Linear Module Torque Calculation: Sizing the Drive Correctly

Alinear modulemotor should not be selected from payload alone. The drive must produce enough torque to overcome the linear load, accelerate the moving mass, accelerate the screw, pulley, coupling and other rotating components, and still operate within the motor's allowable speed and thermal limits.

This is why a correctlinear module torque calculationis a sequence rather than a single formula. The same 20 kg payload can require very different motor torque depending on screw lead, pulley radius, acceleration, mounting orientation, cycle time and transmission efficiency.

This guide develops the calculation from the load side to the motor shaft, then shows how to check peak torque, continuous torque, reflected inertia and safety margin before selecting the motor and drive.

What Torque Are We Actually Calculating?

For a rotary motor driving a linear axis, the motor produces torque while the mechanism converts that torque into linear thrust. The required motor torque normally contains several parts:

• Torque required to overcome the external linear load
• Torque required to accelerate the translating mass
• Torque required to accelerate rotating components
• Friction, seal, preload and transmission losses
• Additional torque caused by gravity on a vertical axis
• Process torque or thrust during pressing, clamping or machining

Motor sizing principle:calculate the torque throughout the complete motion cycle. Maximum torque checks acceleration and short-duration demand; RMS or continuous torque checks thermal loading.

Step 1: Build the Linear Force Requirement

Before converting force into motor torque, determine the force the axis must generate.

For a simplified horizontal axis:

F_total= m × a + F_resistance+ F_process

where:

• m= total translating mass in kg
• a= linear acceleration in m/s²
• F_resistance= guide, seal, cable and other mechanical resistance in N
• F_process= external process force in N

For a vertical axis

Gravity must also be included. During upward acceleration, a simplified force expression is:

F_up= m(g + a) + F_resistance+ F_process

During downward acceleration, gravity may assist the motion instead of opposing it. The complete cycle should therefore be calculated segment by segment rather than using one torque value for both directions.

Do not forget the mass of the moving axis itself

In XY or XYZ systems, the motor often moves much more than the workpiece. A lower axis may carry another module, motor, mounting plate, cable carrier, end effector and workpiece at the same time.

The translating mass used in the calculation should include everything that actually accelerates with that axis.

Step 2: Convert Linear Force Into Drive Torque

The conversion depends on the drive mechanism. A ball screw, timing belt and rack-and-pinion system do not use the same torque relationship.

Ball screw drive

For a screw with leadpand mechanical efficiencyor, the approximate torque required to generate linear forceFis:

T_linear= F × p ÷ (2π × h)

Use consistent SI units: force in newtons, lead in meters per revolution, and torque will be in N·m.

A larger screw lead produces more linear travel per revolution, but it also requires more torque for the same thrust. A smaller lead reduces thrust torque but requires higher motor speed for the same linear velocity.

Timing belt drive

For a belt pulley with effective pitch radiusr:

T_pulley= F × r ÷ h

The radius must be the effective pitch radius used by the belt, not simply the outside radius of the pulley.

Rack-and-pinion drive

At the pinion shaft, the same basic relationship applies:

T_pinion= F × r_p÷ h

wherer_pis the pinion pitch radius. If a gearbox is installed between the motor and pinion, the torque and speed must then be reflected through the gear ratio and gearbox efficiency.

Torque and Thrust: Two Sides of the Same Transmission

Torque is measured on the rotating side of the mechanism; thrust is measured on the linear side.

For a ball screw, the relationship can also be rearranged to estimate available thrust from motor torque:

F ≈ 2π × η × T ÷ p

This relationship is useful for understanding why a small-lead screw can produce high thrust from modest motor torque. It should not be used by itself for final motor sizing because it does not include all acceleration, inertia, preload and thermal requirements.

Step 3: Calculate the Speed Requirement

A motor can have enough torque but still be unsuitable if the required speed is outside its operating range.

Ball screw speed

If the required linear speed isvand screw lead isp:

n_screw= 60 × v ÷ p

wherevis in m/s,pis in m/rev andnis in rpm.

For example, an axis moving at 0.5 m/s with a 10 mm lead screw requires:

n = 60 × 0.5 ÷ 0.01 = 3000 rpm

The screw itself must also be checked for allowable rotational speed, critical speed and other manufacturer limits, particularly on long strokes.

Belt or pinion speed

For a pulley or pinion:

ω = v ÷ r

and rotational speed can be converted from angular velocity as required. If a gearbox is used, motor speed is the output-shaft speed multiplied by the reduction ratio.

