Axis Settling Time & Positioning Accuracy Calculator
Calculate settling time using damping ratio, stiffness, mass, end velocity, acceleration and required positioning accuracy.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
System properties
Motion profile
Performance requirements
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
settling time formula:
ωn = √(k / m)
ccrit = 2√(k · m)
If ζ is provided directly:
ζ = ζinput
ceffective = ζ · ccrit
If damping coefficient is used instead:
ζ = c / ccrit
ceffective = c
tstop = v / a
x0 = v / ωn
Decay time:
- For ζ < 1: tdecay = -ln(εreq / x0) / (ζ · ωn)
- For ζ = 1: tdecay = -ln(εreq / x0) / ωn
- For ζ > 1: λ1 = -ωn(ζ – √(ζ² – 1))
- For ζ > 1: tdecay = ln(x0 / εreq) / (-λ1)
tsettle = tstop + tdecay
Achieved positioning accuracy formula:
- For ζ < 1: ε = |x0 · e-ζωntdecay|
- For ζ = 1: ε = |x0 · e-ωntdecay|
- For ζ > 1: ε = |x0 · eλ1tdecay|
Damping ratio formula:
- ζ = ζinput, when damping ratio is entered directly
- ζ = c / ccrit, when damping coefficient is used
where:
- m — mass (kg)
- k — stiffness (N/m)
- c — damping coefficient (N·s/m)
- ζ — damping ratio (-)
- ωn — natural frequency (rad/s)
- ccrit — critical damping (N·s/m)
- s — travel distance (m)
- v — end velocity (m/s)
- a — max acceleration (m/s²)
- trequired — required time (s)
- εreq — required positioning accuracy (m)
- ε — achieved positioning accuracy (m)
- x0 — initial residual displacement estimate (m)
- λ1 — slow overdamped decay pole (1/s)
When to use this calculator
settling time formula:
ωn = √(k / m)
ccrit = 2√(k · m)
If ζ is provided directly:
ζ = ζinput
ceffective = ζ · ccrit
If damping coefficient is used instead:
ζ = c / ccrit
ceffective = c
tstop = v / a
x0 = v / ωn
Decay time:
- For ζ < 1: tdecay = -ln(εreq / x0) / (ζ · ωn)
- For ζ = 1: tdecay = -ln(εreq / x0) / ωn
- For ζ > 1: λ1 = -ωn(ζ – √(ζ² – 1))
- For ζ > 1: tdecay = ln(x0 / εreq) / (-λ1)
tsettle = tstop + tdecay
Achieved positioning accuracy formula:
- For ζ < 1: ε = |x0 · e-ζωntdecay|
- For ζ = 1: ε = |x0 · e-ωntdecay|
- For ζ > 1: ε = |x0 · eλ1tdecay|
Damping ratio formula:
- ζ = ζinput, when damping ratio is entered directly
- ζ = c / ccrit, when damping coefficient is used
where:
- m — mass (kg)
- k — stiffness (N/m)
- c — damping coefficient (N·s/m)
- ζ — damping ratio (-)
- ωn — natural frequency (rad/s)
- ccrit — critical damping (N·s/m)
- s — travel distance (m)
- v — end velocity (m/s)
- a — max acceleration (m/s²)
- trequired — required time (s)
- εreq — required positioning accuracy (m)
- ε — achieved positioning accuracy (m)
- x0 — initial residual displacement estimate (m)
- λ1 — slow overdamped decay pole (1/s)
How to interpret the result
Settling time is defined as the sum of the time needed to stop the axis motion and the time needed for the residual displacement to decay to the required positioning accuracy.
Achieved positioning accuracy is defined as the residual displacement after the calculated decay time. It depends on the initial residual displacement estimate, natural frequency, damping ratio and decay model selected from the damping regime.
Damping ratio is defined as either the entered damping ratio value or the ratio between damping coefficient and critical damping. A damping ratio below 1 gives an underdamped response, a damping ratio within ±0.05 of 1 gives a near-critical response, and a damping ratio above 1.05 gives an overdamped response.
- Safe — used when damping ratio is at least 0.6, settling time is not greater than required time, achieved positioning accuracy is not greater than required positioning accuracy, and the unsafe or warning conditions are not triggered.
- Limit — used when settling time is greater than required time, or achieved positioning accuracy is greater than required positioning accuracy, or damping ratio is below 0.6.
- Warning — used when settling time is more than 1.2 times the required time, or achieved positioning accuracy is more than 1.2 times the required positioning accuracy, or damping ratio is below 0.4.
- Unsafe — used when damping ratio is below 0.2 or overshoot is greater than 80%.
- Invalid — used when the input constraints are not satisfied, including non-positive mass, non-positive stiffness, non-positive travel distance, negative end velocity, non-positive acceleration, non-positive required time, non-positive required accuracy, impossible stopping condition, zero damping ratio, or required accuracy greater than or equal to the initial residual displacement estimate.
The result is used to evaluate whether the axis can meet the required time and positioning accuracy after the commanded motion profile.
Calculation example
Example:
A user wants to check whether a linear axis can stop and settle within the required positioning time after a short travel move.
- m = 10 kg
- k = 4000 N/m
- ζ = 0.7
- s = 0.5 m
- v = 0.2 m/s
- a = 1 m/s²
- trequired = 1 s
- εreq = 0.001 m
Result: tsettle ≈ 0.365 s, ε ≈ 0.001 m and ζ = 0.7.
Assumptions and limitations
- The axis is represented by a mass, stiffness and damping ratio model.
- Natural frequency is calculated from stiffness divided by mass.
- Required positioning accuracy must be smaller than the initial residual displacement estimate from end velocity divided by natural frequency.
- Motion stop time is calculated from end velocity divided by max acceleration.
- For damping ratio below 1, decay is calculated with the underdamped exponential term.
- For damping ratio equal to 1 within the implemented tolerance, decay is calculated with the critical damping exponential term.
- For damping ratio above 1, decay is calculated using the slow overdamped decay pole.