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Concept
Overconstrained mechanism
Summary
In mechanical engineering, 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 between the links.
If the links of the system move in three-dimensional space, then the mobility formula is
where N is the number of links in the system, j is the number of joints, and fi is the degree of freedom of the ith 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
If a system of links and joints has mobility M = 0 or less, yet still moves, then it is called an overconstrained mechanism.
The reason of over-constraint is the unique geometry of linkages in these mechanisms, which the mobility formula does not take into account. This unique geometry gives rise to "redundant constraints", i.e. when multiple joints are constraining the same degrees of freedom. These redundant constraints are the reason of the over-constraint.
For example, as shown in the figure to the right, consider a hinged door with 3 hinges. The mobility criterion for this door gives the mobility to be −1. Yet, the door moves and has a degree of freedom 1, as all its hinges have colinear axes.
The figure on the left shows a two-hinged trunk lid. The calculated mobility for the lid relative to the car body is zero, yet it moves as its hinges (which are pin joints) have colinear axes. In this case, the second hinge is kinematically redundant.
A well-known example of an overconstrained mechanism is the parallel linkage with multiple cranks, as seen in the running gear of steam locomotives.
Sarrus mechanism consists of six bars connected by six hinged joints.
A general spatial linkage formed from six links and six hinged joints has mobility
and is therefore a structure.
The Sarrus mechanism has one degree of freedom whereas the mobility formula yields M = 0, which means it has a particular set of dimensions that allow movement.
Official source
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