Question: I have a friction damping system which is exited by a harmonic force FE (depicted on the left side).
Is there a way to convert the friction damper to a linear or nonlinear damper, such that the damping at a given excitement frequency is equal?

I am only considering sliding friction.

A reasonable approximation would be sufficient as well.
Any papers or articles on the topic would also be highly appreciated.

[![enter image description here](https://i.stack.imgur.com/YkzOK.png)](https://i.stack.imgur.com/YkzOK.png)

Answer: 
In general this is impossible, because for a given value of $F\_e$, the friction damper dissipates a fixed amount of energy per cycle of vibration *independent of the vibration frequency.*

This is (nonlinear) *hysteretic* damping, not (nonlinear) *viscous* damping.

The only way to approximate this with a viscous damper would be to make $C$ a function of both the $F\_e$ and the frequency $\omega$, which won't produce a useful equation of motion except in the special case where the machine only operates at one fixed frequency $\omega$.

Aside from that issue, a general way to make the approximation is to model one cycle of the stick-slip motion of the friction damper and find the energy dissipated during the cycle. Then choose $C$ to dissipate the same amount of energy.

For a simple slip-stick damper you can do this from first principles, though the details are messy, and you need the complete equation of motion of the system - you haven't specified how the mass and/or stiffness are connected to the damper.

A more general approach is to use the so-called Harmonic Balance Method to produce a numerical approximation. There are many variations on the basic idea (and many research papers describing them!) but one implementation is the NLVib function in Matlab.
The problem when trying to replace coulomb damping (friction) with viscous damping is that it produces a relatively constant force ($- \mu \cdot N$). (the sign is opposite to the sign of the velocity).

Viscous damping on the other hand is proportional to the velocity. $c\cdot \dot{x}$.

Therefore, its not always possible to substitute the one for the other.

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A way to represent the equation of motion is
$$m \ddot {x} + kx = -\mu \cdot m\cdot g \frac{|\dot{x}|}{\dot{x}}$$

The presence of the absolute value makes this DE difficult to solve analytically, so the most common solution is numerical.

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Regarding references:

* S. Graham Kelly - Mechanical Vibrations: Theory and Applications SI edition, has a section (3.7) devoted in coulomb damping. It discusses the formulation and also the solution.

Inman (Engineering Vibration) and Rao (mechanical vibrations) also have sections with coulomb damping.