Calculation
D = 1 / √[1 + (m·π / ln(A₁/A₂))²]
The rate at which successive excursions decay reveals damping even before the complete process model is known.
Zacher/Reuter 2024, Abschnitt 3.5, Gleichungen 3.69 bis 3.72: logarithmisches Dekrement
Second-order damping from a decay curve
The rate at which successive excursions decay reveals damping even before the complete process model is known.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Damping ratio D — Dimensionless model parameter between 0 and 1. Small values mean slow decay and strong overshoot; use D in the second-order step-response calculator or a control-loop simulation.
- Earlier amplitude A₁ — Magnitude of the earlier excursion relative to the steady final value, not the absolute sensor reading. Read the vertical distance between peak and final value from the recorded curve.
- Later amplitude A₂ — Magnitude of a later excursion relative to the same final value and in the same unit as A₁. A₂ must be smaller than A₁; do not enter signs above or below the final value.
- Number of half cycles m — Number of half oscillation periods between A₁ and A₂. Adjacent excursions on opposite sides give m=1; two peaks on the same side give m=2. Enter an integer.
Example
For A₁=10, A₂≈3.723 and m=1, ln(A₁/A₂)≈0.988; hence D≈0.300. The response is underdamped with marked overshoot.
Assumptions and limits
Linear time-invariant underdamped second-order system with 0<D<1 and an exponentially decaying free or step response. The final value, measurement scale and system parameters remain constant; severe noise, nonlinearity and additional dominant poles are excluded.
Understand Second-order damping from a decay curve
This calculator evaluates the decay of a measured oscillation. Two excursions and their separation in half cycles yield damping ratio D, a key parameter of an oscillatory second-order system.
What does this quantity describe?
Damping ratio D relates actual damping to critical damping. In the range 0<D<1 used here, the system oscillates around its final value. Its envelope decays exponentially, so the natural logarithm of the amplitude ratio gives logarithmic decrement.
Formula and variables
D = 1 / √[1 + (m·π / ln(A₁/A₂))²]
δ=ln(A₁/A₂)D=1/√[1+(m·π/δ)²]For consecutive peaks on the same side, m=2
| Symbol / input | Meaning |
|---|---|
| Damping ratio D | Dimensionless model parameter between 0 and 1. Small values mean slow decay and strong overshoot; use D in the second-order step-response calculator or a control-loop simulation. |
| Earlier amplitude A₁ | Magnitude of the earlier excursion relative to the steady final value, not the absolute sensor reading. Read the vertical distance between peak and final value from the recorded curve. |
| Later amplitude A₂ | Magnitude of a later excursion relative to the same final value and in the same unit as A₁. A₂ must be smaller than A₁; do not enter signs above or below the final value. |
| Number of half cycles m | Number of half oscillation periods between A₁ and A₂. Adjacent excursions on opposite sides give m=1; two peaks on the same side give m=2. Enter an integer. |
Choose the inputs correctly
A₁ and A₂ are magnitudes of the vertical distances from two peaks to the same steady final value. They may be measured in kelvin, millimetres or volts but must use the same unit. m counts intervening half cycles: adjacent excursions on alternating sides have m=1; two maxima on the same side have m=2.
How to use the calculator
First determine the steady final value of the trace. Select two clear excursions, subtract the final value and record positive magnitudes only. Count the half cycles between them. More widely separated peaks often reduce relative reading error provided system behaviour remains unchanged.
Worked example
An earlier distance from final value is 10 mm and the adjacent excursion on the opposite side is 3.723 mm. With m=1, the ratio is 2.686, logarithmic decrement is about 0.988 and D≈0.300.
Understand the result and units
D near zero means slow decay and pronounced overshoot. As D increases, oscillations decay faster; D=1 is critical damping and outside this measurement method. Enter the identified value in the second-order response calculator for model comparison and simulation.
Only ratio A₁/A₂ enters the calculation, so any common amplitude unit cancels. D and m are dimensionless.
Typical applications
Use it for laboratory tests, identification of mechanical oscillators or control loops, and checking a second-order approximation against a recorded step or free response.
Assumptions, limits and common mistakes
The method assumes dominant linear second-order behaviour, constant parameters and a stable final value. Coulomb friction, saturation, changing frequency, drift, additional poles or severe noise can make peak ratios inconsistent.
Common mistake: Do not use absolute sensor readings instead of distances from the final value. For peaks on opposite sides, enter positive magnitudes and m=1 rather than a negative amplitude. m counts half cycles, not samples.
Frequently asked questions
What is “Determine damping ratio from a decay curve” used for?
Use it for laboratory tests, identification of mechanical oscillators or control loops, and checking a second-order approximation against a recorded step or free response.
Where do the input values come from?
A₁ and A₂ are magnitudes of the vertical distances from two peaks to the same steady final value. They may be measured in kelvin, millimetres or volts but must use the same unit. m counts intervening half cycles: adjacent excursions on alternating sides have m=1; two maxima on the same side have m=2.
What does the result not cover?
The method assumes dominant linear second-order behaviour, constant parameters and a stable final value. Coulomb friction, saturation, changing frequency, drift, additional poles or severe noise can make peak ratios inconsistent.