Machine Kinematics
solved, then drawn

A mechanism drawing can show a machine doing something it cannot do.

Sketch a four-bar the way anyone sketches one — crank pin where the angle says, rocker interpolated smoothly between its limits because that is what the motion looks like — and the coupler has to change length by 12% as it goes round. Every individual frame is a perfectly plausible picture of a linkage. The animation is of a machine that would tear itself apart. Nothing here is drawn that way: every position on this site is the output of a solve on the loop-closure equations, so a configuration the mechanism cannot reach cannot appear in a figure — the build stops instead.

A four-bar at 60°, solvedGround 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.ABO₂O₄crank (input)couplerrocker (output)crank rocker · residual 0.0e+0positioned by solving, not by drawing
Fig. 1 A four-bar linkage, positioned by Newton–Raphson against its constraint equations to a residual of about 10⁻¹⁶. Turn the crank: every frame on that slider was solved and asserted when the page was built, so the reader is stepping through configurations the mechanism actually has rather than through an interpolation between two of them.

26 fields, in 5 groups

436 essays · one card each, and the essay each field opens with

Loops

A closed chain of links, and the question every one of them starts with: can this move at all, and if so how. Five fields, from counting freedoms to platforms with three legs.

Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not. 15 essays

What can move

Before a mechanism does anything it must be able to. Mobility is countable, and the count can be wrong — which is how some of the most useful mechanisms ever built were nearly ruled out.

opens with “What decides whether it moves” A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°. 15 essays

Linkages

Four bars and four pins is the smallest interesting machine there is. Everything about which link turns, how hard it pushes and where it jams follows from the four lengths.

opens with “Four bars and four pins” Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up. 15 essays

The paths points trace

A point on a coupler draws a curve of degree six. Choosing the linkage that draws the curve you want is the oldest hard problem in the subject, and straight lines were the hardest of all.

opens with “What a coupler point draws” Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built. 15 essays

Out of the plane

In space a body has six freedoms and a pin takes five away, so a closed loop needs seven joints before it moves at all. The mechanisms that move with four are not curiosities — one of them is in every car ever built.

opens with “Six freedoms, not three” Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach. 16 essays

Several legs, one platform

Everything else here has one path from the ground to the moving part. Give it three, and the easy problem and the hard one change places — and a new kind of singularity appears in the middle of the workspace, where the machine can move with every motor locked.

opens with “The easy problem and the hard one change places”

Motion asked for, rather than allowed

A linkage gives the motion its lengths permit. Teeth and cams give the motion specified — and the cost is paid in accelerations nobody asked for and in profiles that cannot be cut.

The involute, unwound. Hold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn. 16 essays

Teeth

A gear tooth is not a shape somebody liked. It is the curve that keeps the velocity ratio constant while the contact point slides, and there is essentially only one answer.

opens with “Why a tooth is an involute” A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding. 15 essays

Prescribed motion

A linkage gives the motion its geometry allows. A cam gives the motion you asked for — and the cost is paid in accelerations you did not ask for.

opens with “Prescribing motion”

The problem backwards

Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's — and how many answers it has is a question about polynomials.

Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off. 15 essays

The problem backwards

Given the lengths, find the motion is the reader's problem. Given the motion, find the lengths is the designer's, and it is why coupler-curve atlases were printed and sold. The constructions are exact; whether what they produce can be built is a separate question.

opens with “The problem the other way round” Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are. 15 essays

How many answers

Newton finds a configuration. It cannot tell you how many there are, and a search that has stopped finding new ones is not a proof that there are no more. Written as polynomials, a mechanism's closure conditions have a number of solutions that is a property of its shape — and this is where the site stops counting by looking.

opens with “Two circles, four answers”

As built, and as taught

Every length here has been a number and every pin a point. Give them ranges and holes, put the result in a machine somebody owns, and see which of the published claims survive — then measure the finished machine and ask which of its numbers a measurement could ever have recovered.

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. 26 essays

As built

Every length here has been a number and every pin a point. Give the lengths ranges and the pins holes and the answers change shape: a curve becomes a band, and some of the mechanisms this site admires most stop working altogether.

opens with “A length is a range” 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. 15 essays

Machines you have met

A suspension, a steering linkage, a scissor lift, a cabinet door, a latch, a chain and a rocker arm. Every one of them is sold with a number, and every one of those numbers is a claim this site can check — which is how the field is built: take the mechanism people have actually met, solve it, and see what kind of number the catalogue was quoting.

opens with “The number on the box” 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. 17 essays

Drawn wrongly

The mechanisms that are illustrated confidently and incorrectly, the ratios quoted from the wrong formula, and the pictures that would not move if they were built.

opens with “The ratio that is not a number” The identification Jacobian of a four-bar, read by protractor. One row for every number the instrument reads and one column for every parameter that might be wrong. Each cell is the derivative of that reading with respect to that parameter, drawn to the right of its centre line when positive and to the left when negative, with the largest entry in the whole matrix at 4.09e-1. 14 rows against 4 columns: far more equations than unknowns, which is what makes an identification a least-squares problem rather than a solve, and what makes the question of which combinations of columns cancel a real one. These are the same derivatives the tolerance field computes one at a time — the same matrix read down instead of across. 32 essays

