Second-Order Step Peak Time
Estimate the time to the first peak after a standard second-order step input.
Try this formula
Calculate time to the first response peak or choose another supported quantity.
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Try this formula
Calculate time to the first response peak or choose another supported quantity.
Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.
Time to the first response peak
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. Use the standard zero-free second-order step response with zero initial conditions.
Difficulty
Intermediate
Read time
5 min
Prerequisites
Basic algebra and introductory dynamic systems
Source
Donney Fan — MECH 466 — Automatic ControlReference equation: Second Order System — Peak Time. FormuLab wrote new explanations, assumptions, applications, and numerical examples.
Definition
The first peak occurs after half a damped oscillation. Additional poles, zeros, or nonzero initial conditions can shift that time, so the formula belongs to the standard second-order response model.
Variables & Units
| Symbol | Meaning | Units |
|---|---|---|
| Time to the first response peak | ||
| Undamped angular natural frequency | ||
| Damping ratio |
Real-World Applications
Step response
Estimate the first peak in a standard underdamped response.
Transient comparison
Compare response speed between model parameters.
Worked Example
Problem
For ωn = 10 rad/s and ζ = 0.2, find the first peak time.
Given
Solution
Final Answer
Contents
Info
Tools
Source & review
Reference equation: Second Order System — Peak 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.
- • Use the standard zero-free second-order step response with zero initial conditions.
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