Machine Kinematics
solved, then drawn
Concept

Feature tolerance — where it appears

A tolerance placed on where a feature sits rather than on a distance between features. It is how real parts are dimensioned, and analysing a mechanism this way changes the band whenever two features share a setup, because a common displacement cancels out of the distance between them.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Where a length comes from. A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. Each carries an error of 0.010, of which 60% is common to both because they were bored in one setup — the whole pattern shifts by that much and the distance between the holes does not change. What survives is the independent part, 0.0063 at each hole, combining to 0.0089 on the length. A drawing that tolerances the length at ±0.010 is describing a part nobody makes, and it is out by a factor of 0.894 — optimistic below a shared fraction of one half and pessimistic above it.

Where a length comes from

A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. What reaches the length is only the part of each hole's error the two do not share — so a drawing that tolerances the length is describing a part nobody makes, and is out by a factor of √2 in one direction or by everything in the other.

metrology · Parameter Where the two analyses cross. The band on the output angle at a crank angle of 57°, with every hole on every part given a position error of 0.010, against how much of that error each pair of holes shares. The flat line is what an analysis on the four lengths gives, which is the same number whatever the answer to that question. They cross at 0.53 and nowhere else: below it the lengths-only answer is optimistic, reaching 1.414× at holes located independently, and above it pessimistic, reaching zero when the error is entirely common and the distance between two holes is perfect however badly the pair is placed. The crossing is at one half because two holes contribute √2 and the surviving fraction is √(1 − shared).

Where the two analyses cross

Tolerance the lengths and you get one number whatever the shop does. Tolerance the holes and you get a curve, running from 1.414 times that number when nothing is shared to zero when everything is. They meet at a shared fraction of exactly one half, and the crossing does not depend on the machine, the tolerance or which length is being asked about.

metrology · Parameter The same part, dimensioned two ways. Three holes at 0.00, 1.40, 3.50, each located to ±0.010 by the process. Dimensioned as a chain, each hole is placed from the last, so the two gaps are ±0.010 each and the errors accumulate over the whole span to ±0.0141. Dimensioned from a baseline, every hole is placed from H0, so the span is ±0.010 and the gap between H1 and H2 — which touches the datum at neither end — is ±0.0141. The two schemes are exactly √2 apart in opposite places. Nothing about the part decides this and nothing about the process does; the drawing does, and a tolerance analysis that starts from the lengths has already thrown the information away.

The same part, dimensioned twice

Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.

metrology · Parameter Which feature to hold tight. A budget of 0.020 divided between the four lengths in inverse proportion to their sensitivities, and the same budget divided between the features those lengths are derived from — a frame jig-bored in one setup at 0.90 shared, a crank drilled twice at 0.10, a coupler at 0.60 and a rocker at 0.30. The tightest tolerance moves from b to a. And the null result matters as much: with one process for the whole machine the two allocations are identical to the last digit, because the transmission factor is then common and divides out. Feature-based tolerancing changes the answer when the parts are made differently and not merely when they are made.

Which feature to hold tight

A budget divided between four lengths in inverse proportion to their sensitivities gives one answer. The same budget divided between the features those lengths are derived from gives the same answer exactly — unless the parts are made differently, in which case the tightest tolerance moves from the rocker to the crank and the frame's loosens by a factor of two.

metrology · Parameter The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer.

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

practice · Overconstraint

Named alongside it

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

ToleranceDerived lengthSetup errorWorst caseCalibrationClearanceDatum schemeRoot-sum-squareSensitivityConstraintDesign ruleDynamics

All concepts