Coupler Curves of a Four-Bar Linkage

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration simulates a planar four-bar linkage mechanism and computes the path traced by a point on its coupler bar.


The four bars of the linkage are

• the ground bar (whose length you can vary by dragging the locator),

• the crank,

• the rocker, and

• the coupler bar connecting the crank and rocker.

The linkage is supposed to be a Grashof linkage [1], allowing at least one link (the crank) to make a complete revolution. Rotate the crank to see how the coupler curve is generated. A warning is displayed if the geometry of the linkage does not meet the Grashof condition.


Contributed by: Erik Mahieu (November 2012)
Open content licensed under CC BY-NC-SA



Grashof's law [1] states that the sum of the longest and the shortest link must be less than or equal to the sum of the remaining two links. This condition must be met to allow at least one link to make a complete revolution.

The four-bar linkage of this Demonstration has four joints () and a total of four links (), of which one is grounded (). According to Gruebler's equation for planar mechanisms [2], the number of degrees of freedom is .

The Freudenstein equation [3] is used to compute the kinematics of a four-bar linkage.

The Hoekens linkage [4] generates an almost rectilinear coupler curve. The Chebyshev lambda mechanism is very similar. See the bookmarks, snapshots 1 and 2, and the Demonstration "Chebishevs Lambda Mechanism" by Nikita Panyunin.


[1] "Grashof's Law." eNotes: Mechanical Engineering.

[2] A. Tchako. "Degree of Freedom." Union College Mechanical Engineering (MER 312: Dynamics and Kinematics (of Mechanisms)). (2008)

[3] A. Ghosal, "The Freudenstein Equation: Design of Four-Link Mechanisms," Resonance, 15(8), 2010 pp. 699–710.

[4] Wikipedia. "Hoekens Linkage." (Sept 7, 2012)

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.