Library / Structural dynamics and vibration testing / SDY-05
Structural dynamics and vibration testing
Damping from half-power frequencies
Damping ratio of a lightly damped, well-separated resonance from the width of its peak at the half-power points, where the magnitude has fallen to 1/√2 of the peak.
- Ref
- SDY-05
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- ζ
For a lightly damped, well-separated resonance the width of the receptance peak between the half-power points, where the magnitude is 1/√2 of the peak, is about 2ζfₙ. So ζ ≈ (f₂ − f₁)/(2fₙ). The frequency resolution must be fine enough to resolve the width, and the method fails if neighbouring modes overlap.
Damping ratio of a lightly damped, well-separated resonance from the width of its peak at the half-power points, where the magnitude has fallen to 1/√2 of the peak.
Calculator
- Bandwidth, f₂ − f₁2 Hz
- Quality factor, 1/(2ζ)25
- Resolution needed for about ten points across the bandwidth0.2 Hz
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.
- A peak 2 Hz wide at 50 Hz gives a damping ratio of 0.02 and a quality factor of 25.
- Unless the bins are much finer than 2 Hz, the half-power points cannot be located and the estimate is biased.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| f1 Lower half-power frequency | 40 to 50.9 Hz | 0.001 | 0.11 | 0.109 | +10% | — |
| f2 Upper half-power frequency | 49.1 to 60 Hz | 0.001 | 0.11 | 0.109 | +10% | +255 % |
| fn Undamped frequency estimate | 10 to 200 Hz | 0.005 | 0.1 | 0.095 | +10% | −9.09 % |
Sweep of Upper half-power frequency
| Upper half-power frequency (Hz) | Damping ratio |
|---|---|
| 49.1 | 0.001 |
| 50.46 | 0.01462 |
| 51.83 | 0.02825 |
| 53.19 | 0.04188 |
| 54.55 | 0.0555 |
| 55.91 | 0.06913 |
| 57.27 | 0.08275 |
| 58.64 | 0.09638 |
| 60 | 0.11 |
Sweep one variable
Variables and how each one contributes
Damping ratio
What the equation gives you. Everything else feeds it.
Lower half-power frequency
The frequency below the peak, in Hz, where the response magnitude has fallen to 70.7 % of the peak. Lowering f₁ widens the peak and raises the damping estimate. Raising it by 10 % (to 53.9) gives no valid result at the other default values.
Upper half-power frequency
The frequency above the peak, in Hz, where the response magnitude has fallen to 70.7 % of the peak. Raising f₂ widens the peak and raises the damping estimate. Raising it by 10 % (to 56.1) moves the result by +255 %, from 0.02 to 0.071.
Undamped frequency estimate
The undamped natural frequency of the mode, in Hz, usually taken as the peak frequency. A higher centre frequency for the same bandwidth means less relative damping. Raising it by 10 % (to 55) moves the result by −9.09 %, from 0.02 to 0.01818.
Worked example with the values above
- Bandwidth: f₂ − f₁ = 51 − 49 = 2 Hz.
- Twice the centre frequency: 2fₙ = 100 Hz.
- ζ = 2 / 100 = 0.02.
Assumptions and limits
- A single, well-separated mode dominates the record and the damping is viscous (proportional to velocity) and light.
- Peaks or half-power points are read from clean data with enough resolution.
Common mistakes
- Mixing peak heights with peak-to-peak values, or taking peaks from opposite sides of the zero line.
- Using a record that contains several modes, so the envelope is not a single exponential.
- Using the half-power width when the frequency resolution is too coarse to resolve it.
- Using an angular frequency in rad/s where Hz is expected, or the reverse.
Statistics and uncertainty
Decrements read from only one pair of peaks carry the reading error of both. Averaging over several cycles, or fitting an exponential to the whole envelope, cuts the scatter. Damping is itself an inexact, amplitude-dependent property, so quote the amplitude at which it was measured.
Where it is used
- Identifying damping from an impact or ring-down test on a structure, instrument or component.
- Setting the damping values used in finite-element and modal models.
Maths you need
- Natural logarithm: the inverse of e to a power
- Damping ratio: damping as a fraction of critical
- Free decay: the envelope of an unforced vibration