Machine Kinematics
solved, then drawn
Concept

Follower — where it appears

The part a cam drives, constrained to move along a line or about a pivot. Which kind of surface it presents — a point, a roller or a flat face — changes the shape the cam has to be while leaving the motion it receives unchanged.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. 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 along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

Prescribing motion

A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.

cams · Cam The same rise, three ways. A 20-unit rise over 120°, then a dwell. Above, the displacement — all three do the same job and are hard to tell apart. Below, the acceleration, differentiated from those same curves. Constant acceleration peaks lowest at 5.556e-3 per degree² and arrives at the dwell with a step; cycloidal peaks highest at 8.727e-3 and arrives at zero. The smoothest law has the largest peak, which is the trade the whole of cam design turns on and which the displacement plot gives no hint of.

The law that costs least is not the smoothest

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

cams · Cam A 6-slot Geneva wheel. The driving pin enters a radial slot, carries the wheel through 60°, and leaves. The centre distance is not free: it must be crank ÷ sin(180°/6) = 60.00 so that the pin enters along the slot, with the crank and slot perpendicular. That is the mechanism's one design requirement and it is what makes the driven wheel start and stop from rest — measured here at 1.5e-3 against a peak of 1.000. What it does not fix is the acceleration, which peaks at 1.35 and is why film sprocket holes tear.

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

cams · Cam 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 What each kind of joint takes away. Grübler's formula is M = 3(n − 1) − 2j₁ − j₂, and the 2 and the 1 in it are not conventions. A lower pair — a pin or a slide — holds two bodies together over a surface and leaves one relative freedom, so it costs 2. A higher pair — a cam against a follower, a wheel on a rail — touches at a point, the contact travels along both surfaces, and it costs 1. Five chains, each built and each measured from the rank of its constraint Jacobian, which has never heard of the formula. The last row is the one worth having: count that cam contact as a pin, as is very easily done, and the formula returns 0 where the mechanism has 1. The Jacobian does not move.

What each joint takes away

Grübler's formula has a 2 in it for pins and a 1 for cam contacts, and those numbers are not conventions to be memorised. They are the number of constraints each kind of joint imposes, they are measurable as the rank of a matrix, and miscounting one of them is the commonest way the formula is got wrong.

constraint · Mobility 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 The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

What is still outside

Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.

practice · Practice A flat face on a base circle of 30, where the profile bends tightest. The cycloidal programme — rise 10 over 120°, return over 120° — cut for a flat-faced follower on a base circle of 30, turned to 212.4°, where the profile's radius of curvature is smallest. The contact sits −5.37 from the follower's axis, which is ds/dθ, and the dashed circle is the one that fits the profile there: R₀ + s + s″ = 24.668, and 24.668 fitted through the drawn curve. The face's pressure angle is zero at every angle. What it needs instead is for that radius to stay positive, and it does so on every base circle above 5.332.

A flat face asks for a convex cam

A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.

cams · Cam What preload takes away, and what it leaves. The same four-bar with 0.01 of clearance at each pin, driven so that the load through every joint keeps one sign. The upper line is the lost motion it had before: about 2.80° of crank rotation thrown away on every reversal, unrepeatable, and not removable by calibration. The lower curve is what is left once the pins are held against one side of their holes — a fixed offset of between -0.330° and 0.013°, which is a dimensional error rather than play, so its average of -0.136° comes out in a calibration and only the 0.343° of variation survives. A factor of 8.2, bought with a permanent parasitic load that is not modelled here.

Taking up the play

Preload does not make a clearance smaller. It takes away the clearance vector's direction, which is the property that made the error unrepeatable — so 2.80° of lost motion becomes a 0.343° offset, of which 0.136° is a constant that calibrates out. A factor of eight, bought with a permanent parasitic load that is not modelled here.

practice · Clearance What the valve does, against what the ratio promises. The solved valve lift, and the cam's lift multiplied by the nominal ratio. They are the same curve to the eye and they are not the same curve: the peak is 12.777 mm against a promised 12.842 mm, short by 0.51%. The shortfall is not an error in either number. It is what happens when a ratio measured at one position is applied across a movement, and it is why cam cards and rocker ratios are quoted together.

