Disc cam with flat-faced follower numerical

Theory

For a radial flat-faced follower, the contact point lies at a distance $x = \dfrac{ds}{d\theta}$ from the follower axis, and the radius of curvature of the cam profile is

$$\rho = r_b + s + \frac{d^2 s}{d\theta^2}$$

where $r_b$ is the base circle radius. If $\rho$ becomes zero or negative, the profile has a cusp (undercutting).

Data

$h = 20$ mm, $\beta = 90^\circ = \pi/2$ rad, $r_b = 30$ mm, $\omega = \dfrac{2\pi \times 600}{60} = 62.83$ rad/s.

SHM during rise:

$$s = \frac{h}{2}\left(1 - \cos\frac{\pi\theta}{\beta}\right),\quad \frac{ds}{d\theta} = \frac{\pi h}{2\beta}\sin\frac{\pi\theta}{\beta},\quad \frac{d^2s}{d\theta^2} = \frac{\pi^2 h}{2\beta^2}\cos\frac{\pi\theta}{\beta}$$

(a) Maximum velocity and acceleration

$$\left(\frac{ds}{d\theta}\right)_{max} = \frac{\pi h}{2\beta} = \frac{\pi(20)}{2(\pi/2)} = 20.0\ \text{mm/rad}$$ $$v_{max} = \omega\left(\frac{ds}{d\theta}\right)_{max} = 62.83 \times 0.0200 = 1.257\ \text{m/s}$$ $$a_{max} = \omega^2\left(\frac{d^2s}{d\theta^2}\right)_{max} = \omega^2\,\frac{\pi^2 h}{2\beta^2} = (3948)(0.040) = 157.9\ \text{m/s}^2$$

where $\dfrac{\pi^2 h}{2\beta^2} = \dfrac{\pi^2 (20)}{2(\pi/2)^2} = 40$ mm.

(b) Minimum radius of curvature

Put $x = \pi\theta/\beta$ (from 0 to $\pi$):

$$\rho = 30 + 10(1 - \cos x) + 40\cos x = 40 + 30\cos x\ \text{mm}$$

The minimum is at the end of the rise:

$$\rho_{min} = 40 - 30 = 10.0\ \text{mm}$$

It is positive, so the profile is not undercut.

(c) Face width

The contact point is at a distance $x = ds/d\theta$ from the follower axis. During the rise it moves from 0 out to a maximum of $20.0$ mm (at mid-rise) and back, always on the same side of the axis. So the contact travel is $20.0$ mm. Adding 5 mm at each end:

$$\text{face width} = 5 + 20.0 + 5 = 30\ \text{mm}$$

(If the face is made symmetrical about the follower axis, as in a usual mushroom follower, or if a similar return puts contact on the other side, the width needed is $2(20 + 5) = 50$ mm.)

Answer: (a) $v_{max} = 1.257$ m/s, $a_{max} = 157.9$ m/s$^2$; (b) $\rho_{min} = 10.0$ mm; (c) face width = 30 mm (50 mm if the face is symmetrical about the axis).