Sizing a Motor and Screw as One System

A screw converts motor rotation into carriage travel, but motor and screw cannot be sized independently. Screw lead sets both motor speed and torque; screw diameter and length affect rotational inertia, critical speed and buckling; the motion profile determines acceleration torque and heating. A complete selection therefore traces one consistent set of load, motion, transmission and duty assumptions through every component limit.

Use one requirements-to-validation loop

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Diagram description

A left-to-right workflow begins with application inputs, then calculates forces, defines a speed profile, calculates load and inertia, peak torque and RMS torque, and finishes with motor-catalogue validation. A return arrow emphasizes iteration.

The complete sizing sequence

1. 1. Define motion and duty.Record mass, stroke, orientation, speed, acceleration, deceleration, dwell, reversals, cycles per hour, accuracy, environment and credible load variation.
2. 2. Resolve all linear forces.Include inertia, friction, gravity, preload, seals, cable carriers and process forces with signs for each motion interval.
3. 3. Select a candidate screw and support arrangement.Choose lead, diameter, accuracy, preload, length and bearings provisionally; these values change speed, torque, inertia, life and stability.
4. 4. Convert to motor-side speed, torque and inertia.Include screw efficiency, transmission ratio, screw and coupling inertia, and any gearbox or belt losses.
5. 5. Build torque-time and speed-time profiles.Calculate acceleration, constant-speed, deceleration, dwell and return segments rather than one average operating point.
6. 6. Test candidate motor-drive combinations.Check peak torque at speed, RMS or equivalent thermal demand, overload duration, maximum speed, inertia guidance, voltage, current and regeneration.
7. 7. Verify the entire axis.Check screw critical speed, buckling, life, bearings, coupling, alignment, feedback, mounting, environment, stopping, holding and safety.

Fix units before using any equation

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Diagram description

A motion axis separates linear variables in SI units from rotational variables and adds a duty-cycle section for time, percentage, peak torque and RMS torque.

Worked example: a horizontal 50 kg axis

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Diagram description

Inputs are 50 kilograms moving mass, 300 millimetres stroke, 0.25 metres per second, 2 metres per second squared, 10 millimetres per revolution, 90 percent efficiency and 50 newtons friction. It derives 150 newtons total acceleration force, 0.265 newton-metres screw torque and 1500 revolutions per minute.

1. Acceleration force

F_total = ma + F_friction = (50 × 2.0) + 50 = 150 N

This is a horizontal-axis simplification with no process force. A vertical or inclined axis must include the signed gravity component, and real friction may differ with direction, speed, preload and temperature.

F_total

linear force during positive acceleration, N

m

moving mass, kg

a

linear acceleration, m/s²

F_friction

assumed resisting force, N

Example

At constant speed, the inertial term becomes zero and the simplified linear force is 50 N. During commanded deceleration, the inertial and friction terms must be combined with their correct directions.

2. Screw torque from linear force

T_s = FL ÷ (2πη) = (150 × 0.010) ÷ (2π × 0.90) = 0.265 N·m

This includes translating-mass acceleration through the force term, but it does not include torque to accelerate the rotating screw, coupling or motor rotor. Preload and seal torque may also need separate terms.

T_s

screw torque caused by axial force, N·m

F

axial force, N

L

screw lead, m/rev

η

positive driving efficiency

Example

At constant speed with 50 N friction, the corresponding simplified screw torque is about 0.088 N·m.

3. Screw speed from linear speed

n_s = (v ÷ L) × 60 = (0.250 ÷ 0.010) × 60 = 1500 rpm

A smaller lead reduces force-related torque but raises screw speed for the same linear speed. A larger lead does the opposite. That trade changes critical-speed, inertia, resolution and motor-curve checks.

n_s

screw speed, rpm

v

linear speed, m/s

L

screw lead, m/rev

Example

The direct-coupled motor also runs at 1500 rpm. A belt or gearbox would require its ratio and efficiency to be applied.

