Machine Kinematics
solved, then drawn
Concept

Velocity ratio — where it appears

The ratio of a mechanism's output speed to its input speed, which is a function of position unless something was designed to make it constant. Quoting it as a number is a claim that it does not vary through the cycle, which is true for gears by construction and false for most other mechanisms.

Named by 56 essays across 15 fields — each of them below, with the objects they name alongside it.

The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

A length is a range

Every figure in these essays so far has been drawn from four numbers. No four numbers were ever cut. Give each of them a tolerance of ±0.01 and the rocker's output stops being an angle and becomes a band 0.73° wide at one part of the turn and 0.36° wide at another — and which of those a designer is told depends only on where somebody measured.

practice · Tolerance 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 What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. Of the 14 rows, 6 are quoted and 3 are exact.

The number on the box

Fourteen machines, fourteen quoted numbers, and five different things a quoted number can be. Six of the fourteen name a quantity the mechanism does not have at all; three are exactly right, and all three of them are counts. The field is built on the difference.

applied · Misconception A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.

wrong · Misconception One contact, and the point the normal has to pass through. Two wheels on fixed centres, turning in the ratio 24 : 36, with one flank of each drawn. The contact is found by solving n·(v₁ − v₂) = 0 along the first flank — the two velocities are formed from the two rotations and subtracted, and nothing in that calculation knows where the pitch point is. The pitch point, marked with a cross, is computed separately as the one place where the two bodies' material points have the same velocity. The common normal misses it by 8.44e-15 of a millimetre, which is the law of gearing arriving as a measurement rather than as an assumption. positioned by solving, not by drawing.

The second shape is not a choice

Two bodies on fixed centres, told to stay in contact. Give one of them a shape and the other one's shape is no longer available to be designed — it is the envelope of the first one's positions, there is exactly one of it, and one routine computes it for a gear, a cam and a rotary engine alike.

meshing · Meshing A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000.

Two inputs and one output

Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.

transmission · Transmission A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms.

A ratio is a null space

Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

transmission · Transmission A flank nobody would draw, and the partner it forces. On the left, a driving flank made up on purpose: a rising curve with a nine-cycle wobble on it, chosen to be nothing in particular. On the right, in the driven wheel's frame, the shape that has to mate with it. There was a contact at all 121 sampled positions and the worst residual of the meshing equation was 3.13e-10. Conjugate action does not single out the involute — every profile has a partner that holds the ratio exactly, and the reasons for preferring one tooth form over another are all somewhere else. Generating back the other way returns the original flank to 2.33e-5 mm over 61 points. positioned by solving, not by drawing.

Any shape has a partner

Conjugate action does not pick out the involute. Hand the construction a flank invented on purpose to be nothing in particular and it returns a mate that holds the ratio exactly — so the question a tooth form answers is not whether it can transmit motion, and every real reason for choosing one is somewhere else.

meshing · Meshing A universal joint at 40° input, shafts 25° apart. Two shafts meeting at 25°, joined by a cross whose two pins are at right angles to each other and each at right angles to the shaft it carries. That is the whole geometry, and everything else follows from it. This is a four-joint spatial loop with all four axes through one point: Kutzbach counts −2 and the screw system has rank 3, so it has one degree of freedom and turns. At this instant the output shaft is at 42.79° while the input is at 40°, and the output is turning 1.0124 times as fast — which is not 1, and never is except at the four points of each turn where the curves cross.

The joint that is not constant velocity

A universal joint is the spatial mechanism everybody has met and almost nobody has been told the truth about. Its output shaft runs fast, then slow, twice per revolution, and the amount depends only on the angle between the shafts — which is why cars have two of them and why the second one has to be fitted the right way round.

spatial · Spatial Two surfaces in contact, sliding everywhere but one place. The sliding speed at the contact of an involute pair, formed by taking the velocity of each body's material point at the contact and subtracting. It is zero at exactly one position — the instant the contact is at the pitch point, measured here at 7.30e-15 mm per radian — and grows linearly on both sides of it, at the rate the relative angular velocity says. Gears roll at one point of the tooth and slide everywhere else, which is why a tooth wears into a shape with a band of polish across it rather than uniformly. What is not claimed is any consequence of the sliding: friction, wear and heat need forces, and there are none here.

Rolling at one point only

Two wheels in mesh are usually described as rolling. Their pitch circles are — those are centrodes, and centrodes roll — but the surfaces that are actually touching slide against each other everywhere except at one instant, and the sliding is the largest velocity in the mechanism.

meshing · Meshing The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach.

The lever that is the gearset

The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.

transmission · Transmission A tackle's ratio, differentiated rather than counted. Four parts of line between two blocks 260 mm apart, drawn as one strand over real sheaves. The ratio a tackle is sold with is the number of parts supporting the moving block, and it is the limit of the true velocity ratio rather than its value: the parts are not parallel, so each of them shortens by less than the lift. Differentiating the run's own length gives 3.9471 here, 1.32% short of 4, and the gap closes as the blocks separate. positioned by solving, not by drawing.

A ratio that is a derivative of a length

A tackle is sold by counting the parts of line holding the moving block: four parts, four to one. Differentiate the strand's own length instead and a four-part tackle gives 3.927 with its blocks 220 mm apart and 3.617 at 90 mm — and the integer it is named for is a limit it reaches nowhere.

strands · Strand What a Simpson gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 4 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.

Holding a member chooses the ratio

A gearset offers a plane of motions and a shift element is one linear condition, so a gear is a line in that plane. Enumerate every brake and every clutch and a Ravigneaux's twenty-seven combinations collapse to seven ratios — eighteen of them being the same gear, because locking any two members at all locks the whole gearset solid. Neutral is not one of the seven, because neutral is not a gear.

transmission · Transmission Two poles at 66°, and only one of them is still. The stubs here are accelerations, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.

The other pole

The instantaneous centre is the point of a moving plane that is not moving. There is a second point that is not accelerating, it is somewhere else entirely, and over a full turn of one four-bar the two are never closer than one and a half coupler lengths and get as far apart as twenty-two.

curvature · Curvature Where the coupler is pivoting, at 70°. At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing.

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

curves · Coupler 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 What a Ravigneaux gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 5 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.

Four speeds from two numbers

A Ravigneaux gearset has five tooth counts and gives seven exact ratios. Two of the counts do not appear in any of them — the short planet is an idler and its size is free — and the remaining three enter only through two dimensionless numbers, so the whole set of shift ratios of a four-speed automatic is a function of ring-over-sun and ring-over-the-other-sun. A Simpson three-speed is a function of one number.

transmission · Transmission Two ellipses on their foci, at a ratio of 0.603. Each wheel is an ellipse turning about one of its own foci, with the centres a major axis apart. The focal property does the work: the two radii from the two foci add to the major axis, so the contact stays on the line of centres by construction, and the rolled arc lengths agree to 2.5e-8 of their length. The ratio at this instant is 0.6033; over a turn it runs from 0.538 to 1.857, a range of 3.4490 against the ((1+e)/(1−e))² = 3.4490 the eccentricity predicts. And one turn of one wheel is exactly one turn of the other, which is the condition that makes it a pair of wheels rather than a pair of curves. positioned by solving, not by drawing.

A ratio that is a function of the angle

A gear pair is usually two circles rolling. Ask instead for an output that runs forty per cent fast for half a turn and forty per cent slow for the other half, and the two shapes that deliver it are not a design decision — the demand fixes both pitch curves completely, and the only question left is whether they close.

meshing · Meshing The roll centre through the travel — double wishbone. The roll centre is a construction: the instantaneous centre of the upright, joined to the contact patch, extended to the car's centreline. It is quoted as a height. Over 160 mm of travel it moves 54 mm — 52 mm to 106 mm — The number in a specification is the value at one position of a curve, and the curve is steeper than the thing it is a property of.

A roll centre is not a point

The roll centre is a construction on the instantaneous centre of the wheel's upright, and every step of it is exact. What it is not is a height: over eighty millimetres of bump and droop it moves 54 mm on a wishbone and 131 mm on a strut, and on the strut it goes below the road.

applied · Fourbar The steps, against the ones the rule asks for. The design rule everybody quotes is that the steps between gears should be equal in ratio, so that the engine returns to the same speed after every shift. That makes the sequence geometric, and the ideal step for this spread over this many gears is 1.5324, marked. The steps a gearset actually gives are not free: the whole sequence is a function of the tooth counts, so once the top and bottom are chosen there is nothing left to spend on the middle. The worst step here is off the ideal by 10.8%.

The steps are not free

A gearbox is supposed to have equal steps between its gears, so that the engine returns to the same speed after every shift. A gearset has one or two numbers to spend on three or four gears, so from the third one the steps are a consequence rather than a choice — and asking for them to be equal turns out to be a quadratic whose root is the golden ratio, realised in tooth counts by consecutive Fibonacci numbers.

transmission · Transmission How far the crank turns before the rocker does. With a clearance of 0.01 at each pin, the crank must be turned this far on reversal before the rocker moves at all. Through most of the cycle it is about 2.8°, and at its best 1.44°. At the two positions where the rocker reverses — marked — it is unbounded: the output velocity passes through zero there, so no amount of crank rotation moves the rocker out of its clearance band. The peak in a plot like this is therefore a property of the sampling and not of the mechanism; it reads 402° at 96 samples and grows without limit as the sampling is refined. The number worth quoting is the plateau.

How far the crank turns first

Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.

practice · Clearance One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it.

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

meshing · Meshing A ratio with no steps in it. Two pulleys whose sheaves slide on their shafts, and one belt. Pushing the primary's sheaves together makes the belt ride further out; the secondary's radius is then not a choice, because the belt is a fixed length and its length over two pulleys at a fixed centre distance is a function of both radii. So the secondary's radius here is the root of that equation, solved rather than assumed, and the drawn belt is the length it is supposed to be to 0.0e+0 mm. Ratio 1.000, with the two radii adding to 110.00 — a number the received rule of thumb says should not change and which changes by 2.6% across the travel.

A ratio with no steps in it

Push a variable pulley's sheaves together and the belt rides further out. The other pulley's radius is then not a choice — the belt has a fixed length — so it is the root of an equation, solved rather than set. The rule of thumb that says the two radii add to a constant is true to first order and wrong by 7.4% of the ratio at full shift, and the departure has a closed form.

transmission · Transmission How far a joint-space move bows off the line. Every joint runs from its start value to its end value at a constant rate — the simplest possible move, and the one that can never be refused, because every point along it is a set of joint values and every set of joint values is a pose. The tool does not travel in a straight line while it happens. It bows away by 402 mm over a move of 1223 mm, which is 32.9% of the distance travelled and enough to hit something that the straight line would have missed.

A straight line at constant speed

Run every joint from its start value to its end value and the tool bows 402 mm off the line between them. Insist on the line instead and the arm will follow it — until the path passes near a singularity, where the joint rates a metre a second demands grow as one over the distance, measured at an exponent of −1.010.

serial · Serial The slider turns round at the dead centre. The slider's position against the crank angle, through the dead centre. The curve has a maximum there — that is what a dead centre is — so the slider retreats on both sides of it, and the latch works because a mechanism resting past the top of this curve has to be pushed back up it before anything can move. Set 4° past, the slider is 71 µm below the peak, and because the curve is quadratic there, doubling the setting quadruples the depth.

Locked on purpose

A toggle clamp, a landing-gear downlock and the catch on a folding table are all the same mechanism parked a few degrees past its dead centre, where the slider's motion is second order in the crank's. Seventy-one microns of slider travel undoes a latch set four degrees over — and setting it eight degrees over does not double that, it quadruples it.

applied · Fourbar Two poles at 66°, and only one of them is still. The stubs here are accelerations, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.

