The simplest and most common model of friction is called linear friction, sometimes also called viscous friction or viscous damping.
Then, the force or the torque is proportional to velocity. So, there are two cases. We can consider a linear case where we have friction force proportional to velocity, v, or we can have the rotational case where a frictional torque, tau, is proportional to the rotational speed, omega. If we plot force against velocity, we can represent the characteristic of friction by a straight line of slope B and refer to B as the friction coefficient.
Typically, the slope of the line for positive and negative velocities is the same. Here is the model for a current controlled electric motor that we introduced earlier. It relates the rotation acceleration to the applied current.
And, I’m going to add a frictional torque term here. And, friction is a real issue in most electric motors. The friction is due to the bearings that the shaft rotates within. And, for motors with brushes this represents the frictional force of the brushes rubbing on the motor’s Commutator.
I can rearrange the equation to look like this and then I can transform it and put it into transfer function form. So, here we have the transfer function for an electric motor and it incorporates the motor torque constant, the inertia of the motor and the friction of the motor. Some systems exhibit a nonlinear friction and the most classic case of this is known as Coulomb friction and this is named after a famous French physicist who also did a lot of work in electrostatics and the unit of electric charge is named after this same gentleman.
Like the linear friction we looked at before, the nonlinear friction force always opposes the direction of motion.
Interestingly, Coulomb friction is not defined for the case of zero velocity. We define the amount of friction when the mechanism is moving in the positive direction and when the mechanism is moving in the negative direction, and these values may be the same or they may be different. It depends on the mechanism.
For a real machine, friction is actually a combination of both these two effects, the nonlinear Coulomb friction and the linear friction. Once again, this friction curve is not defined for the case of zero velocity. These two terms may be the same or different and the slope of the line for positive and negative velocities may be the same or it may be different. Once again, it depends on the particular machine.
All mechanical systems exhibit friction and we learn about two broad classes of friction: linear and non-linear.
High school mathematics
This content assumes an understanding of high school-level mathematics, e.g. trigonometry, algebra, calculus, physics (optics) and some knowledge/experience of programming (any language).