Step 4: Include Rotational Inertia

The linear-force calculation accounts for accelerating the translating mass, but the motor must also accelerate rotating components such as:

• Ball screw shaft
• Coupling
• Pulleys
• Pinion
• Gearbox elements
• Motor rotor

The torque required to accelerate rotational inertia is:

T_acc,rot= J × a

whereJis rotational inertia in kg·m²andais angular acceleration in rad/s².

Convert linear acceleration to screw angular acceleration

For a ball screw:

a_screw= 2π × a ÷ p

This shows an important design trade-off: a small screw lead reduces the torque needed to generate thrust, but it increases screw rotational speed and angular acceleration for the same linear motion.

Reflected Inertia of a Translating Load

Motor sizing software often expresses the translating mass as an equivalent rotational inertia at the screw or motor shaft.

For an ideal screw conversion, the equivalent inertia of translating massmat the screw shaft is approximately:

J_trans= m × (p ÷ 2π)²

The rotating inertia of the screw, coupling and other load-side components is then added to obtain the total external inertia at that shaft.

If a gearbox is used, load inertia reflected to the motor decreases with the square of the reduction ratio. Usingi = motor speed ÷ load-shaft speed:

J_reflected= J_load÷ i²

Why inertia matching matters

A very large reflected load inertia compared with motor rotor inertia can make acceleration and servo tuning more difficult. However, there is no single inertia ratio that is correct for every servo or stepper system.

Use the motor manufacturer's allowable or recommended inertia ratio for the selected motor, drive, tuning method and motion profile rather than applying a universal rule.

Step 5: Calculate Peak Acceleration Torque

During acceleration, the motor must overcome the operating load and accelerate all relevant inertias at the same time.

A simplified motor-side peak torque model is:

T_peak= T_load+ J_total× a_motor+ T_other

whereT_othermay include preload, seal drag, gearbox losses and other known resistance not already included in the linear-force term.

If a reduction gearbox is present, first make sure every torque and inertia value has been referred to the same shaft before adding them.

A Worked Ball Screw Torque Example

Consider a horizontalball screw linear modulewith these simplified conditions:

• Moving mass: 20 kg
• Linear acceleration: 2.5 m/s²
• Other linear resistance: 15 N
• Ball screw lead: 10 mm/rev
• Transmission efficiency: 0.90
• Maximum linear speed: 0.5 m/s
• Combined screw/coupling rotational inertia at the screw shaft: 6.0 × 10^-5kg·m²
• Direct drive, no gearbox

1. Linear force during acceleration

F = m × a + F_resistance

F = 20 × 2.5 + 15 = 65 N

2. Torque required for the linear force

T_linear= 65 × 0.01 ÷ (2π × 0.90) ≈ 0.115 N·m

3. Screw angular acceleration

α = 2π × 2.5 ÷ 0.01 ≈ 1571 rad/s²

4. Torque to accelerate screw and coupling inertia

T_rot = 6.0 × 10^-5× 1571 ≈ 0.094 N·m

5. Simplified peak mechanical torque

T_peak≈ 0.115 + 0.094 = 0.209 N·m

6. Speed check

n = 60 × 0.5 ÷ 0.01 = 3000 rpm

This example deliberately stops before selecting a motor. The final design still needs to include actual screw inertia, coupling and rotor inertia, preload or seal torque, motion-cycle RMS torque, motor torque-speed capability, screw speed limits and an appropriate design margin.

Important:a motor rated above 0.209 N·m is not automatically acceptable. It must provide the required torque at 3000 rpm, tolerate the peak acceleration demand and remain within its continuous thermal rating over the complete cycle.

Step 6: Check RMS or Continuous Torque

Peak torque determines whether the motor can execute the most demanding part of the motion. Continuous or RMS torque determines whether it can repeat the cycle without overheating.

For a cycle divided into several torque intervals, RMS torque is commonly calculated as:

T_RMS= √[(T₁²t₁+ T₂²t₂+ ... + T_n²t_n) ÷ (t₁+ t₂+ ... + t_n)]

A complete cycle may include:

• Acceleration
• Constant-speed travel
• Deceleration
• Dwell
• Return acceleration
• Return travel
• Return deceleration
• Holding or process-force period

Because torque is squared in the RMS calculation, both positive and negative torque contribute to thermal loading.

Peak Torque, Rated Torque and Holding Torque Should Not Be Confused

A servo motor may provide high peak torque for a short acceleration period but cannot necessarily sustain that value continuously.

Vertical Axis Torque Requires Extra Care

On a vertical module, gravity creates a continuous directional load.

During upward motion, the drive must lift the mass and accelerate it. During downward motion, gravity assists acceleration and may require the motor to brake or regenerate energy back through the drive.

