---
title: "Disc cam with flat-faced follower numerical"
subject: "Theory of Mechanism and Machine I"
chapter: 3
key: "semester_6/theory_of_mechanism_and_machine_i:flat-faced-follower-cam-radius-curvature"
source: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/flat-faced-follower-cam-radius-curvature
site: BCT Engineering
---

# Disc cam with flat-faced follower numerical

Theory of Mechanism and Machine I · Chapter 3: Cams and Followers

- Asked in: year not given, 8 marks
- Video: [HOW TO DRAW THE CAM PROFILE (FLAT FACED FOLLOWER) II SIMPLE HARMONIC MOTION](https://www.youtube.com/watch?v=NEwzdJg2_R8) — TECH CAD
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/theory_of_mechanism_and_machine_i/flat-faced-follower-cam-radius-curvature
- Whole chapter: https://www.bctengineer.com/md/semester_6/theory_of_mechanism_and_machine_i/chapter-3

## Question

A disc cam with a radial flat-faced follower has a base circle of radius 30 mm and rotates at 600 rpm. The follower rises 20 mm with simple harmonic motion during 90 degrees of cam rotation, then dwells. Find (a) the maximum velocity and acceleration of the follower during the rise, (b) the minimum radius of curvature of the cam profile during the rise, and (c) the minimum width of the follower face if the face must extend 5 mm beyond the extreme contact point on each side.

## Short answer

For SHM rise: v_max = ω·(πh/(2β)) = 1.257 m/s, a_max = ω^2·(π^2 h/(2β^2)) = 157.9 m/s^2. Profile radius of curvature ρ = r_b + s + d^2s/dθ^2, minimum 10 mm at end of rise. Face width: contact offset x = ds/dθ max 20 mm; add 5 mm each side → 30 mm (50 mm if symmetrical about axis).

## Answer

#### 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 $\pi$):

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

| $x$ | $\theta$ | $\rho$ (mm) |
|---|---|---|
| 0 | 0 | 70 |
| $\pi/2$ | 45 deg | 40 |
| $\pi$ | 90 deg | 10 |

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).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. For a radial flat-faced follower, the radius of curvature of the cam profile is:
   A. rho = r_b + s + d^2s/dtheta^2
   B. rho = r_b + ds/dtheta
   C. rho = r_b - s + d^2s/dtheta^2
   D. rho = s + ds/dtheta
   Answer: A. rho = r_b + s + d^2s/dtheta^2. If it reaches zero or goes negative the profile has a cusp (undercutting). (Key point: Radius of curvature formula)

2. With h = 20 mm, beta = 90° and 600 rpm, v_max of the follower is:
   A. 0.628 m/s
   B. 2.513 m/s
   C. 157.9 m/s
   D. 1.257 m/s
   Answer: D. (ds/dtheta)max = pi*h/(2*beta) = 20 mm/rad, so v_max = 62.83 x 0.020 = 1.257 m/s. (Key point: Maximum follower velocity)

3. Here rho = 40 + 30cos(x) mm during the rise. Where is the radius of curvature smallest and what is it?
   A. At the start, 70 mm
   B. At mid-rise, 40 mm
   C. At the end of rise, 10 mm
   D. At the end of rise, 70 mm
   Answer: C. cos x = -1 at the end of the rise (x = pi), giving rho_min = 40 - 30 = 10 mm, which is positive so no undercutting. (Key point: Where rho is minimum)

4. The contact point moves up to 20 mm from the follower axis. With 5 mm extra at each end the minimum face width is:
   A. 20 mm
   B. 25 mm
   C. 40 mm
   D. 30 mm
   Answer: D. The contact point stays on one side of the axis and travels 20 mm, so width = 5 + 20 + 5 = 30 mm. (Key point: Face width)

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