Free Settling Time Calculator
Calculate the settling time of a second-order system from natural frequency and damping ratio.
Settling Time
0.7824 s
Settling Time vs Natural Frequency (omega_n)
Second-Order Settling Time
Settling time is the time required for the system response to remain within a specified percentage band (typically 2% or 5%) of the final value.
Formula
ts ≈ -ln(criterion) / (zeta × omega_n)
For the common 2% criterion: ts ≈ 4 / (zeta × omega_n). For 5%: ts ≈ 3 / (zeta × omega_n). The decay rate sigma = zeta × omega_n determines how quickly oscillations die out.
Example Calculation
A second-order system with omega_n = 10 rad/s, zeta = 0.5, 2% criterion.
- 01sigma = 0.5 × 10 = 5 s⁻¹
- 02ts = -ln(0.02) / 5 = 3.912 / 5 = 0.782 s
- 03Approximate: 4 / 5 = 0.800 s
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