Module 4 · Lesson 4.2
Measuring damping from a decay trace
The logarithmic decrement, derived and then used — and an honest account of why the number it gives is the least reliable input to any dynamic model.
Why this matters
Mass can be calculated. Stiffness can be calculated. Damping cannot — it has to be measured, or taken from measurements on structures believed to be similar.
That single fact shapes a great deal of practice. It is why design damping values are tabulated by structural type rather than derived, why resonant response predictions carry wide uncertainty, and why a vibration serviceability assessment is often presented as a range rather than a number.
By the end of this lesson you should be able to
- Derive the logarithmic decrement from the free-vibration solution
- Convert a logarithmic decrement to a damping ratio, exactly and approximately
- Explain why several cycles must be used rather than one
- State what a decay test does and does not establish
From first principles
The logarithmic decrement
We want to show: Show that successive peaks of a damped free vibration fall by a constant RATIO, and use that to extract the damping ratio from a measured trace.
The decaying exponential envelope has a property that makes this measurement possible: over any fixed interval of time it falls by the same fraction, wherever you start. Successive peaks are separated by exactly one damped period, so the ratio between them is the same every cycle — and that single ratio contains the damping.
Try it
Estimating damping from a decay trace
This is the measurement an engineer actually makes on site. Choose two peaks and read the damping off them.
| Peak | Measured (mm) | Ratio to previous |
|---|---|---|
| 0 | 40.00 | — |
| 1 | 33.57 | 1.1914 |
| 2 | 26.89 | 1.2487 |
| 3 | 23.08 | 1.1650 |
| 4 | 18.75 | 1.2310 |
| 5 | 15.41 | 1.2164 |
| 6 | 13.15 | 1.1719 |
| 7 | 10.50 | 1.2531 |
- Cycles between the peaks, n
- 1
- Amplitude ratio
- 1.1914
- δ = (1/n) ln(ratio)
- 0.175
- ζ = δ/√(4π² + δ²)
- 0.0279
- ζ ≈ δ/2π (approximation)
- 0.0279
- True value
- 0.0300
- Error in the estimate
- -7.1%
Over one cycle the amplitude falls by only 17.2%. With a 2% reading error on each peak, that is why the estimate is 7% out. Increase the separation between the peaks.
What this measurement cannot tell you
- It measures the damping at the amplitude of the test. Real damping is amplitude-dependent, and a small-amplitude test on a building will report less damping than the same building shows in a storm.
- It assumes viscous damping. Friction in cladding, joints and finishes is not viscous, and fitting an exponential to it gives a number that works only over the range measured.
- The very late peaks are the least reliable, because the signal has fallen towards the noise floor of the instrument.
What this shows: The logarithmic decrement works because consecutive peaks fall by a constant RATIO. Using peaks many cycles apart is not a refinement — at realistic damping levels it is the difference between a usable estimate and noise.
Try it
Where the energy goes
Release the mass from rest and watch the amplitude fall — then watch the energy account that explains why.
- Displacement
- Envelope
- Kinetic
- Strain
- Dissipated (cumulative)
- Total mechanical
Show the numbers behind this plot
- Initial energy E₀
- 493 J
- Energy remaining now
- 100.0%
- Amplitude remaining now
- 100.0%
- Logarithmic decrement δ
- 0.315
- Cycles to half amplitude
- 2.2
- Cycles to 1% amplitude
- 14.6
Energy goes as amplitude squared, so when the amplitude has halved, three quarters of the energy has already gone. That is why a structure that still looks like it is moving may have almost nothing left in it — and why a damping estimate taken from the tail of a decay trace is the least reliable part of the record.
What this shows: Amplitude decays because the damper is converting mechanical energy into heat. Energy falls as the SQUARE of amplitude, so the energy is gone long before the motion looks small.
Worked example
Reading damping off a measured trace
Given
- A decay record from a floor after a heel-drop test
- First peak: 0.412 mm
- Sixth peak: 0.198 mm
- Measured frequency 5.8 Hz
Find
The damping ratio, and what can and cannot be concluded from it.
Assumptions
- Peaks are read one damped period apart, so five cycles separate them
- The response is dominated by a single mode
Practice
The first peak of a decay trace is 12.0 mm and the fifth is 5.4 mm. What is the logarithmic decrement?
Practice
For that trace, what is the damping ratio as a percentage?
Practice
A structure has 2% damping. Over how many cycles must peaks be separated for the amplitude ratio to reach 2.0?
Check yourself
A decay trace shows peaks falling by a constant AMOUNT each cycle — 10 mm, 8 mm, 6 mm, 4 mm — rather than by a constant ratio. What does this indicate?
Check yourself
Two identical structures are tested. One shows 12 visible cycles before the motion dies away, the other 3. What differs?
Worked example
Damping from a decay trace, five cycles apart
Given
- A heel-drop test on a composite floor
- The first measured peak is 12.0 mm
- The peak five cycles later is 7.4 mm
Find
The damping ratio
Worked example
How long does it take to stop?
Given
- The same floor, measured at ζ = 1.54 %
- A second floor after fit-out, at ζ = 5 %
Find
The cycles needed for the amplitude to fall to a tenth, and to a half
Summary
- δ = (1/n)ln(xᵢ/xᵢ₊ₙ) — the log of the amplitude ratio, per cycle
- ζ = δ/√(4π² + δ²) exactly, and ζ ≈ δ/2π below about 20% damping
- The measurement needs no knowledge of mass or stiffness, which is what makes it practical
- Use enough cycles for the amplitude ratio to exceed 2 — about six at 2% damping
- A constant amplitude RATIO confirms viscous damping; a constant difference denies it
- A decay test measures damping at the amplitude tested, in the mode excited, in the condition tested
- Service damping is usually higher than a bare-structure test reports
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint