Machine Kinematics
solved, then drawn
Concept

Approximation — where it appears

A construction that reproduces a wanted motion to within a measurable error rather than exactly. On this site the error curve is treated as the interesting object, since its shape says where the approximation was made to be good.

Named by 34 essays across 8 fields — each of them below, with the objects they name alongside it.

How straight, over how much of the stroke. The deviation from a straight line, as a fraction of the traced span, against how much of each mechanism's stroke is used. The vertical axis covers fifteen decades. Watt's and Chebyshev's linkages are excellent over a short stroke and degrade as more is used; Peaucellier's sits at the bottom of the plot at every fraction, because it is not an approximation. The gap at full stroke is about fourteen orders of magnitude, and it is the difference between a mechanism that is nearly right and one that is right.

The straight-line problem

Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.

curves · Coupler Four tangents, and the two signs that choose between them. Two circles admit four common tangents, and a strand takes whichever one its two wrap senses name. Same sense at both ends — both centres on the same side of the strand — gives the two outer tangents, each 169.0444 mm long. Opposite senses give the two that cross between the circles, each 154.9193 mm. There is no search and no case analysis anywhere in this: the run's length is √(D² − Δ²) with Δ the signed radius difference, and changing one sense changes Δ from -18 to 70. positioned by solving, not by drawing.

Where a strand leaves a body

A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346.

strands · Strand Peaucellier's cell: exact straight-line motion from pin joints. The rhombus and the two long arms hold |OP| · |OQ| constant at 16 = 5² − 3², which is inversion in a circle about O. Inversion carries circles through the centre to straight lines, and the link CQ makes Q run on exactly such a circle — so P travels on a line, with no approximation anywhere. Measured over 160 solved positions the deviation is 6.5e-16 of the span, which is arithmetic noise rather than a small error.

Peaucellier and the exact answer

Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and solved here it comes out straight to 10⁻¹⁶ of its span.

curves · Coupler 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 The shape is what the demanded arm implies. A strand leaving a convex body runs along a tangent, so the only thing about the body the strand can feel is the perpendicular distance from the axis to that tangent — the shape's support function h(ψ). Ask for a rate and you have asked for h, because turning the body by dψ pays out h dψ; the shape then comes back from h with no solve at all, as h(ψ)û + h′(ψ)û⊥. This one was asked for h₀(1 + 0.3 cos 2ψ) with h₀ = 34 mm. The strand drawn here leaves the axis at a perpendicular distance of 24.1286 mm, which is what the demand asks for at this angle. positioned by solving, not by drawing.

The drum that is not round

The only thing about a body a strand can feel is the perpendicular distance from the axis to the tangent it leaves along. Ask for a rate and you have asked for that distance at every angle — and the shape comes back from it with no solve at all, unless the demand exceeds 1/(n²−1), at which point there is no shape.

strands · Strand How wrong the trapezoid is, arm angle 15.6°. The difference between the outer wheel's angle and the angle that would put all four wheels on one circle. It is zero at straight ahead by construction — both wheels point forwards — and it reaches 2.06° at 35° of lock. "One hundred per cent Ackermann" names a condition this linkage meets at 1 angle and nowhere else, and no four-bar can do better than a handful: the condition is not a rational function of the crank angle, and the linkage is.

The steering that is never right

For four wheels to roll without scrubbing, the two front wheels must point at different angles, and the relation between them is a cotangent condition no four-bar can satisfy. The trapezoid under every car meets it at straight ahead and, if the arm angle is chosen well, at exactly one other angle — 0.34° out at worst instead of 2.06°.

applied · Fourbar 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 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 A chain is a strand with a smallest piece. 53 teeth driving 11 at a pitch of 12.7 mm. The taut strand is drawn as this field computes it; inside each pitch circle is the polygon the pins actually sit on. The two agree on the straight spans and not on the wraps, and the gap is what the applied field's chordal action is: the effective radius swings between R cos(π/n) and R within every tooth, which is 4.05% on the small sprocket and 0.176% on the large one. A strand has no such number, because a strand has no pitch. positioned by solving, not by drawing.

