BIZOMAT MAKEOS · Mechanics
Ball-Screw Linear Axis Sizing
Can this ball screw and motor move the load at the speed and acceleration required, and how long will the screw last?
Sizing of a ball-screw driven axis over a move cycle: screw speed, axial forces while accelerating, running and decelerating, motor torque in each phase from the handbook driving-torque relation plus rotor and screw inertia, RMS torque over the cycle, critical speed and buckling load for the end fixity, cubic-mean load and rating life in hours.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load and motion | ||
| Moving mass | kg | Mass of the carriage and load. |
| Guide friction coefficient | Friction coefficient of the linear guides. | |
| External axial force while moving | N | Process or external force opposing motion. |
| Maximum speed | mm/s | Top linear speed. |
| Acceleration (and deceleration) time | s | Time to reach top speed (the same for stopping). |
| Constant-speed time | s | Time at constant speed in each move. |
| Dwell time per cycle | s | Stationary time per cycle. |
| Screw and drive | ||
| Lead | mm | Screw lead (travel per revolution). |
| Root diameter | mm | Root diameter of the screw. |
| Length between supports | mm | Unsupported length between bearings. |
| End fixity | choice | Mounting of the screw ends. |
| Screw efficiency | % | Ball-screw efficiency (about 90%). |
| Motor and coupling inertia | kg·m² | Rotor plus coupling inertia. |
| Dynamic load rating | kN | Dynamic load rating of the ball nut. |
| Load factor | Load factor for vibration and shock. | |
| Required life | h | Life you need in hours of this cycle. |
| Limits | ||
| Allowed share of critical speed | % | Share of critical speed allowed. |
| Allowed share of buckling load | % | Share of buckling load allowed. |
| Screw modulus | GPa | Elastic modulus of the screw. |
| Screw density | kg/m³ | Density of the screw. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
Load and motion
What this means: Moving mass
- What it is
- Everything the ball screw pushes along the axis: carriage, table, fixtures and the heaviest workpiece or payload. It sets both guide friction and the force needed to accelerate.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 1000000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More mass raises the friction force, the accelerating force and the acceleration torque.
- Get it right
- Include the payload at its heaviest, not just the bare carriage; weigh parts or take masses from the CAD model.
What this means: Guide friction coefficient
- What it is
- Rolling or sliding friction coefficient of the linear guides carrying the carriage. Multiplied by the weight of the moving mass, it gives the guide drag the screw must overcome.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.01 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds μ m g to the axial force in every phase of the move.
- Get it right
- Take it from the guide maker’s data for your guide type and preload; seals and wipers add drag on top.
What this means: External axial force while moving
- What it is
- Steady axial force the axis works against while it travels, such as a cutting, pressing or spring force acting along the screw axis. It is added to guide friction in every phase.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the running force, all three torques and the cubic-mean load, so life falls.
- Get it right
- Use the force along the screw axis only; forces across the axis load the guides, not the screw.
What this means: Maximum speed
- What it is
- Top linear speed of the carriage during the constant-speed part of each move. With the lead it sets how fast the screw turns.
- Units and format
- Enter a number in mm/s: millimetres per second. The form accepts 0.000001 mm/s to 100000 mm/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 mm/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Screw speed is speed ÷ lead, so a faster axis approaches the critical speed and needs more acceleration torque.
- Get it right
- This is linear carriage speed in millimetres per second, not motor speed and not metres per minute.
What this means: Acceleration (and deceleration) time
- What it is
- Time the carriage takes to ramp from rest to top speed. The tool uses the same time to slow down again, giving a symmetric trapezoidal move.
- Units and format
- Enter a number in s: seconds. The form accepts 0.000001 s to 100 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.2 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Shorter times raise acceleration, peak torque and the accelerating axial force.
- Get it right
- A shorter ramp means higher acceleration; check it against what the motor and mechanics can really achieve.
What this means: Constant-speed time
- What it is
- Time spent at top speed in each move, between the end of acceleration and the start of deceleration.
- Units and format
- Enter a number in s: seconds. The form accepts 0 s to 10000 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Longer constant-speed time weights the RMS torque and life towards the running force.
- Get it right
- Work it out from the stroke: the ramps cover speed × ramp time between them, the rest of the stroke is at constant speed.
What this means: Dwell time per cycle
- What it is
- Time the axis stands still in each cycle, for loading, machining at rest or waiting, before the next move starts.
- Units and format
- Enter a number in s: seconds. The form accepts 0 s to 100000 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More dwell lengthens the cycle, lowering RMS torque and spreading revolutions over more hours of life.
- Get it right
- Count all stationary time in one repeat of the cycle; it lowers RMS torque but does not change peak torque.
Screw and drive
What this means: Lead
- What it is
- Distance the nut and carriage travel for one revolution of the screw. On a single-start screw it equals the pitch; on multi-start screws it is pitch × starts.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 200 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer lead lowers screw speed but raises the torque needed for each newton of axial force.
- Get it right
- Do not confuse lead with pitch on multi-start screws; take it from the screw designation or catalogue.
What this means: Root diameter
- What it is
- Diameter at the bottom of the ball groove, the smallest solid section of the screw shaft. The tool uses it for stiffness, critical speed, buckling and screw inertia.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 500 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Critical speed rises with the root diameter and buckling load with its fourth power.
