Machine Kinematics
solved, then drawn
Concept

Average — where it appears

A single number standing in for a quantity that varies through a mechanism's cycle. The interesting question about a quoted one is what it averages over, since a mean ratio and an instantaneous ratio can differ by several per cent.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.

wrong · Misconception Two ways of adding four tolerances. The output band from ±0.01 on each of four lengths, combined two ways. The upper curve is worst case — every error at its extreme and conspiring — and the lower is root-sum-square, which treats the four as independent random errors. RSS is smaller everywhere, by between 1.42 and 1.95, and with four contributions the most it can ever be is √4 = 2. That factor is not a saving found in the geometry; it is bought with the assumption that the four errors are independent, and one fixture that locates two of the holes takes it straight back.

Worst case and the square root

Four tolerances can be added as a straight sum or as a root-sum-square, and the second is smaller by between 1.42 and 1.96 through this linkage's cycle. The ceiling is √4 = 2 and no geometry can beat it. That factor is not found in the mechanism — it is bought entirely with an assumption of independence, and one fixture that locates two holes takes it straight back.

practice · Tolerance Tolerancing the holes rather than the lengths. The output band from ±0.01, computed on the four link lengths and then on the features those lengths are derived from. The ground length is the distance between two frame holes, and what happens to it depends entirely on how they were made: located separately, two independent errors combine to √2 times one and the lengths-only answer is optimistic by 11%; bored in one setup, the common part of the machine's error cancels out of the distance between them and the lengths-only answer is pessimistic by 34%. Neither is a correction to apply; it is a question for the machine shop, and this site can compute the sensitivity and not the answer.

Tolerancing the holes

A drawing does not tolerance link lengths. It tolerances holes, and the lengths are derived from them — so what a length's tolerance really means depends on whether its two holes were bored in one setup. Located separately, a lengths-only stack-up is optimistic by 11%; bored together, it is pessimistic by 34%. Neither is a correction to apply blind.

practice · Tolerance

Named alongside it

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

ConstraintAllocationDerivativeFour-barIndependenceJacobianSensitivityStrokeThresholdToleranceDatumDead centre

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