From Linear Force to Motor Torque

A linear-axis motor is not selected from payload mass alone. The motor responds to the force required in each part of the move, the screw lead and losses that convert that force into torque, and the inertia of every rotating part that must speed up or slow down. A useful calculation therefore begins with a signed motion profile and ends with a torque-versus-time and speed-versus-time record—not one average torque value.

Translate the motion profile into force intervals

Open full image

Diagram description

Three aligned qualitative plots show velocity rising, remaining constant and falling; acceleration stepping from positive to zero to negative; and the mass-times-acceleration force following the same signed steps. The graphic states that the complete drive force also includes friction, gravity components and process loads.

Signed axial force for interval i

F_axis,i = m a_i + F_friction,i + F_gravity,i + F_process,i + F_other,i

Define each added term as a force that the drive must overcome in the selected positive direction. For an axis inclined by angle θ above horizontal, one common convention gives a gravity term m g sin θ when positive travel is uphill. Friction opposes actual or impending motion, so its sign normally reverses when velocity reverses. Cable carriers, seals, preload, cutting force and spring force belong in the ledger when they are real contributors.

F_axis,i

net axial drive force required in interval i, N

m

moving mass, kg

a_i

signed linear acceleration in interval i, m/s²

F_friction,i

signed guide, seal and mechanism resistance, N

F_gravity,i

signed component of weight along the axis, N

F_process,i

signed external working load, N

F_other,i

other documented interval-specific force, N

Example

For a horizontal 100 kg carriage accelerating at +2.0 m/s² against 100 N friction and a 700 N process load, the required positive force is 100×2.0 + 100 + 700 = 1000 N.

Convert axial force at the screw, then add rotational demand

Open full image

Diagram description

A motor drives a screw and carriage while a 1000 newton arrow marks the axial load. The calculation panel uses force times 0.010 metre per revolution divided by two pi and 0.90 efficiency, producing 1.77 newton-metres. A note warns that higher lead requires higher torque for the same axial force.

Positive driving torque at the screw

T_screw,load = F_axis L ÷ (2π η_drive)

This relation expresses work per revolution: ideal rotary work 2πT equals linear work FL. The efficiency term accounts for the selected screw's positive, motor-to-load driving direction. Use lead as travel per revolution in metres per revolution when torque is in newton-metres. The equation does not add screw, coupling or rotor acceleration torque.

T_screw,load

torque at the screw due to axial force, N·m

F_axis

signed axial drive force for the interval, N

L

screw lead, m/rev

η_drive

positive efficiency for the stated load and direction

Example

For 1000 N, 0.010 m/rev and η = 0.90, T = (1000×0.010)/(2π×0.90) = 1.768 N·m, which rounds to the diagram's 1.77 N·m.

Direct-drive speed and angular acceleration

n_s = 60v ÷ L; ω_s = 2πv ÷ L; α_s = 2πa ÷ L

These are direct-coupled kinematic relations. A small lead reduces force-related torque but increases screw speed and angular acceleration for the same linear motion. A belt or gearbox changes the motor-side values by its defined ratio, while its losses and inertia must be added separately.

n_s

screw rotational speed, rpm

ω_s

screw angular speed, rad/s

α_s

screw angular acceleration, rad/s²

v

linear speed, m/s

a

linear acceleration, m/s²

L

screw lead, m/rev

Example

At v = 0.40 m/s with a 0.010 m/rev lead, the screw reaches 2400 rpm. At a = 2.0 m/s², its angular acceleration magnitude is about 1256.6 rad/s².

Torque to accelerate directly coupled rotating parts

T_rot,i = J_rot,total α_s,i; T_motor,i = T_screw,load,i + T_rot,i + T_other,i

For a direct drive, J_rot,total can include motor rotor, coupling, screw and other parts rotating at screw speed. Use signed angular acceleration. Do not include the translating mass both as m a in the force equation and again as a reflected-inertia acceleration term; those are alternative domains for the same translational kinetic effect. Manufacturer sizing tools may organize the terms differently, so reconcile definitions before comparing results.

