Machine Kinematics
solved, then drawn
Concept

Dwell — where it appears

An interval of a mechanism's cycle over which the output does not move at all, produced by an arc concentric with the axis the driving part turns about. It is the property a great many machines are built around — a film frame held still, a tool paused at the end of a stroke — and the arc that produces it has to be concentric to the accuracy the dwell is quoted at.

Named by 23 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 Lost motion on a 24-tooth ratchet. A pawl can only drop into a tooth, so an input that has moved by less than one tooth pitch has moved the output by nothing at all. On 24 teeth the pitch is 15.0°, and that is the worst case with one pawl. Two pawls offset by half a pitch halve it and three thirds it, because whichever pawl is over a root drops first. None of this is a manufacturing question: the numbers are the same on a perfectly made ratchet, which is what separates them from the lost motion an as-built measurement would.

The resolution is the pitch

A ratchet's step and a ratchet's error are the same number. Nothing about how well it is made improves that, more pawls divide it by a whole number, and the obvious remedy — cut more teeth — runs into a wall that is geometric rather than practical: at a tooth depth of 0.16 radii the construction stops at twenty-nine.

timing · Intermittent The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell on this site. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake this site has already made once. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell on this site is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

timing · Intermittent 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 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 30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 and a half tooth pitches. The heavier line at each pallet is the locking face, here an arc about the arbor; the lighter one is the impulse face the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0320° from it. Dragging the pallet through its whole engagement moves the wheel by 0.106° of recoil. Of the 6.0° the wheel turns each beat, 65.9% is drop and does nothing.

Where the tooth lets go

A pair of pallets spanning a whole number of tooth pitches and a half advances the escape wheel exactly half a pitch every beat, and that half pitch divides into the impulse and the drop with nothing left over. Drop is not chosen. It is whatever the impulse leaves, and on a thirty-tooth wheel it is two thirds.

timing · Intermittent Watt's chain. Six links and 7 pin joints, so Grübler counts 3(6 − 1) − 2(7) = 1 and the rank of the Jacobian measures 1. Two of the six links must carry three joints; they are shaded. Here they share a joint, which makes this a Watt chain. That adjacency is the entire classification, and it is read off the graph rather than off the picture.

Six bars, and what the extra dyad buys

Every linkage so far has had four bars, because four is the smallest closed chain that moves. The next one up is six, not five, and six is where the subject stops being one family and becomes a taxonomy — two chains, distinguished entirely by whether the two links carrying three joints happen to touch.

linkages · Synthesis A dwell is a measurement, not a stop. The Stephenson six-bar's output against a full turn of the crank, with the four-bar it is built on for comparison. Inside a band of ±1° the six-bar's output holds still for 145.8° of crank and the four-bar's for 41.9°. The dwell comes from a stretch of the coupler curve that fits a circle of radius 3.006 to within 2.96e-3, and the arm from the coupler point is that radius. Nothing here stops; it moves less than the band.

A dwell made from a curve

Parts of a coupler curve are very nearly circular arcs. Put a link of the arc's own radius on the coupler point and its far end stands almost still while the point runs along it, so the output dwells — 146° of crank inside a one-degree band, against 42° for the four-bar it is built on. A dwell linkage does not stop. It moves less than the tolerance, and how much less is a number.

linkages · Synthesis 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°.

An offset trades the rise for the return

Moving a roller follower's line of travel off the cam's centre lowers the pressure angle on one stroke by raising it on the other. When the rise and the return are mirror images the best offset is zero. When the cam rises in 90° and returns over 170°, an offset of 5.88 takes the worse stroke from 27.37° to 21.81° by handing the return 6.48° it did not need.

cams · Cam Crank angle θ and crank angle −θ, joined. The coupler curve alone, with 12 chords, each joining the point the tracing point reaches at a crank angle θ between 0° and 180° to the point it reaches at −θ on the same assembly. Every chord is square to the dashed axis, to 7.5 × 10⁻¹⁶ in the cosine, and every midpoint lies on it. Measured at 48 pairs, the reflection of one point misses the other by at most 4.2 × 10⁻¹⁵. The crank's own angle is the pairing: the two places the curve crosses its axis are crank angles 0° and 180°, the only angles equal to their own negatives.

A symmetric curve from a lopsided machine

A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.

curves · Coupler Where enumerating the corners stops being affordable. The two routes to a tolerance band, costed against the number of toleranced lengths. Enumerating every extreme combination is 2ⁿ mechanisms at every crank position; differentiating the constraints is n linear solves. A four-bar is 16 corners and a Watt six-bar is 128, which is still cheap — 142 solves for one position — and the curve is the point rather than either number: at twenty parameters, which is an ordinary spatial mechanism, the corner route is a million mechanisms and the derivative route is twenty. Both are drawn because the corner route is not merely slower, it is the one that assumes nothing, and its answer is what the cheap route has to be checked against.

