Library / Structural dynamics and vibration testing / SDY-39
Structural dynamics and vibration testing
Rayleigh mass coefficient from two target frequencies
Mass-proportional Rayleigh coefficient α that gives chosen damping ratios at two chosen frequencies.
- Ref
- SDY-39
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- α [1/s]
Requiring Rayleigh damping to give ζ₁ at ω₁ and ζ₂ at ω₂ leads to two linear equations in α and β. The mass coefficient is α = 2ω₁ω₂(ζ₁ω₂ − ζ₂ω₁)/(ω₂² − ω₁²). If the two target ratios are equal, α = 2ζω₁ω₂/(ω₁ + ω₂). The target frequencies must be distinct and positive.
Mass-proportional Rayleigh coefficient α that gives chosen damping ratios at two chosen frequencies. It is the first row of the two-frequency calibration.
Calculator
- Stiffness coefficient β0.002 s
- Damping at the geometric mean of the targets0.04
- 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.
- Targets of 5 % at 10 rad/s and 5 % at 40 rad/s give α = 0.8 1/s.
- With equal targets the damping falls below them between the two frequencies, to 4 % at the geometric mean.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (1/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ζ1 Damping ratio at ω₁ | 0.005 to 0.2 | −0.16 | 4 | 4.16 | +10% | +13.3 % |
| ω1 Lower target angular frequency | 2 to 35 rad/s | 0.1905 | 1.867 | 1.676 | +10% | +7.84 % |
| ζ2 Damping ratio at ω₂ | 0.005 to 0.2 | 0 | 1.04 | 1.04 | +10% | −3.33 % |
| ω2 Upper target angular frequency | 15 to 200 rad/s | 0.6 | 0.9524 | 0.3524 | +10% | +1.85 % |
Sweep of Damping ratio at ω₁
| Damping ratio at ω₁ | Mass coefficient (1/s) |
|---|---|
| 0.005 | −0.16 |
| 0.02938 | 0.36 |
| 0.05375 | 0.88 |
| 0.07813 | 1.4 |
| 0.1025 | 1.92 |
| 0.1269 | 2.44 |
| 0.1512 | 2.96 |
| 0.1756 | 3.48 |
| 0.2 | 4 |
Sweep one variable
Variables and how each one contributes
Mass 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 +7.84 %, from 0.8 to 0.8627 1/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 +13.3 %, from 0.8 to 0.9067 1/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 +1.85 %, from 0.8 to 0.8148 1/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 −3.33 %, from 0.8 to 0.7733 1/s.
Worked example with the values above
- Numerator: 2ω₁ω₂(ζ₁ω₂ − ζ₂ω₁) = 2 × 10 × 40 × (1.5) = 1,200.
- Denominator: ω₂² − ω₁² = 1,500.
- α = 0.8 1/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