The inner contact surface of the sprag is composed of three tangent circles with their own radius and center positions, and the switching of the circles in contact depends on the wedging position.

Their values depend on the position of the contact point A and change to the corresponding value when sprags roll across the junction point of tangent circles. Focusing on the varied contact radius on inner surface of the sprag, a novel nonlinear iteration method should be proposed to compute the normal contact force during whole engagement.

First, a judgment is always carried out within each time step to determine whether the contact point has crossed the near junction point of tangent circles. If not, the calculation of normal contact force continues to proceed.

Ford Sr (1886-1967), is related to ideas about mutually

tangent circles that were studied by, among others, Apollonius of Perga in the third century BC and by Rene Descartes in the 17th century (Wikipedia, 2015).

(It is assumed that the

tangent circles intersect.) Without lose of generality, the radius of each circle (disk) is assumed to be 1.

According to Ibn Sinan's autobiography, his work on the second level of analysis, which treated the conditions under which solutions could exist, was found in his lost treatise

Tangent Circles. "On Analysis and Synthesis" contains his work on the third level of analysis.

* Researchers discovered a remarkable formula relating the curvatures and coordinates of

tangent circles packed within a circle (159: 254 (*)).

These are four mutually tangent circles. Generalisation is quite easy.

These six points of tangency support a second set of four tangent circles, a general fact which Coxeter (1989) ascribes to a schoolteacher named Beecroft.

Some readers may be curious to know how the four tangent circles come to satisfy a celebrated equation found by Descartes and rediscovered by Beecroft and Soddy.

Given four mutually tangent circles with curvatures a, b, c, and d, the Descartes circle equation specifies that ([a.sup.2] + [b2.sup.] + [c.sup.2] + [d.sup.2]) = 1/2 [(a + b+ c + d).sup.2].

The new formula looks like the original Descartes equation for four mutually tangent circles, provided the coordinates of the centers are expressed as so-called complex numbers.