A chain is not a strand

A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.

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 Two ways to stop an axle moving sideways. A Panhard rod is one link from the body to the axle, so the axle's end follows an arc and the whole car shifts sideways as the suspension moves: 3.56 mm at 80 mm of travel, and always in the same direction, so it happens twice per bounce. A Watt's linkage keeps the same point on a path that is straight to 33.7 µm — 106 times better, and it is drawn on the same axis, which is why it looks like the zero line.

Holding an axle still

A Panhard rod moves the axle 3.56 mm sideways over 80 mm of travel and a Watt's linkage moves it 34 microns — a hundred times better, and by a higher power. The Panhard's error is quadratic in the travel and the Watt's is fifth order, which is a much stronger statement than "the Watt is better" because it says how the comparison changes with the suspension.

applied · Coupler 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 Exactness is not bought with links. Five straight-line mechanisms, each measured over its own working arc — walked out to its dead centres and back a tenth — and each plotted at its own bar count. Watt's four bars are wrong by 9.0 per cent of the stroke and Chebyshev's by 12.4; Peaucellier's seven are exact. There is nothing in between, and adding bars to an approximation does not walk down the axis: the compiled machine is exact for the same reason Peaucellier is — an exact algebraic relation — and its extra bars buy generality rather than accuracy.

Exact costs more than close

Watt's four bars are wrong by nine per cent of their stroke and Chebyshev's by twelve. Peaucellier's seven are exact to 4 × 10⁻¹⁶, and a compiled machine is exact to 4.8 × 10⁻¹⁴ in five. There is nothing in between — adding bars to an approximation does not walk down the axis, and the four-bar in every beam engine ever built is on the wrong end of it.

computing · Compute What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing.

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

strands · Strand The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

bodies · Body 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times.

The wrap that walks along the axis

Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

strands · Strand 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 Two cones, 20 teeth and 40, at 90°. The axial section, which is where every bevel quantity is read. Two cones share an apex and roll on one another along the element drawn heavy; their half-angles are 26.57° and 63.43°, adding to the shaft angle, and the ratio of their sines is the tooth-count ratio exactly. The dashed arc is the sphere of radius 22.36 on which a bevel tooth's profile actually lies. The two short lines perpendicular to the common element are the back cones; each is heading for its own axis at a distance r/cos δ from the pitch circle — 11.18 and 44.72 — and that distance is the pitch radius of the spur gear the tooth is really cut to. Both back cones lie on one line, because there is only one perpendicular to the pitch element at that point, and the two heavy stubs straddling it are the two teeth — each one addendum out from the pitch circle and 1.25 in.

A tooth that lives on a sphere

Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

gears · Tooth An inner cable in its sheath, pulled and pushed. A sheath routed through two bends — the first turning 90° on a radius of 70, the second turning back 90° on 55 — drawn with its bore exaggerated to a clearance of 7 so the inner's two positions can be seen. Pulled, the inner is taut and takes the shortest path the tube allows, hugging the inside of each bend; pushed, it is pressed against the outside. Both paths are strands over pulleys of radius R ∓ c at the bend centres. At this clearance the pulled inner is 20.53 shorter than the centreline and the pushed one 23.63 longer, against c times the total turning, 21.99. At a real clearance of 0.25 the pulled inner is short by 0.7834 against 0.7854. Dragging changes the first bend's angle.

A strand in a tube

A Bowden cable's inner runs inside a sheath with a little clearance, and pulled it takes the shortest path the tube allows — a strand over pulleys of radius R − c at every bend. So its lost motion is the clearance times the total angle the sheath turns through: no bend radius in it, no route shape, and the S-bend that turns nowhere net loses as much as the U that turns back. Steering a handlebar changes it by exactly c times the change in angle.

strands · Strand 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 A two-arm lamp balanced at every pose. A two-arm desk lamp: a lower arm 0.4 m long pivoted on its base, an upper arm carrying a 0.8 kg head 0.35 m out. A parallelogram — the lower arm, a short crank at the base, a link parallel to the lower arm, and the upper arm's first few centimetres — keeps the base crank parallel to the upper arm, so the upper arm's spring can run from a post above the base pivot to that crank and see the upper arm's angle alone. The lower arm's spring runs from the same post to the lower arm. Both springs have no free length and stiffnesses matched to their arms' weights. At every pose the lamp's total potential energy is the same: here it differs from the pose with both arms upright by 3e-16 of the head's moment, and the joint torques are 0 and 0 N·m — rounding. Dragging swings the lamp.

