Damping Ratio Calculator Formula

Understand the math behind the damping ratio calculator. Each variable explained with a worked example.

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Formulas Used

Damping Ratio (zeta)

zeta = abs(ln_os) / sqrt(pow(pi, 2) + pow(ln_os, 2))

Damping Type

classification = abs(ln_os) / sqrt(pow(pi, 2) + pow(ln_os, 2))

Variables

VariableDescriptionDefault
overshoot_pctMeasured Percent Overshoot(%)20
ln_osDerived value= log(overshoot_pct / 100)calculated

How It Works

Damping Ratio from Overshoot

The damping ratio determines the character of a second-order system response. It can be extracted from a measured step response by noting the percent overshoot.

Formula

zeta = ln(%OS/100) / sqrt(pi² + ln²(%OS/100))

  • zeta < 1: underdamped (oscillatory)
  • zeta = 1: critically damped (fastest non-oscillatory)
  • zeta > 1: overdamped (sluggish, no oscillations)

    • Worked Example

    A step response shows 20% overshoot.

    overshoot_pct = 20
    1. 01ln(0.20) = -1.6094
    2. 02zeta = 1.6094 / sqrt(9.8696 + 2.5902)
    3. 03zeta = 1.6094 / sqrt(12.4598) = 1.6094 / 3.5299
    4. 04zeta = 0.4559

    Frequently Asked Questions

    What damping ratio is considered optimal?

    For most control systems, zeta between 0.6 and 0.8 is considered good. This gives a fast response with acceptable overshoot (less than 10%). zeta = 0.707 gives the fastest response without resonance peak amplification.

    Can I measure damping from the decay of oscillations?

    Yes. The logarithmic decrement delta = ln(x_n / x_{n+1}) relates to damping as zeta = delta / sqrt(4*pi² + delta²). This method works well for lightly damped systems.

    What if there is no overshoot?

    If the step response has no overshoot, the system is critically damped (zeta = 1) or overdamped (zeta > 1). In this case, the overshoot method cannot be used directly.

    Learn More

    Guide

    Understanding Vibration Analysis: Natural Frequency, Damping, and Resonance

    Learn the fundamentals of mechanical vibration analysis. Covers natural frequency, damping ratio, resonance, forced vibration, vibration isolation, and practical measurement techniques.

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