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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.

**D14**

D07**D08**

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.

**C06**

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.

Go to:

1. Generic
Slider Crank (**C02**)

2. Inversions
of the Slider Crank (**C02**, **C05**, **D2**)

3. Offset
Slider Cranks (**D01**, **D03**)