Machine Kinematics
solved, then drawn
Concept

Motion law — where it appears

The displacement schedule a cam is asked to produce, as a function of cam angle. It is the specification, and every complaint about a cam — its accelerations, its pressure angles, its size — is a consequence of the law chosen.

Named by 22 essays across 5 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 Every way of stopping, on the same four questions. Six mechanisms that all turn a continuous input into an output that moves and then waits. Index is how far the output steps. Moving is the fraction of the input's turn the output is actually going for; the rest is dwell. From rest says whether the output starts and stops at zero velocity, and acceleration whether its acceleration is a number at all. Every entry is computed from the mechanism's own library, which matters for two of them: a Geneva's moving fraction is (n − 2)/2n and not 1/n, and its entry rate is zero in closed form rather than to the accuracy of a sampled sweep. The three rows whose acceleration is not a number are not badly made — they are mechanisms whose output velocity has a step, and no tolerance improves that.

The mechanism that waits

Every mechanism in this collection so far moves whenever its input moves. A ratchet, an indexer and an escapement do not: they are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face.

timing · Intermittent 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 quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse.

The gear with its teeth cut away

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

timing · Intermittent 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 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.

A lift is a size and a law is a shape

A cam's motion law is a dimensionless function of a dimensionless argument, so everything the cam field says about laws — which has the least acceleration, which has an impulsive jerk, which one costs least — transfers between cams of any size. What does not transfer is anything with the lift or the base circle in it.

cams · Cam 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 A yoke 70 wide on a constant-breadth cam, at 60°. The cycloidal programme — rise 10 over 120°, return over 120° — cut for flat faces on a base circle of 30 and held between two parallel faces 70 apart, turned to 60.0°. The upper face rests at 5.000 above the base circle, which is the programme's lift there. The lower face touches the cam too: the distance across the cam along the follower's line is 70.000 at this angle and at every other, so the follower is driven up by one face and down by the other with no spring and nothing to spare.

A cam that holds its follower both ways

Two parallel faces joined into a yoke can drive a follower out and back with no spring, but only if the cam's breadth along the follower's line is the same at every angle — s(θ) + s(θ + 180°) constant. That makes the second half-turn the first one reflected. The cam field's standing programme misses by 9.502, and a second disc that frees the programme needs a yoke at least 60.66 wide.

cams · Cam 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 A roller in the groove of a quick-rise cam at 600 rpm. The cycloidal programme — rise 20 over 90°, return over 170° — cut as a groove on a prime circle of 40 for a roller of 10, driving a follower of 0.5 kg pressed toward the cam by 20 N, and turned anticlockwise to 59.9° at 600 rpm. The groove must supply m·s″·ω² + F along the follower's travel, −66.6 N at this angle, so the roller bears on the outer wall. The marks on the groove are the four places in a turn where that force changes sign and the roller changes walls: 47.87°, to the outer; 87.13°, to the inner; 151.35°, to the outer; 193.65°, to the inner. The stretches of deceleration begin to cross over at 267.6 and 505.5 rpm.

A roller in a groove changes walls with the speed

A groove holds its roller between two walls and chooses between them by the sign of one force, the follower's mass times its acceleration plus the load pressing it in. Below a threshold speed that force never changes sign and the roller stays on the inner wall; each stretch of deceleration adds two crossovers above its own threshold. On a quick-rise cam at 20 N and half a kilogram the count goes from none to two at 268 rpm and to four at 506. A cycloidal law brings the roller across smoothly; a constant-acceleration law throws it across by a step of 3,200 N at 3,000 rpm.

cams · Cam A Geneva's acceleration against three cam laws, at the same index angle. The output's acceleration through one index, divided by the step over the square of the input angle it takes — the unit in which a cam law's acceleration coefficient is quoted — so every law is one fixed curve whatever the step. The dark curve is a Geneva of 6 slots: its step is 360°/6 and its input turns 120° while indexing, and a cam indexer is given exactly those two numbers. The Geneva's coefficient is 5.653 against cycloidal 6.283, modified sine 5.528 and simple harmonic 4.935, so at 6 slots modified sine and simple harmonic have the lower peak. The Geneva's curve starts and ends away from nought, at tan(π/6) = 0.577 in absolute terms: its acceleration steps the instant the pin enters. Dragging the slot count moves only the Geneva.