For a ball screw vertical axis, a simplified upward thrust is:

F_up= m(g + a) + F_resistance+ F_process

The corresponding screw torque is then calculated from the screw lead and efficiency.

Do not rely on motor torque alone for safe holding

If an uncontrolled drop would create a hazard, the machine requires an appropriate holding and safety strategy. A standard motor brake should only be used within its specified function and rating; safety-related load holding may require additional measures.

How Gear Reduction Changes Torque and Inertia

A gearbox can increase available output torque and reduce reflected load inertia at the motor, but it also increases motor speed and introduces efficiency losses, backlash and additional inertia.

With reduction ratioi = motor speed ÷ output speed, a simplified relationship is:

T_output≈ T_motor× i × n_gear

and:

J_{load,reflected}= J_load÷ i²

A gearbox can therefore make inertia matching easier, but the motor must rotate faster to achieve the same output speed.

Where Should a Safety Factor Be Applied?

A design margin is normally required because calculations cannot perfectly represent every real machine condition.

Possible uncertainties include:

• Variation in friction
• Future tooling mass
• Grease and seal resistance
• Manufacturing tolerances
• Supply-voltage conditions
• Temperature
• Unmodeled cable drag
• Process-force variation

However, using one arbitrary multiplier for every application is not good engineering. The correct margin depends on the motor type, drive, duty cycle, mechanical uncertainty and the manufacturer's selection criteria.

Apply margin after the load model is complete; do not use a large safety factor to compensate for missing inputs.

Motor Torque Must Be Checked Against the Torque-Speed Curve

Motor torque is not always constant across the entire speed range.

Stepper motor torque generally falls as speed rises. Servo motors have continuous and peak operating regions that depend on speed, drive voltage, thermal conditions and the selected motor-drive combination.

Therefore, motor sizing requires a point-by-point check:

• Required maximum speed is within the motor and transmission limits.
• Required peak torque is available at the corresponding speed.
• RMS torque is within the continuous operating region.
• Regenerative energy during deceleration or vertical lowering is acceptable to the drive system.
• Required inertia ratio is within the motor manufacturer's guidance.

Common Torque-Calculation Mistakes

Using only F = mg

This calculates weight, not the complete dynamic requirement. A horizontal axis does not normally use the full payload weight as its drive force, while a vertical axis must include gravity plus acceleration and other resistance.

Ignoring acceleration

Acceleration often determines peak torque in high-cycle automation. A module that moves slowly can still need substantial peak torque if it must reach speed in a very short time.

Ignoring screw or pulley inertia

On high-acceleration systems, rotating inertia can be a significant part of the motor torque requirement.

Using millimeters directly in SI formulas

A 10 mm screw lead must be entered as 0.01 m/rev when force is in newtons and torque is required in N·m.

Adding torque values from different shafts

Motor-side and load-side torque cannot be added directly when a gearbox is present. Reflect all values to the same shaft first.

Checking peak torque but not RMS torque

The motor may execute one cycle successfully but overheat after repeated production cycles.

Applying a universal inertia-ratio rule

Allowable inertia ratio depends on the selected motor, drive and tuning strategy. Use the actual manufacturer's selection criteria.

A Practical Drive-Sizing Workflow

1. Define the motion profile.Record stroke, move time, acceleration, constant-speed time, deceleration and dwell.
2. Calculate translating mass.Include carriage-mounted tooling and upper axes.
3. Calculate the linear force in each motion segment.Include gravity, resistance and process forces.
4. Convert force to screw, pulley or pinion torque.
5. Calculate required rotational speed.
6. Calculate rotating and reflected inertia.
7. Add acceleration torque.
8. Determine peak torque across the cycle.
9. Calculate RMS or continuous torque.
10. Check inertia ratio using the selected motor manufacturer's criteria.
11. Check the motor torque-speed curve.
12. Verify screw, belt, bearing, gearbox and coupling limits.
13. Add an appropriate engineering margin.

What Data Should You Send for Motor Sizing?

For QRXQ to evaluate a linear module motor and drive, provide:

• Drive type: ball screw, timing belt, rack-and-pinion or other
• Stroke
• Moving mass
• Mounting orientation
• Maximum speed
• Acceleration or target move time
• Complete cycle time
• External process force
• Required holding force
• Motor or servo brand preference
• Gearbox ratio if used
• Duty cycle and daily operating time

Final sizing question:do not ask only “How much motor torque do I need?” The complete question is “What peak torque, continuous torque, speed and inertia capability does the motor-drive system need for this motion cycle?”

Once those four requirements are known, the motor can be matched to the linear module with much greater confidence and without relying on oversized power as a substitute for calculation.