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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

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  • $\begingroup$ Are you lookinh for a theoretically beautiful solution, or a quick and dirty real life thingy? $\endgroup$
    – Jpe61
    Oct 1 at 15:22
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    $\begingroup$ I'm looking for a theoretical solution but it does not have to be exact or beautiful ;) $\endgroup$
    – M. Martin
    Oct 1 at 16:05
  • $\begingroup$ Dang, I'm a hands on kinda guy, not much to give on the formulas & math... real life solutions do exist, such as the ones I mentioned in @petew's answer, and very "tunable" ones too (adjustable multi-bypass). Unfortunately I do not know how to describe them by means of formulae 😔 $\endgroup$
    – Jpe61
    Oct 1 at 17:29
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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.

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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.

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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

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

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    $\begingroup$ Theory makes this way more difficult than what you have shown here to be possible 😃 For more advaced version pls see multi-bypass shock absorbers used in offroad race cars like trophy trucks and alike. $\endgroup$
    – Jpe61
    Oct 1 at 15:27

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