What this calculator computes
A four-bar linkage has a fixed ground link d, an input crank a, a coupler b and an output rocker c, joined by four pin joints. It has a single degree of freedom: choose the input angle θ and everything else follows. Despite its simplicity it generates an enormous range of motions, from windscreen wipers and suspension arms to walking robots and the Watt and Chebyshev straight-line mechanisms.
| Quantity | Meaning |
|---|---|
| Grashof class | Which links can rotate fully, from the link lengths alone |
| Input range | Full 360° if the input is a crank, otherwise the arc between toggle positions |
| Rocker swing | Total angle swept by the output link |
| Transmission angle μ | Angle between coupler and rocker, which sets how well force is transmitted |
| Loop-closure residual | How exactly the drawn geometry satisfies both link-length constraints |
The Grashof condition
Sort the four lengths into shortest s, longest l and the other two p, q. The linkage is Grashof when
and then at least one link can make full revolutions relative to the others. Which motion you get depends on where the shortest link is:
| Shortest link | Grashof (s + l < p + q) |
|---|---|
| Input crank a | Crank-rocker: input rotates fully, output oscillates |
| Ground d | Double-crank (drag link): both side links rotate fully |
| Coupler b | Double-rocker: neither side link rotates fully, but the coupler does |
| Output c | Rocker-crank: output rotates fully, input oscillates |
If s + l > p + q the linkage is a non-Grashof triple-rocker: no link can rotate fully. If s + l = p + q it is a change-point mechanism, which passes through a configuration where all links are collinear. There it can switch assembly branch unpredictably, so real designs usually avoid it.
Position analysis
With the ground pivots at O₂ = (0, 0) and O₄ = (d, 0), the crank pin is at A = a(cos θ, sin θ). The second moving pin B is the intersection of a circle of radius b around A and a circle of radius c around O₄. The distance between those centres is
and the loop closes only while |b − c| ≤ e ≤ b + c. The two circle intersections are the open and crossed assembly branches; the preview follows the open one. The classic closed-form alternative is Freudenstein’s equation relating input θ to output ψ:
Writing that equation at three chosen (θ, ψ) pairs gives a linear system for
K₁, K₂, K₃. That is three-precision-point function synthesis, available in
MBSD as mbsd.planar.synthesis.freudenstein_3pt.
Transmission angle
The transmission angle μ is the angle between coupler and rocker:
When μ approaches 0° or 180° the coupler pushes almost along the rocker, so force transmission collapses and friction or tolerances can lock the linkage. A common design rule keeps 40° ≤ μ ≤ 140° over the whole motion. For a crank-rocker the extremes occur at θ = 0° and θ = 180°. The calculator warns when μ drops below 40°.
Coupler curves
Any point rigidly attached to the coupler traces a closed coupler curve. Move the coupler-point offset slider to explore them: figure-eights, cusps, and nearly straight segments that are the basis of straight-line and dwell mechanisms. Coupler curves are where four-bar synthesis gets interesting, and where a full solver with dynamics earns its keep.
The same linkage in Python
The linkage shown by default, checked with the open-source MBSD Core synthesis helpers:
import numpy as np
from mbsd.planar.synthesis import FourBar, rocker_angles
linkage = FourBar(ground=3.2, crank=1.0, coupler=3.0, rocker=2.5)
print(linkage.grashof_class()) # crank-rocker
theta = np.linspace(0.0, 2.0 * np.pi, 721)
psi = rocker_angles(linkage, theta)
print(f"rocker swing: {np.degrees(np.ptp(psi)):.1f} deg") # 49.6 degFor forces, joint reactions and time histories, build the linkage from bodies
and pins with Mechanism.planar(), or open it in the full browser solver.