= Damped torsional oscillation
{title2=$I\ddot\theta+b\dot\theta+\kappa\theta=0$}
A <damped torsional oscillation> is rotation about a fiber's fixed reference orientation with a restoring <torque> and dissipative <torque>. In the ideal viscous model, <damping> contributes $-b\dot\theta$ with constant $b\geq0$. This is a rotational <damped harmonic oscillator>. For known constant <moment of inertia> $I$ and <torsion constant> $\kappa$, define $\gamma=b/(2I)$. In the oscillatory regime $\gamma^2<\kappa/I$,
$$
\theta(t)=A e^{-\gamma t}\cos(\omega_dt+\varphi),\qquad \omega_d^2=\frac\kappa I-\gamma^2,\qquad T_d=\frac{2\pi}{\omega_d}.
$$
$A$ and $\varphi$ are set by the initial state. Successive positive peaks measured from the same <equilibrium> have magnitudes $p_n$ and $p_{n+1}$; their ratio gives $\delta=\ln(p_n/p_{n+1})=\gamma T_d$. Consequently a measured damped <oscillation period> and decay together recover
$$
\boxed{\kappa=I\left[\left(\frac{2\pi}{T_d}\right)^2+\gamma^2\right]=\frac I{T_d^2}(4\pi^2+\delta^2).}
$$
Using only $4\pi^2I/T_d^2$ underestimates the <torsion constant> in this model. The toy curve in <torsion constant> sets $I=2.0\times10^{-6}\,\mathrm{kg\,m^2}$, $\kappa=1.0\times10^{-8}\,\mathrm{N\,m/rad}$, and $\gamma=0.020\,\mathrm{s^{-1}}$. It gives $T_0\simeq88.86\,\mathrm s$, $T_d\simeq92.64\,\mathrm s$, and $\delta\simeq1.853$. Ignoring the decay recovers $9.2\times10^{-9}\,\mathrm{N\,m/rad}$, an $8\%$ underestimate; including it recovers the stipulated <torsion constant>.
These equations assume a single rotational mode, a stationary zero, <linear elasticity>, and viscous <damping>. Internal fiber losses need not behave as a constant viscous coefficient: Fleischer and colleagues model their precision apparatus using a loss angle. Their response is evidence for specifying a damping model, not for applying the viscous correction universally. https://arxiv.org/pdf/2206.02890[Fleischer et al. (2022), §I, equation (1): thermal torque noise and internal damping represented by a loss angle]{ref}
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