- Get it right
- Use the root diameter from the catalogue, not the nominal or ball-centre diameter, which are larger.
What this means: Length between supports
- What it is
- Unsupported length of screw between its end bearings, or from the fixed bearing to the nut at its furthest position on a fixed–free screw.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Critical speed and buckling load both fall with the square of this length.
- Get it right
- Use the longest unsupported span that occurs during travel, not the overall screw length.
What this means: End fixity
- What it is
- How the two ends of the screw are held by their bearings. The choice sets the factors used for critical speed and for buckling.
- Units and format
- Choose one of 4 options from the list: Fixed – fixed; Fixed – supported; Supported – supported; Fixed – free.
- Example
- The form opens with “Fixed – supported” selected as a worked example. Choose the option that matches your case.
- Get it right
- A single deep-groove bearing acts as supported; a matched pair of angular-contact bearings acts as fixed.
What this means: Screw efficiency
- What it is
- Share of the motor’s input work that reaches the carriage as useful linear work, after friction losses in the ball nut.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 10% to 99%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lower efficiency raises the torque needed to push the axial force.
- Get it right
- Use the forward-drive efficiency from the screw maker; type the percentage, not a fraction.
What this means: Motor and coupling inertia
- What it is
- Rotational inertia of the motor rotor plus the coupling between motor and screw. The tool adds the screw’s own inertia from its root diameter and length.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0 kg·m² to 100 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0002 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds to the torque needed only while accelerating and decelerating.
- Get it right
- Take the rotor inertia from the motor data sheet; check whether it is quoted in kg·cm² (divide by 10,000 for kg·m²).
What this means: Dynamic load rating
- What it is
- Basic dynamic load rating of the ball nut: the axial load at which it reaches its rated life of one million revolutions. Life is worked out from this rating.
- Units and format
- Enter a number in kN: kilonewtons (1 kN = 1000 N). The form accepts 0.000001 kN to 10000 kN. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15 kN as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Life rises with the cube of the rating.
- Get it right
- Use the dynamic rating Ca, not the static rating C0a, from the screw catalogue.
What this means: Load factor
- What it is
- Multiplier on the cubic-mean axial load that allows for vibration and shock in service before the life is calculated.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 3. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Life falls with the cube of this factor.
- Get it right
- Take it from the ball-screw maker’s guidance for your operating conditions.
What this means: Required life
- What it is
- Running life you need from the screw, in hours of this move cycle repeated, including dwell time.
- Units and format
- Enter a number in h: hours. The form accepts 0.000001 h to 10000000 h. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Base it on real operating hours per year times the years of service you expect.
Limits
What this means: Allowed share of critical speed
- What it is
- The share of the calculated critical (whirling) speed that you allow the screw to reach. It sets the speed limit for the pass or fail check.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 10% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type a percentage of critical speed; the screw maker’s recommendations or your design standard set it.
What this means: Allowed share of buckling load
- What it is
- The share of the calculated Euler buckling load that you allow the accelerating axial force to reach. It acts as a safety margin against buckling.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 10% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Type a percentage of buckling load, not a safety factor: a share of p % equals a safety factor of 100 ÷ p.
What this means: Screw modulus
- What it is
- Elastic (Young’s) modulus of the screw shaft material, used for the bending stiffness in the critical-speed and buckling checks.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 500 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Critical speed rises with √E and buckling load in proportion to E.
- Get it right
- Enter gigapascals; 1 GPa = 1,000 MPa. Take it from the material data sheet.
What this means: Screw density
- What it is
- Density of the screw shaft material. It sets the screw’s mass for its rotational inertia and for the whirling (critical speed) calculation.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 20000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A denser screw has a lower critical speed and more inertia.
- Get it right
- Use the material data sheet value for the screw steel or alloy.
Method and formulas
- Screw speed n = v ÷ lead; acceleration a = v ÷ t_a; axial force = external + μ m g ± m a.
- Drive torque T = F × lead ÷ (2π e) (handbook ball-screw driving torque) plus (J_motor + J_screw) α when accelerating.
- RMS torque = √[Σ T² t ÷ cycle time]; critical speed and buckling as the screw-limits tool (same end-fixity factors).
- Cubic-mean load weighted by travel; life = (C_a ÷ f_w F_m)³ × 10⁶ revolutions.
- Example: 200 kg at 500 mm/s on a 10 mm lead needs 2.3109 N·m peak and 1.0149 N·m RMS.
Limitations
- Horizontal axis; vertical axes add the load weight to every phase.
- Screw inertia uses the root diameter (slightly low).
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Choose a ball screw and servo motor for a linear axis
- Check an existing axis for a faster cycle
- Find whether speed, buckling, torque or life limits the axis
Who it is for
- Machine designers
- Automation engineers
- Design engineers
What you get
- Screw speed, forces and motor torques
- Critical speed and buckling checks
- Cubic-mean load and life
Interpreting the result
- If critical speed limits, use a coarser lead, a stiffer end support, or a rotating nut.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2561–2563. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Useful alongside · Library collection
Power screws and linear motion: leadscrews, ball screws, linear guides and cams
In Machine Elements and Mechanical Design
What it covers: Turning rotation into accurate straight-line motion is central to machines and automation. How to size leadscrews and ball screws, choose guides and actuators, and design cam motion.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.