Numbers that were measured

Every field before this one takes the numbers on the drawing as given: chosen by a designer, cut by a machinist, and thereafter known. They are not known. Run the same kinematics with the parameters as the unknowns and the motion as the data and a new question appears with an exact answer — which of them can be recovered at all. A four-bar read by a protractor is a three-parameter machine however long you measure it, a coupler curve is drawn by three different linkages, a length is not toleranced because it is two holes that are, and the standard description of a robot arm breaks where the arm does not.

opens with “A dimension is a measurement”

Where the premise is tested

The fields whose object breaks one of the assumptions the loop fields share: no loop at all, a state with a discrete part, an input that is not one number, a constraint that does not integrate, no mechanism in the description at all, a shape that is not a choice, a member that is not rigid, an assembly whose size is a parameter, a constraint that only pushes where what a part may do is a cone rather than a subspace — and the connections themselves as the unknown, where the answer is a census rather than a configuration, and the joint itself as the object, where what a pair permits turns out to be a group rather than a number, and the demand itself given as an equation, where the mechanism is compiled from a polynomial instead of searched for, and the links themselves given a width, where the constraint is an inequality between parts rather than an equation in the loop and the plane the mechanism is drawn in turns out to be a stack.

elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₂ shoulder. 19 essays

One path to the tool

Every mechanism here so far has been a loop, and a loop is why a configuration on this site is a solve. An arm has no loop: its pose is a product of six transforms with nothing to converge and nothing to refuse. The difficulty does not go away — it moves to the other end, where one tool pose has eight answers and a straight line can cost more than the machine has.

opens with “The chain that does not close” 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. 16 essays

Motion that stops

Every other mechanism here moves whenever its input does. A ratchet, an indexer and an escapement 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. Whether one holds is the sign of a lever arm; what it wastes is two thirds of its own travel.

opens with “The mechanism that waits” 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. 16 essays

More than one input

Every mechanism here so far has had one input, and its output has been a function of it. A differential does not: its cage turns at the mean of two wheels, and neither of them decides anything alone. What a gear train has is not a ratio but a relation — a plane of permitted motions — and every number a gearbox is sold with is that plane cut by a brake, a clutch or a choice of which shaft is driven.

opens with “Two inputs and one output” A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0. 16 essays

Wheels, and where they may not go

A rolling wheel forbids a velocity, not a position. It may not go sideways and it can still be parked anywhere, at any angle — and the gap between those two sentences is worth an exponent, a shortest path that is a whole circle, and a ball that comes back turned by the area it went round.

opens with “A constraint that takes nothing away” The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing. 16 essays

The motion, not the mechanism

Everything else here is about a machine. This is about the motion a machine makes, near one instant — where the moving plane is turning, which of its points are going straight, which of them could be replaced by a single pivot and for how long. Two linkages with nothing in common that agree on those numbers are interchangeable, and the mechanism turns out to be one of infinitely many ways of producing a motion rather than the thing the subject is about.

opens with “The mechanism drops out” 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. 16 essays

The shape is the unknown

Everywhere else here a body has a shape and the question is where it goes. Put two bodies on fixed centres, require them to stay in contact, and the second shape stops being a choice: it is the envelope of the first one's positions, and there is exactly one of it. One routine cuts a gear tooth out of a straight edge, a cam out of a roller and a rotary engine's rotor out of its housing.

opens with “The second shape is not a choice” One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing. 16 essays

Members that pull

Every link on this site so far has been rigid: it holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a chain, a rope and a tendon are none of that. A strand has no shape of its own — where it runs is decided by the bodies it touches — and it constrains one way only, so it does nothing at all until it is taut, and a mechanism made with one has a different mobility in different places.

opens with “A member with no length of its own” Six assemblies, one routine, three disagreements and one accident. Every row is the same three steps: write down the constraint Jacobian, take its rank, and subtract it from the number of unknowns. The representations differ — bars between points, bodies joined by pins, panels joined by creases, one cell of a pattern that repeats for ever — and the routine does not. The counted column is the arithmetic on the numbers of bodies and joints; the measured column is the nullity of the matrix. They agree on the lazy tong and on the kagome cell and disagree on the other four, most sharply on the deployable ring, which the count declares immobile and which is sold as a mechanism that opens. The right-hand column is the reason: constraints that repeat what another constraint has already said, which the count has no way of seeing and the rank cannot help seeing. The fourth row is worth reading twice: the count says nothing can move and nothing can, so the two agree — and they agree for the wrong reason, because that pattern's flat state shows four freedoms and not one of them is a motion. 15 essays

Many of one thing

Every mechanism here so far is a chain: a handful of links, one or two loops, and a mobility somebody can check on the back of an envelope. A scissor lift, a folded sheet and a deployable ring are one small unit repeated, and three things change at once. The count of bodies is a parameter; mobility becomes the rank of a matrix rather than an arithmetic; and a unit that moves can be rigid the moment it is joined to another of itself.

