Designing a Cam and Follower That Prints

A cam turns a shaft's rotation into whatever back-and-forth motion you can draw around it. A follower rides against the cam, and the cam's shape sets the follower's position at every angle. A cam that works needs two things: a motion law that starts and stops the follower smoothly, and a pressure angle under about 30° so the follower slides instead of jamming.

Engine camshafts open valves this way, and cams drove water-powered automata as early as the third century BC. In a face cam like the one built here, the follower's pin runs in a groove cut into the cam, so the cam pushes it out and pulls it back without a spring.^[1]

This article works through the motion laws and the pressure angle, then builds a parametric cam and follower in Rig Cad: you choose the motion program and the laws, the model refuses programs that would bind, and all three parts print without supports.

Everything the follower does is one function: its lift $s$ at each cam angle $\theta$. This cam runs the classic four-part program:

• Rise: over the rise angle the follower climbs by the full lift $h$.
• High dwell: it holds at full lift.
• Return: over the return angle it comes back down.
• Low dwell: it rests for whatever is left of the turn.

The centre of the follower's pin then sits at a distance $R_b + s(\theta)$ from the axle, where $R_b$ is the base radius. Traced around the cam, those points form the pitch curve, and the circle of radius $R_b$ it starts from is the base circle.^[1] The lift and the four angles are the whole design. The cam's outline follows from them.

Within a rise, the motion law is the shape of the climb. With $u$ running from 0 to 1 across the rise angle $\beta$, the model offers four:

Table: The four motion laws. Peak velocity is in units of $h/\beta$ and peak acceleration in units of $h/\beta^2$, with $\beta$ in radians.

The dwells on either side of a rise have zero velocity and zero acceleration, so what matters is how each law meets them. A cam should keep the follower's position, velocity and acceleration continuous through the whole turn, which keeps the jerk finite; a jump in acceleration is a hammer blow to the follower and everything it drives.^[2] Linear motion fails first: its velocity switches on and off at once, so its acceleration is, in theory, infinite at both ends. Simple harmonic motion starts and stops smoothly in velocity but still jumps in acceleration. Cycloidal and 3-4-5 motion meet the dwells with zero acceleration, which is why the model defaults to cycloidal.^[3]

Smoothness is not free. A law that starts and stops gently has to move faster in the middle to cover the same lift in the same angle; cycloidal motion peaks at twice the average velocity. At speed that becomes force, because peak acceleration grows with the square of the cam speed $\omega$ and falls with the square of the rise angle, scaled by the law's coefficient $C_a$ from the table:

Linear motion has no finite value here, which is why it is left out of the calculator below.

A cam can only push along the normal to its surface at the point of contact. The pressure angle $\phi$ is the angle between that push and the direction the follower is free to move.^[4] Part of the push drives the follower along its guide; the rest presses it sideways into the guide, and friction there grows with it. For a follower in line with the axle, like this one,

with $ds/d\theta$ in length per radian of cam rotation. Keep $\phi$ under about 30° for a follower that slides, or 35° for one on a pivot; past that the side load climbs quickly and the follower can jam.^[4][2]

The formula shows both fixes for a cam that pushes too steeply: a bigger base radius, or more angle for the rise so the follower climbs more slowly. It also shows the price of a smooth law. The laws that treat the dwells gently climb fastest in the middle, so they need a bigger cam for the same lift and angle.

Table: The default program, 10 mm of lift over 120° of rise and 120° of return, under each law.

The model works this out as you change it. Pressure angle now applies the formula at the current cam angle, Peak pressure angle reports the worst point of the whole program, and validators refuse any rise or return that would pass 30°, with a message saying which change would fix it. The peak comes from a close fit to the exact maximum, within about 0.15° anywhere in the slider range.

The model is three parts: the base, the cam and the follower. Each is one branch of the scene tree, and every dimension in it is an expression of the inputs.