Six things a centre is not

The instantaneous centre is the most over-read object in this subject. Six claims about it are in circulation, three are false, two are true of something else, and one is nearly right — and each of them comes with a measurement of how badly it goes wrong.

wrong · Misconception The radius a winch works at is not a property of the winch. A drum of core radius 30 mm filling with line 6 mm thick, one layer at a time. Line speed per turn is 2πr, and r is the radius of the layer being wound rather than anything about the machine: it runs from 207.35 mm on the first layer to 395.84 mm on the 6th, a factor of 1.909. A winch quoted at one speed is quoted at one layer, and which one is rarely said. positioned by solving, not by drawing.

The radius a winch works at

Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.

strands · Strand 404 ways to gear 60 to 1. Each dot is a pair of wheels and a pair of pinions whose ratios multiply to exactly 60, with wheels of 40 to 120 teeth and pinions of 6 to 16 leaves. The line is the hyperbola every exact answer must lie on; the dots are the ones that lie on it at whole numbers of teeth, and there are 404 of them. In a single pair there are 0 — a ratio of sixty needs a wheel of 360 teeth against a pinion of six, and nobody cuts that. Exactness is decided in integers here and never by comparing floating-point ratios, which is the one way a search like this quietly returns answers that are merely close.

A clock is a factorisation

A going train's job is a ratio and its parts are tooth counts, so whether a clock can be built is whether a number factorises inside the counts a wheel-cutting engine will cut. Sixty has 404 answers in two pairs and none at all in one. The ratio between a sidereal day and a mean one has none in any number of pairs, and the best two-pair train is out by a sixth of a second a day.

gears · Tooth Every pin position, and what it costs — 3 mm between the doors. One cell per candidate pin position: the door is swung through 95° from each and the worst clearance recorded. Light cells clear; dark cells mean the door would pass through the partition or through the door beside it, by up to 18 mm. The horizontal line is the door's own front face. With the doors touching, the boundary sits exactly on it — because the front corner's sideways rate is the pin's distance behind it, and that is zero only there. Each millimetre of gap buys a few millimetres of depth, and at 3 mm the deepest pin that still clears is 10 mm behind the face.

Where a hinge pin can go

A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.

applied · Coupler Eleven lobes from twelve pins, at 46° of eccentric. The pins are circles and the disc is their envelope. One turn of the eccentric moves the disc back by one lobe pitch — a reduction of 11 : 1 from a ring, a disc and an offset bearing, with no gear teeth anywhere — and the profile's 11 lobes are counted off the generated curve rather than put there. The roots sit at 50.00 mm and the tips at 60.00, which are R − r ∓ e: the eccentricity is the lobe height, twice over. positioned by solving, not by drawing.

Eleven lobes from twelve pins

A ring of round pins, a disc on an eccentric, and a reduction of eleven to one with no gear teeth anywhere. The disc's profile is not designed: one pin generates one lobe of it, the other ten lobes are the same curve, and the count that decides the ratio is a count of lobes on a shape nobody drew.

meshing · Meshing One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing.

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

strands · Strand One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent The number on the tackle is a limit. The velocity ratio of a 4-part tackle as its blocks separate, measured by differentiating the strand's own length. It is 3.6174 at 90 mm and 3.99992 at 6898 mm, and it reaches 4 nowhere. The shortfall is geometric and has nothing to do with friction: each part of line makes an angle with the lift, and a part at angle β shortens by cos β of the movement. A tackle used at close quarters — which is when a tackle is useful — is the case furthest from its own rating.

Six things a strand is not

A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.

wrong · Misconception Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be.

Three mechanisms, one subtraction

A micrometer's differential screw, a chain hoist's differential pulley and a robot joint's compound epicyclic look nothing like each other and are the same device. Each takes two nearly equal quantities and returns their difference, each buys its enormous ratio with that difference, and each carries the same conditioning number — |a/(a−b)| — measured here by perturbing the mechanisms rather than by quoting the formula.

transmission · Transmission Which teeth one tooth ever meets: 20 on 40. Follow one tooth of the pinion round and mark every wheel tooth it touches. It does not touch them all. It touches 2 of 40, which is z₂ divided by the greatest common divisor of the two counts — 20 here — and it goes on touching the same ones for as long as the gears are in mesh. The pattern repeats after 2 turns of the pinion. Adding one tooth to the pinion makes the counts coprime and takes the count from 2 to 40. The marks are produced by walking the mesh, and the count they give is compared with the gcd rather than derived from it.

