Library / Structural dynamics and vibration testing / SDY-40
Structural dynamics and vibration testing
Rayleigh stiffness coefficient from two target frequencies
Stiffness-proportional Rayleigh coefficient β that gives chosen damping ratios at two chosen frequencies.
- Ref
- SDY-40
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- β [s]
The stiffness coefficient is β = 2(ζ₂ω₂ − ζ₁ω₁)/(ω₂² − ω₁²). For equal targets this is β = 2ζ/(ω₁ + ω₂). Together with α it gives exactly the two target damping ratios at the two target frequencies and a smooth curve between and beyond them. The stiffness term raises damping at high frequency, and can over-damp modes above ω₂.
Stiffness-proportional Rayleigh coefficient β that gives chosen damping ratios at two chosen frequencies. It is the second row of the two-frequency calibration.
Calculator
- Mass coefficient α0.8 1/s
- Damping at twice the upper target frequency0.085
- Frequency of minimum damping, √(α/β)20 rad/s
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 two targets give β = 0.002 s; at twice the upper frequency the damping is already 8.5 %.
- A stiffness coefficient that is too large damps high modes heavily and may force a smaller time step in explicit analysis.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ζ2 Damping ratio at ω₂ | 0.005 to 0.2 | −4e-4 | 0.01 | 0.0104 | +10% | +13.3 % |
| ω2 Upper target angular frequency | 15 to 200 rad/s | 4.762e-4 | 0.004 | 0.003524 | +10% | −7.41 % |
| ζ1 Damping ratio at ω₁ | 0.005 to 0.2 | 0 | 0.0026 | 0.0026 | +10% | −3.33 % |
| ω1 Lower target angular frequency | 2 to 35 rad/s | 0.001333 | 0.002381 | 0.001048 | +10% | −1.96 % |
Sweep of Damping ratio at ω₂
| Damping ratio at ω₂ | Stiffness coefficient (s) |
|---|---|
| 0.005 | −4e-4 |
| 0.02938 | 9e-4 |
| 0.05375 | 0.0022 |
| 0.07813 | 0.0035 |
| 0.1025 | 0.0048 |
| 0.1269 | 0.0061 |
| 0.1512 | 0.0074 |
| 0.1756 | 0.0087 |
| 0.2 | 0.01 |
Sweep one variable
Variables and how each one contributes
Stiffness coefficient
What the equation gives you. Everything else feeds it.
Lower target angular frequency
The lower target angular frequency, in rad/s. Changes where the damping is pinned down. Raising it by 10 % (to 11) moves the result by −1.96 %, from 0.002 to 0.001961 s.
Damping ratio at ω₁
The target damping ratio at the lower frequency. Raises the coefficient that sets the low-frequency damping. Raising it by 10 % (to 0.055) moves the result by −3.33 %, from 0.002 to 0.001933 s.
Upper target angular frequency
The upper target angular frequency, in rad/s. It must be larger than the lower one. Changes where the damping is pinned down. Raising it by 10 % (to 44) moves the result by −7.41 %, from 0.002 to 0.001852 s.
Damping ratio at ω₂
The target damping ratio at the upper frequency. Raises the coefficient that sets the high-frequency damping. Raising it by 10 % (to 0.055) moves the result by +13.3 %, from 0.002 to 0.002267 s.
Worked example with the values above
- Numerator: 2(ζ₂ω₂ − ζ₁ω₁) = 2 × (2 − 0.5) = 3.
- Denominator: ω₂² − ω₁² = 1,500.
- β = 0.002 s.
Assumptions and limits
- Classical damping in the form C = αM + βK, with α and β constant.
- The two target frequencies are distinct and positive, and the damping between them is accepted to dip below the targets.
Common mistakes
- Assuming Rayleigh damping holds the target ratio across the band: it dips between the two targets and rises sharply outside them.
- Mixing α in 1/s with a frequency in Hz.
- Choosing target frequencies that do not bracket the modes that matter.
Statistics and uncertainty
The calibration is exact for the two chosen points, but the damping in between and outside them is a model result and not a measurement. Check it against measured modal damping across the band.
Where it is used
- Setting damping in transient and modal finite-element analyses from two measured damping ratios.
- Checking the predicted damping of every mode in a band after calibration.
Maths you need
- Damping matrix: C in M ü + C u̇ + K u = F
- Modal damping ratio: damping of a single mode as a fraction of critical