The lamp that stays where it is put

A desk lamp that stays wherever it is left is balanced, not held: its springs store exactly the energy its weight gives up, at every pose. For one arm that takes a spring with no free length, fixed above the pivot, whose energy has a cos θ term that cancels the weight's to 10⁻¹⁵. A spring with a free length L₀ leaves an unbalance of exactly L₀/s of the weight's torque, and a stiffness a fraction ε off leaves 2ε of its moment. For two arms it takes a parallelogram: without one, the upper spring sees the angle between the arms and the lamp's energy varies by twice the head's moment; with one, both springs sit at the base and the lamp is flat everywhere.

applied · 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 Two wheels on one frame, on a rough road. A rigid frame on two wheels of 50 mm radius, 100 mm apart — 0.5 of the road's wavelength — on the road of 10 mm amplitude and 200 mm wavelength. The dashed curve is the path every wheel's centre follows on this road, whatever the frame does; the frame only fixes where on it the second wheel is. Over each wavelength of travel each wheel counts 204.847 and 204.847 mm, the length of one period of that path, against the 200 mm travelled; the frame pitches by up to 11.53°. Dragging changes the wheelbase.

A second wheel cannot correct the first

A wheel on a rough road counts the length of its centre's path, not the distance travelled. Put two on one rigid frame and average them, and the error does not shrink at any wheelbase: both centres run on the same curve, fixed by the road and the wheel's size, and each wavelength of travel carries each of them once round it. A half-wavelength wheelbase smooths the count's ripple and rocks the frame instead. What does work is two wheels of different sizes. A bridging wheel's error falls as the inverse square of its radius, so two radii measure their own error: wheels of 0.6 and 1.2 m that overcount by 0.339% and 0.099% extrapolate to 0.018%, knowing nothing about the road.

rolling · Rolling A tilting head balanced with one parallelogram and two springs. A desk lamp with a lower arm 0.4 m long, an upper arm of 0.35 m and a 0.5 kg head that tilts on the upper arm's end, its weight 0.08 m from that wrist. A parallelogram in the lower arm keeps a bracket at the elbow upright. Four springs of no free length: the lower arm's from a post on the base, the upper arm's from the elbow's bracket, and two for the head — one from the top of the elbow's bracket to a point on the head, which sees the head's angle from the vertical, and one across the wrist from the upper arm, which cancels the term the first brings in the angle between upper arm and head. At this pose, lower arm 0°, upper arm 80°, head 130° from the upward vertical, the lamp's energy differs from the first pose's by 5e-15 of the head's moment, and the torque at the wrist is 9e-10 N·m. Dragging moves all three joints.

Two springs stand in for a parallelogram

A lamp's head tilts on the end of its arm, and its weight's moment is a cosine of the head's angle from the vertical, so a spring can balance it only if the spring sees that angle. A second parallelogram carries the vertical out to the wrist and one spring does it. So do two springs with one parallelogram, or six with none: each spring that reaches across a joint brings a term in the joint's angle, and another spring across that joint takes it away. A spring across the wrist alone balances the head at one arm angle and leaves sin(Δ/2) of its moment over a range Δ.

applied · Fourbar A quintic of eighteen terms, and the nearest curve that costs a few. The quintic x^5 + 0.4x^2y − y = 0, whose equation expands into 18 cosines of the arm's two angles, inside the arm's reach (dashed). Drawn over it, the curve of the same degree with at most 4 terms that comes nearest to it inside the reach, found by trying every set of terms of that size: it keeps 0,1|1,0, 1,-2|2,-1 and is at most 2.77e-2 from the quintic and the quintic at most that from it — 1.4% of the reach. The slider steps through the budgets at which the nearest curve improves.