Seven lengths and a hundred corners

Nothing in a tolerance analysis is about four. A Watt six-bar has seven lengths, its corner enumeration is 128 mechanisms rather than 16, and the two routes still agree to a hundredth of a per cent — but the costs have separated — 142 solves against seven. At twenty parameters, which is an ordinary spatial mechanism, it is a million against twenty.

practice · Tolerance 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 A dwell six-bar built on the vertex of a symmetric coupler curve. The four-bar with ground 3, crank 1, and coupler, rocker and arm all 2.5, with the angle at the rocker pin set to 120.8024°, so that its coupler curve is its own mirror image about the dashed line. The curve crosses that line at crank angle 0°, and a link of the osculating radius there, 5.7691, runs from the tracing point to a pin at the centre of curvature; an output link of 3 from a third ground pivot holds that pin, square to the dwell link at the vertex. Faintly, the dwell link and output 50° of crank either side. The output swings 7.43° over a whole turn, and near the vertex its angle changes only at sixth order in the crank's.

The flattest dwell is not the longest

A coupler curve that is its own mirror image has no odd terms in its distance from a circle centred on the mirror line, so one angle of the coupler can remove the fourth-order term and leave a dwell of sixth order, with no search of the curve. A six-bar built there dwells for 58.7° of crank inside 0.1% of its swing. Turned two degrees away from that angle it dwells for 80.1°, and the searched six-bar for 24.4°.

curves · Coupler A roller on an arm of 50 pivoted on the left, at 35.8° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, followed by a roller on an arm of 50 whose pivot is on the left of the cam, at (−49.92, 42.79). The pivot angle balances the two strokes, and the cam is turned anticlockwise to 35.8°. The faint arc is the path the roller's centre swings through. The contact normal leans 23.48° from the roller's direction of motion, which is the pressure angle, and off the drawn pitch curve it is 23.48°. At this instant the roller's line of motion passes 2.98 from the cam's centre. Balanced, the arm peaks at 23.48° on both strokes; a sliding follower at its best offset of 5.88 peaks at 21.81°.

An arm is an offset that grows with the lift

Carry a cam's roller on a swinging arm instead of a slide and its pressure angle obeys the offset follower's formula exactly, with the offset replaced by the distance of the roller's line of motion from the cam's centre. That distance turns with the arm. On a cam whose strokes are mirror images the best arm is worse than a centred slide by about 5,000/L² degrees and leans the follower by its own tilt on the dwells. On a quick-rise cam the pivot's angle rebalances either side of the cam for under a degree, where a sliding offset moved to the wrong side costs eleven.

cams · 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 One pin and two, on the same wheel. A 6-slot Geneva wheel at one instant of its index, driven by a crank carrying one pin and by the same crank carrying two. The wheel, the slots and the centre distance are identical; only the pin count differs. Each pin drives the wheel through one slot pitch while the driver sweeps 120°, so 1 pin gives 1 index a driver turn and leaves 67% of it at rest, 2 pins give 2 indexes a driver turn and leave 33% of it at rest. Nothing else about the mechanism changes, which is why the whole question is arithmetic.

Two pins and no dwell at all

Put a second pin on a Geneva's crank and the wheel indexes twice a turn instead of once, at a quarter of the acceleration for the same output rate. Put a third on a six-slot wheel and it never rests; put a third on an eight-slot wheel and two pins meet in two slots and it jams. Both of those look like separate conditions and are one, and the boundary between them is an equation in integers with exactly three solutions.

timing · Intermittent 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 Three laws in the band: a third, a sixth and a sixth. For bands from 3% to 10 ppm of the output's swing, three things measured on the symmetric six-bar itself: the longest dwell at any angle at the rocker pin, the dwell at the sixth-order angle γ = 120.8024°, and how far below γ the best angle sits. The dots are measurements and the lines are the two-term model's predictions from two coefficients read off the output's even part. Over the four tightest bands the fitted exponents are 0.173, 0.172 and 0.333, against a sixth, a sixth and a third. At 10 ppm the best angle is 0.490° below γ against a prediction of 0.488°, and the dwell at γ is 26.43° against 26.23°. The loosest bands sit above the lines, where the terms the model leaves out are no longer small.

The dip that buys the dwell

A symmetric coupler curve's six-bar dwells longest a little below the angle that makes its error sixth order, and three bands gave three numbers. Across three and a half decades of tolerance the best angle's distance from that angle grows as the cube root of the band, both dwells as its sixth root, and their ratio is (27/4)^(1/6) = 1.3747 at every band — because the best machine spends the whole band on one dip and ends its dwell where the output comes back.

curves · Coupler

Named alongside it

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

Motion lawAccelerationthe Geneva mechanismIntermittent motionCamIndexingJerkCoupler curveRoller followerSix-barDesign ruleFollower

All concepts