Reaction wheels are used in many spacecraft for attitude control. A reaction wheel is a momentum exchange device because it controls the spacecraft by exchanging momentum with the rest of the spacecraft. Momentum is exchanged via a motor that is fixed to the spacecraft. As with all rotating parts it is subject to friction. Friction needs to be modeled as part of the design process.
The standard way to model friction is with three terms. One is damping which is proportional to wheel speed. The faster the wheel spins the more friction torque is produced. Ultimately, this limits the net control torque. At some speed the motor is just balancing the friction torque. The second component is Coulomb friction that is constant but flips signs when the wheel speed changes sign. the third is static friction. It is like Coulomb friction but only exists at zero speed.
An alternative friction model is known as the bristle friction model. This models friction as bristles that bend. It also has the same friction components described above but they are applied though the bristle state.
Both models can be made to produce similar results as shown in the following figure.
The static friction is clearly seen. The wheel speeds are nearly the same. The middle plot is of the bristle state. The problem with these models is when the torque is low and the wheel speed passes through zero. We then get limit cycling as shown below.
This is due to numerical error.
We can eliminate the limit cycling by using a very small integration time step with the bristle friction model. An alternative approach is to use the first model and multiply the sum of the static and Coulomb friction with a sigmoid, or s function.
The coefficient of the sigmoid function is beta. Very small betas remove the static friction, and all Coulomb friction, near zero speed. The large betas retain the form of the friction and eliminate the limit cycling!
These models can be found in the Spacecraft Control Toolbox 2015.1 . This particular script will be available in 2015.2 which will be released in July.