Library / Structural dynamics and vibration testing / SDY-34
Structural dynamics and vibration testing
Mobility from a measured receptance
Velocity per unit force obtained from a displacement-per-force frequency response by multiplying by iω.
- Ref
- SDY-34
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- |Y| [(mm/s)/N]
Mobility is velocity per unit force. Because velocity is the time derivative of displacement, Y = iωH under an exp(+iωt) convention: the magnitude is ω|H| and the phase advances by 90°. Here the receptance H is in mm/N, so the mobility is in (mm/s)/N. The derived values give the real and imaginary parts.
Velocity per unit force obtained from a displacement-per-force frequency response by multiplying by iω. This entry returns the magnitude; the phase advances by 90°.
Calculator
- Real part of Y20 (mm/s)/N
- Imaginary part of Y40 (mm/s)/N
- Phase of Y, phase of H plus 90°63.43 °
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.
- At 20 rad/s the mobility is 44.72 (mm/s)/N.
- Each differentiation raises the weight of high frequencies, so mobility looks flatter than receptance across a spectrum.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing ((mm/s)/N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ω Angular frequency | 1 to 200 rad/s | 2.236 | 447.2 | 445 | +10% | +10 % |
| Re H Real part of the receptance | −5 to 5 mm/N | 20 | 102 | 81.98 | +0.2 mm/N | +8.07 % |
| Im H Imaginary part of the receptance | −5 to 5 mm/N | 40 | 107.7 | 67.7 | +0.2 mm/N | −3.67 % |
Sweep of Angular frequency
| Angular frequency (rad/s) | Mobility magnitude ((mm/s)/N) |
|---|---|
| 1 | 2.236 |
| 25.88 | 57.86 |
| 50.75 | 113.5 |
| 75.63 | 169.1 |
| 100.5 | 224.7 |
| 125.4 | 280.3 |
| 150.3 | 336 |
| 175.1 | 391.6 |
| 200 | 447.2 |
Sweep one variable
Variables and how each one contributes
Mobility magnitude
What the equation gives you. Everything else feeds it.
Angular frequency
The angular frequency, in rad/s (2π times the frequency in Hz). Scales the converted magnitude in proportion (mobility) or with its square (accelerance). Raising it by 10 % (to 22) moves the result by +10 %, from 44.72 to 49.19 (mm/s)/N.
Real part of the receptance
The real part of the receptance, in mm/N: displacement per unit force. Raises the magnitude of the converted response. Raising it by 0.5 (to 2.5) moves the result by +20.4 %, from 44.72 to 53.85 (mm/s)/N.
Imaginary part of the receptance
The imaginary part of the receptance, in mm/N. Raises the magnitude of the converted response. Raising it by 0.5 (to −0.5) moves the result by −7.8 %, from 44.72 to 41.23 (mm/s)/N.
Worked example with the values above
- Receptance magnitude: |H| = √(2² + −1²) = 2.2361 mm/N.
- Multiply by ω = 20 rad/s.
- |Y| = 44.721 (mm/s)/N.
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