Machine Kinematics
solved, then drawn
Series

Fourbar — the series

30 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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°.

    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.

    part 1 · linkages
  2. Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

    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 these essays refuse to print without measuring — so every linkage here is also asked for all 360 positions and required to agree.

    part 2 · linkages
  3. The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

    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.

    part 2 · linkages
  4. Slider-crank at 50°. Crank 1, connecting rod 3. The slider's travel is 2.0000 — exactly twice the crank throw, which is the one thing about this mechanism that does not depend on the rod length. Everything else does: the rod length decides how far the piston's motion departs from a sine wave, and that departure is the second harmonic every engine balancer has to deal with.

    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.

    part 2 · linkages
  5. Where the output stops, and why the return is quicker. A crank-rocker's output reaches an extreme exactly when the crank and coupler line up — stretched out, so O₂ to B is 4.50, or folded back, so it is 2.50. Nothing about the rocker enters the condition, which is why the limits can be written down rather than searched for. Those two crank angles are 40.8° and 228.5°, so the crank spends 187.7° going one way and 172.3° coming back while the rocker covers the same 40.0° both times. The ratio is 1.0894 predicted and 1.0894 measured over 7200 swept positions — a shaper cuts on the slow stroke and returns on the fast one, and this number is what the proportions are chosen to get.

    The return stroke is quicker

    A crank-rocker's output stops at two definite places, and the crank angles at which it does are calculable without touching a solver. The interesting number is not where they are but how far apart — because the crank turns at a constant speed and the output covers the same swing twice in unequal times.

    part 3 · linkages
  6. One chain, four mechanisms. The same four bars and the same four pins in every panel. What changes is which link is bolted to the bench, and that is not a property of the chain — it is a decision about where the bench is. The four mechanisms are crank rocker, double crank, crank rocker, double rocker: one input turns fully in some and rocks in others, and what each one is for is different. What cannot change is the shape of the closed loop, and the two diagonals measure that without reference to which link is held still: swept independently, all four visit the same locus of diagonal pairs to within 2.9e-3, half the sampling resolution. This is why the Whitworth quick-return and the oscillating-cylinder engine are not merely similar to a slider-crank; they are one.

    One chain, four mechanisms

    Which link of a four-bar is bolted to the bench is not a property of the chain. It is a decision about where the bench is, and making a different one gives a mechanism that looks and behaves completely differently while being, as a chain, the same object — which is why the Whitworth quick-return and the oscillating-cylinder engine are both a slider-crank.

    part 3 · linkages
  7. The wheel is the coupler — double wishbone. The suspension solved at 0 mm of bump, with the whole travel ghosted behind it. The two arms are the cranks and the upright between them is the coupler; the wheel is bolted to that coupler, so camber is the coupler's rotation and nothing else. Camber here is 0.00° and the contact patch has moved 0.0 mm across the road. The cross is the instantaneous centre of the upright, found from the solved velocity field; the roll centre is where the line from it to the contact patch crosses the car's centreline, and it is at 73 mm here.

    The wheel is the coupler

    A double wishbone is a four-bar standing on end whose coupler carries a wheel, so camber is coupler rotation and scrub is a coupler point's path. Both are computable, and the second one comes out with the opposite sign from the model every suspension book uses — by more than the whole scrub.

    part 4 · applied
  8. 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.

    part 5 · applied
  9. 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°.

    part 6 · applied
  10. 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.

    part 7 · applied
  11. Eight four-bars, one from each region the three signed sums cut. One linkage from each of the eight sign patterns of T₁ = g + c − a − b, T₂ = g + b − a − c and T₃ = b + c − a − g, each solved at a crank angle in the middle of its range. The thick arc round the left pivot is where the input's pin can go and the arc round the right pivot is where the output's can: crank-rocker + + +, input 100% of a turn and output 22%; double crank − − +, input 100% of a turn and output 100%; rocker-crank − + −, input 22% of a turn and output 100%; double rocker + − −, input 23% of a turn and output 21%; 0–π rocker + + −, input 76% of a turn and output 72%; π–π rocker + − +, input 84% of a turn and output 51%; π–0 rocker − + +, input 84% of a turn and output 85%; 0–0 rocker − − −, input 67% of a turn and output 91%. Grashof's condition names four of these and calls the other four one thing; the arcs show that the four triple rockers differ in which way along the ground line each rocker swings through.

