Library / Mechanical / MEC-06
Mechanical
Damped natural frequency
How damping slightly lowers the oscillation frequency.
- Ref
- MEC-06
- Rev
- v1
- Field
- Mechanical
- Result
- fd [Hz]
fd is the damped frequency in hertz, fn the undamped natural frequency and ζ the damping ratio (0 ≤ ζ < 1).
Damping drains energy from a vibration. For light damping the oscillation still happens at nearly the natural frequency; as damping approaches critical, the oscillation slows and stops.
Calculator
- Fall below the natural frequency2.02 %
- Amplitude ratio per cycle0.2773
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- For typical structures (ζ of 1–5%) the damped and undamped frequencies are practically identical.
- The oscillation disappears at ζ = 1 (critical damping), which is the design target for gun recoil and gauge pointers.
- The amplitude shrinks each cycle by e^(−2πζ/√(1−ζ²)), a quick way to estimate ζ from a decay record.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| fn Natural frequency | 0.1 to 100 Hz | 0.09798 | 97.98 | 97.88 | +10% | +10 % |
| ζ Damping ratio | 0.01 to 0.7 | 2.271 | 3.18 | 0.9089 | +undefined% | — |
Sweep of Damping ratio
| Damping ratio | Damped frequency (Hz) |
|---|---|
| 0.01 | 3.18 |
| 0.09625 | 3.165 |
| 0.1825 | 3.127 |
| 0.2687 | 3.063 |
| 0.355 | 2.973 |
| 0.4412 | 2.854 |
| 0.5275 | 2.702 |
| 0.6138 | 2.511 |
| 0.7 | 2.271 |
Sweep one variable
Variables and how each one contributes
Damped frequency
What the equation gives you. Everything else feeds it.
Natural frequency
Undamped natural frequency. Directly proportional.
Damping ratio
Damping as a fraction of critical damping (c/ccrit). Small for ζ below 0.3 (a 5% fall at ζ = 0.3), then drops quickly as ζ approaches 1.
Worked example with the values above
- 1 − ζ² = 1 − 0.2² = 0.96, and its square root is 0.9798.
- fd = fn × 0.9798 = 3.18 × 0.9798.
- fd = 3.116 Hz.
Assumptions and limits
- Linear viscous damping.
- Underdamped system (ζ below 1).
- One degree of freedom.
Common mistakes
- Using the formula when ζ ≥ 1; there is no oscillation.
- Confusing the damping ratio with the damping coefficient c.
- Entering ζ as a percentage.
Statistics and uncertainty
Damping ratios are hard to measure; 20–30% scatter is normal. Because ζ enters squared, a 30% error in a small ζ barely changes fd, but near ζ = 0.7 and above the frequency becomes sensitive to it.
Where it is used
- Gun recoil mechanisms and vehicle shock absorbers.
- Structural vibration tests.
- Instrument and sensor response.
Maths you need
- Natural frequency: undamped oscillation rate
- Square roots: ζ is under a root