Overconstrained mechanism

A Sarrus linkage.

An overconstrained mechanism is a linkage that has more degrees of freedom than is predicted by the mobility formula. The mobility formula evaluates the degree of freedom of a system of rigid bodies that results when constraints are imposed in the form of joints connecting the links.

If the links of the system move in three dimensional space, then the mobility formula is

$M=6(N-1-j)+\sum_{i=1}^j f_i,$

where N is the number of links in the system, j is the number of joints, and f_i is the degree of freedom of the i^th joint.

If the links in the system move planes parallel to a fixed plane, or in concentric spheres about a fixed point, then the mobility formula is

$M=3(N-1-j)+\sum_{i=1}^j f_i.$

If a system of links and joints has mobility M=0 or less, yet still moves, then it is called an overconstrained mechanism.

Sarrus linkage

A well-known example of an overconstrained mechanism is the Sarrus mechanism, which consists of six bars connected by six hinged joints.

A general spatial linkage formed from six links and six hinged joints has mobility

$M = 6(N - 1 - j) + \sum_{i=1}^j f_i = 6(6-1-6) + 6 = 0,$

and is therefore a structure.

The Sarrus mechanism has mobility M=1, rather than M=0, which means it has a particular set of dimensions that allow movement.^[1]

Bennett's linkage

Another example of an overconstrained mechanism is Bennett's linkage, which consists of four links connected by four revolute joints.

A general spatial linkage formed from four links and four hinged joints has mobility

$M = 6(N - 1 - j) + \sum_{i=1}^j f_i = 6(4-1-4) + 4 = -2,$

which is a highly constrained system.

As in the case of the Sarrus linkage, it is a particular set of dimensions that makes the Bennett linkage movable.^[2] Below is an external link to an animation of Bennett's linkage.

References

1. ^ K. J. Waldron, Overconstrained Linkage Geometry by Solution of Closure Equations---Part 1. Method of Study, Mechanism and Machine Theory, Vol. 8, pp. 94-104, 1973.
2. ^ J. M. McCarthy and G. S. Soh, Geometric Design of Linkages, 2nd Edition, Springer 2010

External links

• Select this link for an animation of Bennett's linkage.
• Mobility of Overconstrained Parallel Mechanisms
• Four- and six-bar function cognates and overconstrained mechanisms