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Mathematics of the Slider Cranks >
4. Topological Extensions of the Slider Crank
The lengths, a and b, of the links, CA and AB can be varied arbitrarily (except that for the crank to turn all the way around we must have b > a). In addition, the links do not need to be straight as long as they are rigid. In this sense the slider crank is topological. Further in this direction, we can, in theory, make the path CE, along which the slider is constrained, any smooth curve we want. But in practice most constraint curves would not produce a useful mechanism nor even a mechanism whose crank would be able to turn though a complete revolution. However, there are several mechanism in the Reuleaux collection which are topological extensions of the slider crank with constraints other than a line or an offset line.
In the case, D14, the constraint curve is the arc of a circle with center at a point P. Thus, we can add a link from P to the slider B and the motion of the mechanism would not be changed. But, in that case, we could now remove the constraint curve completely and let the link to P constrain the slider. But then the slider is not sliding along anything and we have a four bar linkage. Thus the analysis of this mechanism is the same as a four-bar linkage.
The mechanism D07 is an inversion of the slider crank with the slider constraint CE fixed. Note that the slider B is now virtual. The point A is now a slider moving along a circle arc with center at B. This slider arrangement replaces the link joining A to B.
Now image straightening the arc constraint in this model. As the constraint on A becomes more and more straight the virtual point B goes further and further down until we obtain D08. Here the virtual B is at infinity. This mechanism is sometimes the Scotch Yoke Mechanism.
Another topological variation of the slider crank is to curve all the links so that the links lie in concentric spherical shells.
The analysis for the regular slider crank does not apply in this case because that analysis used several results from plane geometry. To analyze this spherical slider crank (C06) we must use results about spherical triangles. In this mechanism the crank is attached at A and thus it makes sense to use the as our parameter.
Then ABC is a spherical triangle and we can apply the Spherical Law of Cosines, which asserts:
where the lengths of the links are measured in radians. (The radian measure of a great circle arc on a sphere is the radian measure of the angle subtended by the arc from the center of the sphere; or, if the radius of the spherical shell is R then the radian measure of a great circle arc of length l is equal to l/R.) For more information on spherical trigonometry, see Chapter 20 of
Henderson, David W., and Taimina, David. Experiencing Geometry: Euclidean and Non-Euclidean With History. Upper Saddle River, NJ: Prentice Hall, 2005.
1. Generic Slider Crank (C02)
2. Inversions of the Slider Crank (C02, C05, D2)
3. Offset Slider Cranks (D01, D03)