Cam Profile Designer

Watch a cam turn and you see almost nothing happen. A lopsided disc spins on a shaft, and a rod resting on its edge rises, waits, falls and waits again. That small dance is the whole point. A cam converts steady rotation into a motion with a schedule, and the schedule is written into the edge of the disc.

Cams are old. A cam-shaped swing arm worked the trigger of Chinese crossbows around 600 BC, Hellenistic water-driven automata had continuously turning cams by the third century BC, and from the end of the tenth century European millwrights used them to drive hammers, fulling mills and paper mills. The modern examples are less romantic and far more numerous: the camshaft that opens an engine’s valves, the switch timers that once ran washing machines, the programming cams that moved each tool of an automatic lathe. In every one of them somebody had to decide not only how far the follower moves, but how it gets there.

Rise, dwell, return

The usual way to describe a disc cam is its displacement diagram. Plot the follower’s lift against the angle the cam has turned. A rise carries the follower away from the shaft, a dwell holds it still, and a return brings it back. A packaging machine might need a part pushed out during the first third of a turn, held while glue sets, and pulled back before the next part arrives.

The smallest circle that touches the cam is the base circle. Add the roller’s radius and you get the prime circle, the path of the roller’s centre while the follower sits at its lowest. Lay the displacement diagram around that circle, angle by angle, and you get the pitch curve: the path the roller centre must follow relative to the cam.

The diagram, though, only says where the follower is. A machine cares just as much about how quickly that position changes, and how quickly that rate changes. Velocity is the slope of the displacement curve. Acceleration is the slope of velocity, and it sets the force needed to move the follower’s mass. Jerk is the slope of acceleration. Engineers who design cams track all four, and the cam literature even has a name for the four curves together: the s, v, a, j diagrams.

Why the motion law matters more than the lift

The simplest possible rise is a straight line on the displacement diagram, a constant velocity. It is the example most textbooks draw first, and it is a trap. The follower sits still during the dwell, then must instantly move at full speed. A step in velocity means an acceleration without bound at that joint, an ideal hammer blow, twice per rise. The contents page of Norton’s cam handbook puts a chapter called “Unacceptable Cam Curves” straight after the introduction.

The simple harmonic law fixes the velocity. Its displacement follows half a cosine wave, the same shape as the textbook spring-and-mass oscillation, so the follower starts and stops gently. Its acceleration does not. At each end of the rise the harmonic law still demands a sizeable acceleration, while the dwell next to it demands none. The acceleration steps, and a step in acceleration is an unbounded spike in jerk. The physics of jerk describes exactly this: a jump in acceleration behaves like a Dirac delta in jerk, scaled to the size of the jump.

The cycloidal law cures that. Its displacement is the path of a point on a rolling wheel, one arch of the curve x equals r times t minus sine t, scaled to fit the rise. Velocity and acceleration both start and end at zero, so the follower leaves and joins a dwell with no step in either. Only jerk still steps at the joints, by a finite amount. A 2024 study that shaped a polynomial cam for a footwear test rig calls continuous velocity and acceleration the fundamental law of cams, and the cycloidal law obeys it.

Polynomials give the same control in a different form. The 3-4-5 polynomial, ten x cubed minus fifteen x to the fourth plus six x to the fifth, has zero velocity and zero acceleration at both ends; its jerk steps like the cycloidal law’s. The 4-5-6-7 polynomial adds one more condition and brings jerk to zero at the ends as well, so nothing steps at a dwell at all. Computer graphics people know these two as smootherstep and the next member of the same family. Cam designers reach for them because jerk excites vibration and wear, and because a follower that cannot keep up with a jerky profile chatters.

Smoothness is not free. For the same 25 mm rise over 120 degrees at 120 rpm, the peak accelerations come out at 4.44 m/s squared for the harmonic law, 5.20 for the 3-4-5 polynomial, 5.65 for the cycloidal law and 6.76 for the 4-5-6-7 polynomial. Each smoother law buys its continuity with a taller peak, and with more force on the follower spring at speed. The constant velocity law has the lowest velocity of all, 0.15 m/s, and the worst acceleration, which is unbounded.

