Glossary

Four-bar linkage

Also called: planar four-bar · four-bar mechanism · 4R linkage

A planar four-bar linkage is a closed kinematic chain of four rigid links connected by four revolute (pin) joints. It is the simplest mechanism with a single degree of freedom, the canonical object of mechanism synthesis, and the building block from which slider-cranks, scotch yokes, and most balance-shaft drives are derived. Three of the four classical defect filters (Grashof, transmission angle, branch) are stated directly on its geometry.

Anatomy

A planar four-bar consists of:

Ground link (the fixed frame) of length g.
Crank or input link of length a, pinned to ground at point A.
Coupler of length b, the floating link that carries the rigid body whose motion the designer is shaping.
Rocker or output link of length c, pinned to ground at point B.

Four revolute joints connect them: A (ground–crank), C (crank–coupler), D (coupler–rocker), B (rocker–ground). The mechanism has one degree of freedom — drive the crank, every other position is determined algebraically by the constraint Jacobian.

Why it is the canonical mechanism

Mechanism synthesis exists because rotating shafts are cheap, but the motions actual products need (oscillation, sweep, complex coupler curves, intermittent dwells) are not rotations. The four-bar is the simplest machine that converts continuous rotation into something else.

It is also the vocabulary of synthesis. Grashof’s condition is stated on the four link lengths. Burmester theory computes the four-bar that hits prescribed coupler positions. The transmission angle filter is the angle at the coupler-rocker joint. Every classical synthesis paper from the 1960s onward starts from the four-bar.

Slider-cranks (engine pistons), scotch yokes (textile machines, balance shafts), and crank-rockers (suspension links, oscillating pumps) are all four-bars in disguise — replace one revolute with a prismatic, or stretch one link to infinity, and the math carries over.

Branches and assembly modes

Given the same four link lengths and the same crank position, a four-bar can be assembled in two distinct configurations — mirror images about the line connecting the moving joints. These are the two branches (or circuits) of the mechanism. Most synthesis problems require staying on a single branch as the crank rotates, because flipping branches mid-stroke means the linkage physically passes through a singular configuration — which it cannot do without disassembly.

The branch defect is the situation where a synthesised candidate requires the assembly to flip branches between precision points. Detecting this defect algebraically is one of the four classical filters, and it does the bulk of the work in our synthesis pipeline — on the practica problem it rejected 62,000 of 72,000 candidates.

Sub-classes (Grashof)

If Grashof is satisfied, the four-bar’s behaviour is determined by which link is the shortest:

Crank-rocker — input rotates fully, output oscillates. The most common engineering configuration.
Double-crank (drag-link) — both moving links rotate fully.
Double-rocker — both moving links oscillate; the coupler rotates fully.

If Grashof is violated, every link is a rocker — triple-rocker — useful only for limited-range oscillating mechanisms.

Coupler curves

The trace of any point on the coupler link as the crank rotates is the coupler curve of that point. Coupler curves are the source of the four-bar’s expressive power — they can be complex, can include cusps and double-points, and can be shaped (within limits) to match a target path via path-generation synthesis.

Burmester theory works on positions and orientations of the coupler body. Path-generation synthesis works on a target curve traced by a single coupler point. Velocity synthesis works on the velocity ratio between input and output. We ship a chapter for each in the synthesis suite.

Why the four-bar is not the answer to every problem

It has one input. It has one output. Most real production mechanisms need more — multi-loop linkages, gears, cams, parallel manipulators. We start from the four-bar because it teaches the language. Production problems inherit it.