2nd-Order Laplace Transfer Function H(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²) Poles & Overshoot Calculator

Analyze 2nd-order Laplace poles, damped frequency ω_d, % Overshoot, and 2% settling time.

Step Response Percentage Overshoot (%OS)
20.535 %
Damped Natural Frequency (ω_d) 10.716 rad/s
2% Settling Time (t_s = 4 / ζωₙ) 0.741 seconds
First Peak Time (t_p = π / ω_d) 0.293 seconds
Inputs
⚡ Scenarios:
rad/s
Undamped natural frequency ωₙ in rad/s.
ζ
Damping ratio ζ (0 < ζ < 1 underdamped, ζ = 1 critical, ζ > 1 overdamped).
Results & Graph
Step Response Percentage Overshoot (%OS)
20.535 %
Damped Natural Frequency (ω_d)
10.716 rad/s
2% Settling Time (t_s = 4 / ζωₙ)
0.741 seconds
First Peak Time (t_p = π / ω_d)
0.293 seconds

Step Response Percentage Overshoot (%OS)

Graph 1 • 2nd-Order Laplace Poles & ζ • Domain Architecture & Live Telemetry Maps Natural Frequency Rad Sec, Damping Ratio Zeta through 2nd-Order Laplace Poles & ζ Formula.
Step Response Percentage Overs: 20.535 % 2nd-Order Laplace Poles & ζ • Invariants & Metrics Step Response Percentage Ove 20.535 % Damped Natural Frequency (ω_ 10.716 rad/s 2% Settling Time (t_s = 4 / 0.741 seconds First Peak Time (t_p = π / ω 0.293 seconds
Graph 2 • Step Response Percentage Overshoot (%OS) Across Natural Frequency Rad Sec Sweep (40% → 160%) Evaluates ω_d = ωₙ·√(1 − ζ²); %OS = 100 × exp(−ζ·π / √(1 − ζ²)); SettlingTime_ across 7 operating points (current input highlighted).
0 % 11.29 % 22.59 % 20.54 4.80rad/s 20.54 7.20rad/s 20.54 9.60rad/s 20.54 12rad/s 20.54 14.40rad/s 20.54 16.80rad/s 20.54 19.20rad/s
Graph 3 • Joint Sensitivity: Natural Frequency Rad Sec × Damping Ratio Zeta Compares Step Response Percentage Overshoot (%OS) when Damping Ratio Zeta is shifted +25% (solid) vs. −25% (dashed).
0 % 17.83 % 35.66 % --- Damping Ratio Zeta @ 75% 11.80 -30% 11.80 -15% 11.80 +0% 11.80 +15% 11.80 +30%
Summary: With ωₙ=12 rad/s and ζ=0.45, s-plane poles sit at −5.4 ± j10.72 rad/s, producing 20.53% overshoot, peak time 0.293 s, and 2% settling time 0.741 s.
Summary: With ωₙ=12 rad/s and ζ=0.45, s-plane poles sit at −5.4 ± j10.72 rad/s, producing 20.53% overshoot, peak time 0.293 s, and 2% settling time 0.741 s.
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Formula & Mathematical Methodology: 2nd-Order Laplace Poles & ζ Formula

ωd = ωₙ·1 − ζ²
%OS = 100 × exp(−ζ·π / 1 − ζ²)
SettlingTime2% = 4 / (ζ·ωₙ)
ω_d = ωₙ·√(1 − ζ²); %OS = 100 × exp(−ζ·π / √(1 − ζ²)); SettlingTime_2% = 4 / (ζ·ωₙ)
  • naturalFrequencyRadSec: Natural Frequency Rad Sec (rad/s)
  • dampingRatioZeta: Damping Ratio Zeta (ζ)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: 2nd-Order Laplace Poles & ζ

Analyze 2nd-order Laplace poles, damped frequency ω_d, % Overshoot, and 2% settling time.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: ω_d = ωₙ·√(1 − ζ²); %OS = 100 × exp(−ζ·π / √(1 − ζ²)); SettlingTime_2% = 4 / (ζ·ωₙ).
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes 2nd-Order Laplace Poles & ζ via ω_d = ωₙ·√(1 − ζ²); %OS = 100 × exp(−ζ·π / √(1 − ζ²)); SettlingTime_2% = 4 / (ζ·ωₙ).

Frequently Asked Questions

How does the 2nd-Order Laplace Transfer Function H(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²) Poles & Overshoot Calculator compute the solution?

It evaluates ω_d = ωₙ·√(1 − ζ²); %OS = 100 × exp(−ζ·π / √(1 − ζ²)); SettlingTime_2% = 4 / (ζ·ωₙ) directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Numerical Analysis & Calculus

2nd-Order Laplace Transfer Function H(s) = ωₙ²/(s² + 2ζωₙs + ωₙ²) Poles & Overshoot Calculator: inputs, method, and how to read the result

Analyze 2nd-order Laplace poles, damped frequency ω_d, % Overshoot, and 2% settling time.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Natural Frequency Rad Sec: Undamped natural frequency ωₙ in rad/s.
  • Damping Ratio Zeta: Damping ratio ζ (0 < ζ < 1 underdamped, ζ = 1 critical, ζ > 1 overdamped).

What the calculation includes

With the default values shown, Step Response Percentage Overshoot (%OS) is 20.535 %. The calculation follows 2nd-Order Laplace Poles & ζ Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Damping Ratio Zeta from 0.45 ζ to 0.5 ζ changes Step Response Percentage Overshoot (%OS) from 20.535 % to 16.303 %. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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