Library / Structural dynamics and vibration testing / SDY-49
Structural dynamics and vibration testing
Mean square over a log–log PSD segment
Mean square acceleration in a band where the spectrum follows the straight-line power law S = S₁(f/f₁)^b on log–log axes, as in a standard test specification.
- Ref
- SDY-49
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- I [(m/s²)²]
In a band where the spectrum follows the straight line S = S₁(f/f₁)b on log–log axes, as in a standard test specification, the mean square is I = S₁f₁[(f₂/f₁)b+1 − 1]/(b + 1). At b = −1 the limit is S₁f₁ ln(f₂/f₁). A slope of 3 dB per octave corresponds to b ≈ 1 and −3 dB per octave to b ≈ −1. Take the square root of I for the RMS.
Mean square acceleration in a band where the spectrum follows the straight-line power law S = S₁(f/f₁)^b on log–log axes, as in a standard test specification. At b = −1 the limit is S₁f₁ ln(f₂/f₁). Take the square root for the RMS.
Calculator
- RMS acceleration, √I0.7447 m/s²
- PSD at the upper frequency, S₁(f₂/f₁)^b0.01 (m/s²)²/Hz
- Slope in dB per octave, 10 b log₁₀2−3.01 dB/oct
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 segment carries a mean square of 0.5545 (m/s²)², an RMS of 0.7447 m/s².
- A steeper positive slope shifts the power to the top of the band: the same S₁ with a slope of +1 would give 3 (m/s²)².
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing ((m/s²)²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| b Log–log slope | −3 to 3 | 0.1875 | 25.5 | 25.31 | +0.12 | +8.8 % |
| S1 PSD at f₁ | 0.001 to 0.2 (m/s²)²/Hz | 0.01386 | 2.773 | 2.759 | +10% | +10 % |
| f2 Upper frequency | 11 to 500 Hz | 0.03812 | 1.565 | 1.527 | +10% | +6.88 % |
| f1 Lower frequency | 2 to 35 Hz | 0.1869 | 0.5883 | 0.4013 | +10% | +2.44 % |
Sweep of Log–log slope
| Log–log slope | Mean square acceleration ((m/s²)²) |
|---|---|
| −3 | 0.1875 |
| −2.25 | 0.2634 |
| −1.5 | 0.4 |
| −0.75 | 0.6627 |
| 0 | 1.2 |
| 0.75 | 2.357 |
| 1.5 | 4.96 |
| 2.25 | 11.02 |
| 3 | 25.5 |
Sweep one variable
Variables and how each one contributes
Mean square acceleration
What the equation gives you. Everything else feeds it.
PSD at f₁
The PSD at the lower band edge, in (m/s²)²/Hz. Raises the mean square in direct proportion. Raising it by 10 % (to 0.044) moves the result by +10 %, from 0.5545 to 0.61 (m/s²)².
Lower frequency
The lower band edge, in Hz, where the PSD equals S₁. Raises the mean square in direct proportion to the band edge, at fixed slope. Raising it by 10 % (to 11) moves the result by +2.44 %, from 0.5545 to 0.568 (m/s²)².
Upper frequency
The upper band edge, in Hz. Extends the band and adds area when the slope is above −1. Raising it by 10 % (to 44) moves the result by +6.88 %, from 0.5545 to 0.5926 (m/s²)².
Log–log slope
The log–log slope of the segment: 3 dB per octave is about 1, −3 dB per octave is about −1 and flat is 0. Steeper positive slopes put more power at the top of the band. Raising it by 0.5 (to −0.5) moves the result by +44.3 %, from 0.5545 to 0.8 (m/s²)².
Worked example with the values above
- Band ratio: f₂/f₁ = 40 / 10 = 4; exponent b + 1 = 0.
- Slope b = −1 uses the logarithmic limit: ln(4) = 1.3863.
- I = 0.04 × 10 × that = 0.55452 (m/s²)².
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