Library / Structural dynamics and vibration testing / SDY-02
Structural dynamics and vibration testing
Damping ratio from the ring-down decrement
Viscous damping ratio of an underdamped free vibration from its logarithmic decrement per cycle.
- Ref
- SDY-02
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- ζ
The exact inversion ζ = δ/√(4π² + δ²) turns a logarithmic decrement into the viscous damping ratio of an underdamped free vibration. For light damping it is close to δ/(2π). The result applies to a single mode with viscous damping.
Viscous damping ratio of an underdamped free vibration from its logarithmic decrement per cycle. The inversion is exact for a single viscous mode.
Calculator
- Light-damping approximation, δ/(2π)0.04775
- Amplitude ratio per cycle, e^(−δ)0.7408
- Cycles to halve the amplitude, ln 2 / δ2.31
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.
- A decrement of 0.3 gives a damping ratio of 0.04769, close to the light-damping value of 0.04775.
- The two agree to within a percent or so for damping ratios below about 0.1, and drift apart as the damping grows.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| δ Logarithmic decrement per cycle | 0.01 to 2 | 0.001592 | 0.3033 | 0.3017 | +10% | +9.97 % |
Sweep of Logarithmic decrement per cycle
| Logarithmic decrement per cycle | Damping ratio |
|---|---|
| 0.01 | 0.001592 |
| 0.2587 | 0.04115 |
| 0.5075 | 0.08051 |
| 0.7562 | 0.1195 |
| 1.005 | 0.1579 |
| 1.254 | 0.1957 |
| 1.502 | 0.2326 |
| 1.751 | 0.2685 |
| 2 | 0.3033 |
Sweep one variable
Variables and how each one contributes
Damping ratio
What the equation gives you. Everything else feeds it.
Logarithmic decrement per cycle
The logarithmic decrement per cycle, a plain number: the natural log of the ratio of successive peaks. More decrement per cycle means more damping. Raising it by 10 % (to 0.33) moves the result by +9.97 %, from 0.04769 to 0.05245.
Worked example with the values above
- Square the decrement: δ² = 0.09.
- Add 4π² = 39.478 and take the square root: 6.2903.
- ζ = 0.3 / 6.2903 = 0.047692.
Assumptions and limits
- A single, well-separated mode dominates the record and the damping is viscous (proportional to velocity) and light.
- Peaks or half-power points are read from clean data with enough resolution.
Common mistakes
- Mixing peak heights with peak-to-peak values, or taking peaks from opposite sides of the zero line.
- Using a record that contains several modes, so the envelope is not a single exponential.
- Using the half-power width when the frequency resolution is too coarse to resolve it.
- Using an angular frequency in rad/s where Hz is expected, or the reverse.
Statistics and uncertainty
Decrements read from only one pair of peaks carry the reading error of both. Averaging over several cycles, or fitting an exponential to the whole envelope, cuts the scatter. Damping is itself an inexact, amplitude-dependent property, so quote the amplitude at which it was measured.
Where it is used
- Identifying damping from an impact or ring-down test on a structure, instrument or component.
- Setting the damping values used in finite-element and modal models.
Maths you need
- Natural logarithm: the inverse of e to a power
- Damping ratio: damping as a fraction of critical
- Free decay: the envelope of an unforced vibration