opens with “Many loops, one freedom” Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing. 16 essays

Contacts that only push

Every constraint here so far has been an equation. A part resting against another part is not one: the contact says do not come closer and nothing at all about going away, so what the part may do is a cone rather than a subspace and a freedom stops being two-sided. Six constraints fix a body in space and six contacts fix nothing — the number is seven — and whether a part is held stops being a rank and becomes a question about where the origin sits inside a hull.

opens with “A constraint that only pushes” One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry. 16 essays

The chain before the lengths

Every field before this one is handed a mechanism and asked what it does. Here the connections are the unknown: which graphs of links and pins are mechanisms at all, how many there are, and which of them are the same mechanism drawn twice. The search space is finite and every quantity is a count — one four-link chain, two six-link, sixteen eight-link, two hundred and thirty at ten — and the counting rule everything else on the site rests on turns out to admit eight graphs for every one that deserves it.

opens with “The mechanism is the graph” Six surfaces, six groups, six pairs. The six lower pairs, drawn as the surfaces they are. A lower pair is two bodies touching over a surface rather than at a point or along a line, and that is the same thing as saying the surface slides on itself — so what the joint permits is the surface's own symmetry group. Each caption is computed from the surface's normals and not from the pair's name: a plane gives three freedoms and planar motion, a sphere gives three and spherical motion, a plain cylinder gives two, a shaft with collars gives one rotation, a prism gives one translation, and a thread gives one screw whose pitch comes back as the thread's own lead. Eleven surfaces were tried and six groups came out, which is where the number in every textbook's table comes from. 15 essays

What a joint is

Every field before this one declares its joints and then counts what they take away. A count cannot tell a pin from a slide: both are one, and the two mechanisms you get by swapping them are not related at all. What a joint permits is a set of displacements closed under composition — a group — and the six lower pairs turn out to be the six groups a surface can have as its own symmetry. The same instrument, run on a whole mechanism, separates the two kinds of overconstraint by an integer.

opens with “A joint is a surface that slides on itself” What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else. 16 essays

The curve as an equation

Every other field here is handed its demand geometrically: three positions, a sampled path, a ratio. This one is handed a polynomial, and the mechanism is compiled from it rather than searched for — every monomial becomes a cosine, every cosine becomes a link, and the linkage closing is the equation being satisfied. The answer is exact everywhere the machine moves, and what it costs is not accuracy but size.

opens with “A demand that is an equation” a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.2561 here, between rocker · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour. 16 essays

Links with a width

Every link in every other field is a distance between two points, and a distance cannot collide with anything because it is not anywhere. Give each one a body and the constraint stops being an equation in the loop and becomes an inequality between pairs of parts — not local, not smooth, and not in the plane the mechanism is drawn in. Not one machine here fits in a single plane, a clearance turns out to be a function with corners, a sampled sweep needs a bound to prove it missed nothing, and a four-bar turns all the way round exactly when it cannot be built with a bearing at each pivot.

opens with “A link that takes up room”

Four ways in

everything the 26 field cards do not reach

Every essay

The whole collection, grouped by field.

436 essays

The ladders

Each idea the site argues about, with its essays ordered by how deep they go.

by depth

The objects

Every mechanism, quantity and construction the essays name, and what else names it.

the index

Every mechanism, counted

Each closed loop the site solves, with its mobility counted by formula and measured from the rank — computed while this page was built.

where the formula is wrong

Threads running through

themes, not chapters

Solved, not drawn

Every mechanism on this site is positioned by solving its loop-closure equations, not by placing the links where they look right. A figure that could not be solved is not published, which means a linkage that would jam cannot be drawn moving.

210 essays

Two routes to the same number

Mobility from Grübler's formula and mobility from the rank of the constraint Jacobian are computed independently and must agree. Where they disagree the formula is wrong and the mechanism is more interesting.

319 essays

The crank is the argument

A mechanism is not a state, it is a relation between an input and an output. The figures are draggable because turning the crank is the explanation rather than a decoration on it.

38 essays

The singularities are the point

Dead centres, toggle positions and the configurations where a mechanism locks or changes branch are treated as the subject rather than as edge cases. They are where the mechanical advantage goes to infinity and where the machine stops working.

130 essays

Exact and approximate

Watt's straight line is not straight and Peaucellier's is. The difference is measurable, it took ninety years to close, and the error curve of an approximation is a more interesting object than the approximation.

193 essays

The constant that is not constant

A velocity ratio quoted as a number is a claim that it does not vary through the cycle. For gears that claim is true and is the whole reason the involute exists; for most other mechanisms it is false, and the variation is measurable.

91 essays

Somebody had a problem

Every mechanism in the canon was invented to solve something specific — a piston that had to move straight, an indicator that had to trace, a film that had to stop thirty times a second. The problem explains the shape.

125 essays

And the indexes

the same collection cut other ways

All essaysFieldsThreadsLaddersThe mechanism ledgerRefutationsConceptsFigure librarySearch