Library / Structural dynamics and vibration testing / SDY-48
Structural dynamics and vibration testing
Band-limited RMS from PSD samples
RMS acceleration over a band from three spectral density samples, using the trapezoidal rule between them.
- Ref
- SDY-48
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- arms [m/s²]
The mean square acceleration is the area under the PSD, and the RMS is its square root. With samples at three frequencies the trapezoidal rule gives a_rms ≈ √Σ[(Si + Si+1)/2](fi+1 − fi). It assumes linear interpolation of the PSD between the points; for a log–log specification use the power-law segment entry instead. For band-edge clipping insert the interpolated boundary values.
RMS acceleration over a band from three spectral density samples, using the trapezoidal rule between them. The area under the spectrum is the mean square. This entry uses three points.
Calculator
- Mean square, area under the PSD0.8 (m/s²)²
- Bandwidth, f₃ − f₁30 Hz
- RMS in g, divided by 9.806650.09121 g
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.
- The band 10 to 40 Hz carries a mean square of 0.8 (m/s²)² and so an RMS of 0.8944 m/s² (0.0912 g).
- The upper segment is wider, so the PSD value at the third point matters most to the total.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f3 Frequency 3 | 21 to 200 Hz | 0.4796 | 2.366 | 1.887 | +10% | +7.24 % |
| S2 PSD at frequency 2 | 0 to 0.2 (m/s²)²/Hz | 0.5916 | 1.83 | 1.239 | +10% | +2.77 % |
| S3 PSD at frequency 3 | 0 to 0.2 (m/s²)²/Hz | 0.7071 | 1.581 | 0.874 | +10% | +1.86 % |
| S1 PSD at frequency 1 | 0 to 0.2 (m/s²)²/Hz | 0.866 | 1.323 | 0.4569 | +10% | +0.312 % |
| f1 Frequency 1 | 1 to 19 Hz | 0.7874 | 0.9899 | 0.2025 | +10% | −1.26 % |
| f2 Frequency 2 | 11 to 39 Hz | 0.781 | 0.9434 | 0.1624 | +10% | −1.26 % |
Sweep of PSD at frequency 3
| PSD at frequency 3 ((m/s²)²/Hz) | RMS acceleration (m/s²) |
|---|---|
| 0 | 0.7071 |
| 0.025 | 0.866 |
| 0.05 | 1 |
| 0.075 | 1.118 |
| 0.1 | 1.225 |
| 0.125 | 1.323 |
| 0.15 | 1.414 |
| 0.175 | 1.5 |
| 0.2 | 1.581 |
Sweep one variable
Variables and how each one contributes
RMS acceleration
What the equation gives you. Everything else feeds it.
Frequency 1
The first frequency, in Hz. Moves the lower edge of the band. Raising it by 10 % (to 11) moves the result by −1.26 %, from 0.8944 to 0.8832 m/s².
Frequency 2
The second frequency, in Hz. Moves the middle or upper edge of the band. Raising it by 10 % (to 22) moves the result by −1.26 %, from 0.8944 to 0.8832 m/s².
Frequency 3
The third frequency, in Hz: the upper edge of the band. Widens the band and adds area. Raising it by 10 % (to 44) moves the result by +7.24 %, from 0.8944 to 0.9592 m/s².
PSD at frequency 1
The PSD at the first frequency, in (m/s²)²/Hz. Adds to the area under the first segment. Raising it by 10 % (to 0.011) moves the result by +0.312 %, from 0.8944 to 0.8972 m/s².
PSD at frequency 2
The PSD at the second frequency, in (m/s²)²/Hz. Adds to the area under both segments. Raising it by 10 % (to 0.033) moves the result by +2.77 %, from 0.8944 to 0.9192 m/s².
PSD at frequency 3
The PSD at the third frequency, in (m/s²)²/Hz. Adds to the area under the last segment. Raising it by 10 % (to 0.033) moves the result by +1.86 %, from 0.8944 to 0.911 m/s².
Worked example with the values above
- First segment: (0.01 + 0.03)/2 × (20 − 10) = 0.2.
- Second segment: (0.03 + 0.03)/2 × (40 − 20) = 0.6.
- arms = √(0.8) = 0.89443 m/s².
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