Four terms of eighteen draw the quintic

A quintic whose equation costs a compiled machine eighteen terms is drawn to 1.4% of the arm's reach by four of them and to 0.02% by ten, found by trying every set. The four are not its four largest. They are a line through the pivot bent by the square of the radius, the family the cheapest curves come from, and a curve close to that family is nearly free. An ellipse, a parabola, a lemniscate and a folium are not close to it, and each costs its whole equation to come within four per cent.

computing · Compute A wheel on a real road rides the long waves and bridges the short ones. Three metres of a class C road — the ISO 8608 average — built from 80 sines of random phase from 5 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are. The second line is the lowest point of a wheel of 0.6 m radius rolled along it: where it lies on the road the wheel touches, and where it lifts off the wheel bridges a trough. On this stretch it touches 12% of the way. A wheel counts its centre's path, so the texture it bridges no longer adds to its count, and the waves it follows still do.

A real road sets its own power

On a road of one sine, a wheel's overcount falls as the inverse square of its radius, and two wheels extrapolate it away. A real road is a spectrum. Over the band road standards describe, every wheel smaller than the road's tightest trough — about a metre and a half on an average road — counts the road's whole length, and its size makes no difference at all. With the texture below that band, bigger wheels do bridge more, but the overcount falls as the two-thirds power of the radius, a power the road's spectrum sets: 2(3 − w)/(5 − w) for a spectrum falling as n^−w. The inverse-square recipe removes a sixth of the error.

rolling · Rolling 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 Where the head spring hangs decides whether the arm's spring must push. The tilting-head desk lamp with one parallelogram: lower arm 0.4 m, upper arm 0.35 m, a 0.5 kg head with its weight 0.08 m beyond the wrist, and four springs of no free length. The spring from the top of the elbow's bracket to the head is hung 0.03 m along the head (the slider). Solved for an exact balance, its stiffness is 131 N/m, the wrist spring's 687, the lower arm's 561, and the upper arm's −434, negative: a spring that would have to push (drawn red). The upper arm's spring pulls exactly when the head attachment point is beyond 0.0615 m, 76.9% of the way to the head's centre of mass.

A spring that must push is hung in the wrong place

Balancing a lamp with a tilting head exactly, with springs of no free length, sometimes asks one spring for a negative stiffness, a spring that would have to push. It looked like a property of the lamp: a light upper arm under a heavy head might admit no design in which every spring pulls. It is a property of one attachment point. The upper arm's spring pulls exactly when the head spring meets the head beyond a point that is always short of the head's centre of mass, 77% of the way on this lamp, and at most 99% even with a head fifty times the arm's mass. Hang the head spring at its centre of mass and every spring pulls on every lamp.

applied · Fourbar Three wheels on one road, each bridging a different share of its texture. Two metres of a class C road built from 110 sines of random phase from 1 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are, and the lowest points of three wheels of 0.3, 0.6, 1.2 m radius rolled along it together. Where a wheel's line lies on the road the wheel touches it; where it lifts off, the wheel bridges a trough. On this stretch they touch 4%, 2%, 1% of the way. Each counts its own centre's path, so the three counts differ by how much texture each bridges, and the differences carry the road's spectrum.

Three wheels read the road they measure

A wheel's overcount on a real road falls as a power of its radius that the road's spectrum sets, so two wheels cannot extrapolate it away without being told the power. Three can fit it. Rolled together over 150 m of a class C road textured to 1 cm, wheels of 0.3, 0.6 and 1.2 m read a power of 0.64 against the law's 2/3, read the road's class from its level, and extrapolate the distance to 2 × 10⁻⁵ where the smallest wheel alone is out by 10⁻³. The standard band's floor, which no practical wheel bridges, turns out to lie on the same law and is extrapolated with it. The method is worth five times on a smooth road and five hundred on a rough one, and near the end of the road's texture it reads the end instead.

