Library / Structural dynamics and vibration testing / SDY-38
Structural dynamics and vibration testing
Modal damping from Rayleigh coefficients
Damping ratio that Rayleigh damping, C = αM + βK, gives to a mode of angular frequency ω.
- Ref
- SDY-38
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- ζr
With Rayleigh damping C = αM + βK, a mode of angular frequency ω has damping ratio ζ = α/(2ω) + βω/2. The mass term α/(2ω) dominates at low frequency and falls with frequency; the stiffness term βω/2 dominates at high frequency and rises. The sum has a minimum at ω = √(α/β). Coefficients tuned at two frequencies can give quite different damping elsewhere.
Damping ratio that Rayleigh damping, C = αM + βK, gives to a mode of angular frequency ω. The mass term dominates at low frequency and the stiffness term at high frequency.
Calculator
- Mass part, α/(2ω)0.02
- Stiffness part, βω/20.004
- Frequency of minimum damping, √(α/β)44.72 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.
- At 20 rad/s the model gives 2.4 % damping, of which 83.3 % comes from the mass term.
- The damping is lowest near 44.7 rad/s and rises on either side, so check every mode in the band against the damping you intended.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωr Modal angular frequency | 1 to 200 rad/s | 0.01792 | 0.4002 | 0.3823 | +10% | −5.91 % |
| α Mass coefficient | 0 to 5 1/s | 0.004 | 0.129 | 0.125 | +10% | +8.33 % |
| β Stiffness coefficient | 0 to 0.01 s | 0.02 | 0.12 | 0.1 | +10% | +1.67 % |
Sweep of Modal angular frequency
| Modal angular frequency (rad/s) | Modal damping ratio |
|---|---|
| 1 | 0.4002 |
| 25.88 | 0.02063 |
| 50.75 | 0.01803 |
| 75.63 | 0.02041 |
| 100.5 | 0.02408 |
| 125.4 | 0.02827 |
| 150.3 | 0.03271 |
| 175.1 | 0.03731 |
| 200 | 0.042 |
Sweep one variable
Variables and how each one contributes
Modal damping ratio
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 +8.33 %, from 0.024 to 0.026.
Stiffness coefficient
The stiffness-proportional Rayleigh coefficient, in s. Raises damping mostly at high frequency. Raising it by 10 % (to 4.4×10−4) moves the result by +1.67 %, from 0.024 to 0.0244.
Modal angular frequency
The modal angular frequency, in rad/s. Moves the mode along the damping curve. Raising it by 10 % (to 22) moves the result by −5.91 %, from 0.024 to 0.02258.
Worked example with the values above
- Mass part: α/(2ω) = 0.8 / (2 × 20) = 0.02.
- Stiffness part: βω/2 = 4×10−4 × 20 / 2 = 0.004.
- ζ = 0.024.
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