Why is it that on a damped harmonic oscillator or a pendulum in a fluid, the Stokes drag law in the fluid only applied to small amplitude oscillation compared to large amplitudes oscillations?

Dose for a large amplitudes have different effect on the velocity compared to small amplituded?

note: I have recently come across some new information, about to why this is the case the issue is the paper I have found only give a brief outline on the reasoning, and I would like to have more literature to read. I have googled the method with little success and was wondering if anyone could maybe expand a bit more on what has been outlined in the paper.



If you have a masse which oscillate $x(t)=a\sin (\omega t)$ , the speed is $v(t)=\omega a\cos (\omega t)$and so the maximum is ${{v}_{\max }}=\omega a$

To apply Stokes law, a necessary condition is that the Reynolds number must be small compared to $1$ : $\operatorname{Re}=\frac{\mu {{v}_{\max }}d}{\eta }=\frac{\mu \omega ad}{\eta }\ll 1$ and so $a\ll \frac{\eta }{\mu \omega d}$


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