= Oscillator parameters from static and resonant displacement
For a lightly damped <forced harmonic oscillator> with known force amplitude $F_0$, measure the static displacement $x_s$, the resonance <angular frequency> $\omega_r$ and the peak displacement $x_p$. The response is $|x_0|=F_0/\sqrt{(k-m\omega^2)^2+b^2\omega^2}$. To leading order at large <quality factor>,
$$
k\simeq F_0/x_s,\qquad m\simeq k/\omega_r^2,\qquad b\simeq F_0/(\omega_r x_p),\qquad Q\simeq x_p/x_s.
$$
The exact displacement resonance is $\omega_r^2=k/m-b^2/(2m^2)$, whereas the undamped natural <angular frequency> is $\omega_0=\sqrt{k/m}$. A lightly damped half-power bandwidth obeys $\Delta\omega\simeq b/m$ and $Q\simeq\omega_0/\Delta\omega$. Half power corresponds to displacement $x_p/\sqrt2$, not $x_p/2$. Graph readings should be reported as estimates rather than extra significant digits from the model inversion.
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