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... control system is: c(t) = 1 + 0.25e^-50t - 1.25e^-10t


Second Order Control System

Question:

What are the damping ratio (ζ) and undamped natural frequency (ωₙ) if the output of a control system is:
c(t) = 1 + 0.25e-50t - 1.25e-10t

1. Identify the Exponential Terms

The response contains two exponential terms:

  • e-50t
  • e-10t

These correspond to the system poles.

s₁ = -50
s₂ = -10

2. Form the Characteristic Equation

For poles s₁ and s₂:

(s - s₁)(s - s₂) = 0

Substitute the values:

(s + 50)(s + 10) = 0

Expanding:

s² + 60s + 500 = 0

3. Compare with Standard Second-Order Form

s² + 2ζωₙ s + ωₙ²

Comparing with:

s² + 60s + 500

We get:

2ζωₙ = 60
ωₙ² = 500

4. Calculate Undamped Natural Frequency

ωₙ = √500
ωₙ = 22.36 rad/s

5. Calculate Damping Ratio

ζ = 60 / (2 × 22.36)
ζ ≈ 1.34

Final Results

  • Undamped Natural Frequency (ωₙ) = 22.36 rad/s
  • Damping Ratio (ζ) = 1.34

Conclusion: Since ζ > 1, the system is overdamped (non-oscillatory response).

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