Diagram: the program settings feed the Cam node, whose pitch curve is opened, stroked into a band and extruded to cut the track out of the cam disc. The cam angle turns the cam and places the follower, and the program also sizes the base.

flowchart TD
  P["Program: base radius, lift, angles, motion laws"] --> C["Cam node: pitch curve"]
  C --> O["Open the closed curve"]
  O --> S["Stroke it by pin radius + half the clearance"]
  S --> E["Extrude the band"]
  E --> K["Cam: disc and knob, minus track, bore and mark"]
  A["camAngle"] --> K
  A --> F["Follower: bar and pin at the current lift"]
  P --> B["Base: plate, axle, guide rails"]

• The Cam node draws the pitch curve. It takes the base radius, the lift, the three angles and a motion law for each phase, and outputs the pitch curve as a path, here with 720 segments. The program runs clockwise from the +X axis, so turning the cam counterclockwise plays it forward, and cam angle 0° is the start of the rise.
• The track is the pitch curve made wide. One path operation opens the closed curve, a second strokes it by the pin radius plus half the clearance, and the band is extruded and subtracted from the disc above a 2.4 mm floor. Every point of the track wall is the same distance from the pitch curve, so the gap around the pin is the same all the way round. Sketches to Solids covers offsetting 2D shapes.
• The cutter is oversized. The extruded band runs 1 mm above the cam's top face so the cut passes cleanly through it, the rule from Oversize the Cutter.
• Tight bends would spoil the track. Offsetting a curve by more than its radius of curvature folds the offset over itself; for a roller follower this is undercutting, and the cam stops reproducing the pitch curve.^[5] Across the whole slider range, the tightest bend in the pitch curve of any of the three smooth laws is at least 1.3 times the half-width of the track, so both walls follow the pitch curve exactly. Linear motion's corners are the exception: at each one, one wall of the track rounds over the corner.
• The follower is solved, not animated. Its position is the base radius plus the lift the program gives at the current cam angle, from the same motion-law formulas the Cam node uses, so the pin sits in the middle of the track at every angle.
• A handle and a mark. An angular control point on the knob turns the cam in the viewport, and a groove in the top of the knob points at the follower at 0°. Expressions and Variables in Rig Cad covers readouts, validators and handles, and Beyond Sliders explains why a model should refuse settings it cannot build.

A cam that fits at one angle can still bind at another. Each modelling stage went in with verification requirements that have to keep holding: no overlap between the parts, each part a single watertight solid, and overall sizes tied to their formulas. Then the cam was checked at eight angles through a full turn, and the settings were pushed to the ends of their ranges, with a clearance and interference check each time.

Table: Smallest gap between the pin and the track wall.

No case found any overlap between two parts, and the gaps are what the construction promises: half the clearance on each side of the pin.

All three parts print without supports: the base flat, the cam with its knob up, and the follower flipped so its pin points up. At the default size that is about 52 g of filament and five and a quarter hours (base 26 g, cam 23 g, follower 2.4 g). The only bridge is the roof of the cam's blind axle bore, 8.4 mm across.

The default 0.4 mm clearance leaves 0.2 mm on each side of the pin in the track, and around the axle. If your printer makes holes and slots tight, raise it; Clearances for 3D-Printed Parts That Have to Move has a coupon that finds the right value for your machine. The axle and the pin print upright, with layer lines across them. That is fine here: they are short, thick pegs under light load, and nothing in the mechanism flexes.

Nothing needs glue or screws. Gravity holds the cam on its axle and the follower in its guide, and because the track holds the pin on both sides, the follower comes back on the return without a spring.

Try a few changes in the model above. Switch both motion laws to Linear (constant velocity) and the peak pressure angle drops from 17.8° to 10.8°, but the follower's velocity now switches on and off at the ends of each stroke. Shorten the rise to 60° and the model refuses it at the default size; raise the base radius to 28.5 mm first and it goes through. Set the high dwell to zero and the rise turns straight into the return.

The ideas behind the model have their own articles: Designing a Geneva Drive That Prints for the other classic way to turn rotation into timed motion, Expressions and Variables in Rig Cad for the readouts, validators and drag handle, Beyond Sliders for building a parameter space that refuses bad combinations, and Clearances for 3D-Printed Parts That Have to Move for the gap the track depends on.