\[ \delta=\frac{\omega_{\max}-\omega_{\min}}{\langle\omega\rangle_t},\qquad \langle\omega\rangle_t=\frac{1}{T}\int_0^T\omega\,\mathrm dt \]
\[ \delta_{\mathrm{Tredgold}}=\frac{2 k e r}{\bigl(J_O + \tfrac12 m e^2\bigr)\,\langle\omega\rangle_t^2} \]
Degree of irregularity
δ from trace vs δbook (textbook)
Geometry & inertia
0.45
0.12
0.80
0.080
Spring & motor
42
0.00
Initial state
0.00
8.00
Simulation
State
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$\dot\theta$ vs $\theta$ $[0,\,2\pi]$▼
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$\dot\theta$ vs $t$ on $[0,\,T]$▼
$T$ = first time $\theta$ advances by $2\pi$ on the trace; marker uses $t \bmod T$.
Energy vs $\theta$ $[0,\,2\pi]$▼
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$M_{\mathrm{res}}=-k e(r+e\sin\theta)\cos\theta$ vs $\theta$▼
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$\int_0^\theta M_{\mathrm{res}}\,\mathrm{d}\varphi$ vs $\theta$▼
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$f(\theta)=e\cos\theta\bigl(k r+\sin\theta\,(k e - m e\,\omega_{\mathrm{med}}^2)\bigr)$▼
scroll: zoom · drag: pan · dbl-click: reset — $\omega_{\mathrm{med}}$ = RK probe mean (phase dashed line)