Library / Structural dynamics and vibration testing / SDY-33
Structural dynamics and vibration testing
Magnitude-squared coherence
Fraction of the output power at one frequency that is linearly related to the input, between 0 and 1.
- Ref
- SDY-33
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- γ2
Coherence γ² = |Syx|²/(SxxSyy) is the fraction of the output power at one frequency that is linearly related to the input. It lies between 0 and 1. A value near 1 means the measurement is clean; lower values point to noise, leakage, nonlinearity or another source. A single unaveraged record gives a value of exactly 1 wherever it is defined, so average several records.
Fraction of the output power at one frequency that is linearly related to the input, between 0 and 1. A value near one means the response is caused by the measured input.
Calculator
- Share of the output not related to the input, 1 − γ²0.1111
- Ratio of H1 to H2 magnitude (equals γ²)0.8889
- Cross-spectrum magnitude squared8 (m·N/Hz)²
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 coherence of 0.889 means about 88.9 % of the output power is explained by the input at this frequency.
- Coherence above about 0.9 is generally taken as good for frequency response work; dips at anti-resonances are normal because the output is small there.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Syy Output autospectrum | 0.5 to 20 m²/Hz | 0.4 | 16 | 15.6 | +10% | −9.09 % |
| Sxx Input autospectrum | 0.1 to 5 N²/Hz | 0.1778 | 8.889 | 8.711 | +10% | −9.09 % |
| Re Syx Real part of the cross-spectrum | −5 to 5 m·N/Hz | 0.4444 | 3.222 | 2.778 | +0.2 m·N/Hz | +10.5 % |
| Im Syx Imaginary part of the cross-spectrum | −5 to 5 m·N/Hz | 0.4444 | 3.222 | 2.778 | +0.2 m·N/Hz | +10.5 % |
Sweep of Output autospectrum
| Output autospectrum (m²/Hz) | Coherence |
|---|---|
| 0.5 | 16 |
| 2.938 | 2.723 |
| 5.375 | 1.488 |
| 7.813 | 1.024 |
| 10.25 | 0.7805 |
| 12.69 | 0.6305 |
| 15.13 | 0.5289 |
| 17.56 | 0.4555 |
| 20 | 0.4 |
Sweep one variable
Variables and how each one contributes
Coherence
What the equation gives you. Everything else feeds it.
Real part of the cross-spectrum
The real part of the averaged cross-spectrum of output against input at one frequency, in m·N/Hz. Raises the magnitude of the estimate. Raising it by 0.5 (to 2.5) moves the result by +28.1 %, from 0.8889 to 1.139.
Imaginary part of the cross-spectrum
The imaginary part of the cross-spectrum, in m·N/Hz. It sets the phase of the estimate. Raises the magnitude of the estimate. Raising it by 0.5 (to 2.5) moves the result by +28.1 %, from 0.8889 to 1.139.
Input autospectrum
The averaged input autospectrum, in N²/Hz. It must be positive. A larger input spectrum lowers the estimate. Raising it by 10 % (to 1.1) moves the result by −9.09 %, from 0.8889 to 0.8081.
Output autospectrum
The averaged output autospectrum, in m²/Hz. More output power raises the H2 estimate and lowers the coherence for the same cross-spectrum. Raising it by 10 % (to 9.9) moves the result by −9.09 %, from 0.8889 to 0.8081.
Worked example with the values above
- Cross-spectrum magnitude squared: 2² + 2² = 8.
- Product of autospectra: 1 × 9 = 9.
- γ² = 0.88889.
Assumptions and limits
- Averaged spectra from the same records, windows and frequency bin, with a stationary, linear system.
- At least several averages are used, because a single unaveraged record gives a coherence of one at every frequency.
Common mistakes
- Using H1 where the force channel is noisy, which biases the peak low, or H2 where the output is noisy.
- Reading frequency response values where the coherence is low.
- Forgetting the 90° and 180° phase changes when converting between receptance, mobility and accelerance.
Statistics and uncertainty
Spectral estimates carry random error that falls as the square root of the number of averages and bias that depends on leakage and noise. Quote the number of averages, and read the coherence before relying on any frequency response value.
Where it is used
- Estimating frequency response functions from impact hammer and shaker tests.
- Converting between displacement, velocity and acceleration responses for modal fitting and coupling.
Maths you need
- Cross-spectrum: the averaged product of one signal and the conjugate of another
- Complex numbers: real and imaginary parts, magnitude and phase
- Frequency response function: output per unit input at each frequency