Machine Kinematics
solved, then drawn
Concept

Bearing pedestal — where it appears

The structure carrying a ground pivot, standing up from the base plate and so present in every plane at once. It is the one obstacle a planar linkage cannot design around by choosing heights, and it is what limits how wide a four-bar's links may be.

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

crank rocker: the closest pair at one position. The site's standard four-bar: ground 4, crank 1, coupler 3.5, rocker 3. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.5438 here, between coupler · frame. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Two bars that have to cross

The site's own four-bar has its coupler inside its frame by a full link width for the whole of a turn. It is not an impossible mechanism; it is a mechanism that cannot be built in one plane — and the plane it has been drawn in for twenty-three fields was a convenience nobody had to pay for.

bodies · Body Turning all the way round, against being made of something. Seven four-bars, classified by Grashof's inequality on their four lengths and then asked a question Grashof cannot answer: with a bearing pedestal at each ground pivot, how wide may the links be? The two instruments have nothing in common — one is an inequality on four numbers, the other counts sign changes of (B − A) × (G − A) over a sweep — and they agree about something Grashof was not for. Every four-bar that turns all the way round sweeps a link straight over a ground pivot, so its closest approach is exactly zero and no positive width is admissible; not one of the rockers does, and they take widths up to 0.20 of their shortest link.

The crank that cannot turn all the way

Grashof's inequality says which four-bars turn fully. Ask instead how wide their links may be with a bearing at each ground pivot, and the same inequality answers the opposite question: every four-bar that turns all the way round sweeps a link straight over one of its own pivots.

bodies · Body One link, five bodies. Five bars with the same two pins, offset by -0.24, -0.12, 0, 0.12, 0.24 of the link's length, drawn to a common scale. Every one of them holds its two pins exactly the same distance apart, so every one of them is the same link: put any of them into a mechanism and the mechanism solves to the same joint positions at every configuration. Nothing in the kinematics of this collection — no loop equation, no velocity, no coupler curve, no mobility count — can tell them apart. What they do not have in common is which ground they occupy on the way from one pin to the other.

A link may be bent

A link is two pins at a fixed distance and the metal between them is a free choice. Bending it moves no joint of the mechanism by more than 10⁻¹³ and moves the clearance by a tenth of a link length — enough to build a machine that a straight bar refuses, and worth exactly nothing against a bearing pedestal the link sweeps over.

bodies · Body

Named alongside it

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

InterferenceLink bodySigned clearanceKinematic chainPivot crossingBearingBossChromatic numberConflict graphCrank-rockerDesign ruleGrashof's condition

All concepts