Machine Kinematics
solved, then drawn
Concept

Cam profile — where it appears

The surface a cam follower rides on, obtained as the envelope of the follower under the motion the cam's programme prescribes. It is the only surface on a cam that gets manufactured, which is why a check on the curve the roller's centre travels on can pass while the profile itself is wrong.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

A 34-unit roller on a 20-unit base: the cam cannot be made. The outer curve is the pitch curve — where the roller's centre has to go, which is just the base radius plus the follower's displacement, plotted in polar coordinates. The inner curve is what has to be cut: the pitch curve offset inward by the roller's radius along its own normal. Where the pitch curve bends more tightly than the roller is wide, that offset crosses itself, and the metal the profile needs has already been removed. The tightest radius of curvature here is 19.43 at 85°, against a roller of 34 — so the cam is undercut by 14.57 and no amount of care in manufacture recovers it. The pressure angle, meanwhile, peaks at a perfectly comfortable 21.2°: this cam fails a test it never looked like failing.

The cam that cannot be cut

A cam has to be big enough for two reasons, and they are not the same reason. One is that the follower will jam in its guide if the pressure angle is steep. The other is that the roller will gouge the profile if the curvature is tight — and there is a combination where the pressure angle is comfortable and the cam still cannot be manufactured at all.

cams · Cam A cam profile is an envelope, and its curvature obeys the same law. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius. That offset is the conjugate law of this field with one centre of curvature sent to infinity, and it says ρ_cut = ρ_pitch − r. Measured off the drawn polyline at six angles, the worst departure is 4.8e-7: at 95° the pitch curve has radius 29.52 and the cut profile 21.52, against 21.52 predicted. It is also why undercutting is a curvature condition rather than an accident: where ρ_pitch falls below the roller radius the offset turns itself inside out.

A profile is an envelope

A cam's surface is not a curve somebody drew. It is the envelope of the roller as the roller runs round the pitch curve, and its curvature is the pitch curve's less the roller radius — a law that this site's cam library was breaking for six months, in the one curve that gets manufactured and the only one no check looked at.

cams · Cam A cam is the envelope of a roller. The follower's roller drawn in the cam's frame at 15 angles of the cam: the roller slides in a straight guide and the cam turns under it, so from the cam's point of view the roller travels round it on a path set by the lift law. The cam's surface is the envelope of those circles — the same computation as a rack generating a tooth, with a circle instead of a straight edge and a lift law instead of a rolling condition. The cams field builds the same surface by offsetting the roller-centre path inward by the roller radius, and the two agree to 4.1e-4 mm. positioned by solving, not by drawing.

A cam is a conjugate pair

A cam is built by offsetting the path of the follower's centre inward by the roller radius. It can also be built by asking what shape stays in contact with a circle that slides in a stated way — the same computation that cuts a gear tooth — and the two surfaces agree to sixteen millionths of a millimetre.

cams · Cam A flat follower needs a face, and how wide is not a matter of taste. The cam profile the cycloidal programme produces on a base circle of 30, at 70° of rotation, with the follower's face across the top and the contact marked. The contact is not on the follower's axis: it sits at ds/dθ from it, so it wanders sideways as the cam turns and the face has to reach from the most negative value of that to the most positive. Here that is 19.099 — measured off the drawn profile as 19.099, the two routes agreeing to 1.0e-16. A face cut to the lift, or to the base circle, or to whatever looked right, is a face the cam runs off.

A follower needs a face

A flat-faced follower does not touch the cam on its own axis. The contact wanders sideways as the cam turns, by exactly ds/dθ, and a face cut to the lift or to the base circle or to whatever looked right is a face the cam runs off.

cams · Cam A flat face on a swinging arm, and the cam it asks for. The cam a flat-faced follower needs when its face is carried on an arm pivoted 300 from the cam's centre, with a base circle of 20 and a lift of 20, drawn in the cam's own frame at three cam angles. The heavy line is the face, the dashed line is the perpendicular from the cam's centre to it, and the dot is the contact. Its distance from the foot of that perpendicular is the offset, which for a sliding face would be the programme's own velocity and here is a quantity with the arm in it. Over the turn it runs from -20.41 to 17.95, so the face must be 38.4 wide.

A flat face on an arm is worse

Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.

cams · Cam

Named alongside it

The objects these essays reach for when they reach for this one.

EnvelopeFollowerPressure angleBase circlePitch curveRoller followerUndercuttingAccelerationConjugate-actionConjugate pointConstraintContact normal

All concepts