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: 
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.

---

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.

---

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.
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)