    Eight kinds of four-bar

    Grashof's condition gives a four-bar one of four names and calls every linkage that fails it a triple rocker. Three signed sums of the lengths give eight, and a census of four thousand random linkages finds every one moving exactly as its signs say — because the planes where those sums vanish are the only places a four-bar's motion can change its kind.

    part 8 · linkages
  12. Every pivot on these arcs gives a 60° swing at a time ratio of 1.2. The rocker's pivot O₄ is fixed and its two limit positions B₁ and B₂ are 60° apart. At a limit the crank and coupler are in line, so the crank pivot sees the chord B₁B₂ at the angle δ = 180°(Q − 1)/(Q + 1) = 16.36°, and the points that see a chord at a fixed angle are arcs of two circles. The thick stretches are the 828 sampled pivots that give a crank-rocker with exactly this swing and ratio; the rest of each circle gives a linkage of another kind. The linkage drawn is the member whose worst transmission angle is largest, at both limits: ground 1.222, crank 0.478, coupler 1.130 and rocker 1, with its worst transmission angle 40.32°.

    A swing and a time ratio

    A shaper's specification gives the rocker's swing and how much quicker the return must be than the cut, and those two numbers do not fix a linkage. They leave a one-parameter family of crank-rockers on the arcs of two circles, every member exactly right, and the transmission angle chooses between them — which is also what decides that a 60° swing cannot return more than 1.207 times as fast and keep 40°.

    part 9 · linkages
  13. Four offset slider-cranks, one from each region the limit leaves. One slider-crank from each region of T₂ = b − a + e and T₃ = b − a − e, with the slide the dashed vertical line a distance e from the crank's pivot and the ground line dashed across. The thick arc round the pivot is where the crank's pin can go and the thick stretch of the slide is where the slider can: crank-rocker + +, crank 1, rod 3.5, offset 0.8, the crank reaching 100% of a turn and the slider between 2.37 and 4.43 on each side of it; double rocker − −, crank 3.5, rod 1.2, offset 0.6, the crank reaching 22% of a turn and the slider between 2.22 and 4.66 on each side of it; 0–π rocker + −, crank 2, rod 2.5, offset 1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line; π–π rocker − +, crank 2, rod 2.5, offset −1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line. These are the four of the eight four-bar kinds in which T₁ is positive.

    Four kinds of slider-crank

    An offset slider-crank is a four-bar whose output bar and ground have grown without bound, and in that limit the three signed sums that sort four-bars into eight kinds lose one of their signs. Four kinds survive. A census of four thousand finds every slider-crank moving as its region predicts, the textbook condition for a full crank turn turns out to be one region exactly, and each of the four kinds that vanish is carried, at a length that can be written down, into the survivor that shares its other two signs.

    part 10 · linkages
  14. A drag link driving a crank-rocker at a phase of 284.8°. A drag link, ground 1, crank 2.5, coupler 2.25, output 1.5, whose output crank carries the crank of a crank-rocker with a 60° swing, ground 1.2223, crank 0.4783, coupler 1.1298, output 1, turned 284.8° ahead of it, drawn at an input angle of 40°. The two cranks on the middle pivot are one rigid part. The thick arc at the right is the rocker's swing. At this phase the whole machine returns 2.71 times as fast as it works; the crank-rocker alone, driven at constant speed, returns 1.2 times as fast.

    A drag link ahead of a crank-rocker

    A crank-rocker with a 60° swing keeps a transmission angle of 40° only up to a time ratio of 1.207, and at a ratio of 2 no crank-rocker keeps even 20°. Drive its crank from the output of a drag link, whose cranks both turn but not at the same speed, and a pair in which each stage keeps 40° returns 2.71 times as fast as it works. The phase between the two stages decides almost all of it: the same two linkages give anything from 1.003 to 2.71.

    part 10 · linkages
  15. 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.

    part 10 · applied
  16. Grashof's classification, predicted and then swept. Four sets of link lengths. For each, Grashof's condition predicts from the lengths alone whether the input can rotate a full turn, and the solver then attempts all 180 positions and reports how many assembled. The prediction and the measurement agree in every case, which is what licenses quoting the classification for a mechanism nobody has swept.

    Grashof is a shape test

    The oldest classification in the subject compares sums of lengths, so it is unchanged by making the machine bigger — which means a protractor recovers it exactly without recovering a single length. What it does not recover is the margin, and the margin is what says whether the classification is safe.

    part 11 · linkages
  17. 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.

    part 11 · linkages
  18. The chain's whole configuration space, drawn on its two angles. A slider-crank asks one thing of its two angles: a cos θ + b cos φ = e, with θ the crank's angle and φ the rod's. Each panel is the square of those two angles from −π to π, with the curve that equation cuts out, followed by arclength on the equation alone. The number under each panel is how many whole turns θ, φ and φ − θ make around a circuit. A curve that crosses the square from side to side carries a turn in that angle; a closed loop inside the square carries none. The two regions where a member turns have two circuits each and the two where none does have one, which is what the four-bar regions these came from predict.

