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Physics · 术语表

什么是 Canonical second-order form？

Definition 7.8 University Physics — Year 1 · 第 7 章 — Transient Regimes: First and Second Order

A quantity $x(t)$ is a second-order linear system when

\frac{\dd^2 x}{\dd t^2} + \frac{\omega_0}{Q}\,\frac{\dd x}{\dd t} + \omega_0^2\,x = \omega_0^2\,x_\infty ,

with $\omega_0 > 0$ the natural angular frequency, $Q > 0$ the quality factor (dimensionless), and $x_\infty$ the steady state. One also writes $\omega_0/Q = 2\xi\omega_0$ with $\xi = 1/2Q$ the damping ratio.

The two faces of the second-order system: the series RLC circuit (_0 = 1/√LC, Q = √L/C/R) and the mass–spring–damper (_0 = √k/m, Q = √km/). Same equation, same three regimes. — The two faces of the second-order system: the series RLC circuit ( $\omega_0 = 1/\sqrt{LC}$ , $Q = \sqrt{L/C}/R$ ) and the mass–spring–damper ( $\omega_0 = \sqrt{k/m}$ , $Q = \sqrt{km}/\alpha$ ). Same equation, same three regimes.

Step response of a second-order system from rest, for three quality factors. Small Q: a slow creep; Q = 1/2: the quickest return without overshoot; large Q: ringing that lasts about Q oscillations. — Step response of a second-order system from rest, for three quality factors. Small $Q$ : a slow creep; $Q = \tfrac12$ : the quickest return without overshoot; large $Q$ : ringing that lasts about $Q$ oscillations.

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