Exploring the dynamics of a simple system involving Coulomb friction

The work presented here consists in an exhaustive study of a simple mass-spring system involving Coulomb friction. The aim was to gain some insight into the behaviour of a chain of masses in frictionnal contact. If it is simple to explicit the analytical solution of a single mass system, the analytical solution for a two mass system is already far more complicated. We thus consider two masses linked by a spring in bilateral contact with Coulomb friction and submitted to an external force applied onto one of the masses. The existence, uniqueness and regularity of the dynamics of the system is established through a recent paper [

1

]. Once the uniqueness is ensured it is simple to exhibit the explicit solution for certain values of the external force (i.e. when the amplitude of the force is either small or large). When the amplitude of the external force belongs to a certain intermediate range the dynamics turns out to be more interesting. The solution can be calculated analytically, however as the computation becomes rapidly tiresome, we use a symbolic calculus tool to compute a solution corresponding to a given external force.

We thus observe that:

for a given value of the amplitude of the external force, the two masses oscillate for a certain time before coming to rest (these oscillations are obviously not periodic because of the non linearity due to the Coulomb friction),

for increasing values of the amplitude of the external force, the number of oscillations that the masses carry out before coming to a halt, grows each time that the amplitude of the external force passes through a critical value,

these critical values of the amplitude of the external force accumulate as the amplitude of the force reaches the upper bound of the interesting range.</lt>

Extending these results to more than two masses may prove to be not such a simple task but some progress has been made and shall be presented.