Second-Order Settling Time (2% Approximation)
Estimate settling time from the exponential decay rate of a standard second-order response.
Try this formula
Calculate approximate two-percent settling time or choose another supported quantity.
Show calculatorHide calculator
Try this formula
Calculate approximate two-percent settling time or choose another supported quantity.
Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.
Approximate two-percent settling time
Enter the known values to see the result.
Use a stable, linear, underdamped second-order model with 0 < ζ < 1. Natural frequency is angular frequency in rad/s, not cycles per second. This is the common engineering approximation for a two-percent settling band, not an exact last-crossing calculation. Use a standard second-order step response; check the actual response if extra poles or zeros are important.
Difficulty
Intermediate
Read time
5 min
Prerequisites
Basic algebra and introductory dynamic systems
Source
Donney Fan — MECH 466 — Automatic ControlReference equation: Second Order System — Settling Time. FormuLab wrote new explanations, assumptions, applications, and numerical examples.
Definition
The response envelope decays at a rate set by damping ratio times natural frequency. Four approximate decay time constants provide a useful two-percent settling estimate, but the exact last crossing depends on the oscillation.
Variables & Units
| Symbol | Meaning | Units |
|---|---|---|
| Approximate two-percent settling time | ||
| Damping ratio | ||
| Undamped angular natural frequency |
Real-World Applications
Transient estimates
Estimate a standard model’s settling time.
Model comparison
Compare decay speed while varying frequency or damping.
Worked Example
Problem
For ζ = 0.2 and ωn = 10 rad/s, estimate the two-percent settling time.
Given
Solution
Final Answer
Contents
Info
Tools
Source & review
Reference equation: Second Order System — Settling Time. FormuLab wrote new explanations, assumptions, applications, and numerical examples.
Assumptions
- • Use a stable, linear, underdamped second-order model with 0 < ζ < 1.
- • Natural frequency is angular frequency in rad/s, not cycles per second.
- • This is the common engineering approximation for a two-percent settling band, not an exact last-crossing calculation.
- • Use a standard second-order step response; check the actual response if extra poles or zeros are important.
Languages
FormuLab is a study reference. Verify formulas, units, assumptions, and course-specific conventions before relying on them.