Second-Order System Calculator
A second-order system with ζ = 0.5 and ωₙ = 4 rad/s overshoots by 16.3034 %, peaks at 0.907 s and settles within 2 % in about 2 s.
16.303353482158 %
Full answer, then cut to fewer decimals (not rounded):
- 16.303353482158 %
- 16.30335348215 %
- 16.3033534821 %
- 16.303353482 %
- 16.30335348 %
- 16.3033534 %
- 16.303353 %
- 16.30335 %
- 16.3033 %
- 16.303 %
- 16.30 %
- 16.3 %
- Damping ratio ζ
- 0.5
- Natural frequency ωₙ
- 4 rad/s
- Damped frequency ω_d = ωₙ√(1 − ζ²)
- 3.464101615 rad/s
- Peak time tₚ = π/ω_d
- 0.9068996821 s
- Rise time t_r = (π − θ)/ω_d
- 0.6045997881 s
- Settling time, 2 % (4/ζωₙ)
- 2 s
- Settling time, 5 % (3/ζωₙ)
- 1.5 s
- Poles
- −2 ± j3.4641
- Response
- underdamped
Step-by-step solution
- Mₚ = e^(−ζπ/√(1 − ζ²)) = e^(−1.813799364) = 0.1630335348
- tₚ = π ÷ ω_d = 0.9068996821 s; t_s(2 %) = 4 ÷ (ζωₙ) = 2 s
Second-Order System Formula
Mₚ = e^(−ζπ/√(1−ζ²)), tₚ = π/ω_d
ω_d = ωₙ√(1 − ζ²), t_r = (π − arccos ζ)/ω_d, t_s ≈ 4/(ζωₙ) (2 %)
Worked Example
A system with ζ = 0.5 and ωₙ = 4 rad/s (s² + 4s + 16):
- Mₚ = e^(−ζπ/√(1 − ζ²)) = e^(−1.813799364) = 0.1630335348
- tₚ = π ÷ ω_d = 0.9068996821 s; t_s(2 %) = 4 ÷ (ζωₙ) = 2 s
Natural frequency ωₙ: 4 rad/s · Damped frequency ω_d = ωₙ√(1 − ζ²): 3.464101615 rad/s · Peak time tₚ = π/ω_d: 0.9068996821 s · Rise time t_r = (π − θ)/ω_d: 0.6045997881 s · Settling time, 2 % (4/ζωₙ): 2 s · Settling time, 5 % (3/ζωₙ): 1.5 s · Poles: −2 ± j3.4641 · Response: underdamped
GATE tip
Overshoot depends on ζ alone: 0.4 → 25.4 %, 0.5 → 16.3 %, 0.6 → 9.5 %, 0.707 → 4.3 %, 0.8 → 1.5 %. Memorise a few and you can work backwards from a stated overshoot.
More Control Systems Calculators
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Routh-Hurwitz Calculator
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number of sign changes in the first column = number of roots in the right half-plane
Example: The characteristic equation s⁴ + 2s³ + 3s² + 4s + 5 = 0: Result: Unstable: 2 roots in the right half-plane.
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Steady-State Error Calculator
e_ss = A/(1 + Kₚ), A/K_v, A/Kₐ
Kₚ = lim G(s), K_v = lim sG(s), Kₐ = lim s²G(s) as s → 0
Example: G(s) = 10 / [s(s + 1)(s + 5)] with a unit ramp input: Steady-state error (ramp) e_ss = 0.5.
Frequently asked questions
How do I read ζ and ωₙ from a transfer function?
Write the denominator as s² + 2ζωₙs + ωₙ². For s² + 4s + 16: ωₙ = 4 and 2ζωₙ = 4, so ζ = 0.5.
Why are settling times approximate?
t_s = 4/(ζωₙ) uses the decaying envelope e^(−ζωₙt) and ignores the oscillation, so the true time can be a little shorter.
Sources
- The International System of Units (SI), 9th edition (SI Brochure), Bureau International des Poids et Mesures .
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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Scientific Converters. (2026, October 15). Second-Order System Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/second-order-system-calculator/
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