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CURVE OF THE SLIDERCRANK MECHANISM


Curve studied by Bérard in 1820 and RuizCastizo
in 1889.
Other name: RuizCastizo's quartic. 
The curve of the slidercrank mechanism is the
locus of a fixed point M on the plane linked to the bar [PQ]
(called the connecting rod) of an articulated mechanism (OPQ),
O being fixed and Q being constrained to move on a line (D)
(the drawer, or piston). In other words, the curve of the
slidercrank mechanism is the locus of a point linked to a segment line
of constant length joining a circle (C) and a line (D).
Let A(0, a) be the projection of O
on (D), OP = b, PQ = c.
Using lower case letters for the affix of a point, we have: With , and we get: Cartesian parametrization: , where with following the position of Q with respect to (OP). Specifically, the motion of Q is not sinusoidal. Bicircular quartic (?). Cartesian equation when M is on the line (PQ) (i.e. l = 0): . 
The curve is not empty if and only if a £ b + c, and in that case it is connected iif b £ a + c (?).
When M is on the connecting rod, the equation above shows that the curve is then a polyzomal curve, medial between two ellipses: , and .
In particular:
 when a = 0 ((D) passes by O) and k = 1, these two ellipses are concentric circles: the associated curves are the quartics of Bernoulli. See also on this page the base and rolling curve of the planar movement on the associated plane.  
 when c = a + b and k = 1, these two ellipses are tangent circles: the associated curves are the double heart curves.  
 when a = 0 and b = c, the curve of the slidercrank mechanism is composed of a circle and an ellipse (in fact, we find the construction of an ellipse with a strip of paper). 

This device allows for a linear, almost sinusoidal motion;
on the right is the representation of the movement of Q for a
= 0, b = 1, c = 3.
See also an application to Mercedes's windscreen wiper. 
If the circle (C) is replaced by any conic, we get all the polyzomal curves.
If the line (D) is replaced by a circle, we get a curve of the threebar mechanism.
If the connecting rod is no longer constrained to have its end sliding on a line but is only constrained to slide while passing through a fixed point, we get the conchoids of circles.
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© Robert FERRÉOL 2017