Library / Structural dynamics and vibration testing / SDY-03
Structural dynamics and vibration testing
Ring-down time to a chosen amplitude fraction
Time for the decaying envelope of an underdamped oscillator to fall to a chosen fraction p of its starting size, from the damping ratio and the undamped angular frequency.
- Ref
- SDY-03
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- t [s]
The envelope of an underdamped free vibration falls as e−ζωₙt, so the time to reach a fraction p of its starting size is −ln p divided by ζωₙ. A smaller p, less damping or a lower frequency all lengthen the ring-down. Use it to decide how long to wait between impacts.
Time for the decaying envelope of an underdamped oscillator to fall to a chosen fraction p of its starting size, from the damping ratio and the undamped angular frequency.
Calculator
- Time to half amplitude, ln 2/(ζωₙ)0.6931 s
- Decay rate, ζωₙ1 1/s
- Number of cycles in the ring-down7.32
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.
- With 5 % damping at 20 rad/s, the vibration falls to 10 % of its starting size after 2.303 s.
- Doubling the damping ratio halves the ring-down time, so the effect of damping is direct.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωn Undamped angular frequency | 1 to 200 rad/s | 0.2303 | 46.05 | 45.82 | +10% | −9.09 % |
| ζ Damping ratio | 0.01 to 0.9 | 0.1279 | 11.51 | 11.39 | +10% | −9.09 % |
| p Remaining envelope fraction | 0.01 to 0.9 | 0.1054 | 4.605 | 4.5 | +10% | −4.14 % |
Sweep of Damping ratio
| Damping ratio | Ring-down time (s) |
|---|---|
| 0.01 | 11.51 |
| 0.1212 | 0.9495 |
| 0.2325 | 0.4952 |
| 0.3438 | 0.3349 |
| 0.455 | 0.253 |
| 0.5663 | 0.2033 |
| 0.6775 | 0.1699 |
| 0.7888 | 0.146 |
| 0.9 | 0.1279 |
Sweep one variable
Variables and how each one contributes
Ring-down time
What the equation gives you. Everything else feeds it.
Remaining envelope fraction
The fraction of the starting envelope that is left, between 0 and 1 (0.1 means 10 % remains). A smaller remaining fraction takes longer to reach. Raising it by 10 % (to 0.11) moves the result by −4.14 %, from 2.303 to 2.207 s.
Damping ratio
The viscous damping ratio of the mode (a plain number, 0 to 1). Raising the damping shortens the ring-down time. Raising it by 10 % (to 0.055) moves the result by −9.09 %, from 2.303 to 2.093 s.
Undamped angular frequency
The undamped natural angular frequency, in rad/s (2π times the frequency in Hz). A higher frequency means more cycles per second and a faster decay. Raising it by 10 % (to 22) moves the result by −9.09 %, from 2.303 to 2.093 s.
Worked example with the values above
- Decay rate: ζωₙ = 0.05 × 20 = 1 1/s.
- Natural log of the fraction: −ln(0.1) = 2.3026.
- t = 2.3026 / 1 = 2.3026 s.
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