A four-bar linkage is the simplest planar mechanism that turns a rotation into a different, non-uniform motion: four links, four revolute joints, one drive. This page states which quantities are computed, where they come from, and where hand calculation ends.
One mechanism, several names
Four-bar linkage, four-bar mechanism and quadrilateral linkage all name the same thing. Four-bar chain refers to the chain before one of its links is declared the frame - which one it is decides the type and turns one chain into several mechanisms. The frame counts as a link; anyone counting only the three moving ones is puzzled by the name and gets the degree of freedom wrong. The underlying systematics are in Kerle/Pittschellis/Corves, Einführung in die Getriebelehre, 3rd edition 2007, chapter 2.
Why a single drive is enough
The degree of freedom of a planar mechanism follows from the counting formula of Grübler and Chebyshev: F = 3·(n − 1) − Σu, with n the number of links including the frame and u the number of freedoms a joint blocks - a revolute joint blocks two. For the four-bar that gives F = 3·(4 − 1) − 4·2 = 1. A single drive therefore determines the motion completely; the mechanism is constrained. If a constraint is missing, F exceeds the number of drives and the position is no longer determinate; if one is superfluous, the chain is over-constrained. (Kerle, chapter 2.3, pages 25 to 30.)
The type follows from four lengths
Whether and which link revolves fully is decided by Grashof’s condition: the shortest plus the longest link must not together exceed the other two. If it holds strictly, fully rotatable joints exist, and the position of the shortest link determines the type:
- Shortest link at the frame: crank-rocker - the crank revolves, the opposite link rocks.
- Shortest link is the frame itself: double crank - both links at the frame revolve.
- Shortest link is the coupler: double rocker - both links at the frame rock while the coupler revolves.
- Condition violated: triple rocker - no link revolves.
- Equality instead of a strict inequality: change-point mechanism - it passes through the straight-line position, but which branch it takes afterwards is no longer determined.
See Kerle, page 38, equations (2.14) to (2.16). The condition applies to the four-bar revolute chain; the slider-crank chain has an inequality of its own, and a five-bar has no Grashof type at all.
The transmission angle tells you whether it runs easily
The transmission angle μ is the angle at the corner between coupler and rocker. At μ = 90° the entire coupler force goes into the motion; at μ = 0° the mechanism is in a dead-centre position and jams. It can be computed in closed form from the four lengths:
with a the crank, b the coupler, c the rocker and d the frame (Kerle, page 161, equation 6.5). Both values belong to the positions in which crank and frame are aligned; in between the angle is larger. What matters is therefore not the angle in one position but the smallest one over the whole revolution. As an empirical value for slow-running mechanisms Kerle gives 40 to 50 degrees.
Every crank position has two assembly configurations
Crank, coupler and rocker form a triangle with the frame, and a triangle can be closed to either side from the same side lengths. Each crank position therefore has two assembly configurations. An assembled mechanism stays in its branch - except in the change-point case, where the two coincide. Anyone assembling a coupler curve by hand from the law of cosines has to carry that branch along; otherwise a sign error jumps to the other path in mid-revolution.
The coupler curve and Roberts’ theorem
The coupler curve is the path of a point rigidly attached to the coupler - the actual reason for building linkages: it can deliver stretches that are nearly straight or nearly at rest without any part being guided (Kerle, chapter 4.1.5, page 110). By Roberts’ theorem the same coupler curve is generated by three different four-bar linkages, whose frame triangle is similar to the coupler triangle. Anyone happy with a curve but unhappy with the space it needs has two further mechanisms to choose from (Kerle, chapter 6.3.1, pages 185 to 188, table 6.2).
Why a four-bar is not solved in one go
There is no formula for the position that one simply fills in. The loop equation of the chain is non-linear; it is solved through vector loops and the Newton-Raphson method, and the Jacobian of that method is also where dead-centre positions show up as a singularity (Kerle, chapter 4.1, pages 101 to 112). For a four-bar, compasses and the law of cosines still do. As soon as one link is added or a prismatic joint is involved, that ends - and for the course over a full revolution nobody computes it by hand.
What you can compute it with
That is exactly what Kinematics Studio does: you build the chain from links and joints, the solver resolves the constraint equations across the whole drive range and returns positions and paths, the transmission angle, the Grashof type, the instant centre with fixed and moving centrode, plus drive torque and joint forces. A four-bar is merely the simplest case; slider-cranks, five-bars and chains of your own work the same way.
Related, and often the shorter route: the crank-rocker in particular, the slider-crank mechanism with piston and connecting rod, and the cam design to VDI 2143 when the motion is to be prescribed freely instead of following from four lengths.
Frequently asked questions
Four-bar linkage, four-bar chain, quadrilateral linkage - the same thing?
Yes. All of these name the closed chain of four links and four revolute joints. Anyone counting only the moving links arrives at three: the frame counts as a link, otherwise the degree-of-freedom calculation does not work out. The term four-bar chain is used for the chain before one of its links is declared the frame.
How do I tell whether the crank makes a full revolution?
By Grashof's condition: the sum of the shortest and the longest link must be smaller than the sum of the other two. If it holds, at least one link revolves fully, and which one depends on where the shortest link sits. At equality the mechanism can change branch and is no longer determinate in the straight-line position; if the condition is violated, every link merely rocks. The condition applies to the four-bar revolute chain, not to the slider-crank.
How small may the transmission angle become?
For slow-running mechanisms Kerle gives 40 to 50 degrees as an empirical value. Below that nothing breaks, but joint forces, friction and the sensitivity to joint clearance grow quickly. It is therefore a guide value and not a passed verification - and what matters is the smallest value over the whole revolution, not the one in the position currently on screen.