4. Angular acceleration and rotating-inertia torque

α = 2πa ÷ L = 1256.6 rad/s²; T_rot = J_rot,total α = 0.00030 × 1256.6 = 0.377 N·m

The illustrative 0.00030 kg·m² includes the assumed screw-plus-coupling inertia and candidate motor rotor inertia. The moving mass is not added again here because its acceleration already appears in the 150 N linear force.

α

motor and screw angular acceleration, rad/s²

J_rot,total

directly coupled screw, coupling and rotor inertia, kg·m²

T_rot

torque to accelerate those rotating parts, N·m

Example

The positive acceleration torque before margin is approximately 0.265 + 0.377 = 0.642 N·m. A 1.5 illustrative design factor applied after this sum gives about 0.963 N·m for candidate screening. Final margins must have a documented basis.

5. Reflected translating-load inertia

J_trans = m(L ÷ 2π)² = 50(0.010 ÷ 2π)² = 0.000127 kg·m²

For an ideal kinematic reflection, external inertia at the motor is the translated load term plus screw, coupling and other driven rotating inertia. Efficiency belongs in the torque and power-loss model; follow the selected manufacturer's sizing convention.

J_trans

moving mass reflected to the screw shaft, kg·m²

m

moving mass, kg

L

screw lead, m/rev

Example

Adding the assumed screw and coupling inertia gives about 0.000327 kg·m² external to the motor. Dividing by the illustrative 0.00010 kg·m² rotor inertia gives a 3.27:1 external-load-to-rotor ratio; only the exact motor-drive maker can determine whether that is acceptable.

6. Example cycle timing and RMS torque

T_RMS = √[(0.642²×0.125 + 0.088²×1.075 + (−0.465)²×0.125 + 0²×0.675) ÷ 2.000] ≈ 0.209 N·m

Accelerating to 0.25 m/s at 2 m/s² takes 0.125 s and travels 0.015625 m. The same deceleration distance leaves 0.26875 m at constant speed, or 1.075 s. With 0.675 s dwell, the example cycle is 2.000 s. The deceleration value combines negative inertia torque with the assumed opposing friction direction. Real return motion and holding torque must be included when present.

T_RMS

example equivalent cycle torque, N·m

T_i

signed motor torque for interval i, squared in the formula

t_i

duration of interval i, s

Example

The candidate still needs at least the calculated peak torque at 1500 rpm inside its permitted intermittent region and about 0.209 N·m inside the relevant continuous thermal region, with justified margins and manufacturer overload rules.

Validate a candidate against every operating condition

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Diagram description

A motor and drive connect through a coupling to a screw axis. Six panels call out electrical matching, torque-speed margin, coupling selection, mechanical mounting, feedback and controls, and thermal and environmental conditions.

Commissioning evidence to retain

1. 1. Preserve the calculation package.Store inputs, units, assumptions, formulas, source documents, software version and rejected candidates.
2. 2. Verify configuration.Record motor and drive codes, feedback, current and speed limits, control mode, brake logic and travel protection.
3. 3. Test progressively.Run reduced-energy checks before maximum speed, acceleration, load and duty.
4. 4. Measure performance.Capture speed, current or torque estimate, following error, settling, vibration, temperature and stopping behavior.
5. 5. Test worst defined cases.Include maximum load, warm operation, low supply where relevant, reversal and credible disturbances.
6. 6. Set acceptance limits.Compare measurements with documented requirements, not merely whether the axis moves.

Use Understanding Motor Torque–Speed Curves for the candidate-envelope checks, Reading Motor Specs Before Buying Components for rating conditions, and Connecting and Mounting a Motor Correctly for installed-axis verification. Keep the sizing model alive after purchase so measured commissioning data can confirm or correct its assumptions.

Selection boundary and evidence to keep with the decision

Treat “Sizing a Motor and Screw as One System” as a selection problem, not only a definition. Record the required result, alternatives considered, the controlling assumption and the condition that would change the choice.

