Calculation
ymax=y₀+Δy∞·[1+e^(−πD/√(1−D²))]
The calculator directly shows the maximum reached by an oscillatory system after a step—important for limits and safety margins.
Zacher/Reuter 2024, Kapitel 8: Überschwingweite und Dämpfung
PT2 overshoot: calculate first peak
The calculator directly shows the maximum reached by an oscillatory system after a step—important for limits and safety margins.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- First peak ymax — Largest output during the first overshoot. Compare it with the permissible process, component or actuator range.
- Damping ratio D — Dimensionless decay measure of the second-order model from identification or model parameters. Small values cause large overshoot; valid range is 0<D<1.
- Output before the step y₀ — Actual steady output immediately before the input change, taken from the measurement log or operating point.
- Final output change Δy∞ — Final steady output minus y₀. Together with D, this sets the absolute first-peak level.
Example
D=0.5, y₀=0 and Δy∞=10 give 16.30% overshoot and therefore ymax≈11.6303.
Assumptions and limits
Normalised linear second-order system without zeros or dead time, underdamped case 0<D<1 and positive step. Saturation or nonlinear friction may change the real peak.
Understand PT2 overshoot: calculate first peak
Determine the first maximum output of an underdamped second-order step response from damping ratio, initial value and final change.
What does this quantity describe?
The calculator directly shows the maximum reached by an oscillatory system after a step—important for limits and safety margins. This calculator represents a clearly bounded technical relationship between the displayed quantities. The definition helps put inputs into the same reference state before interpreting the result.
Formula and variables
ymax=y₀+Δy∞·[1+e^(−πD/√(1−D²))]
| Symbol / input | Meaning |
|---|---|
| First peak ymax | Largest output during the first overshoot. Compare it with the permissible process, component or actuator range. |
| Damping ratio D | Dimensionless decay measure of the second-order model from identification or model parameters. Small values cause large overshoot; valid range is 0<D<1. |
| Output before the step y₀ | Actual steady output immediately before the input change, taken from the measurement log or operating point. |
| Final output change Δy∞ | Final steady output minus y₀. Together with D, this sets the absolute first-peak level. |
Choose the inputs correctly
First peak ymax: Largest output during the first overshoot. Compare it with the permissible process, component or actuator range. Damping ratio D: Dimensionless decay measure of the second-order model from identification or model parameters. Small values cause large overshoot; valid range is 0<D<1. Output before the step y₀: Actual steady output immediately before the input change, taken from the measurement log or operating point. Final output change Δy∞: Final steady output minus y₀. Together with D, this sets the absolute first-peak level.
How to use the calculator
Select the target quantity, enter the other known values with units, then check the result against the worked example and model limits.
Worked example
D=0.5, y₀=0 and Δy∞=10 give 16.30% overshoot and therefore ymax≈11.6303.
Understand the result and units
The calculator directly shows the maximum reached by an oscillatory system after a step—important for limits and safety margins. Read the result as a model value for the selected operating point and check units, sign, order of magnitude and application boundary conditions.
Use the displayed units and convert afterwards. Prefixes such as k-, m- and µ- are common sources of mistakes.
Typical applications
PT2 overshoot: calculate first peak: Such basic calculations support plausibility checks, early component selection and preparation of a complete verification.
Assumptions, limits and common mistakes
Normalised linear second-order system without zeros or dead time, underdamped case 0<D<1 and positive step. Saturation or nonlinear friction may change the real peak.
Common mistake: A formally correct result can still be unsuitable when load case, reference state or units do not match the application.
Frequently asked questions
How do I check the result?
Compare unit and order of magnitude with a second calculation and vary inputs one at a time.
Are assumptions automatically satisfied?
No. The calculator exposes a model; real boundary conditions require separate review.
Can I mix arbitrary units?
Only when the calculator converts within the same physical quantity type.