Continuous

SecondOrderSystem Block

Standard second-order transfer function: G(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²). Models damped oscillatory systems.

# SecondOrderSystem Block

Description

Standard second-order system transfer function:

G(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²)

This block models damped oscillatory systems commonly found in mechanical vibrations, electrical circuits, and control systems. The natural frequency and damping ratio provide intuitive tuning parameters.

Mathematical Model

The second-order system describes a linear time-invariant (LTI) system with characteristic equation:

s² + 2ζωₙs + ωₙ² = 0

Poles:

If ζ < 1 (underdamped): s = -ζωₙ ± jωₙ√(1-ζ²)
If ζ = 1 (critically damped): s = -ωₙ (repeated)
If ζ > 1 (overdamped): s = -ζωₙ ± ωₙ√(ζ²-1)

Parameters

naturalFrequency

Natural frequency in rad/s. Determines the undamped natural frequency of the system.

Type: number
Symbol: ωₙ
Range: 0.01 to 100000
Default: 10
Step: 0.1
Tooltip: Undamped natural frequency in rad/s

dampingRatio

Damping ratio that determines the nature of the system response.

Type: number
Symbol: ζ
Range: 0 to 10
Default: 0.7
Step: 0.05
Tooltip: Damping ratio: ζ < 1 underdamped, ζ = 1 critically damped, ζ > 1 overdamped

Behavior by damping ratio:

ζ < 1: Underdamped (oscillatory response with overshoot)
ζ = 1: Critically damped (fastest non-oscillatory response)
ζ > 1: Overdamped (slow response, no oscillation)

Remarks

Common Applications: Mass-spring-damper systems, RLC circuits, vehicle suspension
Step Response: Overshoot percentage = 100 × exp(-ζπ/√(1-ζ²))
Rise Time: Decreases with higher ωₙ
Settling Time: Determined by ζ × ωₙ