Machine Kinematics
solved, then drawn
Concept

Friction — where it appears

The resistance to sliding at a contact, which lets a contact push anywhere inside a cone about its normal rather than only along it. It decides how few contacts can hold a part, but only in proportion to how hard they are pressed, and its coefficient has to be measured on real surfaces.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

What is still outside

Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.

practice · Practice The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

Where the boundary moved again

The practice field's inventory ended by naming what the work after it should do first: take the feature positions as the variables and derive the lengths. That is done, and it turned out not to be an extension of the tolerance field but half of a different one — because where a length comes from and what a measurement determines are the same question.

practice · Practice Two jaws on a polygon, and the cones they can push within. A regular polygon of 5 sides and circumradius 1, gripped by two contacts with a friction coefficient of 0.3. Each wedge is the cone of directions its contact can push in, half-angle atan 0.3 = 16.7° about its normal. Two contacts hold the part — resist every load, given a squeeze — exactly when the line between them lies inside both cones. An odd polygon has no parallel faces, and its most nearly opposed faces are 36.0° short of parallel, so the line can be inside both cones only if their half-angle is at least half that: a friction of tan(π/10) = 0.3249. Here the part is not held. Dragging changes the number of sides.

Two jaws and a coefficient

Every hold in this field has been frictionless, and two of its essays say what that leaves out. Put friction back and a part is held by two contacts in the plane instead of four and by three in space instead of seven; a disc, which no frictionless fixture can hold at all, is held by two jaws at any friction above nought; and an odd polygon needs a friction of exactly tan(π/2n). What friction takes away is the thing that made a frictionless hold a fact about geometry: a frictional hold is a hold per unit of squeeze.

holding · Restraint 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 A pin with friction lets the cam's force miss the pivot, by the friction circle. A rope pulled to the right, gripped by a cam whose pivot pin has a radius of 0.16 (drawn larger) and friction 0.5, so the pin's force on the cam may lie anywhere tangent to or inside its friction circle, radius 0.072. The dashed lines from the contact touch that circle; the cam's force on the rope can lie anywhere between them, where a frictionless pin would allow only the line through the centre. The shaded wedge is the contact's friction cone, 21.80° either side of the normal for μ = 0.4. At a lean of 23° the cleat holds a forward pull (it holds up to 24.14°, against 21.80° with a frictionless pin) and lets go when pulled back (it locks backwards below 2.56°).

A stiff pin buys grip with release

A cam cleat on a frictionless pin holds a rope while the line from the contact to the pivot leans no more than the contact's friction angle. Give the pin friction and the cam's force may miss the pivot by the pin's friction circle, so the cleat holds to a lean steeper by exactly the angle that circle subtends: tan γ ≤ μ + (ρ/h)√(1 + μ²). But the same friction makes a cleat leaning less than arctan(ρ/h) lock against a backward pull as well, and refuse to let go. Both edges of the working window rise, the lower at a slope of one and the upper at √(1 + μ²): the grip gained is almost all paid for in release.

holding · Restraint A cleat's lever, and the squeeze the load has left it. A cam cleat at a lean of 10°, its pivot pin with friction 0.5 and a radius a tenth of the pivot's height (drawn larger, with its friction circle), and a release lever on the cam. Along a load path the rope's pull rises to one, falls to nought and rises again, and the squeeze follows it between the band's edges, 4.50 and 7.62 times the pull. Here the pull is 1, the squeeze 4.50, and the lever must turn the cam with a moment of 0.412 times the pivot's height to free the rope.

A cleat set hard lets go easiest

A cam cleat on a frictionless pin is freed by a lever with no effort at all. With pin friction the lever must beat h(T − N tan γ) + ρ√(T² + N²), and since a stiff pin lets the squeeze N sit anywhere in a band, the effort is set by where in the band it sits: nought at the top, twice the pin's friction moment at the bottom. A rising load drags the cam in to the bottom, so a cleat holding the load it was set by is the hardest to free — 0.412 times the pull times the pivot's height at a lean of 10° — while one jammed hard by hand, or one whose load has eased, frees itself. Below the backward-lock lean nothing jumps in the effort. What changes is the state: the slack cleat keeps the squeeze its largest pull left.

holding · Restraint 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 normalClearanceDesign ruleLockDynamicsEscape coneFixtureHysteresisLever armLinear programPositive spanTolerance

All concepts