Library / Structural dynamics and vibration testing / SDY-47
Structural dynamics and vibration testing
Response PSD from a single input PSD
Power spectral density of a response from the squared magnitude of the frequency response and the input power spectral density, for a linear time-invariant system with one stationary input.
- Ref
- SDY-47
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- Syy [(m/s²)²/Hz]
For a linear, time-invariant system with one stationary input and no extra noise, the response PSD is Syy = |H|²Sxx: the input PSD multiplied by the squared magnitude of the frequency response. Here H is the accelerance in (m/s²)/N and Sxx the one-sided force PSD in N²/Hz, giving an acceleration PSD. The phase of H does not matter.
Power spectral density of a response from the squared magnitude of the frequency response and the input power spectral density, for a linear time-invariant system with one stationary input.
Calculator
- Magnitude of the accelerance, |H|0.02236 (m/s²)/N
- Gain in dB, 20 log₁₀|H|−33.01 dB
- Acceleration spectral density, √Syy0.04472 (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 force PSD of 4 N²/Hz through this response gives an acceleration PSD of 0.002 (m/s²)²/Hz.
- The result is quadratic in the response magnitude: halving |H| quarters the PSD.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing ((m/s²)²/Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Re H Real part of the accelerance | −0.1 to 0.1 (m/s²)/N | 4e-4 | 0.0404 | 0.04 | +0.01 (m/s²)/N | +100 % |
| Im H Imaginary part of the accelerance | −0.1 to 0.1 (m/s²)/N | 0.0016 | 0.0416 | 0.04 | +0.01 (m/s²)/N | −20 % |
| Sxx One-sided force PSD | 0.5 to 20 N²/Hz | 2.5e-4 | 0.01 | 0.00975 | +10% | +10 % |
Sweep of One-sided force PSD
| One-sided force PSD (N²/Hz) | Response PSD ((m/s²)²/Hz) |
|---|---|
| 0.5 | 2.5e-4 |
| 2.938 | 0.001469 |
| 5.375 | 0.002688 |
| 7.813 | 0.003906 |
| 10.25 | 0.005125 |
| 12.69 | 0.006344 |
| 15.13 | 0.007562 |
| 17.56 | 0.008781 |
| 20 | 0.01 |
Sweep one variable
Variables and how each one contributes
Response PSD
What the equation gives you. Everything else feeds it.
Real part of the accelerance
The real part of the accelerance, in (m/s²)/N. Raises the response PSD with its square. Raising it by 0.01 (to 0.03) moves the result by +100 %, from 0.002 to 0.004 (m/s²)²/Hz.
Imaginary part of the accelerance
The imaginary part of the accelerance, in (m/s²)/N. Raises the response PSD with its square. Raising it by 0.01 (to 0) moves the result by −20 %, from 0.002 to 0.0016 (m/s²)²/Hz.
One-sided force PSD
The one-sided force PSD, in N²/Hz. Raises the response PSD in direct proportion. Raising it by 10 % (to 4.4) moves the result by +10 %, from 0.002 to 0.0022 (m/s²)²/Hz.
Worked example with the values above
- Squared magnitude: |H|² = 0.02² + −0.01² = 5×10−4 ((m/s²)/N)².
- Multiply by the force PSD 4 N²/Hz.
- Sᵧᵧ = 0.002 (m/s²)²/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