T_rot,i

rotating-inertia torque in interval i, N·m

J_rot,total

sum of directly coupled rotating inertias, kg·m²

α_s,i

signed screw angular acceleration, rad/s²

T_motor,i

calculated electromagnetic motor torque demand, N·m

T_other,i

documented additional motor-side torque, N·m

Example

If the directly coupled rotating inertia is an illustrative 0.00025 kg·m², accelerating at 1256.6 rad/s² adds +0.314 N·m; the same term becomes −0.314 N·m during equal deceleration.

Build a signed segment table before choosing a motor

Carry the full profile into motor and drive verification

What the next calculation must preserve

1. 1. Pair torque with speed.Every interval needs motor torque, motor speed and duration; a peak torque without its speed cannot be checked against a torque-speed envelope.
2. 2. Preserve signs.Signed torque and speed reveal motoring, braking and potentially regenerative quadrants even though later RMS arithmetic squares torque.
3. 3. Include the complete cycle.Return moves, holds, clamps, settling and dwell belong in the repeated sequence when they consume torque or cooling time.
4. 4. Separate peak and thermal questions.The largest instantaneous demand and the cycle's heating-equivalent demand must each pass their own manufacturer limits.
5. 5. Check the drive and supply.Peak and continuous current, voltage, feedback, overload duration and regenerative-energy handling can reject a motor-drive combination that passes a simple torque calculation.
6. 6. Validate the mechanics.Screw critical speed, buckling, life, bearings, coupling, alignment and mounting remain independent limits.

Use Sizing a Motor and Screw as One System for the complete requirements-to-commissioning loop, Reading Motor Specs Before Buying Components for rating conditions, and Understanding Motor Torque–Speed Curves to test each torque-speed point. Preserve this signed segment table as the calculation input and later compare it with measured commissioning traces.

Selection boundary and evidence to keep with the decision

Treat “From Linear Force to Motor Torque” 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: 6. Validate the mechanics. Screw critical speed, buckling, life, bearings, coupling, alignment and mounting remain independent limits. Revisit the decision when the load, environment, duty, interface, regulation or maintenance plan changes.

One-minute questions

Questions to answer before you apply this lesson

What should I select or check first for From Linear Force to Motor Torque?

1. Pair torque with speed. Every interval needs motor torque, motor speed and duration; a peak torque without its speed cannot be checked against a torque-speed envelope.

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?

Friction is not always one constant number. Starting friction can differ from running friction, preload and seals can make resistance direction-dependent, and lubrication and temperature can change it.

How do I verify the result in the real system?

6. Validate the mechanics. Screw critical speed, buckling, life, bearings, coupling, alignment and mounting remain independent limits.

Sources and further reading

1. Examples of Calculating Driving TorqueTHK · 2026-10-12Primary manufacturer reference reviewed for positive screw efficiency and the axial-force-to-driving-torque relation, including a worked ball-screw example.
2. Ball Screw Product InformationTHK · 2026-10-12Primary manufacturer material reviewed for the distinction between positive and reverse efficiency and the relationship between thrust and torque.
3. Sigma-7-Series AC Servo Drive Rotary Servomotor with 400 V-Input Power Product ManualYaskawa Electric Corporation · 2026-10-12Primary product manual reviewed for horizontal and vertical ball-screw force, motor-shaft torque, speed, inertia and capacity-selection formulas.
4. Sigma-X Series Rotary Servomotor Product ManualYaskawa Electric Corporation · 2026-10-12Current primary manual reviewed for ball-screw load torque, motor-shaft inertia and product-specific operating conditions.
5. 9 Things to Consider When Specifying Servo MotorsYaskawa America, Inc. · 2026-10-12Manufacturer paper reviewed for motion-profile, load, mechanism, efficiency and environment inputs that must precede motor selection.
6. Ball / Lead Screw Sizing ToolOriental Motor U.S.A. Corp. · 2026-10-12Manufacturer sizing workflow reviewed as an independent comparison for load, speed, acceleration torque, inertia, cycle and stopping inputs.