Damped natural frequency
ω_d = ω_n√(1−ζ²)
Valid for an underdamped second-order system.
Second-Order System Calculator
Calculate standard second-order response metrics from natural frequency and damping ratio.
Access: Free to use, no installation, and No account required.
Inputs
Rise time and peak time are reported for the underdamped case where those closed-form relationships apply. Settling-time formulas are standard approximations.
Results
Second-order response relationships
For 0 < ζ < 1, the damped natural frequency is ωd = ωn√(1−ζ²). Percent overshoot follows the standard exponential damping relationship, while the approximate 2% settling time is Ts ≈ 4/(ζωn).
Model boundary
These metrics describe the canonical second-order model. Real systems with zeros, higher-order poles, nonlinearities, saturation, delays, discrete sampling, or noisy feedback require a more complete simulation.
Engineering reference
Second-Order System Calculator: background and worked detail
Use the Second-Order System Calculator to estimate damping class, damped natural frequency, percent overshoot, peak time, rise time, and approximate 2% and 5% settling times for the canonical second-order system.
Standard second-order form
Percent overshoot
M_p = 100e^(−πζ/√(1−ζ²))
The standard unit-step overshoot relation for 0 < ζ < 1.
2% settling estimate
T_s ≈ 4/(ζω_n)
A common envelope approximation for underdamped responses.
How damping ratio shapes the response
Many control-system transients can be approximated by a dominant second-order denominator. Natural frequency sets the time scale, while damping ratio controls oscillation and overshoot. For 0 < ζ < 1, the poles form a complex-conjugate pair.
The familiar overshoot and settling-time formulas are model-specific approximations. Added zeros, higher-order poles, saturation, dead time, sampling, nonlinearities, actuator limits, and noise can make the real response substantially different.
ωn = 5 rad/s, ζ = 0.7
Enter natural frequency 5 rad/s and damping ratio 0.7.
- The damped natural frequency is about 3.57 rad/s.
- Percent overshoot is about 4.60%.
- Peak time is about 0.880 s and the 2% settling-time estimate is about 1.14 s.
Result: This is a well-damped underdamped response often used as a reference point for transient design.
Validation checks
Critical and undamped cases bracket the behaviour and are checked directly.
Critical damping
Case: Set ζ = 1.
Expected: The system should classify as critically damped and report zero overshoot.
Higher natural frequency
Case: Double ωn while holding ζ fixed.
Expected: Characteristic times such as peak and settling time should approximately halve.
Assumptions and model boundaries
Assumptions
- The system is represented by the normalized canonical second-order form.
- Transient metrics refer to a unit-step-like response.
- The standard closed-form underdamped relationships are used where applicable.
Limitations
- Does not account for zeros, additional poles, delay, nonlinearities, saturation, sampling, or actuator limits.
- Settling-time formulas are approximations and should be validated against the complete system response.
Second-Order System Calculator FAQ
What damping ratio gives no overshoot?
For the ideal second-order model, ζ ≥ 1 gives no oscillatory overshoot. ζ = 1 is critically damped.
Is 0.7 damping ratio always best?
No. Around 0.7 is a common compromise between speed and overshoot, but the correct target depends on bandwidth, robustness, noise, actuator limits, and system requirements.
Can I tune a PID controller from this page?
This page characterizes a second-order response. Use the full Control Systems Workbench or simulator to design and tune feedback controllers.
Where this calculation comes from
Shared with the Control System Simulator, which simulates the full closed loop rather than the standard-form response alone.
Open the source workbench →Read calculation and source methodology →