Overconstrained mechanism

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A Sarrus linkage.
Watt II overconstrained mechanism - ver.1
Watt II overconstrained mechanism - ver.2

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 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

 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[edit]

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[edit]

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] [3] Below is an external link to an animation of Bennett's linkage.

Assembly of cognate linkages[edit]

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[edit]

  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
  3. ^ 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.
  4. ^ P.A. Simionescu & M.R. Smith (2000) "Applications of Watt II function generator cognates", Mechanism and Machine Theory, 35(11), p. 1535–1549.
  5. ^ 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.

External links[edit]