Axis Settling Time & Positioning Accuracy Calculator
Calculate total stop-and-settle time and residual positioning error for a damped axis model using equivalent mass, stiffness, damping ratio, stop velocity, deceleration, allowed time, and in-position tolerance.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Equivalent axis dynamics
Commanded stop
Positioning requirement
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
Axis settling time and positioning-error model
The calculator represents the moving axis as a linear single-degree-of-freedom mass-spring-damper system. The equivalent moving mass m, stiffness k, and damping ratio ζ define the second-order dynamics. The commanded stop is modeled as constant deceleration from the entered velocity to zero.
Natural frequency and damping
The undamped natural frequency is:
ωn = √(k / m)
where ωn is in rad/s, k is the equivalent stiffness in N/m, and m is the equivalent moving mass in kg.
The corresponding natural frequency in hertz is:
fn = ωn / (2π)
The critical damping coefficient and effective viscous damping coefficient are:
ccrit = 2√(km)
c = ζccrit = 2ζ√(km)
with c in N·s/m. The exponential decay rate is:
σ = ζωn
For an underdamped system, ζ < 1, the damped natural frequency used in the response calculation is:
ωd = ωn√(1 − ζ2)
The calculator evaluates underdamped, critically damped, and overdamped responses separately rather than restricting the calculation to ζ < 1.
Constant-deceleration stop
For velocity magnitude v at the start of deceleration and constant deceleration magnitude a:
tstop = v / a
sstop = v2 / (2a)
The relative compliance coordinate is defined as the displacement of the equivalent moving mass relative to the commanded position. During deceleration it satisfies:
ë + 2ζωnė + ωn2e = a
with initial conditions:
e(0) = 0 and ė(0) = 0.
The static acceleration-deflection scale is:
estatic = a / ωn2 = ma / k
Underdamped response during deceleration
For ζ < 1, the normalized response used to obtain the relative displacement and velocity is:
y(t) = 1 − e−σt [cos(ωdt) + (σ / ωd)sin(ωdt)]
ẏ(t) = (ωn2 / ωd) e−σtsin(ωdt)
At the end of the commanded stop:
estop = estaticy(tstop)
ėstop = estaticẏ(tstop)
Free vibration after the command stops
After tstop, commanded acceleration becomes zero and the residual error decays according to the free response of the equivalent second-order system.
For the underdamped case:
e(τ) = e−στ [A cos(ωdτ) + B sin(ωdτ)]
where:
A = estop
B = [ėstop + σestop] / ωd
For critical damping, the calculator uses:
e(τ) = e−ωnτ(A + Bτ)
with A = estop and B = ėstop + ωnestop.
For an overdamped system, ζ > 1, the response is represented by two real negative poles. Defining:
q = √[(ζ − 1)(ζ + 1)]
r1 = −ωn / (ζ + q)
r2 = −ωn(ζ + q)
the free response has the form:
e(τ) = C1er1τ + C2er2τ
Settling criterion
The required in-position tolerance εreq is an absolute position-error limit. The post-stop settling time tdecay is the earliest time after the commanded stop for which the response remains inside that tolerance:
max |e(u)| ≤ εreq for all u ≥ tdecay
The calculator searches this condition numerically using the future maximum of the regime-specific free response. Total stop-and-settle time is:
tsettle = tstop + tdecay
For an allowed total time trequired, the time available for free vibration decay is:
tavailable = max(0, trequired − tstop)
The reported residual positioning error εactual is the maximum remaining error from that time onward:
εactual = max |e(u)| for u ≥ tavailable
The requirement passes only when both conditions are satisfied:
tsettle ≤ trequired
εactual ≤ εreq
The calculator also reports the time margin Δt = trequired − tsettle and the accuracy margin Δε = εreq − εactual.
When to use this calculator
When to use this calculator
Use this calculator when a positioning axis can be approximated by an equivalent linear mass-stiffness-damping system and you need to determine whether the axis can stop and remain within a specified position tolerance within an allowed time.