    Three rotations, and four benches

    The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.

    part 12 · linkages
  19. A six-bar followed round one circuit, with the links that turn against the frame coloured. A Watt six-bar — a four-bar of ground 4, crank 3.17, coupler 3.10 and rocker 1.90, with a dyad of 3.83 and 3.03 hung from its rocker to a third ground pivot — followed once round one of its 4 circuits. Links drawn in ink come back to where they started; links drawn in colour make a whole turn against the frame. On this circuit that is rocker + dyad link + output, and they turn together, so every one of the fifteen relative rotations between pairs of links is either nought or that one turn. The pins' paths are dashed.

    Fifteen pairs and one cut

    A six-bar has fifteen pairs of links, and each pair has a relative rotation. Followed round every circuit of more than four thousand random Watt and Stephenson six-bars, those fifteen rotations come from six whole-number windings. Almost always they take two values, so the links split into a group that turns once against the rest. How many inversions can be driven is then the number of links touching that cut: nought, three, four or five, never one, two or six. One chain in ten can have different cuts on different circuits, and a Stephenson chain can turn its rocker twice against its dyad.

    part 13 · linkages
  20. 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.

    part 14 · linkages
  21. 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.

    part 15 · applied
  22. 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.

    part 15 · linkages
  23. 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.

    part 16 · applied
  24. A centred slide's time ratio is its drag link's split of half a turn. The output angle of a drag link against its input angle over one turn: ground 1, input crank 4.019, coupler 3.920, output crank 1.339, the double crank that splits half a turn of its output most unevenly while keeping a worst transmission angle of 40.0°. A centred slider-crank driven by its output reaches its two dead centres at output angles exactly half a turn apart, whatever its rod, so its forward and return strokes take the input angles between the two dotted levels and outside them. At a phase of 66.3° between the stages they are 104.5° and 255.5° of input, a time ratio of 2.443. The largest over every phase angle is 2.443, against 2.304 for the best link of the quarter-step grid: no rod can raise it.

    A centred ram cannot be timed past its drag link

    A shaper's drag link and slider-crank were held to a combined time ratio of 2, and the centred slide's rod turned out to have a best length. Asking for a quicker return was supposed to shorten that rod and narrow the offset slide's lead. The first happens; the second runs the other way. A centred slide's dead centres are half a turn of its crank apart whatever its rod, so its ratio is the drag link's split of half a turn and nothing else: 2.304 on the grid, 2.443 off it, and no rod moves it. Below that ceiling the centred slide's best cut worsens from 1.118 to 1.466 and the offset slide's lead grows from 1% to 28%. Above it only an offset slide with a short rod reaches, and no slide keeping 45° reaches 3.

    part 16 · linkages
  25. 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.

    part 17 · applied
  26. The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out.

    The lamp needs three springs

    A desk lamp with a tilting head, balanced exactly with springs of no free length, was built with four springs whose peak pulls add to 162 N. Searched over every attachment point inside the same outline, the least total is 63.4 N, and it is reached with three: the head spring hung at the threshold where the upper arm's spring stops being needed, 77% of the way to the head's centre of mass. Beyond the threshold a second spring is the first one split in two, and splitting a spring never makes it smaller. A design with no parallelogram and six springs is never close: 213 N at its best.

    part 18 · applied
  27. A ratio of 3 costs both stages forty degrees. The largest quick-return ratio of a shaper's drive against the worst transmission angle it must keep, from 30° to 60°: an offset slider-crank alone, 1.448, 1.360, 1.286, 1.224, 1.172, 1.128, 1.092; a drag link alone ahead of a centred slide, 2.867, 2.632, 2.443, 2.284, 2.130, 1.968, 1.809; and the two in series, both stages keeping the angle, 3.975, 3.427, 3.006, 2.696, 2.432, 2.184, 1.957, against the product of the separate ceilings, 4.153, 3.580, 3.143, 2.796, 2.496, 2.221, 1.976. The train reaches 3 at 40.0°. Marked: 40°, where the train reaches 3.006.

