Library / Structural dynamics and vibration testing / SDY-01
Structural dynamics and vibration testing
Logarithmic decrement across several cycles
Natural logarithm of the ratio of an initial displacement peak to the peak n complete cycles later, divided by n.
- Ref
- SDY-01
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- δ
The logarithmic decrement is the natural logarithm of the ratio of two peaks n complete cycles apart, divided by n. For a single viscous mode each cycle reduces the amplitude by the same factor e−δ. Use peak magnitudes on the same side of the zero line and in the same unit.
Natural logarithm of the ratio of an initial displacement peak to the peak n complete cycles later, divided by n. It measures how fast a free vibration dies away, per cycle.
Calculator
- Amplitude ratio per cycle, e^(−δ)0.7071
- Fraction of amplitude lost per cycle0.2929
- Damping ratio, δ/√(4π² + δ²)0.05508
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.
- Peaks of 8 mm and 2 mm, 4 cycles apart, give a decrement of 0.3466, so each cycle keeps 70.7 % of the amplitude.
- The matching damping ratio is about 0.0551, which is typical of a lightly damped structure.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Number of complete cycles | 1 to 20 | 0.06931 | 1.386 | 1.317 | +1 | −20 % |
| An Peak n cycles later | 0.1 to 7.9 mm | 0.003145 | 1.096 | 1.092 | +10% | −6.88 % |
| A0 Initial displacement peak | 2.5 to 50 mm | 0.05579 | 0.8047 | 0.7489 | +10% | +6.88 % |
Sweep of Number of complete cycles
| Number of complete cycles | Logarithmic decrement |
|---|---|
| 1 | 1.386 |
| 3.375 | 0.4108 |
| 5.75 | 0.2411 |
| 8.125 | 0.1706 |
| 10.5 | 0.132 |
| 12.88 | 0.1077 |
| 15.25 | 0.0909 |
| 17.63 | 0.07865 |
| 20 | 0.06931 |
Sweep one variable
Variables and how each one contributes
Logarithmic decrement
What the equation gives you. Everything else feeds it.
Initial displacement peak
The first displacement peak, in mm. Use the size of the peak, not its sign, and take both peaks on the same side of the zero line. A larger starting peak with the same later peak means faster decay. Raising it by 10 % (to 8.8) moves the result by +6.88 %, from 0.3466 to 0.3704.
Peak n cycles later
The displacement peak n complete cycles later, in mm, on the same side as the first. A larger later peak means slower decay and a smaller decrement. Raising it by 10 % (to 2.2) moves the result by −6.88 %, from 0.3466 to 0.3227.
Number of complete cycles
The number of complete cycles between the two peaks (a whole number). Counting several cycles averages out reading error. Dividing by more cycles lowers the decrement per cycle for the same two peaks. Raising it by 1 (to 5) moves the result by −20 %, from 0.3466 to 0.2773.
Worked example with the values above
- Ratio of the peaks: 8 / 2 = 4.
- Natural logarithm: ln(4) = 1.3863.
- Divide by n = 4: δ = 0.34657.
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