Refer to the Parameters table and the following figures for the definition of various geometrical parameters. Additional internal parameters are defined as follows.

The notch width, $w$, is divided to the 'a-side' and 'b-side' such that $w\=\mathrm{w\_\_a}\+\mathrm{w\_\_b}$ with $\mathrm{w\_\_a}$ and $\mathrm{w\_\_b}$ set by the notch angles $\mathrm{\α\_\_a}$ and $\mathrm{\α\_\_b}$:

$\mathrm{w\_\_a}\=\frac{w\tan \left(\mathrm{\α\_\_a}\right)}{\tan \left(\mathrm{\α\_\_a}\right)\+\tan \left(\mathrm{\α\_\_b}\right)}$        $\mathrm{w\_\_b}\=\frac{w\tan \left(\mathrm{\α\_\_b}\right)}{\tan \left(\mathrm{\α\_\_a}\right)\+\tan \left(\mathrm{\α\_\_b}\right)}$

The horizontal distance between the notch vertex and the vertical centerline of the ball when it is at its highest point is given by

$\mathrm{x\_\_0}\=-\frac{w\tan \left(\mathrm{\α\_\_a}\right)}{\tan \left(\mathrm{\α\_\_b}\right)\+\tan \left(\mathrm{\α\_\_a}\right)}\+r\cos \left(\mathrm{\α\_\_a}\right)\+\frac{1}{2}\frac{-2\sin \left(\mathrm{\α\_\_a}\right)r-r\sin \left(\mathrm{\α\_\_b}\+2\mathrm{\α\_\_a}\right)\+\sin \left(\mathrm{\α\_\_b}\right)r\+w\sin \left(\mathrm{\α\_\_b}\+\mathrm{\α\_\_a}\right)\+w\sin \left(\mathrm{\α\_\_a}-\mathrm{\α\_\_b}\right)}{\sin \left(\mathrm{\α\_\_b}\+\mathrm{\α\_\_a}\right)}$

For a symmetrical notch, i.e. $\mathrm{\α\_\_a}\=\mathrm{\α\_\_b}$, we have $\mathrm{x\_\_0}\=0$.

The maximum vertical displacment of the ball is given by

$\mathrm{\δ\_\_max}\= r-r\sin \left(\mathrm{\α\_\_a}\right)\+\frac{1}{2}\frac{\sin \left(\mathrm{\α\_\_b}\right)r-r\sin \left(2\mathrm{\α\_\_a}\+\mathrm{\α\_\_b}\right)-2r\sin \left(\mathrm{\α\_\_a}\right)\+w\sin \left(\mathrm{\α\_\_a}\+\mathrm{\α\_\_b}\right)\+w\sin \left(\mathrm{\α\_\_a}-\mathrm{\α\_\_b}\right)}{\sin \left(\mathrm{\α\_\_a}\+\mathrm{\α\_\_b}\right)\tan \left(\mathrm{\α\_\_a}\right)}$

The variable $\mathrm{\δ}\left(t\right)$ gives the displacement of the ball:

$\mathrm{\delta }\left(t\right)\&equals;\&lcub;\begin{array}{cc}0& s<-\mathrm{w\_\_a}\\ r-\sqrt{r^2-{\left(s\&plus;\mathrm{w\_\_a}\right)}^2}& s<-\mathrm{w\_\_a}\&plus;r\cos \left(\mathrm{\&alpha;\_\_a}\right)\\ r-r\sin \left(\mathrm{\&alpha;\_\_a}\right)\&plus;\frac{s\&plus;\mathrm{w\_\_a}-r\cos \left(\mathrm{\&alpha;\_\_a}\right)}{\tan \left(\mathrm{\&alpha;\_\_a}\right)}& s<\mathrm{x\_\_0}\\ r-r\sin \left(\mathrm{\&alpha;\_\_b}\right)\&plus;\frac{-s\&plus;\mathrm{w\_\_b}-r\cos \left(\mathrm{\&alpha;\_\_b}\right)}{\tan \left(\mathrm{\&alpha;\_\_b}\right)}& s<\mathrm{w\_\_b}-r\cos \left(\mathrm{\&alpha;\_\_b}\right)\\ r-\sqrt{r^2-{\left(s-\mathrm{w\_\_b}\right)}^2}& s<\mathrm{w\_\_b}\\ 0& \mathrm{otherwise}\end{array}$

As the ball comes into contact with sides of the notch, the angle of the reaction force changes. This angle is given by

