Second-Order System Response

System Parameters

ζ — Damping Ratio 0.30

ωn — Natural Frequency (rad/s) 2.0

System Type

Underdamped (ζ < 1)

Underdamped — The system oscillates before settling. Common in lightly damped mechanical and electrical systems.

Presets

Transfer Function G(s) = ωn² / (s² + 2ζωn·s + ωn²)

Damped Frequency ωd = ωn√(1 − ζ²)

Rise Time

— s

Overshoot

— %

Peak Time

— s

Settling Time (2%)

— s

Damped Freq

— rad/s

Parameters & Equations

The second-order model and its closed-form step response for each damping case.

Model

Transfer function G(s) = ω_n² / (s² + 2ζω_ns + ω_n²) ζ — damping ratio · ω_n — natural frequency (rad/s).

Poles s_1,2 = −ζω_n ± ω_n√(ζ² − 1) Underdamped → complex pair · critically damped → double real · overdamped → two real.

Step Response (unit step)

Underdamped (ζ < 1) ω_d = ω_n√(1 − ζ²)
y(t) = 1 − (e^−ζω_nt/√(1 − ζ²)) · sin(ω_dt + φ) φ = atan2(ζ, √(1 − ζ²))

Critically damped (ζ = 1) y(t) = 1 − (1 + ω_nt)·e^−ω_nt

Overdamped (ζ > 1) y(t) = 1 + (s₁e^s₂t − s₂e^s₁t)/(s₂ − s₁)

Parameters

Symbol	Meaning	Default
ζ	Damping ratio (ζ<1 under, =1 critical, >1 over)	0.30
ω_n	Natural frequency (rad/s)	2.0
ω_d	Damped frequency = ω_n√(1−ζ²)	derived

Performance Metrics

Closed-form indicators (ζ < 1) Overshoot = 100·e^{−πζ/√(1−ζ²)} %
Peak time t_p = π / ω_d
Rise time t_r ≈ (π − φ) / ω_d
Settling (2%) t_s ≈ 4 / (ζω_n) Higher ζ → less overshoot but slower; higher ω_n → faster.