\theta_1)+h(0)} +R_1 = (R_2-R_1)\frac{e^{\frac{-1}{\theta_2-\theta_1}}}{e^{\frac{-1}{\theta_2-\theta_1}}+0} + R_1= (R_2-R_1)1+R_1 = R_2\] so \(y_{(R_1, R_2, \theta_1, \theta_2)}(x)\) is a rise function from the low-dwell radius of \(R_1\) at \(\theta_1\) to the high-dwell radius of \(R_2\) at \(\theta_2\).

Also, all order derivatives exist and are continuous on all of \(\mathbb{R}\), and \(y_{(R_1, R_2, \theta_1, \theta_2)}^{(n)}(\theta_1) = 0\) and \(y_{(R_1, R_2, \theta_1, \theta_2)}^{(n)}(\theta_2) = 0\) by the above, so \(y_{(R_1, R_2, \theta_1, \theta_2)}(x)\) satifies The Fundamental Law for Cam Design, as outlined in [2].

[NB: The symmetric fall function is \[\left. \tilde{y}_{(R_1, R_2, \theta_3, \theta_4)}(x) = (R_2-R_1) \frac{h(\theta_4-x)}{h(x-\theta_3) + h(\theta_4 - x)} + R_1\right]\]

Bibliography:
[1] Munkres, James R. (1991). Analysis on Manifolds. (1st ed). Redwood City, Calif.: Addison-Wesley Pub. Co., Advanced Book Program.
[2] Norton, Robert L. (2000). Machine Design : An Integrated Approach. (2nd ed). Upper Saddle River, New Jersey: Prentice Hall.