Machine Kinematics
solved, then drawn
Concept

Lever arm — where it appears

The perpendicular distance from a pivot or axis to the line along which a contact acts, whose sign decides the sense of the moment it produces. For a strand it is the length paid out per radian of the body's rotation, which is the body's radius only when the body is round and turns about its own centre.

Named by 14 essays across 8 fields — each of them below, with the objects they name alongside it.

A 12-tooth ratchet, holding. A ratchet wheel with a 8° tooth face and its pawl. The long line is the face extended; the short one through the contact is the normal the tooth pushes along, and both belong to the wheel and turn with it. The pawl holds, and the reason is which side of those two lines its pivot is on: 1.080 wheel radii from the face's line and 0.495 from the normal's, and on opposite sides of them. Nothing about how hard anything pushes enters the question: a normal is a direction and a moment arm is a length with a sign. Dragging turns the wheel the way it may go, and the pawl's tip is solved onto the wheel's surface at every frame: it is lifted up a tooth's back and dropped into the next notch, once per click.

A joint that works one way

A pawl either holds a ratchet or is levered out of it, and which one happens is decided by two lines through the contact. One of them is the rule a workshop quotes. The other is the boundary that rule leaves out, and a check written to confirm the quoted rule turned out to be incapable of failing.

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

The drum that is not round

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

strands · Strand One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

The angle that holds the lock

A locking face cut exactly concentric with the pallet arbor has no tendency to hold itself: the tooth's push aims straight at the pivot and its moment is zero, so the smallest disturbance opens the lock. Tilt the face and the moment becomes ρ sin δ — and the same tilt, by virtual work, is exactly the rate at which the wheel is driven backwards. Draw and recoil are one angle read twice.

timing · Intermittent The radius a winch works at is not a property of the winch. A drum of core radius 30 mm filling with line 6 mm thick, one layer at a time. Line speed per turn is 2πr, and r is the radius of the layer being wound rather than anything about the machine: it runs from 207.35 mm on the first layer to 395.84 mm on the 6th, a factor of 1.909. A winch quoted at one speed is quoted at one layer, and which one is rarely said. positioned by solving, not by drawing.

The radius a winch works at

Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.

strands · Strand One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing.

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

strands · Strand One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

One test, three mechanisms

Whether a pawl holds, whether an escapement's lock draws itself deeper, and how much a four-bar's coupler can do for its rocker are the same question asked three times: on which side of a pivot does a contact normal pass? All three are one cross product, none evaluates a force, and the three answers are used for completely different things.

timing · Intermittent Every joint's lever arm. The thin lines run from the tool to each joint's axis, meeting it square. Their lengths are what a radian of error at each joint costs the tool in metres — not a rule of thumb but the Jacobian column, which is ω × r and therefore that perpendicular exactly. The shortest of them belongs to the joint nearest the work and the longest to the joint furthest from it, which is why an arm's accuracy is decided at the shoulder and its resolution at the wrist.

Where an error at the shoulder ends up

The same angular error at every joint of an arm, and the tool is out by 0.156 mm because of the shoulder, 0.017 mm because of the wrist roll and exactly nothing because of the last joint. The numbers are not properties of the joints. Each one is the distance from the tool to that joint's axis, measurable off the drawing with a ruler.

serial · Tolerance A tooth flank is the end of an unwound strand. A strand wrapped on a circle of radius 45.105 mm, unwound while kept taut. Its free end traces the involute — the same curve the gears field builds from its own parameterisation, agreeing to 1.5e-14 mm over the whole flank. The strand is the important part of that sentence and not the curve: the taut portion is a tangent to the base circle, its length is the arc it has left, and both of those are statements about a strand rather than about a tooth. At this position the free length is 27.965 mm. positioned by solving, not by drawing.

A tooth flank is an unwound strand

Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.

gears · Tooth An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.

rolling · Rolling A stop on the coupler, and the one direction in which it stops nothing. A four-bar — frame 4, crank 1.2, coupler 3.4, rocker 2.8, crank at 70° — whose coupler is stopped by a contact with the frame at (1.59, 2.16). The dashed line through the stop is its normal; the ringed point is the coupler's pole, where the crank's and rocker's lines meet, at (2.505, 6.883). With the normal 10° off the line to the pole the stop removes the four-bar's one freedom; through the pole it removes nothing, because the contact point is already moving along the stop's face. The graph of the stopped four-bar is the same in every case.

