Undamped Natural Frequency and Damping Ratio

Second Order Control System

For a second-order control system, the undamped natural frequency (ωₙ) and damping ratio (ζ) are usually calculated from the characteristic equation.

1. Standard Second-Order System Form

The standard form of a second-order system is:

s² + 2ζωₙ s + ωₙ² = 0

Where:

• ωₙ = Undamped natural frequency (rad/s)
• ζ = Damping ratio
• s = Laplace variable

2. How to Calculate from a Given Equation

If the characteristic equation is:

s² + a s + b = 0

Compare it with:

s² + 2ζωₙ s + ωₙ²

Then:

Undamped Natural Frequency

ωₙ = √b

Damping Ratio

ζ = a / (2√b)

3. Example

Given:

s² + 6s + 25 = 0

Compare with standard form.

Step 1: Find ωₙ

ωₙ = √25 = 5 rad/s

Step 2: Find ζ

ζ = 6 / (2 × 5)

ζ = 0.6

4. Final Answer

• Undamped Natural Frequency (ωₙ) = 5 rad/s
• Damping Ratio (ζ) = 0.6

5. Shortcut Formula

ωₙ = √(constant term)

ζ = coefficient of s / (2√(constant term))