Which tooth meets which

A tooth on a pinion does not meet every tooth on its wheel. It meets z₂ divided by the greatest common divisor of the two counts, and it meets the same ones for the whole life of the drive. On the default planetary used throughout — sun 24, planets 24 — a planet tooth touches exactly one sun tooth and never touches another, and nothing in the drawing says so.

gears · Tooth 53/11, instant by instant. The ratio a 53-tooth sprocket and a 11-tooth sprocket actually deliver, against the driver's rotation. The quoted 4.818 is the flat line, and it is exactly right as an average — the mean over a turn comes out at 4.8182, which the computation is never told. Within every tooth the ratio swings from 4.756 to 4.948, a fluctuation of 3.99%. The sharp corners are pins seating: the identity of the pin the strand runs from changes 53 times a turn on one sprocket and 11 times on the other, and the two do not coincide.

The chain is a polygon

A chain's pins sit on a polygon, so the radius that matters swings by 1 − cos(π/n) within every tooth: four per cent on an eleven-tooth sprocket. The quoted 53/11 is the mean of that, exactly — and how much of the fluctuation reaches the back wheel is decided by the fractional number of pitches in the taut strand, which nobody adjusts and which can take the drive from perfectly uniform to two per cent.

applied · Tooth Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be.

The reductions a planetary cannot give

One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.

transmission · Transmission A reduction made out of a difference. A compound epicyclic's reduction is 1/(1 − z₁z₄/z₂z₃) — a ratio of two integers, so it is exactly what a catalogue says, at every position, forever. It is the only number in this field that survives being measured, and the reason is that it is a count. What it is not is insensitive: each row's denominator is a difference of two nearly equal products, and the last two rows differ by one tooth on one ring.

A ratio that is a count

A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.

applied · Tooth Reverted trains of 12 : 1. A reverted train has its input and output shafts in line, which means the two stages share one centre distance — so the tooth counts must satisfy z₁ + z₂ = z₃ + z₄ and give the ratio asked for. Two equations in four integers, and there is no reason for a solution to exist. For 12 : 1 there is none at all until the wheels are allowed to reach 63 teeth; below that the coaxial condition and the ratio simply cannot both be met. This is the same kind of arithmetic as the clock trains of the timing field, with one extra equation, and the extra equation is what a shaft position costs.

Two shafts that must be in line

Asking a two-stage gear train for a ratio is easy. Asking it for a ratio and for its input and output shafts to be coaxial is asking for a solution of two equations in four integers, and there is no reason for one to exist. A twelve-to-one reverted train needs a sixty-three-tooth wheel before it has any solution at all — while sixteen to one, a larger ratio, manages with fifty-six.

transmission · Transmission A tooth flank is the end of an unwound strand. A strand wrapped on a circle of radius 45.105 mm, unwound while kept taut. Its free end traces the involute — the same curve the gears field builds from its own parameterisation, agreeing to 1.5e-14 mm over the whole flank. The strand is the important part of that sentence and not the curve: the taut portion is a tangent to the base circle, its length is the arc it has left, and both of those are statements about a strand rather than about a tooth. At this position the free length is 27.965 mm. positioned by solving, not by drawing.

A tooth flank is an unwound strand

Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.

gears · Tooth 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 An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.

rolling · Rolling What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

practice · Tolerance What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. The lit row is the one this essay is about. Of the 14 rows, 6 are quoted and 3 are exact.

Which numbers survived

Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.

applied · Misconception Oldham's coupling: two slides between two offset shafts. An input hub on one axis and an output hub on a parallel axis 0.6 away, joined by a disc that slides in a slot across the input hub and carries a tongue across the output hub at 90° to the slot. At an input angle of 35° the disc has slid 0.491 along the slot and the output hub -0.344 along the tongue, and the output hub has turned through exactly the input's angle, because neither slide can change an orientation. The disc's centre, the dot, runs round the dashed circle of diameter 0.600 — the offset divided by sin 90° — and goes round it 2 times for each turn of the shafts.

