Machine Kinematics
solved, then drawn
Concept

Error accumulation — where it appears

How a tolerance on one unit shows up in an assembly of many. Errors that share a sign add, so the total is the unit count times one unit's; independent errors grow as the square root of the count instead, and the two estimates differ by the square root of the count — which is a factor of 5.66 at thirty-two units.

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

The network where counting works. 5 scissor units, each two bars pinned at their middles, with consecutive units sharing their end pins. Sixteen bodies and twenty-two pins at eight units, and Grübler's 3(n − 1) − 2j gives 4 — three rigid motions and one internal freedom — against a measured nullity of 4 and no redundant constraints at all. That holds at one unit and at sixteen. The tong is here to make the field's point in the direction nobody expects it: a network is not a place where counting fails, it is a place where counting stops being checkable by hand, and one of the two assemblies that spans this plate is counted perfectly. Span 8.1388 mm at this opening, which is 5 times one unit's 1.6278. positioned by solving, not by drawing.

One input at one end

A lazy tong's reach is 2nL cos θ — exactly, to a part in a million million, over eight sizes and sixty openings — so it multiplies its input by the number of units. It multiplies everything else by the same number, including the part of the drawing nobody wanted multiplied: a unit cut a hundredth of a radian out puts a tong of thirty-two units 0.374 out at the far end.

networks · Network The same multiplier, applied to the error. A tong whose units are cut to an angle 0.01 radians away from the drawing. If one unit is out, the span is out by that unit's share and nothing more; if every unit is out the same way — which is what a machine setting or a worn tool produces — the error is multiplied by the unit count, exactly, to 7.6e-14. The third column is what would happen if the errors were independent and equally likely either way: the accumulation goes as the square root of the count instead, and the difference between the two columns at thirty-two units is a factor of 5.66. Which column applies is a question about how the parts were made, not about the mechanism.

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

practice · Tolerance The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them.

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

practice · Tolerance

Named alongside it

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

ToleranceDeployableMobilityNetworkScissor linkageWorst caseAmplificationAssemblyCanonical formCycle basisDesign ruleGrübler's criterion

All concepts