📈

2nd Order System

Continuous

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:

H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}

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)

Inputs & Outputs

Direction	ID	Label	Type	Status
→ In	`in`	Input	number	Required
← Out	`out`	Output	number	Output

Parameters

Parameters	Label	Type	Default	Description
`gain`	Gain (K)	number	`1`	DC Gain of the system
`naturalFrequency`	Frequency	number	`10`	Undamped natural frequency
`frequencyUnit`	Frequency Unit	select	`rad/s`
`parameterMode`	Damping Parameter	select	`dampingRatio`
`dampingRatio`	Damping Ratio ζ	number	`0.7`	Damping ratio: ζ < 1 underdamped, ζ = 1 critically damped, ζ > 1 overdamped
`qFactor`	Q-Factor	number	`0.707`	Quality factor Q = 1 / (2ζ)
`initialCondition`	Initial Condition [y, dy]	vector	`0,0`

Usage Examples

Zie de ingebouwde voorbeelden in de handleiding voor een demonstratie van dit blok.

Remarks & Best Practices

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

Related Components

TransferFunction: General continuous transfer function
LeadLag: First-order lead-lag compensator
PIDController: For closed-loop control of second-order systems