A coupling that only translates

A coupling between two parallel, offset shafts turns its output at exactly the input's speed when, and only when, the relative motion of its two hubs contains no rotation — when it lies in the translation group. Oldham's two slides give that group by construction and so do two equal parallel cranks; a four-bar that is not a parallelogram gives the whole planar group and its output wanders by more than a radian. And Oldham's right angle is not what makes the ratio one: it is what makes the slides slide least.

pairs · Pair A ball bearing turned by its inner race. A bearing of pitch diameter 40 with 8 balls of diameter 8, its outer race held and its inner race turned 60°. Rolling without slipping at both contacts leaves the cage — the ring that keeps the balls apart, marked by the dark tick — turned 24.00°, a share of 0.4000 of the race, and each ball spun 120.0° backwards about its own centre, marked by its own tick. The share is (1 − d/D)/2, less than a half by d/2D; the gear-train solver, handed a planetary with a sun of 32 teeth and a ring of 48, returns 2/5 for its carrier. Dragging turns the inner race.

A bearing is a planetary with no teeth

Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.

rolling · Rolling A double Cardan joint in the W arrangement. Two universal joints on an intermediate shaft, the input arriving at 15° to it and the output leaving at 15°, bent the same way so that the input and output lines meet, with the intermediate shaft's two yokes in one plane. Each pin is found as the line square to its own shaft and to the pin it crosses, so the output's angle is solved rather than taken from a formula. The dashed marks are one on each hub. The shaded square is the plane that swaps the input's approach with the output's departure. At an input angle of 40° the output is at 40.00°, exactly in step.

Constant velocity is a mirror

Between parallel shafts a coupling keeps its output in step exactly when its hubs only translate relative to each other, which is membership of a group. Between shafts that meet, no group will do: a single universal joint at 30° and a correctly phased double Cardan joint both close to the spherical group, and one wanders by 8.2° a turn while the other keeps step to 10⁻¹⁵. What separates them is a mirror that swaps the two hubs. It either fixes every configuration of the joint or fixes none, so one symmetric position certifies a whole turn — which is what the phasing mark on a propshaft checks.

pairs · Pair A pumping unit is a crank-rocker with a drum on the end of its rocker. A representative beam pumping unit, in metres: crank 0.95 on the gearbox shaft, pitman 3.6, walking beam pivoted on the samson post with a back arm of 2.8 to the pitman, and a horsehead on the front, an arc of radius 3.4 about the beam's pivot. The crank, pitman and back arm are a crank-rocker. The cable is fixed at the top of the head, lies on its face and leaves it where the head's tangent is vertical, 3.4 in front of the pivot, then hangs to the rod over the well. Faded: the beam at the bottom and top of the stroke, where crank and pitman are in line. The rod travels 2.357 m, and at the crank angle drawn it is 0.614 m above the bottom. Use the slider to turn the crank.

The horsehead is a drum

An oil-well pumping unit is a crank-rocker, and the curved head on the end of its beam is not decoration. It is a drum: the cable leaves it where its face is vertical, which is the same place at every angle, so the rod moves on an exact straight line and rises by exactly the head's radius times the beam's turn. That makes the unit's torque factor the four-bar's velocity ratio, to 3 × 10⁻⁹, and leaves the crank's position as the only thing that shapes the stroke — and the direction the crank turns as the thing that decides which stroke gets the benefit.

applied · Fourbar How evenly the rocker is driven through the working stroke. The rocker's speed through the working stroke, divided by its mean over the stroke, against the fraction of the stroke's duration, for a crank-rocker with a 60° swing and a time ratio of 1.2 of its own, ground 1.2223, crank 0.4783, coupler 1.1298, output 1. Every curve must start and end at nought, because the rocker stops to reverse; what differs is the middle. The number after each name is the fastest speed over the slowest while the rocker covers the central 80% of its swing. Driven directly at constant speed the crank-rocker gives 1.91 at a time ratio of 1.20. The drag link that gives the highest time ratio, 2.71, gives 2.77: a hump in the middle of the cut. The most even design that still reaches 2, a drag link of ground 1, crank 5, coupler 4.5, output 2.25, gives 1.21 at a ratio of 2.03 — more even than the crank-rocker alone. Dragging moves that design's phase.

