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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))

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