Calculation
Td=(t₂−t₁)/N; ωd=2π/Td; T₀=√(1−D²)/ωd
Peak spacing gives damped oscillation frequency; together with damping ratio it yields the undamped natural time for the second-order model.
Zacher/Reuter 2024, Abschnitt 3.5, Gleichungen 3.59 und 3.68: gedämpfte Eigenkreisfrequenz
Second-order natural time from measured oscillation peaks
Peak spacing gives damped oscillation frequency; together with damping ratio it yields the undamped natural time for the second-order model.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Natural time T₀ — Reciprocal of undamped natural angular frequency of the identified second-order model. Use T₀ with D in step response, settling-time calculation or simulation.
- Damping ratio D — Dimensionless decay measure between 0 and 1. Obtain it from model parameters or the damping-from-decay-curve calculator.
- Time of earlier peak t₁ — Timestamp of a clear maximum or minimum relative to the start of recording. Use the same peak direction for t₁ and t₂.
- Time of later peak t₂ — Timestamp of a later peak in the same direction. t₂ must exceed t₁; peaks farther apart can reduce relative reading error.
- Number of complete periods N — Integer number of complete oscillation periods between t₁ and t₂. Consecutive maxima or consecutive minima give N=1.
Example
For D=0.3, consecutive maxima occur at t₁=1 s and t₂=7.2832 s. Thus Td≈6.2832 s, ωd≈1 rad/s and T₀=√(1−0.3²)/1≈0.9539 s.
Assumptions and limits
Linear time-invariant underdamped second-order system with 0<D<1, constant oscillation period and clearly identifiable same-direction peaks. Dead time shifts both timestamps equally and cancels from their difference; drift, nonlinearity and additional dominant poles are excluded.
Understand Second-order natural time from measured oscillation peaks
This calculator extracts a second important model parameter directly from a measurement trace. Amplitude decay describes damping ratio, while the time between same-direction peaks gives damped oscillation frequency and hence natural time T₀.
What does this quantity describe?
A second-order system has two energy-storage elements. In the underdamped range 0<D<1 its response oscillates at damped natural angular frequency ωd. T₀ is the reciprocal of undamped natural angular frequency ω₀. They are related by ωd=√(1−D²)/T₀.
Formula and variables
Td=(t₂−t₁)/N; ωd=2π/Td; T₀=√(1−D²)/ωd
Measured period: Td=(t₂−t₁)/NDamped natural angular frequency: ωd=2π/TdNatural time: T₀=√(1−D²)/ωd
| Symbol / input | Meaning |
|---|---|
| Natural time T₀ | Reciprocal of undamped natural angular frequency of the identified second-order model. Use T₀ with D in step response, settling-time calculation or simulation. |
| Damping ratio D | Dimensionless decay measure between 0 and 1. Obtain it from model parameters or the damping-from-decay-curve calculator. |
| Time of earlier peak t₁ | Timestamp of a clear maximum or minimum relative to the start of recording. Use the same peak direction for t₁ and t₂. |
| Time of later peak t₂ | Timestamp of a later peak in the same direction. t₂ must exceed t₁; peaks farther apart can reduce relative reading error. |
| Number of complete periods N | Integer number of complete oscillation periods between t₁ and t₂. Consecutive maxima or consecutive minima give N=1. |
Choose the inputs correctly
D is damping ratio from model parameters or amplitude decay. t₁ and t₂ are timestamps of two maxima or two minima, not a maximum and the following minimum. N counts complete periods between them. Adjacent same-direction peaks give N=1.
How to use the calculator
Inspect the measured trace relative to its final value and mark two clear peaks in the same direction. Read both timestamps on one time axis and count complete periods. If possible use several periods while frequency and operating condition remain constant. Determine D separately from decay.
Worked example
For D=0.3, consecutive maxima occur at 1 s and 7.2832 s. Period is approximately 2π s, so ωd≈1 rad/s. Therefore T₀=√(1−0.3²)/1≈0.9539 s.
Understand the result and units
T₀ characterises the model's undamped base dynamics. Together with D it fully parameterises the underdamped second-order element. Use it directly in step response and settling time; smaller T₀ means faster base dynamics.
t₁, t₂ and T₀ are times and convert internally to seconds. ωd is calculated internally in radians per second. D and N are dimensionless.
Typical applications
Use it for model identification from laboratory or plant measurements, mechanical oscillators, oscillatory control loops and preparation for simulation or controller design.
Assumptions, limits and common mistakes
The method assumes linear time-invariant second-order behaviour, constant period and 0<D<1. Additional dominant poles, nonlinear friction, changing parameters, drift or wrongly paired peaks distort T₀. Dead time does not affect peak spacing when both peaks are shifted equally.
Common mistake: Do not combine a maximum with the immediately following minimum; that is only half a period. Do not interpret N as the number of visible peaks. t₂ must be later than t₁. Do not treat damped period as simply 2πT₀ when D is appreciable.
Frequently asked questions
What is “Second-order natural time from measured oscillation peaks” used for?
Use it for model identification from laboratory or plant measurements, mechanical oscillators, oscillatory control loops and preparation for simulation or controller design.
Where do the input values come from?
D is damping ratio from model parameters or amplitude decay. t₁ and t₂ are timestamps of two maxima or two minima, not a maximum and the following minimum. N counts complete periods between them. Adjacent same-direction peaks give N=1.
What does the result not cover?
The method assumes linear time-invariant second-order behaviour, constant period and 0<D<1. Additional dominant poles, nonlinear friction, changing parameters, drift or wrongly paired peaks distort T₀. Dead time does not affect peak spacing when both peaks are shifted equally.