    A ratio of three costs forty degrees

    A shaper's two-stage drive, a drag link ahead of a slider-crank, could not be given a return three times as quick as its cut while its drag link kept 40° and its slide 45°. Drawn as a frontier, the largest ratio against the worst transmission angle both stages keep, a ratio of 3 costs 40.0° in both. The two stages in series reach 95.7% to 99.0% of the product of their separate ceilings at every angle from 30° to 60°, so two stages relax one stage's bound very nearly by a product. Shared unequally, every degree the drag link gives up buys the slide between one and a half and three: the drag link is the stage to let lean. And the trains on the frontier cut badly, the ram's fastest speed in the middle of its cut 3.2 times its slowest at a ratio of 3. On the way, the drag link's own ceiling at 40° turns out to be 2.443, not the 2.365 a single search had found.

    part 19 · linkages
  28. Re-chosen attachment points buy back a third of the tilt at four times the springs. The tilting-head lamp with one parallelogram and the smallest springs of the earlier essay. Every spring 2 cm free and every attachment point re-chosen for the least worst moment, all springs still pulling: 0.0611 N·m, a third less, with peak pulls 29.6, 0.0, 41.6, 195.1 N (lower arm, upper arm, elbow bracket to head, wrist), 266.3 N in all. The upper arm's spring is dotted where it has no stiffness.

    A free length is absorbed where the springs attach

    Give every spring of the smallest balanced desk lamp 2 cm of free length and refit the stiffnesses, and the lamp is left with 0.093 N·m of unbalance, growing in exact proportion to the free length. The spare fourth spring that ideal springs had no use for recovers under three per cent of it, and moving the head spring beyond the threshold only makes it worse. What absorbs a free length is a spring whose length barely changes. Re-chosen attachment points buy back a third of the unbalance that way, and pay for it with four times the pull.

    part 20 · applied
  29. Five degrees below forty buy back an even cut at a ratio of three. At each worst transmission angle kept by both stages of a drag link ahead of an offset slide: the evenness of the cut, the ram's fastest speed over its slowest across the middle four fifths of the working stroke, for the train with the largest time ratio at that angle (dashed), which reaches 3.98, 3.75, 3.52, 3.43, 3.33, 3.24, 3.16, 3.08, 3.01 and cuts at 6.11, 5.21, 4.70, 4.35, 4.14, 3.90, 3.65, 3.46, 3.25; and for the most even train that still reaches a ratio of 3 (solid): 1.22, 1.26, 1.32, 1.42, 1.64, 1.90, 2.53, 2.73, 2.97, at 30, 32, 34, 35, 36, 37, 38, 39, 40°. The shaded band is 1.12 to 1.47, the best ratio-2 trains of the earlier essays.

    Five degrees buy an even cut at a ratio of three

    A shaper's two-stage drive reaches a quick return of 3 only by keeping no more than 40° of transmission angle in both stages, and the one train that does cuts unevenly, its ram 3.25 times faster at one point of the cut than at another. Searched for the most even cut that still returns three times as fast, the trains at 40° do little better, 2.97. But the price falls steeply below it: 1.90 at 37°, 1.42 at 35°, 1.22 at 30°. By 35° a ratio-3 shaper cuts as evenly as the best ratio-2 trains did. The even trains leave the long, nearly folded drag links that buy the most ratio for compact ones, and not one of them takes a longer rod than its angle requires.

    part 20 · linkages
  30. Allowed 100 newtons, the lamp spends them on its attachment points. The tilting-head lamp with one parallelogram, every spring 2 cm free, its attachment points chosen inside the earlier outline for the least worst moment on any joint with the springs' peak pulls capped at 100 N in total. Worst moment 0.0620 N·m; peak pulls 22.5, 28.7, 13.9, 34.9 N (lower arm, upper arm, elbow bracket to head, wrist), 100.0 N in all. Each spring's zigzag is drawn heavier the harder it pulls, and dotted where it pulls less than half a newton.

    The first pull goes to the spring the lamp did without

    Give every spring of the balanced desk lamp 2 cm of free length and let it have more pull, and the worst moment it is left with falls from 0.086 N·m at 62 N to 0.052 at 227 N, the most any attachment points inside the outline can do. Half of that improvement costs the first 18 newtons; the second half costs the next 147. Spent on stiffness alone, with the attachment points held, extra pull buys nothing at all. And the first place it goes is the upper arm's spring, the spare that ideal springs had no use for, hung lower on its bracket while the head spring moves out past the threshold. The wrist spring, which carried almost all of the earlier re-attached design's cost, takes the largest share only after 140 N.

    part 21 · applied

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