Typical applications include compliant servo-driven mechanisms, linear stages, gantries, machine axes, handling equipment, and other positioning systems where structural flexibility can produce residual vibration after the commanded motion has stopped.
The calculation is particularly useful when the available engineering data are equivalent moving mass, equivalent stiffness, damping ratio, velocity at the start of braking, and a constant deceleration limit. From these values the model derives natural frequency, effective damping, stopping time, stopping distance, dynamic compliance error, residual vibration, and total settling time.
Suitable design questions
- Will the axis settle inside the required position band before the allowed time expires?
- How much residual position error is predicted after the available settling period?
- How do equivalent mass, stiffness, or damping ratio influence settling performance?
- What is the natural frequency of the equivalent axis model?
- How much of the total positioning time is consumed by commanded deceleration versus vibration decay?
- What time and positioning-accuracy margins remain relative to the entered requirements?
When not to use it as the final design model
The calculator does not simulate a complete servo controller or a measured closed-loop transfer function. It does not explicitly model controller gains, current or torque saturation, encoder behavior, backlash, Coulomb friction, nonlinear stiffness, multiple structural modes, cross-axis coupling, or jerk-limited acceleration profiles.
If those effects materially influence the machine response, use the result as an equivalent-model screening calculation and verify the final design with an appropriate dynamic simulation, measured response, modal model, or machine test.
How to interpret the result
How to interpret the results
The two primary results are total stop-and-settling time tsettle and maximum residual error εactual.
Total stop + settling time
tsettle is measured from the beginning of commanded deceleration. It contains two parts: the constant-deceleration stop time tstop and the additional decay time tdecay required after the command reaches zero velocity.
A positive time margin Δt = trequired − tsettle means the modeled axis settles before the entered deadline. A negative time margin means the required time is exceeded.
Residual positioning error
εactual is not simply the instantaneous error at one selected time. It is the maximum predicted residual error from the end of the available settling period onward. This distinction is important for an oscillatory underdamped response because the error may cross the tolerance band and later leave it again.
A positive accuracy margin Δε = εreq − εactual means the modeled residual response remains inside the required in-position tolerance. A negative value means the tolerance is exceeded.
PASS and FAIL
PASS means both the allowed-time criterion and the in-position error criterion are satisfied. FAIL means at least one of these two requirements is not satisfied.
A PASS should be interpreted as: the evaluated stop-and-settle criterion is satisfied by the entered equivalent dynamic model. It is not a general certification of machine accuracy, servo stability, structural safety, or performance under every operating condition.
Secondary dynamic results
- tstop is the commanded constant-deceleration stopping time.
- tdecay is the additional post-stop time required for residual vibration to remain inside the tolerance band.
- sstop is the commanded stopping distance from the entered initial velocity and deceleration.
- Residual peak after stop is the largest predicted free-response error from the instant the commanded stop ends.
- Peak dynamic error during deceleration is the largest compliance error generated while the constant-deceleration command is active.
- ωn and fn describe the natural frequency of the equivalent mass-stiffness system.
- Damping regime identifies the response as underdamped, critically damped, or overdamped.
- Equivalent viscous damping c is the damping coefficient corresponding to the entered damping ratio, mass, and stiffness.
Increasing stiffness generally raises natural frequency and reduces compliance error. Increasing equivalent moving mass generally lowers natural frequency and increases dynamic deflection. Damping changes how rapidly post-stop motion decays, while deceleration creates a trade-off between shorter commanded stopping time and stronger dynamic excitation.
