免费超调量计算器
计算控制系统阶跃响应的超调百分比。
Percent Overshoot
37.23 %
Percent Overshoot vs Damping Ratio (zeta)
公式
## Second-Order Overshoot Overshoot is the amount by which the step response exceeds the final steady-state value, expressed as a percentage. It depends only on the damping ratio. ### Formula **%OS = 100 × exp(-pi × zeta / sqrt(1 - zeta²))** **Peak time: tp = pi / omega_d**, where omega_d = omega_n × sqrt(1 - zeta²) is the damped natural frequency. For zeta = 0 (undamped), overshoot is 100%. For zeta >= 1 (critically damped or overdamped), there is no overshoot.
计算示例
A system with zeta = 0.3, omega_n = 10 rad/s.
- 01omega_d = 10 × sqrt(1 - 0.09) = 10 × 0.9539 = 9.539 rad/s
- 02%OS = 100 × exp(-pi × 0.3 / 0.9539)
- 03%OS = 100 × exp(-0.9874) = 100 × 0.3730 = 37.30%
- 04Peak time = pi / 9.539 = 0.3293 s
常见问题
What damping ratio gives 5% overshoot?
Setting %OS = 5 and solving: zeta = -ln(0.05) / sqrt(pi² + ln²(0.05)) = 2.996 / sqrt(9.870 + 8.976) = 2.996 / 4.340 = 0.690. So zeta ≈ 0.69 gives about 5% overshoot.
Can overshoot be eliminated?
Yes, by making zeta >= 1 (critically or over-damped). However, this makes the system slower. In practice, designers often accept 5-10% overshoot for faster response (zeta ≈ 0.6-0.8).
Is overshoot always bad?
Not always. Some systems (like position servos) may tolerate small overshoot if it means faster response. But for processes like temperature control or chemical dosing, any overshoot can be harmful or dangerous.
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