When the index law becomes a choice

A Geneva's motion law is forced by its slot count and a cam indexer's is chosen, so the fair comparison gives the cam the Geneva's own index angle and step. On peak acceleration the cam wins only below a slot count that depends on the law — 5.19 for cycloidal, 6.23 for modified sine, 8.06 for simple harmonic — and above it the Geneva does. What no slot count removes is the step: the pin arrives with an acceleration of exactly tan(π/n).

timing · Intermittent The roller's flight across a 0.05 mm clearance at every crossover. The cycloidal quick-rise programme carrying 0.5 kg against 20 N at 600 rpm, with 0.05 mm between the roller and the wall it is not bearing on. At each sign change of the groove's force the roller leaves its wall, and each curve is the gap it opens, from nought to the far wall at the top of the plot: leaves at 47.9° and lands 6.6° later at 80 mm/s; leaves at 87.1° and lands 7.1° later at 63 mm/s; leaves at 151.4° and lands 14.6° later at 35 mm/s; leaves at 193.6° and lands 13.4° later at 42 mm/s. Dragging the speed moves the crossings, lengthens or shortens each flight, and just above a stretch's threshold shows a flight that never reaches the far wall.

The time a crossover takes

A roller in a groove changes walls where the groove's force changes sign, and it gets there by flying across the clearance. How hard it lands depends on the clearance through an exponent the motion law decides — two thirds where the force passes through nought, one half where it steps — a flight that ends in a dwell lands at √(2cF/m) whatever the speed, and just above each threshold the force changes sign and the roller never arrives at all.

cams · Cam The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 0° of driver, with the driver's locking disc of radius 27.0 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, outside the 6 concave locking arcs cut into it, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 236.3°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot.

The disc decides the pin count

A Geneva's pin count is usually bounded by the slots: p indexes must not overlap, so fewer than 2n/(n−2) pins fit. The other half of the mechanism has its own inequality and nobody had measured it. The locking disc must be cut away wherever the wheel passes through it, that cut-away is wider than the index sweep at every slot count, and it is the binding condition everywhere — one pin only, from four slots upward.

timing · Intermittent 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 A curved slot bends at its mouth with the crank's circle, or against it. A six-slot Geneva wheel whose slots are curved: mouth curvature 0.5 of the crank's. The driver's pin runs round the dashed circle and enters each slot along its mouth, which is radial so that the wheel starts from rest. At entry the pin's acceleration is square to the slot, and the part of it the slot's own bend takes up does not turn the wheel. Solved over the index, the wheel starts with an acceleration of 0.289 and stops with a deceleration of 0.866, in units of the crank's speed squared, against 0.577 at both ends for a straight slot; the two always add to 1.155. Its peak is 1.649 against the straight slot's 1.350.

A curved slot moves the step

A Geneva's pin arrives with an acceleration of tan(π/n) of the crank's speed squared, and no choice of entry direction removes it. A choice of slot shape can: bend the slot's mouth with the crank's own circle and the slot absorbs the pin's acceleration instead of handing it to the wheel. But the pin leaves by the mouth it came in by, and there the same bend adds to the step. Solved at four slot counts and seven curvatures, the entry and exit steps are tan(π/n)(1 ∓ κ̂) and always sum to 2 tan(π/n). A curved slot moves the step from one end of the index to the other, every bend raises the peak, and no slot tried beat the straight one.

timing · Intermittent A roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°.

Six things a cam drive is not

A follower that is better off offset, a flat face with no size limit, a swinging arm that improves any follower, a groove that keeps its roller on one wall, a smooth law that is gentle everywhere, and a camshaft that delivers the law it was cut to. Six things people say about cam drives, each the natural reading of something true, each answered by a number the cam essays measured. They fail the same way: each reads the cam at first order, in positions and slopes, and the drive is decided at second order, in accelerations and curvatures.

wrong · Misconception A pin steered off its circle carries the wheel on any law. A 4-slot Geneva wheel and its driver. The pin rides in a radial guide on the driver and a stationary cam sets its radius, so its path through the index (solid) is not the crank's circle (dashed). Here it is cut for the modified trapezoid law over the circle's own window of 90° of driver: the wheel starts with an acceleration of nought and stops with nought, against tan(π/4) = 1.000 at both ends for the crank pin, and peaks at 3.112 against the crank pin's 5.407, in units of the driver's speed squared. The pin's radius moves between 0.667 and 0.708 on a crank of 0.707.

