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Ball Screw Calculator

Critical speed, buckling, dynamic load rating, life and drive torque in one check.

Size a ball screw for a linear axis the way catalogue engineers do. Enter the axial load, speed, lead, screw diameters, unsupported length and end support, and the calculator checks the d·n speed limit, whirling (critical) speed, Euler buckling and the dynamic load rating needed for your life target.

  • Live results
  • Full step-by-step working
  • PASS / FAIL checks
  • Free PDF report

Inputs

From catalogue; ≈ 0.85 × d₀ if unknown
Distance between bearings (or bearing to nut at max stroke)

Results

Screw speed
1500 rpm
PASS
Critical speed
3213 rpm
PASS
Allowable axial load
20543 N
PASS
Required Ca
17.38 kN
Driving torque
2.653 N·m
Step-by-step calculation (9 steps)
  1. 1Screw rotational speed
    Formulan = v·60 000 / l
    Solutionn = 0.25 × 60 000 / 10
    Resultn = 1500 rpm
  2. 2d₀·n speed limit
    Formulad₀·n ≤ 70000
    Solutiond₀·n = 25 × 1500
    Resultd₀·n = 37500 ≤ 70000PASS
  3. 3Critical (whirling) speed
    Formulanc = 0.8 × (60/2π) × (λ / Lb²) × (d₁/4) × √(E/ρ)
    Solutionnc = 0.8 × 9.549 × (15.42 / 1²) × (0.0213/4) × √(206×10⁹ / 7850) [Fixed-supported, λ = 3.927²]
    Resultnc = 3213 rpm; n / nc = 0.467PASS
  4. 4Root section second moment of area
    FormulaI = π·d₁⁴ / 64
    SolutionI = π × 21.3⁴ / 64
    ResultI = 10104 mm⁴
  5. 5Buckling load (Euler) and allowable
    FormulaPcr = η₁·π²·E·I / Lb², Pallow = 0.5·Pcr
    SolutionPcr = 2 × π² × 206 000 × 10104 / 1000² = 41085 N
    ResultPallow = 20543 N ≥ 1500 NPASS
  6. 6Required life in revolutions
    FormulaL = 60·n·Lh
    SolutionL = 60 × 1500 × 10000
    ResultL = 9×10⁸ rev
  7. 7Required dynamic load rating
    FormulaCa,req = fw·Fa·(L / 10⁶)^(1/3)
    SolutionCa,req = 1.2 × 1500 × (900)^(1/3)
    ResultCa,req = 17379 N (17.38 kN)
  8. 8Driving torque (constant speed)
    FormulaT = Fa·l / (2π·η)
    SolutionT = 1500 × 10 / (2π × 0.9) = 2653 N·mm
    ResultT = 2.653 N·m
  9. 9Driving power
    FormulaP = T·2π·n / 60
    SolutionP = 2.653 × 2π × 1500 / 60
    ResultP = 416.7 W

Engineering takeaways

  • Pick a catalogue size with dynamic rating Ca ≥ 17.38 kN, then enter it to confirm the life.
  • Torque shown is for constant speed. Add acceleration torque from rotating inertia, nut preload torque and support bearing friction when sizing the motor.
  • Life uses the maximum load. For a real duty cycle, a cubic mean load gives a longer, more realistic life.

E = 206 GPa, ρ = 7850 kg/m³. Critical speed includes a 0.8 safety factor; buckling allowable includes 0.5.

The PDF report contains every input, formula, substitution and check shown above, in LTM calculation-sheet format.

Worked Example

The calculator then gives the constant-speed drive torque and power so you can move straight on to servo motor sizing. All four end-support cases are covered: fixed-free, supported-supported, fixed-supported and fixed-fixed.

The calculator above opens with this example. Change any input to run your own case.

Inputs

Max axial load Fa1500 N
Load factor fw1.2 normal operation
Max linear speed v0.25 m/s
Lead l10 mm
Nominal (ball centre) diameter d₀25 mm
Root diameter d₁21.3 mm
Unsupported length Lb1000 mm
End supportFixed-supported
Ball screw efficiency η0.9
Required life10000 h
Catalogue dynamic rating Ca (0 = skip)0 N
d₀·n limit70 000 standard ground

Results

Screw speed1500 rpm PASS
Critical speed3213 rpm PASS
Allowable axial load20543 N PASS
Required Ca17.38 kN
Driving torque2.653 N·m

Ball Screw Formulas Used

Screw speed

n=v⋅60 000ln = \frac{v \cdot 60\,000}{l}
  • vlinear speed (m/s)
  • llead (mm)

Critical speed (0.8 safety)

nc=0.8⋅602π⋅λLb2⋅d14Eρn_c = 0.8\cdot\frac{60}{2\pi}\cdot\frac{\lambda}{L_b^2}\cdot\frac{d_1}{4}\sqrt{\frac{E}{\rho}}
  • λ(βL)²: 3.52 fixed-free, 9.87 supported-supported, 15.4 fixed-supported, 22.4 fixed-fixed
  • d₁root diameter
  • Lbunsupported length

Buckling (0.5 safety)

Pallow=0.5⋅η1 π2EILb2,I=πd1464P_{allow} = 0.5\cdot\frac{\eta_1\,\pi^2 E I}{L_b^2},\qquad I = \frac{\pi d_1^4}{64}
  • η₁0.25, 1, 2, 4 for the four support cases

Required dynamic load rating

Ca,req=fw Fa(60 n Lh106)1/3C_{a,req} = f_w\,F_a\left(\frac{60\,n\,L_h}{10^6}\right)^{1/3}
  • fwload factor
  • Lhrequired life (h)

Drive torque

T=Fa l2π ηT = \frac{F_a\,l}{2\pi\,\eta}
  • ηball screw efficiency, 0.85 to 0.95

Basis: JIS B 1192 / ISO 3408 rating life; Euler buckling; shaft whirling theory. Results are for preliminary design and should be confirmed against supplier data and the governing standard.

Ball Screw Calculator FAQ

How do I calculate ball screw critical speed?

Critical speed is the first bending natural frequency of the rotating screw. It rises with root diameter and falls with the square of unsupported length. Catalogues apply a 0.8 safety factor; the calculator includes it, so operating speed should stay below the value shown.

Which end support should I choose?

Fixed-supported is the most common: angular contact bearings at the motor end, a deep groove bearing at the far end. Fixed-fixed doubles stiffness and raises critical speed for long, fast screws. Fixed-free is only suitable for short, slow screws.

What is the d·n value of a ball screw?

d·n is the nominal diameter in mm multiplied by speed in rpm. It limits ball recirculation speed. Standard precision ground screws are usually limited to about 70 000, rolled screws to about 50 000 and high-speed nuts to 100 000 or more.

How do I choose the ball screw lead?

A larger lead gives more speed per revolution and lowers screw rpm, which helps critical speed and d·n. A smaller lead gives higher thrust per N·m of motor torque and better resolution. Balance motor speed and torque against the required axis speed.

Does the torque include acceleration?

No. The value is the steady driving torque against the axial load. Add the acceleration torque of the screw, coupling and load inertia, plus nut preload and bearing friction, when sizing the motor. Use the servo motor sizing calculator for that step.

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