Wiener–Hopf method
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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
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.
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
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
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] [3] Below is an external link to an animation of Bennett's linkage.
Assembly of cognate linkages
Overconstrained mechanisms can be also obtained by assembling together cognate linkages; when their number is more than two, overconstrained mechanisms with negative calculated mobility will result. [4] [5] The companion animated GIFs show two types of overconstrained mechanisms obtained by assembling together function cognates of the Watt II type.
References
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External links
- Select this link for an animation of Bennett's linkage.
- Mobility of Overconstrained Parallel Mechanisms
- ↑ 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.
- ↑ J. M. McCarthy and G. S. Soh, Geometric Design of Linkages, 2nd Edition, Springer 2010
- ↑ Dai, J.S., Huang, Z., Lipkin, H., Mobility of Overconstrained Parallel Mechanisms, Special Supplement on Spatial Mechanisms and Robot Manipulators, Transactions of the ASME: Journal of Mechanical Design, 128(1): 220-229, 2006.
- ↑ P.A. Simionescu & M.R. Smith (2000) "Applications of Watt II function generator cognates", Mechanism and Machine Theory, 35(11), p. 1535–1549.
- ↑ P.A. Simionescu & M.R. Smith (2001) "Four- and six-bar function cognates and overconstrained mechanisms", Mechanism and Machine Theory, 36(8), p. 913–924.