Library / Structural dynamics and vibration testing / SDY-45
Structural dynamics and vibration testing
One-sided windowed acceleration periodogram
One-sided power spectral density at bin k from the magnitude of the unnormalised transform of a windowed acceleration record, divided by the sample rate and the sum of the squared window values.
- Ref
- SDY-45
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Saa [(m/s²)²/Hz]
A one-sided power spectral density from a windowed record is Saa[k] = 2|Xw[k]|²/(fₛΣw²), where Xw is the unnormalised transform of the windowed acceleration. The factor 2 folds the negative-frequency half onto the positive half and applies only to bins that have a distinct negative partner: use 1 at DC and at Nyquist for an even transform length.
One-sided power spectral density at bin k from the magnitude of the unnormalised transform of a windowed acceleration record, divided by the sample rate and the sum of the squared window values.
Calculator
- Spectral density without the one-sided factor0.01667 (m/s²)²/Hz
- Density in dB re 1 (m/s²)²/Hz−14.77 dB
- Square root, amplitude spectral density0.1826 (m/s²)/√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 bin of magnitude 2 m/s² gives a density of 0.03333 (m/s²)²/Hz, or 0.1826 (m/s²)/√Hz.
- A single periodogram is a noisy estimate of the true PSD; average several segments (Welch’s method) to reduce its scatter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing ((m/s²)²/Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| |Xw| Magnitude of the windowed transform at the bin | 0.1 to 50 m/s² | 8.333e-5 | 20.83 | 20.83 | +10% | +21 % |
| fs Sample rate | 10 to 10,000 Hz | 2.667e-4 | 0.2667 | 0.2664 | +10% | −9.09 % |
| Σw2 Sum of the squared window samples | 0.5 to 3,000 | 3.333e-5 | 0.2 | 0.2 | +10% | −9.09 % |
Sweep of Magnitude of the windowed transform at the bin
| Magnitude of the windowed transform at the bin (m/s²) | Acceleration PSD ((m/s²)²/Hz) |
|---|---|
| 0.1 | 8.333e-5 |
| 6.337 | 0.3347 |
| 12.57 | 1.318 |
| 18.81 | 2.949 |
| 25.05 | 5.229 |
| 31.29 | 8.158 |
| 37.52 | 11.73 |
| 43.76 | 15.96 |
| 50 | 20.83 |
Sweep one variable
Variables and how each one contributes
Acceleration PSD
What the equation gives you. Everything else feeds it.
Magnitude of the windowed transform at the bin
The magnitude of the unnormalised transform of the windowed record at the bin, in m/s². Raises the PSD with its square. Raising it by 10 % (to 2.2) moves the result by +21 %, from 0.03333 to 0.04033 (m/s²)²/Hz.
Sample rate
The sample rate, in Hz: samples taken per second. Sets the frequency span and scales the bandwidth in proportion. Raising it by 10 % (to 88) moves the result by −9.09 %, from 0.03333 to 0.0303 (m/s²)²/Hz.
Sum of the squared window samples
The sum of the squared window samples, Σw² (dimensionless). For Hann it is 3N/8. A larger sum lowers the PSD. Raising it by 10 % (to 3.3) moves the result by −9.09 %, from 0.03333 to 0.0303 (m/s²)²/Hz.
Worked example with the values above
- Squared magnitude: |Xw|² = 2² = 4.
- Scaling: fₛΣw² = 80 × 3 = 240.
- S = 2 × 4 / 240 = 0.033333 (m/s²)²/Hz.
Assumptions and limits
- A uniformly sampled, stationary record, with the window and scaling conventions stated for each entry.
- A one-sided spectrum, with the factor 2 applied only to bins that have a distinct negative-frequency partner.
Common mistakes
- Forgetting to divide by the window’s noise bandwidth or power, which scales the spectrum wrongly.
- Applying the factor 2 at DC and at the Nyquist bin.
- Assuming that zero padding adds resolution: it only interpolates the spectrum.
- Treating overlapping segments as independent: they are correlated.
Statistics and uncertainty
Windowing and averaging trade resolution against variance and leakage. Record length sets the bin spacing, and the number of segments sets how much random error remains in the estimated spectrum.
Where it is used
- Setting the record length, window and averaging for a vibration test so that nearby modes can be told apart.
- Scaling windowed FFT output to correct amplitudes and spectral densities.
Maths you need
- Discrete Fourier transform: frequency content of a sampled record
- Window function: a taper applied to reduce spectral leakage
- Power spectral density: power per unit frequency