So a motion law does not remove roughness. It decides where the roughness goes: into a step in velocity, a step in acceleration, a step in jerk, or a taller peak that the follower spring has to carry.

Pressure angle: when the follower starts to jam

Once the motion is chosen, the shape has to deliver it. The cam pushes on the follower along the common normal at the contact point. The follower can only slide along its own axis. The angle between those two directions is the pressure angle, and only its cosine part of the push lifts the follower. The rest presses the follower sideways into its guide, where friction eats it.

That sideways load is why cams jam. A recent kinematic synthesis of an engine cam describes the pressure angle as governing both efficiency and the tendency of the pair to jam, and keeps it under about 30 degrees for translating followers. The Brazilian authors of a set of pressure-angle nomograms report the same 30 degree ceiling as the empirically accepted value for radial cams with translating roller followers, citing standard cam design handbooks. They also report cases where angles up to 48 degrees still ran, so 30 is a working guide rather than a law of nature.

For a roller on a follower whose axis is offset from the cam centre, the pressure angle has a short closed form. Its tangent is the follower velocity per radian of cam angle, minus the offset, divided by the follower’s height above the cam centre. Two design levers fall straight out of it. A larger base circle raises the denominator and lowers the angle everywhere. An offset lowers the angle on the rise and raises it on the return, which is useful when the rise is the hard stroke.

Plotting this equation, the nomogram paper’s first example, a 150 mm harmonic rise over 180 degrees on a 100 mm prime circle with a 20 mm offset, peaks at 18.9 degrees on the rise and 32.1 on the return. The designer above recomputes that example to within a tenth of a degree.

A flat-faced follower behaves differently. If its face is square to its axis, the contact normal always runs along the axis and the pressure angle is zero. The price moves elsewhere. The contact point slides across the face by a distance equal to the follower velocity per radian, so the face must be wide enough to hold the extremes on both sides, or the motion is distorted when the contact runs off the edge.

Undercutting, and how the outline is finally drawn

The cam outline is not drawn directly. For a roller follower, the pitch curve comes first, and the profile is the envelope of all the roller circles placed along it: a curve tangent to every member of the family. In practice that means stepping one roller radius inward along the normal of the pitch curve at each point. For a flat face the profile is the envelope of the face line, and its radius of curvature works out to the base radius plus the lift plus the lift’s second derivative.

Curvature is where cams fail to exist. Where the pitch curve bends convexly with a radius smaller than the roller, stepping inward by the roller radius folds the profile back over itself. No cutter can make that shape, and a cam ground anyway undercuts: the roller no longer follows the intended motion. For the flat face the profile radius must stay positive, or a cusp forms. A vibration-exciter study found that the no-undercut condition, not the pressure angle, set the smallest usable base circle for its aggressive rise.

A worked example ties the pieces together. Take a 40 mm base circle, a 10 mm roller and a 5 mm offset. Rise 25 mm over 120 degrees with the cycloidal law, dwell for 60, return over 120 with the 3-4-5 polynomial, dwell for 60, and turn it at 120 rpm. The follower peaks at 0.3 m/s and 5.65 m/s squared.

The pressure angle reaches 17.1 degrees on the rise, at 56 degrees of cam angle, and 24.2 degrees on the return. The tightest convex bend of the pitch curve is 47.8 mm, far above the roller’s 10 mm, so nothing undercuts. Shrink the base circle to 25 mm and the return climbs to 31.1 degrees, just over the guide.

The designer cannot say everything about a cam. It treats every part as rigid and the cam speed as constant, so it says nothing about spring force, follower float, vibration or contact stress, which are the subjects of whole chapters in the handbooks. It handles translating followers only, not oscillating arms, and its maxima are read from samples every quarter, half or whole degree. What it can do is show, before any metal is cut, whether a chosen motion is smooth, whether the follower will be pushed or jammed, and whether the shape can be made at all.