Four-Bar Linkage Calculator: Grashof Condition, Coupler and Rocker Angles, Transmission Angle and Quick Return
Analyse a four-bar mechanism from its link lengths.
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How this calculator works
Enter the four link lengths: the fixed link, the crank, the coupler and the rocker.
Enter the crank angle for the position you want.
The calculator checks the Grashof condition, names the linkage type, solves the positions and gives the transmission angle, the rocker swing and the quick return ratio.
Formulas used
- s + l\le p + q
- \cos\mu = \frac{r_3^2 + r_4^2 - d^2}{2r_3r_4}
- Q = \frac{\alpha_1}{\alpha_2}
Worked example
With the default numbers in the calculator above:
- Grashof condition: the shortest plus the longest link against the other two: 130 against 150: it is Grashof, so at least one link can turn fully
- Type of linkage: crank-rocker: the crank turns fully and the rocker swings (shortest link: crank)
- Coupler angle (from the fixed link direction, open configuration): 31.629 degrees
- Rocker angle: 103.96 degrees
- Transmission angle (between the coupler and rocker): 72.328 degrees (the useful value, 72.33, should stay above about 40)
- Smallest transmission angle over a whole turn of the crank: 55.15 degrees
- Swing angle of the rocker: 50.803 degrees
- Quick return time ratio (the slower stroke compared with the faster stroke at constant crank speed): 1.022 to 1 (the crank turns 182 degrees one way and 178 the other)
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
What is the Grashof condition?
If the shortest link plus the longest link is no more than the sum of the other two, at least one link can make a full turn.
What is a crank-rocker?
A four-bar linkage where the shortest link is the crank. It turns fully and drives a rocker that swings back and forth.
What is the transmission angle?
The angle between the coupler and the output link. It should stay above about 40 degrees, or the linkage will jam.
What is a quick return mechanism?
One where the output moves faster in one direction than the other for a constant input speed, used in shapers and saws.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any theory of machines text (four-bar linkages and the Grashof criterion).
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