Library / Structural dynamics and vibration testing / SDY-51
Structural dynamics and vibration testing
Miles estimate of resonant RMS acceleration
Approximate RMS absolute acceleration of a lightly damped single-degree-of-freedom system on a base that vibrates with a broadband, locally flat acceleration PSD, from the natural frequency, the quality factor Q = 1/(2ζ) and the base PSD at that frequency.
- Ref
- SDY-51
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- arms [m/s²]
The Miles equation estimates the RMS absolute acceleration of a lightly damped single-degree-of-freedom system on a base with a broadband, locally flat acceleration PSD: arms ≈ √[(π/2) fₙ Q Sb(fₙ)], with Q = 1/(2ζ). It is a resonance approximation that ignores the small contribution from the damping term in the numerator, so it is best for Q above about 10.
Approximate RMS absolute acceleration of a lightly damped single-degree-of-freedom system on a base that vibrates with a broadband, locally flat acceleration PSD, from the natural frequency, the quality factor Q = 1/(2ζ) and the base PSD at that frequency.
Calculator
- Mean square, (π/2) fₙ Q Sb39.27 (m/s²)²
- RMS in g0.639 g
- Damping ratio, 1/(2Q)0.02
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 resonance at 50 Hz with Q = 25 on a base PSD of 0.02 (m/s²)²/Hz responds with 6.267 m/s² RMS (0.639 g).
- Doubling Q raises the response only by √2: a sharper resonance amplifies but not in proportion.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Sb Base acceleration PSD at fₙ | 1e-4 to 1 (m/s²)²/Hz | 0.4431 | 44.31 | 43.87 | +10% | +4.88 % |
| fn Natural frequency | 5 to 2,000 Hz | 1.982 | 39.63 | 37.65 | +10% | +4.88 % |
| Q Quality factor, 1/(2ζ) | 1 to 200 | 1.253 | 17.72 | 16.47 | +10% | +4.88 % |
Sweep of Quality factor, 1/(2ζ)
| Quality factor, 1/(2ζ) | RMS acceleration (m/s²) |
|---|---|
| 1 | 1.253 |
| 25.88 | 6.375 |
| 50.75 | 8.928 |
| 75.63 | 10.9 |
| 100.5 | 12.56 |
| 125.4 | 14.03 |
| 150.3 | 15.36 |
| 175.1 | 16.59 |
| 200 | 17.72 |
Sweep one variable
Variables and how each one contributes
RMS acceleration
What the equation gives you. Everything else feeds it.
Natural frequency
The natural frequency of the resonance, in Hz. Raises the response with the square root of the frequency. Raising it by 10 % (to 55) moves the result by +4.88 %, from 6.267 to 6.572 m/s².
Quality factor, 1/(2ζ)
The quality factor, 1/(2ζ), a plain number. A higher Q means a sharper resonance. Raises the response with the square root of Q. Raising it by 10 % (to 27.5) moves the result by +4.88 %, from 6.267 to 6.572 m/s².
Base acceleration PSD at fₙ
The base acceleration PSD at the natural frequency, in (m/s²)²/Hz, assumed flat nearby. Raises the response with the square root of the PSD. Raising it by 10 % (to 0.022) moves the result by +4.88 %, from 6.267 to 6.572 m/s².
Worked example with the values above
- Product: (π/2) × 50 × 25 × 0.02 = 39.27.
- Take the square root.
- arms = 6.2666 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