二阶系统标准形式
Φ ( s ) = ω n 2 s 2 + 2 ζ ω n s + ω n 2 \Phi (s) = \frac{\omega_n^2}{s^2+2\zeta \omega_n s+\omega_n^2} Φ(s)=s2+2ζωns+ωn2ωn2
极点为：
s = − ζ ω n ± ω n ζ 2 − 1 s=-\zeta \omega_n \pm \omega_n\sqrt{\zeta^2-1} s=−ζωn±ωnζ2−1
当 0 < ζ < 1 0<\zeta<1 0<ζ<1时，称为二阶欠阻尼系统。此时系统极点为一对共轭负根
s = − ζ ω n ± j ω n 1 − ζ 2 s=-\zeta \omega_n \pm j\omega_n\sqrt{1-\zeta^2} s=−ζωn±jωn1−ζ2
对应的大致冲击响应为 具体过程暂无
ϕ ( t ) = e − a t s i n ( ω t + β ) \phi(t)=e^{-at}sin(\omega t + \beta) ϕ(t)=e−atsin(ωt+β)
可见呈阻尼震荡的运动模态
现在分析，在输入 r ( t ) r(t) r(t)为阶跃信号 ε ( t ) \varepsilon (t) ε(t)时，二阶欠阻尼系统的输出信号 c ( t ) c(t) c(t)
C ( s ) = R ( s ) ⋅ Φ ( s ) = 1 s ⋅ ω n 2 s 2 + 2 ζ ω n s + ω n 2 \begin{aligned} C(s) &=R(s)\cdot \Phi(s) \\ &=\frac 1s \cdot \frac{\omega_n^2}{s^2+2\zeta \omega_n s+\omega_n^2} \end{aligned} C(s)=R(s)⋅Φ(s)=s1⋅s2+2ζωns+ωn2ωn2
将系统传递函数分母配方
Φ ( s ) = ω n 2 ( s + ζ ω n ) 2 + ω n 2 ( 1 − ζ 2 ) \Phi (s) = \frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)} Φ(s)=(s+ζωn)2+ωn2(1−ζ2)ωn2
记 ω n 2 ( 1 − ζ 2 ) \omega_n^2(1-\zeta^2) ωn2(1−ζ2)为 ω d 2 \omega_d^2 ωd2 ， C ( s ) C(s) C(s)部分分式分解
b 1 s + a 1 s + b 2 ( s + ζ ω n ) 2 + ω d 2 + b 3 ( s + ζ ω n ) 2 + ω d 2 \frac {b_1}s+\frac{a_1s+b_2}{(s+\zeta\omega_n)^2+\omega_d^2}+\frac{b_3}{(s+\zeta\omega_n)^2+\omega_d^2} sb1+(s+ζωn)2+ωd2a1s+b2+(s+ζωn)2+ωd2