Machine Kinematics
solved, then drawn
Concept

Wrap angle — where it appears

The angle through which a strand lies on a body between arriving and leaving, which decides how much of that body's rotation the strand can carry. Signed by the way the strand goes round, the wraps of a closed run add to a whole number of turns — one for a belt, none for a crossed one.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

A member with no length of its own

Every link of a pin-jointed machine holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a rope and a tendon do neither: a strand's path is decided by the bodies it touches, and its constraint is an inequality that does nothing at all until it is taut.

strands · Strand Four tangents, and the two signs that choose between them. Two circles admit four common tangents, and a strand takes whichever one its two wrap senses name. Same sense at both ends — both centres on the same side of the strand — gives the two outer tangents, each 169.0444 mm long. Opposite senses give the two that cross between the circles, each 154.9193 mm. There is no search and no case analysis anywhere in this: the run's length is √(D² − Δ²) with Δ the signed radius difference, and changing one sense changes Δ from -18 to 70. positioned by solving, not by drawing.

Where a strand leaves a body

A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346.

strands · Strand What the wraps add up to, and what decides it. Four runs this field draws, with every wrap angle signed by the way the strand goes round its body. The right-hand column is their sum divided by a full turn, and it is a whole number every time — the turning number of a closed plane curve, arrived at by adding up a handful of angles that were computed one at a time from tangent lines. It is 1 for a loop that goes round its pulleys once and 0 for a crossed belt, whose two wraps are equal and opposite whatever the two radii are. The serpentine's idler contributes -25.3°, and the total is still exactly one turn: a tensioner lengthens the path without changing what the path is.

The wraps add up to a turn

Every wrap angle in a closed run is computed on its own, from a pair of tangent lines that knows nothing about the others. Signed by which way the strand goes round, they add to exactly one turn — or to exactly nothing, for a crossed belt — and the integer is decided by the route rather than by any of the geometry.

strands · Strand Four pulleys, one idler, and one equation. A closed strand over four fixed pulleys and an idler carried on an arm. The idler sits outside the loop the other four make and bulges the strand out to reach it, so its wrap is signed the other way — -25.34° against the four positive ones — and the five still add to exactly one turn. The arm angle is not a free choice: the strand has a length, and matching it is one scalar equation, whatever the number of pulleys. At this position the run is 1018.046 mm and the idler takes up 0.4387 mm of strand for every millimetre it moves along the bisector of its two spans. positioned by solving, not by drawing.

The tensioner is the unknown

A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm.

strands · Strand A ratio with no steps in it. Two pulleys whose sheaves slide on their shafts, and one belt. Pushing the primary's sheaves together makes the belt ride further out; the secondary's radius is then not a choice, because the belt is a fixed length and its length over two pulleys at a fixed centre distance is a function of both radii. So the secondary's radius here is the root of that equation, solved rather than assumed, and the drawn belt is the length it is supposed to be to 0.0e+0 mm. Ratio 1.000, with the two radii adding to 110.00 — a number the received rule of thumb says should not change and which changes by 2.6% across the travel.

A ratio with no steps in it

Push a variable pulley's sheaves together and the belt rides further out. The other pulley's radius is then not a choice — the belt has a fixed length — so it is the root of an equation, solved rather than set. The rule of thumb that says the two radii add to a constant is true to first order and wrong by 7.4% of the ratio at full shift, and the departure has a closed form.

transmission · Transmission The number on the tackle is a limit. The velocity ratio of a 4-part tackle as its blocks separate, measured by differentiating the strand's own length. It is 3.6174 at 90 mm and 3.99992 at 6898 mm, and it reaches 4 nowhere. The shortfall is geometric and has nothing to do with friction: each part of line makes an angle with the lift, and a part at angle β shortens by cos β of the movement. A tackle used at close quarters — which is when a tackle is useful — is the case furthest from its own rating.

Six things a strand is not

A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.

wrong · Misconception What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing.

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

strands · Strand 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.

A drum is a size, a wrap is a shape

A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.

strands · Strand 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times.

The wrap that walks along the axis

Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

strands · Strand An inner cable in its sheath, pulled and pushed. A sheath routed through two bends — the first turning 90° on a radius of 70, the second turning back 90° on 55 — drawn with its bore exaggerated to a clearance of 7 so the inner's two positions can be seen. Pulled, the inner is taut and takes the shortest path the tube allows, hugging the inside of each bend; pushed, it is pressed against the outside. Both paths are strands over pulleys of radius R ∓ c at the bend centres. At this clearance the pulled inner is 20.53 shorter than the centreline and the pushed one 23.63 longer, against c times the total turning, 21.99. At a real clearance of 0.25 the pulled inner is short by 0.7834 against 0.7854. Dragging changes the first bend's angle.

A strand in a tube

A Bowden cable's inner runs inside a sheath with a little clearance, and pulled it takes the shortest path the tube allows — a strand over pulleys of radius R − c at every bend. So its lost motion is the clearance times the total angle the sheath turns through: no bend radius in it, no route shape, and the S-bend that turns nowhere net loses as much as the U that turns back. Steering a handlebar changes it by exactly c times the change in angle.

strands · Strand The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off.

Which walls a strand is held by

A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

strands · Strand 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.

applied · Fourbar One belt, five steps. Two stepped pulleys 400 mm apart, each with five steps whose radii on the two shafts always sum to 160 mm, giving five speeds a factor of 1.6 apart. The belt is on step 4: 98.46 mm driving 61.54 mm. Solid: an open belt, 1306.066 mm on this step against 1302.655 mm on the middle one. Dashed: a crossed belt, 1367.552327 mm — the same on every step, because a crossed belt's length depends on the two radii only through their sum. Faint circles are the other steps. Dragging moves the belt to another step.

One belt for every step

A lathe changes speed by moving one flat belt between the steps of two stepped pulleys, so every pair of steps has to take the same belt. A crossed belt's length depends on the two radii only through their sum, so steps whose radii sum to the same value take the same crossed belt exactly — to the last bit of the arithmetic, over 266 random pairs. An open belt's length depends on the difference too: at constant sum it grows by d²/C, then d⁴/12C³, so an open belt made for the middle step of a five-speed pair is 12.3 mm short on the outer steps, and those steps must give up 3.74 mm of radius sum to take it.

strands · Strand One open belt on two drums. Two identical drums on parallel shafts 400 mm apart, one reversed against the other, seen from the side; the belt runs between them at a station 60 mm from the middle of its 300 mm travel. The drums are cut so that the speed ratio grows by the same proportion for every millimetre the belt is moved — from 1/2.56 to 2.56 over the travel — and so that one open belt fits at every station. At this station the driving radius is 94.45 mm, the driven 64.85 mm, a ratio of 1.456. The profiles are not cones, but the difference is a millimetre or two on radii of eighty and cannot be seen at this scale; the figures below measure it.

A cone drive is not two cones

Take a stepped pulley to its limit and the belt runs on two drums, shifted continuously along them. Drawn, the drums are straight cones of opposite taper, and a crossed belt fits them everywhere because their radius sum is constant. But straight cones grade the speed unevenly — 0.58% per millimetre of shift in the middle, 0.72% at the ends — and an evenly graded drive needs drums whose radii are logistic curves. And an open belt, the common kind, fits neither: on a lathe-sized drive it is 12.3 mm short at the ends, and two identical drums that take it and keep the grading must each be shaved in proportion to their radius, 2.69 mm at the large end and 1.05 mm at the small.

strands · Strand

Named alongside it

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

StrandDesign ruleApproximationBelt driveTangencyTurning numberVelocity ratioConvex hullInextensibleIdlerRoot-findingTaut

All concepts