A quick return that cuts evenly

A drag link ahead of a crank-rocker buys a shaper its time ratio of 2 with both stages at 40°. The drag link that buys the most ratio drives the cut unevenly — its fastest speed through the middle of the stroke is 2.77 times its slowest — and a different drag link at a different phase reaches 2.03 with a ratio of 1.21, which is more even than the crank-rocker driven alone at constant speed. Up to a ratio of about 2.4 the second stage can improve both specifications at once.

linkages · Fourbar One belt, five steps. Two stepped pulleys 400 mm apart, each with five steps whose radii on the two shafts always sum to 160 mm, giving five speeds a factor of 1.6 apart. The belt is on step 4: 98.46 mm driving 61.54 mm. Solid: an open belt, 1306.066 mm on this step against 1302.655 mm on the middle one. Dashed: a crossed belt, 1367.552327 mm — the same on every step, because a crossed belt's length depends on the two radii only through their sum. Faint circles are the other steps. Dragging moves the belt to another step.

One belt for every step

A lathe changes speed by moving one flat belt between the steps of two stepped pulleys, so every pair of steps has to take the same belt. A crossed belt's length depends on the two radii only through their sum, so steps whose radii sum to the same value take the same crossed belt exactly — to the last bit of the arithmetic, over 266 random pairs. An open belt's length depends on the difference too: at constant sum it grows by d²/C, then d⁴/12C³, so an open belt made for the middle step of a five-speed pair is 12.3 mm short on the outer steps, and those steps must give up 3.74 mm of radius sum to take it.

strands · Strand One open belt on two drums. Two identical drums on parallel shafts 400 mm apart, one reversed against the other, seen from the side; the belt runs between them at a station 60 mm from the middle of its 300 mm travel. The drums are cut so that the speed ratio grows by the same proportion for every millimetre the belt is moved — from 1/2.56 to 2.56 over the travel — and so that one open belt fits at every station. At this station the driving radius is 94.45 mm, the driven 64.85 mm, a ratio of 1.456. The profiles are not cones, but the difference is a millimetre or two on radii of eighty and cannot be seen at this scale; the figures below measure it.

A cone drive is not two cones

Take a stepped pulley to its limit and the belt runs on two drums, shifted continuously along them. Drawn, the drums are straight cones of opposite taper, and a crossed belt fits them everywhere because their radius sum is constant. But straight cones grade the speed unevenly — 0.58% per millimetre of shift in the middle, 0.72% at the ends — and an evenly graded drive needs drums whose radii are logistic curves. And an open belt, the common kind, fits neither: on a lathe-sized drive it is 12.3 mm short at the ends, and two identical drums that take it and keep the grading must each be shaved in proportion to their radius, 2.69 mm at the large end and 1.05 mm at the small.

strands · Strand How evenly the ram is driven through the working stroke. The ram's speed through the working stroke of an offset slider-crank with a time ratio of 1.2 of its own, crank 1, rod 2.719, offset 0.847, divided by its mean over the stroke, against the fraction of the stroke's duration. The number after each name is the ratio of the fastest to the slowest ram speed while the ram covers the central 80% of its stroke. Driven directly at constant speed the slide gives 2.02. The drag link that gives the highest time ratio, 2.71, gives 2.60. The most even design that still reaches 2, a drag link of 4.75, 4, 2.25 on a ground of 1, gives 1.19 at a ratio of 2.03 — the same pattern the shaper's rocker showed, measured on a ram. Dragging moves that design's phase.