Calculation example
Worked example — compliant positioning axis
Consider an axis with an equivalent moving mass of 50 kg and an equivalent stiffness of 200 kN/m. The damping ratio is 0.20. At the start of the commanded stop the axis is moving at 0.50 m/s and decelerates at a constant 5.0 m/s². The complete stop-and-settle process must finish within 0.50 s, and the required in-position tolerance is 0.10 mm.
| Input | Value |
|---|---|
| Equivalent moving mass, m | 50 kg |
| Equivalent stiffness, k | 200 kN/m = 200,000 N/m |
| Damping ratio, ζ | 0.20 |
| Velocity at start of deceleration, v | 0.50 m/s |
| Constant deceleration, a | 5.0 m/s² |
| Allowed stop + settle time | 0.50 s |
| Required in-position tolerance | 0.10 mm = 0.00010 m |
1. Natural frequency
ωn = √(200,000 / 50) = 63.246 rad/s
fn = 63.246 / (2π) = 10.066 Hz
Because ζ = 0.20, the equivalent system is underdamped. Its damped natural frequency is:
ωd = 63.246√(1 − 0.20²) = 61.968 rad/s
2. Equivalent damping
ccrit = 2√(200,000 × 50) = 6,324.56 N·s/m
c = 0.20 × 6,324.56 = 1,264.91 N·s/m
3. Commanded stop
tstop = 0.50 / 5.0 = 0.100 s
sstop = 0.50² / (2 × 5.0) = 0.0250 m = 25.0 mm
The acceleration-deflection scale is:
estatic = 5.0 / 63.246² = 0.00125 m = 1.25 mm
4. Dynamic response and settling
Evaluating the exact underdamped response at the end of the 0.100 s stop gives a relative error of approximately 0.905 mm. The maximum compliance error during the complete deceleration interval is approximately 1.908 mm.
After the commanded stop, the free vibration must decay until every subsequent peak remains within the ±0.10 mm tolerance. The numerical settling search gives:
tdecay ≈ 0.1634 s
Therefore:
tsettle = 0.1000 + 0.1634 = 0.2634 s
With 0.50 s allowed from the beginning of deceleration, 0.40 s is available for post-stop decay. At that point the maximum remaining future error is approximately:
εactual ≈ 0.00535 mm
5. Requirement check
The time margin is:
Δt = 0.5000 − 0.2634 = +0.2366 s
The positioning-error margin is:
Δε = 0.1000 − 0.00535 ≈ +0.09465 mm
Result: PASS. Within this equivalent SDOF model, the axis both settles before the 0.50 s time limit and remains inside the specified 0.10 mm in-position tolerance.
This result verifies only the modeled mass-stiffness-damping response under a constant-deceleration stop. It does not by itself verify controller performance, backlash, higher structural modes, drive saturation, or total machine accuracy.
Assumptions and limitations
Assumptions and limitations
- The moving mechanism is represented by one equivalent linear single-degree-of-freedom mass-spring-damper system.
- Equivalent moving mass, stiffness, and damping ratio are treated as constant throughout the evaluated stop.
- Damping is modeled as linear viscous damping. The model does not explicitly represent Coulomb friction, stiction, hysteresis, or amplitude-dependent damping.
- The commanded braking phase uses constant deceleration magnitude. Jerk-limited, S-curve, multi-segment, torque-limited, and controller-generated deceleration profiles are not reproduced directly.
- At the start of deceleration the relative compliance displacement and relative velocity are assumed to be zero.
- After the commanded stopping time, the command is treated as stationary and the remaining displacement follows the free response of the equivalent second-order system.
- The required positioning tolerance is an absolute displacement limit, not a fixed 2% or 5% settling band.
- Underdamped, critically damped, and overdamped regimes are evaluated, but zero or negative damping is rejected because finite vibration decay would not be obtained by the implemented model.
- Multiple structural modes, mode coupling, flexible links with distributed mass, cross-axis dynamics, and nonlinear structural behavior are not represented.
- Servo-controller gains, loop bandwidth, command filters, feed-forward terms, drive current limits, torque saturation, encoder dynamics, and digital sampling are not explicitly modeled.
- Backlash, clearance, lost motion, bearing play, thermal drift, geometric error, manufacturing tolerances, and calibration error are outside the calculation.
- The reported positioning error is therefore a dynamic compliance result, not a complete machine accuracy or repeatability specification.
Use measured modal data or experimentally identified equivalent parameters when available. For high-accuracy or dynamically critical machinery, verify the final design with appropriate servo simulation, modal analysis, FEA, frequency-response measurements, or physical settling tests.