A steered pin keeps only the window

A Geneva's step at entry and exit belongs to its pin's circle, not to its slots. Carry the pin in a radial guide on the driver and set its radius with a stationary cam, and the wheel can be given any law at all — the pin is simply where the driver's ray meets the slot's line. Cut for a modified trapezoid, a six-slot wheel starts and stops with no step and peaks 14% below the crank pin, while the cam moves the pin by 1.2% of the crank. What the circle cannot hand over is its window: no ray before the crank pin's entry meets a resting slot, so the index can never be longer, and every degree spent holding the wheel in its slot comes out of it as the square.

timing · Intermittent A 45° rise: the slide's cam and the arm's, each as small as it can be. The quick programme with its rise of 20 shortened to 45°, cut for two flat-faced followers at the smallest base that keeps each cam convex, drawn to one scale. Left: a sliding face, base 185.56. Right: a face on an arm pivoted 90 from the cam's centre, lifting against the cam's turning and stood off −44.9, base 23.26, 8.0 times smaller. At that base the arm's cam stays convex for bases up to 1.0% larger, and no further.

A steep rise spends the arm's room

A flat face on an arm that lifts against the cam's turning needed a fifth of a sliding face's base on a 90° rise. Steepen the rise to 45° and the slide's base grows 5.6 times, to 185.6, while the arm's least base grows only 3.2 times, to 23.3. But the arm's least base sits where its cam is about to go concave on the return as well as the rise: the base can be raised by only a hundredth before the return fails. Ask for a tenth's room above the base and the arm's base grows 5.1 times, nearly the slide's rate, and at 35° no stand-off gives that room at all.

cams · Cam On even drums the law lives in the cam's groove. The station the belt must stand at on drums cut for the even proportion, 300 mm long, against the share of the operator's handwheel travel, for the speed law (solid) and for the even proportion, whose groove is straight (dotted). Speed ratios are marked at the right; on the even drums the logarithm of the ratio is proportional to the station, so the groove is the demanded law redrawn. The speed law's groove is steepest where the handwheel starts, 2.95 times the even proportion's slope, and least steep at 0.45 times it; on a barrel of 40 mm pitch radius with a 30° pressure-angle limit it needs 6.1 turns of the handwheel, against 2.07 for the even proportion.

A cam moves the law off the drums

A cone drive cut for a steep speed law needs long drums, because a belt cannot straddle a steep change of ratio across its own width. Cut the drums for the even proportion instead and let a cam between the handwheel and the belt fork place the belt, and every law runs on the same 300 mm. The steepness does not go away. The cam's groove is steeper than the even groove by exactly the factor the law's own drum was longer, 2.95 for the speed law and 44.7 for a fan's, and it is paid in handwheel turns instead of drum: six turns for the speed law against 884 mm of drum, and a fan law that needed thirteen metres becomes buildable.

strands · Strand Three steered pins hand a six-slot wheel from slot to slot. A 6-slot Geneva wheel with no locking disc and a driver carrying 3 pins, each in a radial guide whose radius a stationary cam sets. Each pin is in some slot for 120° of every driver turn: it runs in along a resting slot, indexes the wheel on a modified sine law over 40°, and runs out along the slot it moved. The solid curve is the path every pin follows while it is in a slot; the dashed circle is the crank's. At the moment shown pin 1 is running in, pin 2 is clear of the wheel, pin 3 is clear of the wheel; the wheel is held by one slot.

A wheel held by its slots needs three pins

A steered pin can hold a Geneva's wheel by running along a resting slot, but every pin is in some slot for exactly the crank pin's window, 180° less 360°/n of every turn, because it can leave a slot only by the mouth it will reach. Two windows of less than half a turn can never cover a turn, so two pins cannot hold a wheel without its disc at any slot count. Three can, on six slots and more, and four on four. At the Geneva's own share of index to dwell the index is no harder than the crank pin's, and the slots hold the wheel tighter than any lock in the field. The disc's work moves to the steering cam: on a run the cam's pressure angle is the run's own angle, and the six-slot drive's runs are 40°.

timing · Intermittent

Named alongside it

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

DwellAccelerationJerkthe Geneva mechanismFollowerIndexingCamPressure angleTangential entryBase circlecycloidal motionIntermittent motion

All concepts