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Solving ODE with Coulomb friction

Solving ODE with Coulomb friction

Solving ODE with Coulomb friction


I would like to solve an ODE rapresenting a SDOF spring-mass-damper system with Coulomb friction. The system is subjected to base acceleration ÿ, therefore the equation of motion is the following

mü + cú + ku = -mÿ - µdN sign(ú)

where µd is the dynamic friction coefficient, N the normal force and u is the relative coordinate defined as
u = x-y.
First question, is it ok for the sign() function to depend on ú or should it depend on the absolute velocity of the mass?
The previous equation holds when the mass is moving. But what I would like to model is the whole system behaviour, thus including also the stick-slip behaviour.
I was thinking to model the stick-slip condition by using events but I do not know to define the triggering quantity to move from sticking to slipping.

If I am correct, the motion should start when

mü > µsN

where µs is the static friction. How to define this condition using events?

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