# Friction and damping

Limit on friction force magnitude.

$F \le \mu N$

Variables $F$ and $N$ are the magnitude of the friction force $\vec{F}$ and normal force $\vec{N}$, and $\mu$ is the coefficient of friction.

## States of motion with frictional contact

For persistent contact with friction there are several different states that are possible, as follows.

Motion description
Sticking The available friction force is sufficient to prevent motion between the contact points. Thus, the contact points have zero relative velocity and acceleration, and the applied friction force magnitude might be less than the maximum available.
Transition The available friction force is on the border of allowing motion to occur. Thus, no motion is occurring and the applied friction force magnitude is the maximum available.
Slipping The contact points are moving relative to each other and the friction force is trying as hard as it can to stop the relative motion. Thus, there is non-zero relative motion between the contact points and the applied friction force magnitude is the maximum available.
About to slip The contact points currently have zero relative velocity, but the available friction force is not enough to prevent future motion. Thus, the contact points have non-zero relative acceleration and the applied friction force magnitude is the maximum available.

The descriptions of the different friction motion states can be written mathematically as follows.

Possible states of motion with friction.

relative motion $\vec{F}$ magnitude $\vec{F}$ direction
Sticking $v_{Px} = 0$ and $a_{Px} = 0$ $F \le \mu N$ any direction
Transition $v_{Px} = 0$ and $a_{Px} = 0$ $F = \mu N$ any direction
Slipping $v_{Px} \ne 0$ $F = \mu N$ opposes $\vec{v}_P$
About to slip $v_{Px} = 0$ and $a_{Px} \ne 0$ $F = \mu N$ opposes $\vec{a}_P$

Point $P$ is the contact point, $x$ is the tangential contact direction, $\vec{F}$ is the friction force, $\vec{N}$ is the normal force, and $\mu$ is the coefficient of friction.

## Analysis method for systems that might start to slip

For a system that currently has zero relative contact velocity, we might not know whether it will stick or start to slip. To determine this, we can check the following cases in any order. Exactly one of the checks will pass, and that is the motion state that will physically occur.

Assume Check
Sticking $v_{Px} = 0$ and $a_{Px} = 0$ $F \le \mu N$
About to slip $F = \mu N$ $\vec{F}$ opposes $\vec{a}_P$

Depending on the geometry, the “About to slip” case may need to be split into multiple cases, one for each possible direction of $\vec{F}$. For example, in 2D we might need to test two cases: (1) $F = \mu N$ to the left, and (2) $F = \mu N$ to the right. Both of these have the same condition to check of the force direction opposing the acceleration direction.