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.

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19 essays

ABO₂O₄crank (input)couplerrocker (output)crank rocker · residual 0.0e+0positioned by solving, not by drawing Linkages

Four bars and four pins

The smallest interesting machine there is. Four lengths decide everything about it — which link can turn all the way round, how hard it pushes, where it stops and whether it can be assembled at all — and every one of those is a number that falls out of a solve rather than a judgement about a drawing.

7 figures mobility 0 — a structuremobility 1 — a mechanismtriangle: 2 coordinates, rank 2, 0 freeone bar apart What can move

What decides whether it moves

Before a mechanism does anything it has to be able to. Two bars pinned to a point cannot move; three can. The count that separates them is one subtraction, it is the first thing anybody computes about a machine, and it can be wrong in a way that no amount of care with the arithmetic will catch.

6 figures the traced pointtangencybase circle r = 9.4020 teeth, module 1, 20° pressure anglenormal meets the base circle to 2.3e-5 Teeth

Why a tooth is an involute

A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.

6 figures roller followerbase circle 30cycloidal, rise 20 over 120°, roller 8pressure angle 25.5° Prescribed motion

Prescribing motion

A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.

6 figures -0.25000.2500.5000100200300crank angle (degrees)output ÷ input angular velocitya 20:40 gear pair, 0.500mean 0.000four-bar 4/1/3.5/3-0.40 to 0.33 through one turn Drawn wrongly

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.

5 figures ground 4, crank 1, coupler 3.5, rocker 35 attachment points, 150 solves each The paths points trace

What a coupler point draws

A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.

7 figures GrüblerJacobiantriangulated frame00agreefour-bar11agreeslider-crank11agreePeaucellier cell11agreeparallelogram + third bar01they disagree — the mechanism moves3(n−1) − 2j₁ − j₂ · free coordinates − rank(J)one row where the formula loses What can move

Counting and measuring mobility

Grübler's criterion counts links and joints and never asks how long anything is. The rank of the constraint Jacobian measures the lengths and never asks what a joint is. Two calculations with no inputs in common, producing one number — which is the only arrangement under which agreement is evidence.

6 figures fraction of the input rotation that assemblescrank rocker180/180predicted: full turndouble crank180/180predicted: full turndouble rocker32/180predicted: rocksnon-Grashof (triple rocker)137/180predicted: rocksprediction from the four lengths · measurement from 180 solvesthey agree, and the build requires it Linkages

Grashof, predicted and then swept

Add the shortest link to the longest. If the total does not exceed the other two, some link can turn a full revolution. It is a sentence about four numbers, it was published in 1883, and it is the kind of claim this site refuses to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.

6 figures 10⁻¹⁶10⁻¹³10⁻¹⁰10⁻⁷10⁻⁴10⁻¹0%25%50%75%100%Watt, 1784Chebyshev, 1850sPeaucellier, 1864fraction of the available stroke useddeviation ÷ span120 solved positions per pointfifteen decades, and one flat line The paths points trace

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.

6 figures pitch pointline of actionmodule 1, 20° pressure angle, centre distance 26contact ratio 1.612 Teeth

What happens in a mesh

Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven.

6 figures 0153045600100200300crank angle (degrees)mechanical advantage (clipped at 60)toggle: advantage 1673worst μ = 46°four-bar 3.4/1.2/3/2.4, 720 solved positions222° apart Drawn wrongly

Two things called jamming

A four-bar's mechanical advantage peaks at 1,673 in one configuration and its transmission angle collapses in another, 222° away. Both get described as the mechanism jamming. One is enormous force output and the other is force disappearing into the bearings, and they are opposite situations.

7 figures 05101520050100150lift-0.010-0.00500.0050.010050100150cam angle (degrees)acceleration (per degree²)constant accelerationsimple harmoniccycloidalacceleration differentiated from the displacement above itsmoothest is not gentlest Prescribed motion

The law that costs least is not the smoothest

Constant acceleration gives the lowest peak acceleration of any motion law and an impulsive jerk. Cycloidal motion has finite jerk everywhere and a peak acceleration 57% higher. The trade is real, it is measurable, and the displacement curves that everyone plots give no hint of it.

6 figures the redundant oneGrübler: 3(5−1) − 2(6) = 0Jacobian: 6 − rank 5 = 1 What can move

The mechanism Grübler says cannot move

Three parallel bars between two frames. Five links, six pins, and the criterion every engineering course teaches gives zero degrees of freedom — a structure. It is a mechanism, it is in drafting machines and locomotive coupling rods, and the formula cannot see why.

6 figures 0501001500100200300crank angle (degrees)transmission angle μ (degrees)40° design limitground 4, crank 1, coupler 3.5, rocker 3μ from 54.3° to 100.3° Linkages

The transmission angle

The angle at which the coupler meets the rocker decides how much of an applied force becomes useful output torque and how much goes into the bearings. It is pure geometry, it is computed here from every solved position rather than from a formula, and it is the number a linkage is judged by after Grashof has said it turns.

6 figures 10 teethundercut14 teethundercut17 teethundercut18 teethclean24 teethcleanred: the root circle has risen above the base circlethreshold N = 2/sin²α = 17.097 Teeth

Undercutting, and the seventeen-tooth rule

Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way.

5 figures driver6 slotscentre distance 60.00 = crank ÷ sin(180°/6)index 60° per turn Prescribed motion

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

6 figures PQOCarm 5, rhombus 3, crank 1.6deviation 6.5e-16 of span The paths points trace

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 on this site it comes out straight to 10⁻¹⁶ of its span.

5 figures Astroke = 2.000 = 2 × crankpositioned by solving, not by drawing Linkages

The slider-crank

Replace one pin of a four-bar with a slide and you get the mechanism in every reciprocating engine ever built. Its stroke is exactly twice the crank throw and does not depend on the connecting rod at all. Everything else about the motion depends on the rod, including the part that is always described as a sine wave and is not.

7 figures ringsunhold the ringsun in, carrier out4.000 : 1same directionhold the carriersun in, ring out−3.000 : 1output reverseshold the sunring in, carrier out1.333 : 1same directionWillis: (ω_s − ω_c)/(ω_r − ω_c) = −72/24both derivations agree, and the build requires it Teeth

Epicyclic ratios, two ways

An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so this site computes every ratio by Willis's equation and by the tabular method and requires them to agree.

6 figures

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.

11 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.

6 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.

2 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.

4 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.

4 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.

7 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.

4 essays