The mechanism that worked on paper and jammed in the hand

A motion study that ignored the tolerances of the real parts.

A mechanism that worked perfectly on paper jammed in the hand because a motion study calculated from exact, nominal dimensions assumes every link and every pivot is precisely the length and precisely the position the drawing says, while a real, manufactured linkage never quite is, and a closed loop of rigid links has nowhere for those small real-world differences to go except into binding.

Introduction and overview

A linkage's motion study on paper treats every link length and every pivot position as an exact number, and solving for how the mechanism moves from those exact numbers produces a clean, smooth motion with no hint of difficulty anywhere in its travel. A real linkage is built from parts that are each slightly off their nominal dimension, a link a fraction of a millimetre longer or shorter than drawn, a pivot hole positioned a fraction off centre, well within any normal manufacturing tolerance and individually completely unremarkable. The paper study never sees any of this, since it was never asked to, and the gap between the idealised geometry it solved and the real geometry that actually gets built is exactly where a mechanism that worked flawlessly on paper can jam solidly the first time a hand tries to move it.

This is a specific and easy-to-miss version of the more general habit covered earlier in this era, hand calculations and paper reviews catching real problems cheaply before anything is cut, since a motion study is exactly that kind of paper check, and it earns its keep only if it is actually checking the question that matters, which for a closed-loop linkage is never simply whether the nominal geometry moves smoothly.

The mitred-picture-frame comparison

Four picture-frame corner pieces cut to a mathematically exact forty-five degrees each should, on paper, assemble into a perfectly flat, perfectly square frame with every joint closing cleanly. Cut those same four corners on a real saw, each one landing a fraction of a degree off forty-five purely from ordinary blade wander and measuring error, and the frame no longer closes flat at all, since the four small individual errors do not politely cancel each other out, they accumulate around the closed loop of the frame and force the last joint to either gap open or bind under pressure to force it shut. A closed-loop linkage suffers from exactly the same arithmetic, since its links and pivots also have to form a closed loop that returns to where it started, and small individual manufacturing errors distributed around that loop have nowhere to go except into extra friction, binding, or a joint that simply will not sit where the drawing said it should.

Why a closed loop is less forgiving than an open one

An open chain of links, one end fixed and the other free to move wherever the mechanism happens to put it, tolerates manufacturing error gracefully, since any small deviation in a link length simply moves the free end's final position slightly, with no requirement anywhere in the chain that everything add back up to a specific closed shape. A closed-loop linkage has no such freedom, since every link in the loop has to satisfy the same closure condition simultaneously, all the way around, and a small dimensional error in any one link forces every other link in the loop to absorb that error somehow, through unwanted stress, unwanted friction, or outright binding at whichever joint happens to be least able to flex around the mismatch, the same closed-loop tolerance stack-up that an earlier article in this era found hiding inside a set of mating drawings that each looked correct in isolation. This is precisely why a motion study that treats a closed-loop mechanism's links as mathematically exact is missing the one property, how the loop actually behaves once it is forced slightly out of its ideal closed shape, that decides whether the built version moves freely or seizes.

One figure worth keeping in mind

A closed-loop linkage built from several links, each individually within an entirely normal manufacturing tolerance of a few hundredths of a millimetre, can accumulate a combined loop-closure error many times larger than any single link's own tolerance once every link's small deviation is added around the full loop, and that accumulated error is exactly what a paper motion study working from nominal dimensions has no way of predicting on its own. A mechanism with only a few links tolerates this reasonably well, while a mechanism with many links stacked into one closed loop can accumulate enough error to bind noticeably even when every individual part measured perfectly acceptable on its own, a genuinely counter-intuitive result for anyone assuming that parts which each pass inspection individually must therefore assemble into a mechanism that also works.

What follows from this

A motion study genuinely worth trusting for a closed-loop mechanism has to check the design's sensitivity to small dimensional errors, not merely its behaviour at the exact nominal dimensions, deliberately testing what happens if every link is pushed toward the worst-case edge of its expected tolerance rather than assuming the mechanism only ever needs to work at the perfect values drawn on the page. Building in a small amount of deliberate clearance or compliance somewhere in the loop, a joint slightly looser than strictly necessary, a link with a touch of built-in flex, gives the accumulated error somewhere harmless to go rather than forcing it to resolve as binding at whichever joint happens to be least forgiving.

Deciding in advance which single joint in the loop is allowed to absorb that error is itself worth doing deliberately, since a mechanism with no designated slack point simply binds wherever its worst combination of tolerances happens to land, unpredictably and differently from one built unit to the next, while a mechanism with one joint intentionally left slightly looser than the others concentrates that unpredictability somewhere harmless and consistent, rather than leaving it to be discovered fresh on every single build.

My key error with this

The mechanism was meant to swap a component automatically partway through a test sequence, and in the model it swung through its full travel with clearance everywhere I looked, which I took as evidence that it would swap the component reliably. It jammed the first time a person loaded it, and then intermittently afterwards in a way that took considerably longer to diagnose than the original design had taken to draw. The clearance I had checked was the clearance between nominal parts in their nominal positions, and the real assembly stacked several tolerances in the same direction at once while a human hand introduced a small misalignment that no model had been asked about. What replaced the belief is that clearance in a model is a statement about one specific idealised assembly rather than about the population of assemblies that will actually exist, and that any mechanism which a person loads by hand has to be checked against the worst combination of tolerance and handling rather than against the drawing.