Tools › Mechanical Design › Mechanisms › Linkage and Cam Kinematics › Example
Plate cam, 3 units of lift in 90° by the parabola: pitch circle for a 30° pressure angle (Furman, Problem 3)
| Tool | Linkage and Cam Kinematics |
| Source | Furman, F. DeR. (1921). Cams, Elementary and Advanced. John Wiley and Sons, New York. Art. 29, formula (1), p. 17; Art. 30, Table of Cam Factors, p. 18; Problem 3, Art. 45–46, p. 29. |
| Level | Undergraduate |
Problem
A radial plate cam lifts a roller follower 3 units in 90° of rotation with uniform acceleration and retardation (the parabola), returns it 3 units in the next 90° by the same law, and leaves it at rest for 180°. Find the pitch circle that keeps the pressure angle to 30°, then the peak factors, the pressure angle round the cam and the profile.
Inputs to Enter
- Analysis: cam and follower; segments 90° / +3 in / parabolic, 90° / −3 in / parabolic, 180° / dwell
- Roller follower, no offset; roller radius 0.75 in (assumed, the problem leaves it to the designer); base circle radius 4.37 in, so the prime circle is 5.12 in and the pitch circle through mid-lift 6.62 in
- Cam speed 60 rev/min; pressure-angle limit 30°
Answers
- Cam factor for the parabola at 30°: f = C_v cot 30° = 2 × 1.7321 = 3.464 (book table 3.46)
- Pitch circle radius r = 57.3 h f / b = C_v h / (β tan 30°) = 6.616 in (book: chart length 3 × 3.46 × 4 = 41.52, radius 41.52 / 2π = 6.61)
- Smallest prime circle for 30° by bisection on the scanned pressure angle: 5.116 in, equal to the pitch circle less h/2, because the parabola is steepest at mid-lift; base circle 4.366 in
- On the entered 5.12 in prime circle the pressure angle peaks at 29.98° at 45° of cam angle and at −29.98° at 135° (the book's rounded 6.61 would give 30.02°)
- Peak factors of the parabola C_v = 2, C_a = 4: at 60 rev/min the follower reaches 24 in/s and ±192 in/s²; the acceleration reverses in a step at mid-stroke and at the dwell, so the jerk is infinite there
- Smallest radius of curvature of the pitch curve 4.244 in at 45°, of the cam surface 3.494 in: no undercutting
Notes
Choose Cam and follower in the Analysis select; that analysis opens on this problem, press Submit. The page also lists the pitch-curve and surface coordinates and a CSV. The table of cam factors is reproduced by C_v cot φ for the straight line, the crank curve and the parabola (test_linkage_cam_kinematics.php); one cell, the parabola at 20°, is printed 5.50 where 2 cot 20° = 5.495. The tool's first analysis, the four-bar linkage, opens on an illustrative crank-rocker for which no printed example was found.