A contact turns and slides at once

A cam on its follower, a gear tooth or a pin in a slot takes one freedom away where a pin or a slide takes two, and it turns as well as slides. Give every joint of the four-bar, Watt's and Stephenson's chains a third label and there are 1,317 different chains. The rule that read slides off the graph extends to all of them, agreeing with the constraint rank on every one, and the two obvious shortcuts miss 36 and 769. What the graph cannot record is which way a contact's normal points — and on these chains that never matters, while a single stop added to a four-bar holds everywhere except through one point.

topology · Topology A cam cleat holds while its pivot leans inside the friction cone. A rope along the bottom, pulled to the right, pressed against a fixed backstop by a cam on a frictionless pivot. The cam can push on the rope only along the line from the contact to its pivot, because a body on one pin with one other force is a two-force member. That line leans 15° from the contact's normal; the shaded wedge is the friction cone, 21.80° either side of the normal for a friction coefficient of 0.4. The line is inside the cone, so the cam's force can resist the pull, and it squeezes the rope with 3.73 times the pull. Dragging moves the pivot.

A hold the load squeezes shut

A cam cleat bites harder the harder the rope pulls, and a wedge dragged by a bar into a narrowing gap squeezes the bar harder the harder it is drawn. Neither needs a screw: the load supplies its own squeeze, if the geometry lets it. A cam on a free pivot can push only along the line to its pivot, so it holds exactly while that line leans from the contact's normal by no more than the friction angle. A wedge's squeeze passes through two contacts, and it jams while its angle is no more than the sum of their friction angles — twice the figure usually quoted. And the set of loads either one holds does not shrink towards its threshold. It is lost there, all at once.

holding · Restraint The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 0° of driver, with the driver's locking disc of radius 18.8 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, outside the 6 V notches cut into it, each face tilted 14° off the chord, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 159.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot.

A V holds the wheel to first order

A Geneva wheel locked on concave arcs holds to first order and carries one pin; locked on flat chords it carries two and holds only to second order. Cut the lock as a shallow V instead — two faces tilted a few degrees off the chord — and it keeps both. Its disc is the crank times sin(β + γ), small enough for a second pin up to 34° at four slots and 5.6° at twelve. It rocks as acos(cos γ − c/a) − γ, which is linear in the clearance below 2a sin²γ, and at a running clearance of 0.1 on a crank of 30 it holds five to eight times tighter than the flat. The trade was between two shapes, not a law.

timing · Intermittent A spring that pulls the same at every pose holds the band for less. The five prescribed poses of the rocking slide's body, with the arm point the earlier essays found lightest, (-0.60, -1.20) in the body's frame, and the spring that keeps the block on + − + − + for every working load from nought to the design load. The shaded sliver is every attachment point from which any spring that pulls can choose those walls at all. A constant pull of 44.6 from (3.10, 2.05), no stiffness, pulls at most 44.6 times the unit load.

A constant pull holds the band for less

A spring with free length was the obvious way to make the rocking slide's spring cheaper, and it does nothing: offered one, the lightest design takes none, at every arm point tried. The spring's law never mattered to where it may hang, because the walls depend only on which side of each crank pin the spring's line passes, and that cuts the attachment points to a polygon whatever the spring is. What the law decides is the price. Three poses bind at the lightest attachment point, and a spring whose tension follows its length must pay for the longest of them. A pull that is the same at every pose pays for none of them, and holds the band from nought to the design load for 44.6 units against 52.0. Designing for the band directly halves the hundred units the earlier essay read off by scaling.

synthesis · Synthesis A cam cleat whose pin swings on a sprung link. A cam cleat at a lean of 2°, below its pin's lock lean of 2.56°, its pin carried on a link that swings about a frictionless pivot 2.2 pin heights above the rope, held by a spring. The link's lever is chosen so that, with the spring a quarter as stiff as the rope at that lever, cam and link turning together squeeze the rope 12.53 times the pull, the pin's own bottom edge. Along a path the pull rises to one, falls to nought and rises again. At this step the pull is 0: a fixed pin squeezes 12.53, this link 0.00, and a link at half that ratio 6.27.

A cleat on a swinging pin has two leans

Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.

holding · Restraint

Named alongside it

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

Contact normalStrandVelocity ratioDesign ruleHigher pairTransmission angleApproximationClearanceDrawEccentricEscapementFriction

All concepts