Technology / Technical guide
Logarithmic Decrement: Free Decay & Damping Assumptions
Logarithmic decrement relates successive peaks in a particular decaying oscillation model. Its attraction is that an amplitude ratio removes the unknown overall amplitude. The relationship is only as meaningful as the free-response, single-mode and damping assumptions behind it; a falling sequence of peaks alone does not establish those conditions.
A peak ratio represents decay only under the right model.
Equilibrium, peak direction and elapsed cycle count determine what the amplitude-ratio logarithm represents.
01 / Free decay
A simple damped oscillator separates rate and oscillation.
Consider m ẍ + c ẋ + kx = 0, with constant positive m and k, positive viscous damping c and no forcing.
Consider m ẍ + c ẋ + kx = 0, with constant positive m and k, positive viscous damping c and no forcing. Define ωn = √(k/m), ζ = c/(2√(km)) and ωd = ωn√(1 − ζ²). Require ζ strictly between zero and one.
Its underdamped displacement is x(t) = A exp(−ζωn t) cos(ωd t + φ). A and φ represent the initial condition. The exponential factor describes amplitude decay, while ωd describes oscillation. These are separate terms, not two labels for the same frequency.
02 / Peak direction
Successive matching peaks are one damped cycle apart.
For this model, successive peaks with the same direction occur a damped period Td = 2π/ωd apart.
For this model, successive peaks with the same direction occur a damped period Td = 2π/ωd apart. Their magnitudes are measured relative to the model’s equilibrium. The phase at the extrema repeats each cycle, so the cosine contribution cancels in the ratio.
Adjacent positive and negative extrema are half a cycle apart. Taking their absolute ratio without changing the interval convention would not give the same one-cycle decrement. The direction and cycle count must therefore accompany any stated peak comparison.
03 / Logarithm
The decrement removes the overall amplitude factor.
Define δ = ln(Ak/Ak+1) for magnitudes of successive same-direction peaks.
Define δ = ln(Ak/Ak+1) for magnitudes of successive same-direction peaks. The free-decay model gives δ = ζωnTd = 2πζ/√(1 − ζ²). The ratio and logarithm are dimensionless, while the time constants and frequencies retain their own units.
MIT’s course material presents the logarithmic peak-ratio definition and its extension across several cycles. In the declared model, an n-cycle comparison gives δ = ln(Ak/Ak+n)/n. Using a ratio across more cycles without dividing by n changes the represented decrement.
04 / Toy result
The exact damping relation differs from a small-damping shortcut.
For invented same-direction peaks of 10 and 8 units separated by one cycle, δ = ln(10/8) ≈ 0.2231.
For invented same-direction peaks of 10 and 8 units separated by one cycle, δ = ln(10/8) ≈ 0.2231. Rearranging the model gives ζ = δ/√((2π)² + δ²) ≈ 0.0355. The peak units cancel in the ratio.
For sufficiently small ζ, δ/(2π) approximates ζ because the square-root correction is small. That shortcut is an approximation, not the defining relationship. The example is a calculated oscillator; it supplies no measured damping ratio for a blade, tower or generator.
05 / Broken assumptions
A decaying-looking trace can represent something else.
Several modes can add and cancel, producing beating and irregular peaks.
Several modes can add and cancel, producing beating and irregular peaks. A continuing input can alter the amplitude history. An offset in the equilibrium reference changes peak ratios, and uncertain peak values can make the logarithm sensitive to error.
Nonlinear or amplitude-dependent damping also breaks the constant-viscous model. The logarithm can still be calculated numerically, but it may no longer correspond to a single constant ζ. A mathematical operation existing is different from its physical interpretation remaining valid.
06 / Scope
One modal decay does not describe all equipment behaviour.
A damping ratio belongs to its model, coordinates and response conditions.
A damping ratio belongs to its model, coordinates and response conditions. It is not automatically a fatigue-life value, a vibration limit or proof of structural condition. A mixed-mode operating waveform cannot be treated as the stated free single-mode response merely because some peaks decrease.
No MARSTON measured decrement, damping ratio or diagnostic function is claimed. This page includes no release, excitation, shutdown or field-testing procedure. It explains the analytical relationship and the conditions that make a same-direction peak ratio meaningful.
Related technical guides: Modal Superposition: Coordinates, Coupling & Truncation · Vibration Displacement, Velocity & Acceleration · Aerodynamic Damping: Work Done by Flow over a Motion Cycle.
General background: MIT OpenCourseWare: logarithmic decrement and damped period. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Damping ratio ζ | c/(2√(km)), dimensionless | Positive constant linear model with underdamped response |
| Damped period Td | 2π/ωd in seconds | Successive same-direction peaks are one cycle apart |
| Peak magnitude Ak | Departure from equilibrium | Consistent direction and reference |
| Decrement δ | Natural log of one-cycle peak ratio | Free single-mode constant-viscous assumptions |
| Multi-cycle ratio | ln(Ak/Ak+n)/n | n is the number of elapsed damped cycles |
Engineering questions.
Is decrement measured in seconds?
No. It is a dimensionless logarithm of an amplitude ratio.
Which successive peaks define one cycle?
Peaks in the same direction, measured from a consistent equilibrium.
Can opposite-direction peaks use the same formula?
Their half-cycle separation must be accounted for; they are not the stated one-cycle pair.
Why divide a multi-cycle logarithm by n?
The logarithm spans n cycles rather than one.
Does every falling peak sequence have one damping ratio?
No. Forcing, mixed modes, offsets or nonlinear damping can invalidate the model.
Is the toy result an equipment measurement?
No. It follows from invented peaks within a declared analytical model.
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