Cams and Followers
Motion laws, velocity and acceleration, pressure angle, base circle and follower forces with a worked 20 mm rise.
What a cam does
A cam is a shaped disc or cylinder that moves a follower through a prescribed motion: open and close valves, index a conveyor, trip a sensor, drive a packaging machine. The designer picks the follower motion (rise, dwell, fall, dwell) and works out the cam shape that produces it. A cam gives accurate, repeatable motion in a single mechanical package with no control system, but the shape is fixed (an electronic cam on a servo drive changes the profile in software: see Motion Control and Actuators).
Motion laws
During a rise of height h over a cam rotation β at angular speed ω, the follower displacement s(θ), velocity v = ds/dt and acceleration a = d²s/dt² follow a motion law. The follower force is mass × acceleration, so the peak acceleration and, even more, abrupt changes of acceleration (jerk) govern the stress, wear, noise and vibration. The golden rule of cam design (the fundamental law of cam design): the follower acceleration must be finite and the jerk finite, so that acceleration is continuous at the ends of every segment.
| Law | vmax factor Cv | amax factor Ca | Peak velocity (h = 20 mm, β = 120°, 300 rpm) | Peak acceleration | Remark |
|---|---|---|---|---|---|
| Simple harmonic | 1.57 | 4.94 | 0.47 m/s | 22.2 m/s² | jumps at the ends (infinite jerk) |
| Cycloidal | 2.00 | 6.28 | 0.60 m/s | 28.3 m/s² | zero at the ends, smooth |
| Modified sine | 1.76 | 5.53 | 0.53 m/s | 24.9 m/s² | zero at the ends, lowest peak acceleration of the smooth laws |
| Modified trapezoid | 2.00 | 4.89 | 0.60 m/s | 22.0 m/s² | lowest acceleration of the smooth laws, finite jerk |
| Parabolic (constant acceleration) | 2.00 | 4.00 | 0.60 m/s | 18.0 m/s² | lowest peak acceleration of all, but infinite jerk at three points |
vmax = Cv h ω ÷ β amax = Ca h ω² ÷ β²
Worked example: a 20 mm rise in 120°
Worked example: the cam that lifts the sensor flag on the press
The cam turns at 300 rpm (ω = 31.4 rad/s) and lifts the 0.8 kg follower 20 mm in 120° (2.094 rad, taking 67 ms), with a dwell, a symmetrical fall and a dwell of 60° each.
- Cycloidal law: vmax = 2 × 0.020 × 31.4 ÷ 2.094 = 0.60 m/s; amax = 2π × 0.020 × 987 ÷ 4.386 = 28.3 m/s² (2.9 g).
- Simple harmonic law (the same rise): amax = 22.2 m/s², lower, but it starts at that value: acceleration (and so force) jumps from zero to 18 N instantly at the start of the rise: this knocks, and high-speed cams avoid it.
- Inertia force F = m a = 0.8 × 28.3 = 22.6 N (peak, cycloidal).
- Follower spring: on the falling part of the motion the follower tends to lift off the cam (the cam can only push). The spring must supply at least m a = 22.6 N at that point plus a margin: a spring with a preload of about 15 N and a rate of 8 N/mm gives 175 N at full lift, enough for the acceleration, the friction and a 50 percent safety factor.
- Power: at peak velocity the cam delivers F v = (23 + 40 N load) × 0.60 = 38 W; the average over the cycle is far lower. The motor sees a pulsating torque, so a flywheel or a smooth law helps (Flywheels, Balancing and Rotors).
Pressure angle and base circle
The pressure angle φ is the angle between the follower’s line of motion and the normal to the cam surface at the contact. A large angle means that much of the force pushes the follower sideways against its guide instead of lifting it, raising friction and the risk of jamming. The rule: keep φ below 30° for a translating roller follower (45° for an oscillating follower). For a radial roller follower tan φ = (ds ÷ dθ) ÷ (Rp + s), with Rp the prime radius (base circle plus roller radius). The way to reduce the pressure angle is to enlarge the base circle, to lengthen the rise angle, or to offset the follower.
Worked example: the base circle
- The maximum slope of the cycloidal rise is ds ÷ dθ = 2 h ÷ β = 19.1 mm/rad at mid-rise (s = 10 mm).
- With a prime radius of 30 mm: tan φ = 19.1 ÷ (30 + 10) = 0.477, φ = 25.5°: acceptable (under 30°).
- To hold φ ≤ 30° the prime radius must be at least 19.1 ÷ tan 30° − 10 = 23.1 mm. With a larger radius of 50 mm the angle falls to 17.7°, at the price of a bigger cam.
- A shorter rise angle (60° instead of 120°) doubles the slope and the pressure angle roughly doubles: the faster the rise, the bigger the cam must be.
Follower force, spring and contact stress
The cam contact is a line (roller on cam) with a high Hertz contact stress; for steel on steel the allowable is about 1 000 to 1 400 MPa for hardened surfaces (58 to 62 HRC). The stress is σH = √(F ÷ (π L) × (1 ÷ Req) × Eeq ÷ (1 − ν²)) ÷ … : use the standard line-contact formula σH = 0.564 √(F Eeq ÷ (L Req)) with the equivalent radius of the roller and the cam’s radius of curvature. Concave parts of the profile have a larger contact area; convex parts of small radius (sharp lobes) are worst: the cam radius of curvature must exceed the roller radius (no undercut). Case-harden the cam and use a hard roller, with lubrication.
Design steps and practical rules
- Define the motion (rise, dwell, fall, timing) from the machine cycle.
- Select the motion law (cycloidal or modified sine for most; modified trapezoid for heavy followers at speed).
- Choose the base circle from the pressure angle limit, and check the radius of curvature for undercutting.
- Compute the forces (inertia + load + spring + friction) and the follower spring: keep contact at all speeds, with a margin of at least 1.5.
- Check the natural frequency of the follower train (a stiff, light follower: at least 10 times the cam frequency, preferably more) or the vibration will spoil the motion (Vibration, Isolation and Noise).
- Check contact stress, shaft torque and bearings, then make the profile by CNC and heat treat.
Follower types and cam profiles
| Follower | Description | Notes |
|---|---|---|
| Knife edge | sharp tip | wears fast; theory only |
| Roller | a roller on a pin (cam follower bearing) | the usual choice: rolling contact, low wear |
| Flat-faced | flat disc, often offset | compact, valve trains; needs a convex cam |
| Spherical / mushroom | curved face | less edge loading than flat |
Cams are made as plate (disc) cams (the commonest), cylindrical (barrel) cams with a groove for the follower (positive drive, no return spring needed: desmodromic), face cams, and globoidal indexers for fast, smooth rotary indexing. A grooved cam gives positive motion in both directions at the price of a harder-to-make track and clearance (backlash).
Related
- Previous: Power Screws and Lead Screws. Next: Linkages and Mechanisms.
- Motion Control and Actuators for electronic cams.
- Machine Design: the whole guide.
FAQ
What is the fundamental law of cam design?
The cam–follower acceleration function must be continuous across all boundaries, and the jerk finite. Laws that violate it (simple harmonic with dwell, constant velocity) knock and wear at high speed.
Why does the follower jump off the cam?
The spring is too weak for the acceleration, or the follower system resonates. Increase the spring force or stiffness, lower the speed or choose a smoother law.
How do I reduce the pressure angle?
Increase the base circle radius, increase the rise angle, reduce the rise, or offset the follower.
Cam or servo?
A cam is cheap, fast and robust for a fixed motion at a high rate. A servo with an electronic cam is flexible, quiet and reprogrammable, at higher cost.
References
- Norton, R. L., Cam Design and Manufacturing Handbook, Industrial Press; Norton, R. L., Design of Machinery, McGraw-Hill; Rothbart, H. A., Cam Design Handbook, McGraw-Hill.
- Shigley, J. E., Uicker, J. J., Theory of Machines and Mechanisms, Oxford University Press; Mabie, H. H., Reinholtz, C. F., Mechanisms and Dynamics of Machinery, Wiley.
- VDI 2143 (cam mechanisms: kinematics) and ISO 1101 for the cam profile tolerance.
Last reviewed: October 2026. Motion-law constants are exact; contact and spring values are typical.