$\mathrm{\theta }\left(t\right)\&equals;\&lcub;\begin{array}{cc}\frac{1}{2}\mathrm{\pi }& s<-\mathrm{w\_\_a}\\ \arccos \left(\frac{s\&plus;\mathrm{w\_\_a}}{r}\right)& s<-\mathrm{w\_\_a}\&plus;r\cos \left(\mathrm{\&alpha;\_\_a}\right)\\ \mathrm{\&alpha;\_\_a}& s<\mathrm{x\_\_0}-\frac{1}{2}\mathrm{w\_\_0}\\ \mathrm{\&alpha;\_\_a}\&plus;\frac{2\left(s-\mathrm{x\_\_0}\&plus;\frac{1}{2}\mathrm{w\_\_0}\right)\left(\frac{1}{2}\mathrm{\pi }-\mathrm{\&alpha;\_\_a}\right)}{\mathrm{w\_\_0}}& s<\mathrm{x\_\_0}\\ \mathrm{\pi }-\mathrm{\&alpha;\_\_b}\&plus;\frac{2\left(\mathrm{x\_\_0}\&plus;\frac{1}{2}\mathrm{w\_\_0}-s\right)\left(\mathrm{\&alpha;\_\_b}-\frac{1}{2}\mathrm{\pi }\right)}{\mathrm{w\_\_0}}& s<\mathrm{x\_\_0}\&plus;\frac{1}{2}\mathrm{w\_\_0}\\ \mathrm{\pi }-\mathrm{\&alpha;\_\_b}& s<\mathrm{w\_\_b}-r\cos \left(\mathrm{\&alpha;\_\_b}\right)\\ \arccos \left(\frac{s-\mathrm{w\_\_b}}{r}\right)& s<\mathrm{w\_\_b}\\ \frac{1}{2}\mathrm{\pi }& \mathrm{otherwise}\end{array}$

where $\mathrm{w\_\_0}$ is the transition length. This parameter is needed to avoid discontinuous change in the contact angle.

In the above piecewise equations for $\mathrm{\delta }\left(t\right)$ and $\mathrm{\theta }\left(t\right)$, the variable $s\left(t\right)$is defined as

$s\&equals;-\mathrm{s\_\_rel}\&plus;\mathrm{x\_\_0}-\mathrm{s\_\_0}$

where   $\mathrm{x\_\_0}$ is defined above and $\mathrm{s\_\_0}$ is a user parameter for the horizontal placement of the notch. $\mathrm{s\_\_rel}$ is the relative distance between the two flanges,

$\mathrm{s\_\_rel}\&equals;\mathrm{s\_\_b}-\mathrm{s\_\_a}$

The vertical force (spring/damper and preload) is calculated as

$\mathrm{fs}\&equals;-c\left(\mathrm{\delta }-\mathrm{\&delta;\_\_max}\right)-d\frac{\&DifferentialD;}{\&DifferentialD; t} \mathrm{\&delta;}\&plus;\mathrm{P\_\_0}$

The horizontal force (in the direction of slider motion) is calculated as

$f\&equals;\frac{\mathrm{fs}}{\tan \left(\mathrm{\&theta;}\right)}\&plus;\mathrm{\&mu;}\mathrm{fs}\tanh \left(\mathrm{k\_\_\&mu;}\mathrm{v\_\_rel}\right)$

where

$\mathrm{v\_\_rel}\&equals;\frac{\&DifferentialD;}{\&DifferentialD; t} \mathrm{s\_\_rel}$

Finally the detent force is applied to the base (side a) and the slider (side b)

$\mathrm{f\_\_a}\&equals;-f$

$\mathrm{f\_\_b}\&equals;f$

Note: The radius of the ball has to satisfy the condition below to ensure proper contact with both sides of the groove:

$r<\min \left(\frac{\mathrm{w\_\_a}-\mathrm{sign}\left(\mathrm{\&alpha;\_\_a}-\mathrm{\&alpha;\_\_b}\right) \mathrm{x\_\_0}}{\cos \left(\mathrm{\&alpha;\_\_a}\right)}\&comma;\frac{\mathrm{w\_\_b}\&plus;\mathrm{sign}\left(\mathrm{\&alpha;\_\_a}-\mathrm{\&alpha;\_\_b}\right) \mathrm{x\_\_0}}{\cos \left(\mathrm{\&alpha;\_\_b}\right)}\right)$