# Growth time of static friction

I have a doubt about the best model of friction to use ina dynamic simulation. The most popular model, the Coulomb-Morin's friction, is defined by:

$F =\cases{ -F_d \cdot sign(v) & with$v \neq 0$and$F_d$known real constant\\ F_s & with$v = 0$and$F_s \in \mathbb{R} \; \big\vert \; -F_{s,max} \leq F_s \leq F_{s,max}$}$

This model leads to a instantaneous growth of the static friction: at $v=0$, the force assumes immediately one value in the interval $[-F_{s,max},F_{s,max}]$. This fact causes me some doubts: is the Coulomb's model the most "realistic" one to model the static friction? I know that there are several other models, but the way they deal with the static friction is nearly the same: a discontinuity at $v=0$ or a very steep growth. The finite growth time is often introduced for computational purposes, since this allows to obtain a continuous model. This finite growth time is to consider:

• only a trick to make the function suitable for computer implementation;
• a way to model an experimentally observed effect.

I think that, in the real world, the value of friction should rise very rapidly but with a finite growth time; however, I am not enough expert in tribology to make conclusions.

Thank you in advance for the help.

The actual behavior of forces that may be categorized as friction are very messy indeed whether they are static, dynamic, or a combination of the two. There are no models that suitably allow one to obtain a 'deeper' understanding of friction, only models that partially meet practical needs, in an approximate matter, and mostly they have no basis in fundamental principles. They can however be useful in predicting behavior of complex systems as long as they behave closely to what matters for the application.

For example, the Dahl Friction model has been successfully used for years by aerospace engineers to predict how bearings respond in a closed loop control system.

Paraphrasing George Box: all models are wrong, however some are useful.