Library / Structural dynamics and vibration testing / SDY-50
Structural dynamics and vibration testing
Displacement PSD from acceleration PSD
Power spectral density of displacement from that of acceleration at the same coordinate: divide by (2πf)⁴.
- Ref
- SDY-50
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Sxx [mm²/Hz]
For motion at a single frequency, displacement is acceleration divided by ω² and so the PSD of displacement is that of acceleration divided by ω⁴: Sxx = Saa/(2πf)⁴. The division blows up as f falls, so low-frequency noise and drift dominate a double-integrated record. Exclude DC. Here the acceleration PSD is in (m/s²)²/Hz and the displacement PSD in mm²/Hz.
Power spectral density of displacement from that of acceleration at the same coordinate: divide by (2πf)⁴. It is not defined at zero frequency and magnifies low-frequency noise greatly.
Calculator
- Displacement spectral density, √Sₓₓ0.05066 mm/√Hz
- Division factor, (2πf)⁴1.559×107 (rad/s)⁴
- Velocity PSD, Saa/(2πf)²10.13 (mm/s)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.
- At 10 Hz an acceleration PSD of 0.04 (m/s²)²/Hz corresponds to a displacement PSD of 0.002566 mm²/Hz.
- Halving the frequency multiplies the displacement PSD by 16, which is why low-frequency noise ruins double-integrated acceleration.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm²/Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Frequency | 1 to 200 Hz | 1.604e-8 | 25.66 | 25.66 | +10% | −31.7 % |
| Saa Acceleration PSD at f | 0.001 to 1 (m/s²)²/Hz | 6.416e-5 | 0.06416 | 0.0641 | +10% | +10 % |
Sweep of Frequency
| Frequency (Hz) | Displacement PSD (mm²/Hz) |
|---|---|
| 1 | 25.66 |
| 25.88 | 5.726e-5 |
| 50.75 | 3.869e-6 |
| 75.63 | 7.847e-7 |
| 100.5 | 2.516e-7 |
| 125.4 | 1.039e-7 |
| 150.3 | 5.036e-8 |
| 175.1 | 2.729e-8 |
| 200 | 1.604e-8 |
Sweep one variable
Variables and how each one contributes
Displacement PSD
What the equation gives you. Everything else feeds it.
Acceleration PSD at f
The one-sided acceleration PSD at the frequency, in (m/s²)²/Hz. Raises the displacement PSD in direct proportion. Raising it by 10 % (to 0.044) moves the result by +10 %, from 0.002566 to 0.002823 mm²/Hz.
Frequency
The frequency, in Hz. It must be above zero. Lowers the displacement PSD with the fourth power of the frequency. Raising it by 10 % (to 11) moves the result by −31.7 %, from 0.002566 to 0.001753 mm²/Hz.
Worked example with the values above
- Angular frequency: 2πf = 2π × 10 = 62.832 rad/s; to the fourth: 1.5585×107.
- Divide the acceleration PSD 0.04 (m/s²)²/Hz by it, giving 2.5665×10−9 m²/Hz.
- Convert to mm²/Hz: Sₓₓ = 0.0025665 mm²/Hz.
Assumptions and limits
- A linear, time-invariant system and a stationary random input, with spectra that are one-sided and in consistent units.
- For the Miles estimate: a lightly damped single-degree-of-freedom system and a base PSD that is locally flat around the resonance.
Common mistakes
- Mixing one-sided and two-sided spectra, which differ by a factor of 2.
- Using the Miles equation for heavily damped systems or sharply varying input spectra.
- Dividing by (2πf)⁴ at very low frequency, where drift and sensor noise dominate.
- Treating RMS as a peak value: peaks in a Gaussian random signal are about three times larger.
Statistics and uncertainty
Spectral estimates carry random error that falls with the number of averages, and integrating them adds the errors of every band. Integrals are most sensitive to the frequencies where the spectrum is largest.
Where it is used
- Qualification and fatigue assessment under random vibration specifications.
- Estimating overall RMS, displacement and resonant response from measured or specified PSDs.
Maths you need
- Power spectral density: power per unit frequency
- Root mean square: the square root of the mean square
- Quality factor: Q = 1/(2ζ), the resonance sharpness