Library / Structural dynamics and vibration testing / SDY-41
Structural dynamics and vibration testing
Frequency of minimum Rayleigh damping
Angular frequency at which the Rayleigh damping curve is lowest, from the mass and stiffness coefficients.
- Ref
- SDY-41
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- ωmin [rad/s]
The Rayleigh damping curve ζ(ω) = α/(2ω) + βω/2 has its minimum at ωmin = √(α/β), where the two terms are equal. Its value there is √(αβ). Between two equal target ratios calibrated at ω₁ and ω₂ the minimum lies at √(ω₁ω₂), the geometric mean.
Angular frequency at which the Rayleigh damping curve is lowest, from the mass and stiffness coefficients. Between two equal target ratios the damping dips here.
Calculator
- Frequency in hertz3.183 Hz
- Minimum damping ratio, √(αβ)0.04
- Damping at twice that frequency0.05
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 damping curve is lowest at 20 rad/s (3.18 Hz), with a ratio of 4 %.
- Modes near this frequency are the least damped in the model, so check that they are not the ones that matter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| β Stiffness coefficient | 1e-4 to 0.01 s | 8.944 | 89.44 | 80.5 | +10% | −4.65 % |
| α Mass coefficient | 0.05 to 5 1/s | 5 | 50 | 45 | +10% | +4.88 % |
Sweep of Mass coefficient
| Mass coefficient (1/s) | Frequency of minimum damping (rad/s) |
|---|---|
| 0.05 | 5 |
| 0.6688 | 18.29 |
| 1.288 | 25.37 |
| 1.906 | 30.87 |
| 2.525 | 35.53 |
| 3.144 | 39.65 |
| 3.763 | 43.37 |
| 4.381 | 46.8 |
| 5 | 50 |
Sweep one variable
Variables and how each one contributes
Frequency of minimum damping
What the equation gives you. Everything else feeds it.
Mass coefficient
The mass-proportional Rayleigh coefficient, in 1/s. Raises damping mostly at low frequency. Raising it by 10 % (to 0.88) moves the result by +4.88 %, from 20 to 20.98 rad/s.
Stiffness coefficient
The stiffness-proportional Rayleigh coefficient, in s. Raises damping mostly at high frequency. Raising it by 10 % (to 0.0022) moves the result by −4.65 %, from 20 to 19.07 rad/s.
Worked example with the values above
- Ratio: α/β = 0.8 / 0.002 = 400 (rad/s)².
- Square root.
- ωₘᵢₙ = 20 rad/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