The first stage does not know what it drives

A drag link that evens a shaper's rocker evens a ram driven by an offset slider-crank just as well — within 3.8% of the best that ram can have — once it is turned by the angle between the two machines' stroke starts, to within a degree. The two outputs cover their working strokes alike to 1.5% when they share an own time ratio, so what the first stage has to know about the second is that ratio and where its stroke begins. The exception is a centred slide with a short rod, whose skewed stroke moves the best phase by 12° more than the rule says and makes the train more even than any rocker's.

linkages · Fourbar The drums a ratio law asks for. The two drum radii along the travel for the law speed even, the power λ of the ratio changing evenly (its logarithm at λ = 0) with λ = 1, over the range 1/2.56 to 2.56 in 300 mm, with a crossed belt so that the radii sum to 160 mm everywhere. Faint: the even proportion's logistic drums; dotted: the driven drum turned end for end, which lies on the driving drum only if the two are one shape. For this law they are not: the two shapes differ by up to 30.7 mm. The driving drum departs from the straight cone between its end radii by up to 16.18 mm, and its steepest taper is 30.8° from the drum's axis, against the straight cone's 13.2°.

A drum for every law

A cone drive's drums can be cut for any ratio law along the travel, not only the even proportion a machine tool wants. A crossed belt fits every law whose ratio stays positive, and one open belt fits any of them after a shave that depends only on the ratio at each station, so one table of shaves serves them all. What the law decides is everything else. Only the even proportion keeps the two drums one shape; the speed changing evenly or a fan's power changing evenly each need two different drums, 30 and 55 millimetres apart, and pack the change of speed into the slow end at three and fifty times the even law's rate.

strands · Strand Shortening a centred slide's rod evens the dragged ram until the rod is about two cranks long, and then it stops helping. A centred slider-crank with crank 1 and its rod swept from 20 cranks down to 1.305, where its worst transmission angle reaches 40°, plotted against crank ÷ rod. For each rod, the ratio of the ram's fastest to slowest speed across the central 80% of its working stroke: driven directly, which worsens from 1.701 to 2.752; and with the most even of 821 drag links, at whichever phase angle between the stages suits it, that still gives a combined time ratio of 2, which improves from 1.435 to 1.246 at a rod of 2.0 (worst angle 60°) and then worsens again to 1.401 at the 40° limit. Dashed: the best design found on any offset slide, 1.181, at an offset of 0.6 and a worst angle of 65°.

A centred slide has a best rod, and an offset beats it

Behind a drag link that brings a shaper's ram to a time ratio of 2, a centred slide's rod is a knob with a best setting. Shorten it from twenty cranks and the most even cut improves from 1.435 to 1.246 at a rod of two cranks, then worsens to 1.401 before the transmission angle reaches 40°, because the rod's skew moves the ram's speed peak where a drag link can phase against it only until the peak stops moving and just grows. An offset beats it: at every worst transmission angle up to 65° some offset slide is more even than any centred one, the best at 1.181, and on the most offset slide the rod hardly matters.

linkages · Fourbar Softening a law is drawing a taut string through its tolerance. The speed even law's ln q along the share of the travel (dashed), the tube of speeds within 11% of it with both ends of the range kept exact (shaded), and the law with the least peak grading anywhere in the tube (solid): it climbs from the slow end at the least peak slope the tube allows until it rejoins the demanded law, at 26% of the travel, and from there is the demanded law exactly; it never lags by more than the tolerance. The dotted line is the even proportion. Exact, the law needs 884 mm of travel for a belt that needs 300 mm on the even proportion; softened by 11%, 547 mm.

A tolerance shortens the drum

A cone drive's drums must be long enough that a belt never straddles too large a change of ratio across its own width, so a demanded speed law sets the drums' length through its steepest point. Allowed to deliver speeds within a tolerance of the demand, a law's least steep version is a taut string through a tube: the demanded law with its steep end capped, exact everywhere else. Softening the speed law by 11% shortens its drums from 884 mm to 547, where the even proportion needs 300, and leaves it exact over three quarters of the travel. What softening never does, at any tolerance a designer would accept, is turn the law into the even proportion: that needs 52% for the speed law and 165% for a fan's.

strands · Strand 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

Named alongside it

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

Design ruleStrandApproximationTransmission relationEpicyclicTransmission angleBelt driveFour-barMobilityRatioCentrodeConjugate-action

All concepts