The cam is not the valve

A rocker arm's ratio is the ratio of two moment arms measured at one position, and the rocker swings twelve degrees while the valve opens. The instantaneous ratio runs 1.588 to 1.605, so the peak valve lift is 12.78 mm where the number on the box promises 12.84 — and the shortfall depends on how the rocker was set up, not on the cam.

applied · Cam Where the pad touches, and the rule that fixes it. The contact point's travel across the valve tip through one cam event, for a rocker squared up with the valve shut and for the same rocker squared up at mid-lift. The first wipes 1.35 mm and the second 0.34 mm — a factor of 4.0 for a shim under a stud. The rule an engine builder is taught is that the geometry is symmetric about wherever it is set square, so setting it square at rest puts the entire lift on one side of the symmetry. It is folklore that turns out to be geometry, and the same shim brings the peak lift back to 12.84 mm against 12.78.

Where the pad touches

A rocker's pad slides 1.35 mm across the valve tip through one cam event when the geometry is squared up with the valve shut, and 0.34 mm when it is squared up at mid-lift. The engine builder's rule about shimming a rocker stud is folklore that turns out to be geometry, and the same shim brings the peak lift back to what the ratio promised.

applied · 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 The same cycloidal rise behind a bent joint. A cycloidal rise over 120° of cam angle, with the camshaft driven at constant speed through a Hooke's joint bent at 30°. Across: the fraction of the rise; up: the follower's acceleration in time, divided by its peak on a steady shaft. The joint turns the camshaft unevenly with a period of half a turn, so where the rise falls in that period — the clocking, the cam angle at which the rise begins, measured from where the shaft turns fastest — decides what the follower feels. At the worst clocking, 100°, the peak is 1.352 times the steady one; at the best, 30°, it is 0.828 — lower than on a steady shaft. Every curve still starts and ends at nought. Dragging changes the bend.

A cam behind a bent joint

A cam's motion law is written in cam angle and delivered in time, and the two are the same only on a shaft that turns steadily. Drive the camshaft through a Hooke's joint and the follower's acceleration gains a second term — the law's velocity times the shaft's angular acceleration — that a steady shaft does not have. Where the rise falls in the joint's half-turn then decides everything: at a 30° bend the same cycloidal rise peaks 35% above its steady value or 17% below it. And past a bend of 34.5° a well-clocked cycloidal cam gives its follower less peak acceleration than the constant-acceleration law, the one designed to minimise it.

cams · Cam A cycloidal rise on a chain-driven camshaft. A cycloidal rise over 120° of cam angle, the camshaft driven by a roller chain moving at steady speed over an 11-tooth sprocket. The chain leaves the sprocket at a radius that varies as each pin seats, so the shaft speeds up and slows once per tooth, and its angular acceleration jumps from one sign to the other at every seating. The follower's acceleration (solid) therefore steps at every tooth — 4 or so inside this rise — by up to 0.414 of the steady peak, and the cycloidal law's continuous acceleration, the reason it is chosen, is lost. At the worst clocking against the teeth the peak is 1.165 times the steady shaft's (dashed). Dragging changes the tooth count.

A cam on a chain steps at every tooth

A roller chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so a chain-driven camshaft speeds up through every tooth and its angular acceleration jumps back at every seating. That puts a step into the follower's acceleration once per tooth, whatever the law — a cycloidal rise, chosen for its continuous acceleration, gets four steps inside a 120° rise on an eleven-tooth sprocket, each 0.41 of its peak. The harm falls with the tooth count, as one over it, not up as its square, and the simple harmonic law, whose peak sits where its velocity is nought, is hurt least of all.

cams · Cam The same 45° rise on an arm, with the quick return and with a return chosen for the arm. A cycloidal rise of 20 over 45°, cut for a flat face on an arm pivoted 90 from the cam's centre and lifting against its turning, each at the least base that keeps the cam convex. Left: the quick programme's return, 170° cycloidal, base 23.26, with 1.0% of room before the return goes wrong. Right: a 170° simple harmonic return, base 18.08, with 4.9% of room before the rise limits it.

A gentle return cannot widen a steep rise

An arm lifting against a cam on a 45° rise had its least base pinned by the return: raise it a hundredth and the return went concave. Give it a return chosen for the arm, slower or gentler, and the least base falls from 23.3 to 18.1, a fifth less. But the window above it only grows from 1% to 4.9%, because once the return is out of the way the rise closes the window itself, and the arm's own reach closes the next one. With a tenth's room the best return saves 8% of base, and no return of nine gives a quarter's room. Constant acceleration, the law with the lowest peak deceleration, is the worst return of the three.

cams · Cam

Named alongside it

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

AccelerationMotion lawDwellJerkPressure angleBase circleCamConstraintCam profilecycloidal motionDerivativeEnvelope

All concepts