rolling · Rolling 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 Along one line the nearly free curve stays nearly free. The arm's pivot is fixed at the centre of its reach (dashed); the quintic x⁵ + 0.4x²y − y = 0 is drawn with its own point — where the catalogue put the pivot — moved 0.2 of a link along the line x = 0, which is the same as moving the pivot the other way. Heavy: the arc priced, the part of the quintic within 1.4 of its own point. Thin: the nearest curve built from at most four cosines of the arm's angles, found by trying every set of four: 6.0% of the reach away, the line through the pivot bent by the square of the radius. The equation itself costs 22 cosines here.

The quintic's cheapness runs along a line

A quintic whose equation costs a compiled machine eighteen cosines is drawn to a few per cent of the arm's reach by four of them, and its exact count jumps to 22 or 30 the moment the arm's pivot moves. Priced by the nearest curve instead, the two directions part completely. Along the line x = 0 the nearest four-term curve stays within 7.3% of the reach out to 0.4 of a link and the nearest six-term within 3.4%. Across it the four-term price goes from 3.2% to 18% between a twentieth and a tenth of a link, and from 0.15 on four terms buy nothing over three. The move along adds four cosines of amplitude 0.2 and 0.4 times the move; the move across adds twelve, sixty to ninety times larger in total.

computing · Compute One tyre pressed harder: its patch grows and averages the road it used to ride. 0.6 m of a class C road built from 110 sines from 1 cm to 90 m, heights in millimetres and drawn far taller than they are, under a 0.3 m tyre. The tyre is a circular tread on a bed of vertical springs; at each position its centre height is the one at which the springs carry the load. The grey line is a rigid wheel's lowest point, riding the peaks. The coloured line is the level the loaded tread's patch rests at, 9 mm deflection, a patch 146 mm long on a flat floor: it passes over the short texture as an average of it instead of following its tops.

A load reads the road a radius cannot

Press a tyre harder and its contact patch grows, and a patch averages the road where a rigid wheel rides its peaks. Averaging is linear, so the overcount it leaves falls as the patch's first power and scales with the road's roughness in exact proportion, not as the two-thirds power three wheel sizes read. One 0.3 m tyre at deflections of 3, 9 and 27 mm fits that law and extrapolates the distance to 3 × 10⁻⁷ on a smooth road and 6 × 10⁻⁶ on an average one, against 2 × 10⁻⁵ for three rigid wheels. On the roughest roads its lightest patch lifts off in places, and it does worse than they do.

rolling · Rolling The cheap pivots lie in a valley tilted off the line x = 0. The price of the quintic's arc drawn by the line through the pivot bent by the square of the radius, four cosines, as a share of the arm's reach, at pivots a two-hundredth of a link apart across and a twentieth along, within 0.16 and 0.45 of the catalogue's point. Darkest: within 5%; then within 10% and 20%. The cheapest pivot at each height (dots) lies on a line at dx/dy = -0.172, 9.8° off the line x = 0 (dashed), with prices from 1.4% to 2.7%; on x = 0 itself the price reaches 7.3%.

The quintic's cheap pivots lie off the line

Four cosines draw the quintic within a few per cent of the arm's reach from its catalogue pivot, and the earlier measurement found that cheapness surviving along the line x = 0 and dying across it. Mapped over a grid of pivots, it does neither exactly. The cheap pivots form a narrow valley tilted ten degrees off that line, about a twentieth of a link wide. In the valley four cosines hold the arc to 1.4–2.5% at every height out to 0.4 of a link; on the line itself, at the same heights, they miss by 6–7%. At two valley pivots every set of four cosines and every set of six was tried, and none beat the bent line. The amplitudes of the cosines a move adds predict a tilt a seventh as large, so they do not explain it.

computing · Compute

Named alongside it

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

Design ruleStrandVelocity ratioInextensibleWrap angleBelt driveScale invarianceDerivativeTangencyFour-barRolling constraintChebyshev's linkage

All concepts