# Coulomb's static friction in multidimensional case - decide which mass begins to move

Consider a system of N coupled oscillators, under the effect of elastic forces, damping, dynamic and static friction and an external force; for simplicity, let's suppose $N=3$. The friction model is Coulomb-Morin's friction:

$F =\cases{ -F_d \cdot sign(v) & with$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}$}$

The problem is how to determine the value of $F_s$. In the simple unidimensional case, the reasoning is: if exists a force, with satisfies $-F_{s,max} \leq F_s \leq F_{s,max}$ and is able to keep the mass at $v=0$, then that is the value of $F_s$. In the multidimensional case, I have this situation: at instant time $t=t_0$, we have $v_1(t_0) = v_2(t_0) = 0$. In order to keep the zero velocity, the requested forces are $F_1$ and $F_2$, with $F_1 > F_{s,max}$ and $F_2 > F_{s,max}$. Since these forces exceed the limit of static friction, at least one mass must change its velocity to a nonzero value.

Since the "growth time" of the friction force is zero (due to its discontinuity at $v=0$), none of the forces $F_i$ reaches the limit $F_{s,max}$ before the other; at time $t = t_0 - dt$ we have $F_1 < F_{s,max}$ and $F_2 < F_{s,max}$, while at $t = t_0$ we have $F_1 > F_{s,max}$ and $F_2 > F_{s,max}$.

My question is: which mass begins to move? (only mass 1, only mass 2 or both?) The problem is that there isn't a mass which reaches the "limit value" $F_{s,max}$ before the other, since the two forces change value immediately (discontinuous function). Without a finite growth time of the function, I have no idea how to "decide" which static friction ($F_1$ or $F_2$ "breaks" first. Should I consider a finite growth time for static friction?

I have already written the motion equations, in order to calculate the static friction forces required to keep the zero velocity at $t=t_0$.

• Hello, and welcome to Stack Exchange Physics! Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better. – Daniel Griscom Jan 15 '16 at 11:39
• this is not a homework; I have tried to solve the problem for days with several approaches; for example, I used a smooth approximation of Coulomb friction and solved the system of motion equations with a numerical algorithm. Now, I am trying to face the problem using a discontinuous model of friction; from my point of view, this is not a trivial homework-like question and the problem seems quite specific: how to determine the value of static friction in multidimensional case, considering a discontinuous friction model? – Christian Lorenz Jan 15 '16 at 12:49
• In my opinion, this question is asking about the problem of deciding which mass begins to move, and how one would go about deciding that. That falls under the "asking about a principle" rubric and makes this on topic. – Floris Jan 15 '16 at 14:09