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.
I think a hydraulic device could approximate it.

General concept: For each direction: a spring-loaded check valve (ie with a cracking pressure) in series with a small-diameter-tube would realize a roughly constant pressure difference, as long as there is some very small flow (i.e. cylinder movement). This could then operate a pilot valve, opening the larger main valve, allowing much more flow once the pre-set amount of pressure (i.e. force on the cylinder) is reached.

[![response](https://i.stack.imgur.com/nYqdX.jpg)](https://i.stack.imgur.com/nYqdX.jpg)

(it wouldn't be this neat in a real implementation)