Machine Kinematics
solved, then drawn
Field

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

The number on the box

What it means to check a catalogue figure against the mechanism it describes, and the five verdicts that come out — exact, mean, bounded, operating point, and a number where there is no number.

 A 3-stage stack at 35°. Every bar is the same length and every stage is at the same angle, which is a consequence of the solve rather than an assumption in the drawing: the pins at the crossings and the roller on the ground leave one freedom, and it has been used. The platform is at 1547 mm and the ram — the heavy line from the ground pin — is 428 mm long. Moving the ram one millimetre from here raises the platform 5.0 mm; at the bottom of the travel the same millimetre is worth 17.6 mm.

A roller is not a slider

The mobility of a scissor stack counted two ways and measured once, and the one equation that separates a roller that can turn from a slider block that cannot.

 kelvin: 6 contacts, 6 of the six freedoms taken. A plan view of the arrangement, with each contact drawn where it acts and an arrow along the direction of the force it can carry. A ball's contact force passes through the ball's centre whatever surface it rests on, so the whole constraint system is these points and these directions and nothing else. Written as wrenches — a force along a line, which is a screw of zero pitch — their rank is 6. Six independent wrenches leave nothing free: the part has one place to be, and putting it down twice puts it in the same place.

Six points and no more

Exact constraint read as a screw system — three arrangements that work, one that does not, and why the ball's centre is the only geometry a contact wrench needs.

 The contacts that fight each other — four-legs. With more contacts than freedoms, some weighted sum of the contact forces is zero: those contacts push against each other and not against the part. The weights are the left null space of the wrench matrix and they are drawn here. For four legs on a floor they come out as +−+− — the alternating sum — which is why a table rocks about a diagonal and never sideways, and why the gap under the fourth leg is δ₁ − δ₂ + δ₃ − δ₄ exactly.

The seventh contact

What a redundant contact costs, computed as a left null vector: the gap it leaves, the alternating sum under a four-legged table, and the reason an over-constrained fixture cannot be made better by making it more accurately.

 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 car's front suspension solved as a four-bar: camber through the travel, two incompatible models of where the tyre touches, and the strut that answers the same questions differently.

 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

Constructing a suspension's roll centre from the solved velocity field, sweeping it through the travel, and what it means that a number quoted in every specification is a point on a curve nobody plots.

 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

The steering trapezoid as an approximate function generator: the correct condition, the error curve, the arm angle that minimises the worst error, and the equal ripple that says the optimum has been found.

 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

The dead centre of a slider-crank used as a product rather than avoided: what makes a toggle latch hold, why the security is a second-order quantity, and what a millimetre of the output is worth near the singularity.

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

Where a hinge pin can go

A door swung as a body rather than a line — a map of every pin position against the clearance it leaves, a boundary derived from the velocity field, and the interference question this site could not previously ask.

 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

Two ways of locating a live axle sideways, measured against the same travel: the arc error of a single rod, the inversion of Watt's straight-line linkage, and the order of each error fitted rather than assumed.

 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

Chordal action derived from the pin geometry, the exact phase relation between two sprockets, and a measured mean that comes out at the tooth ratio without ever being told it.

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

A ratio that is a count

Why a gear ratio is the one exact number in the applied field, how a compound epicyclic makes a hundred-to-one reduction out of a difference of two nearly equal products, and what that difference costs in sensitivity.

 What the valve does, against what the ratio promises. The solved valve lift, and the cam's lift multiplied by the nominal ratio. They are the same curve to the eye and they are not the same curve: the peak is 12.777 mm against a promised 12.842 mm, short by 0.51%. The shortfall is not an error in either number. It is what happens when a ratio measured at one position is applied across a movement, and it is why cam cards and rocker ratios are quoted together.

The cam is not the valve

The valve train solved as a linkage: what a rocker arm actually does to a cam's motion law, why the quoted ratio is an operating point, and where the missing lift goes.

 Where the pad touches, and the rule that fixes it. The contact point's travel across the valve tip through one cam event, for a rocker squared up with the valve shut and for the same rocker squared up at mid-lift. The first wipes 1.35 mm and the second 0.34 mm — a factor of 4.0 for a shim under a stud. The rule an engine builder is taught is that the geometry is symmetric about wherever it is set square, so setting it square at rest puts the entire lift on one side of the symmetry. It is folklore that turns out to be geometry, and the same shim brings the peak lift back to 12.84 mm against 12.78.

Where the pad touches

The contact point between a rocker's pad and a valve tip, swept through a cam event: what the wipe is, why setting the geometry square at mid-lift quarters it, and what the same adjustment does to the lift.

 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

The applied field's closing ledger — what the five verdicts came out at, why counts are the only exact survivors, what a good specification would say instead, and what the field did not do.

 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

A beam pumping unit as a crank-rocker with a drum on its rocker: the cable's leave point found by shortest path and fixed to 10⁻¹⁰ m, the pinned beam end's 20 cm of sideways travel, the torque factor as head radius times the four-bar's velocity ratio, the time ratio bought by moving the crank's centre, and why an offset unit must turn one way.

 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 two-arm desk lamp with a parallelogram and two springs of no free length, balanced at every pose; one arm's weight and spring energies cancelling across half a turn; the unbalance a spring's free length leaves against L₀/s; the lamp's energy over its working range without the parallelogram; the joint torques along a swing, with and without it; and the energy spread a spring-rate error leaves against 2ε.

 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 three-joint desk lamp with a tilting head: the lamp balanced with one parallelogram and two head springs; the energy's terms spring by spring for a lamp with no parallelogram; the worst wrist torque a spring across the wrist leaves against the arm's range; the lamp's energy along one motion for six designs; parts, springs, forces and mass for each; a counterweight's mass and what it asks of the arm springs.

 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

Which sign each balancing spring's stiffness takes on a desk lamp with a tilting head: the lamp with its head spring's head attachment set by a slider; the four stiffnesses against that attachment point, closed form and term table; the attachment point at which the upper arm's spring changes sign, as a share of the head's centre of mass, against the head's mass over the arm's; and the design with no parallelogram at three mass ratios, every stiffness positive.

 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

The smallest springs that balance a desk lamp with a tilting head: the lamp as built, at its least total of peak pulls and at its least largest pull; the lower arm's spring against its attachment on the arm for three bracket heights; each spring's peak pull against the head attachment, with the upper arm's spring at the wrist and short of it; the best head attachment against the head's mass over the arm's; and the six designs' springs, totals and largest pulls in one table.

 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

The smallest balanced desk lamp with real springs: the lamp with ideal springs, with 2 cm of free length and with its attachments re-chosen; the worst leftover moment against the free length for three springs and four; each joint's leftover moment across the lower arm's swing; each spring's refitted stiffness against the free length; the worst moment as the head spring moves beyond the threshold; each spring's swing in length at the threshold and re-attached; and the four designs' springs and unbalance in one table.

 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

The tilting-head lamp with 2 cm of free length on every spring, its attachment points chosen for the least worst moment with the springs' total peak pull capped: the lamp at six caps; the frontier of worst moment against pull, with the attachment points held and the earlier designs for comparison; each spring's peak pull along the frontier; the four attachment points that move; each spring's swing in length; and six designs in one table.

All essays