Keep evidence proportional to consequence: 7. Verify the entire axis. Check screw critical speed, buckling, life, bearings, coupling, alignment, feedback, mounting, environment, stopping, holding and safety. Revisit the decision when the load, environment, duty, interface, regulation or maintenance plan changes.

E3 · Educational scene

Explore the decision in three states

Select a state to compare the starting condition, the important change and the evidence needed before acting. The same meaning remains in the static list.

StartWhat changesVerify

Start

Follow a traceable motor-and-screw sizing workflow from linear requirements through force, screw speed, torque, reflected inertia, peak and RMS checks, then verify mechanical, electrical and environmental integration.

What changes

Millimetres hidden in a torque equation create a thousand-fold error. If force is in newtons and torque is required in newton-metres, screw lead must be in metres per revolution. A 10 mm/rev lead is 0.010 m/rev. Preserve units beside intermediate values so the error is visible.

Verify

7. Verify the entire axis. Check screw critical speed, buckling, life, bearings, coupling, alignment, feedback, mounting, environment, stopping, holding and safety.

1. Follow a traceable motor-and-screw sizing workflow from linear requirements through force, screw speed, torque, reflected inertia, peak and RMS checks, then verify mechanical, electrical and environmental integration.
2. Millimetres hidden in a torque equation create a thousand-fold error. If force is in newtons and torque is required in newton-metres, screw lead must be in metres per revolution. A 10 mm/rev lead is 0.010 m/rev. Preserve units beside intermediate values so the error is visible.
3. 7. Verify the entire axis. Check screw critical speed, buckling, life, bearings, coupling, alignment, feedback, mounting, environment, stopping, holding and safety.

One-minute questions

Questions to answer before you apply this lesson

What should I select or check first for Sizing a Motor and Screw as One System?

1. Define motion and duty. Record mass, stroke, orientation, speed, acceleration, deceleration, dwell, reversals, cycles per hour, accuracy, environment and credible load variation.

Which assumption can change the conclusion?

Load, motion profile, geometry, alignment, duty, lubrication and environment match the selected mechanism and ratings.

What can go wrong if I copy the example directly?

Millimetres hidden in a torque equation create a thousand-fold error. If force is in newtons and torque is required in newton-metres, screw lead must be in metres per revolution. A 10 mm/rev lead is 0.010 m/rev. Preserve units beside intermediate values so the error is visible.

How do I verify the result in the real system?

7. Verify the entire axis. Check screw critical speed, buckling, life, bearings, coupling, alignment, feedback, mounting, environment, stopping, holding and safety.

Sources and further reading

1. Servomotor Capacity Selection ExamplesYaskawa Electric Corporation · 2026-10-12Primary application examples reviewed for ball-screw speed, load torque, linear and rotating inertia, motion timing and power calculations.
2. Servo Motor Sizing ConceptsYaskawa America, Inc. · 2026-10-12Primary technical document reviewed for speed, peak torque at speed, RMS torque and inertia ratio as separate selection checks.
3. 9 Things to Consider When Specifying Servo MotorsYaskawa America, Inc. · 2026-10-12Manufacturer paper reviewed for motion-profile, mechanism, load, ball-screw inertia, efficiency and environment inputs.
4. Sigma-X Series Rotary Servomotor Product ManualYaskawa Electric Corporation · 2026-10-12Current primary manual reviewed for horizontal ball-screw load-torque and motor-shaft inertia calculations and product-specific selection conditions.
5. Motor Sizing CalculationsOriental Motor U.S.A. Corp. · 2026-10-12Manufacturer technical reference reviewed for inertia, torque and speed as separate motor-sizing quantities.
6. Ball / Lead Screw Sizing ToolOriental Motor U.S.A. Corp. · 2026-10-12Manufacturer sizing workflow reviewed for motion inputs, screw geometry, efficiency, torque, speed, RMS torque, inertia and stopping-accuracy reporting.
7. Ball Screw Product InformationTHK · 2026-10-12Primary manufacturer material reviewed for screw conversion efficiency, selection context and the need for load, speed, accuracy and life checks.