Duty Cycle, RMS Torque and Reflected Inertia

A motor can survive a short high-torque event yet overheat during repeated operation, or it can have adequate continuous torque but fail to accelerate a high-inertia load. RMS torque and reflected inertia address those different questions. They are valuable screening tools only when every torque segment is expressed at the motor shaft, the complete repeating cycle is included, and the result is checked against the exact motor-drive manufacturer's conditions.

Calculate RMS torque from the complete repeated cycle

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Diagram description

The diagram shows 4 N·m for 1 second, 1 N·m for 2 seconds, −2 N·m for 1 second and 0 N·m for 1 second. Its calculation sums 16, 2, 4 and 0 N²·m²·s, divides by 5 seconds and reports approximately 2.10 N·m.

Piecewise-constant cycle RMS torque

T_RMS = √[Σ(T_i² t_i) ÷ Σt_i]

Calculate all T_i values in one motor-shaft reference frame and include every acceleration, cruise, deceleration, return, settle and dwell interval in the repeating cycle. Torque sign disappears after squaring, so a braking interval can heat the motor even though its mechanical power direction differs from a motoring interval. Use the signed profile separately for energy-flow and regeneration analysis.

T_RMS

root-mean-square motor torque over the repeated cycle, N·m

T_i

motor torque during interval i, N·m

t_i

duration of interval i, s

Σt_i

complete repeated-cycle duration, s

Example

For 4 N·m×1 s, 1 N·m×2 s, −2 N·m×1 s and 0 N·m×1 s, T_RMS = √[(16+2+4+0)/5] = √4.4 = 2.0976 N·m, or 2.10 N·m.

Treat RMS torque as a thermal screen, not a temperature prediction

For many permanent-magnet servo applications within their specified operating region, torque is approximately proportional to torque-producing current, while winding copper loss varies with current squared. That is why time-weighted torque squared is useful. Real temperature also depends on winding resistance as it warms, motor speed, iron and mechanical losses, drive switching, ambient temperature, altitude, mounting heat flow, airflow, cable current limits and the motor's thermal time constants. The exact manufacturer's sizing method and derating rules therefore control the final decision.

Continuous-waveform form

T_RMS = √[(1/T_cycle) ∫₀ᵀ T(t)² dt]

Use this form for ramps, smooth jerk-limited profiles or sufficiently fine sampled data. Sample rapidly enough to retain real peaks and transients, and use the torque-producing quantity required by the motor-drive maker. Filtering that removes short peaks may leave RMS nearly unchanged while invalidating the peak check.

T(t)

motor torque as a function of time, N·m

T_cycle

duration of the repeated cycle, s

∫

integration over the complete cycle

Example

A piecewise-constant waveform reduces to the segment formula above because each interval contributes T_i² multiplied by its duration.

Refer every inertia to the motor shaft before adding it

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Diagram description

A motor directly drives a screw carrying a mass. The panel calculates J_l = m(L/2π)² and gives 5.07×10⁻⁵ kg·m² for 20 kg and 0.010 m/rev. It then shows total direct-drive inertia as motor, screw, coupling and reflected load, acceleration torque as equivalent inertia times angular acceleration, and downstream inertia divided by reduction ratio squared.

Translating mass reflected to a direct-driven screw

J_trans,screw = m(L ÷ 2π)²

This is an ideal kinematic equivalence derived from equal linear and rotational kinetic energy. Use mass in kilograms and lead in metres per revolution to obtain kg·m². It represents the translating mass only; add screw, coupling, pulley, brake and other rotating inertias separately. Do not place mechanical efficiency inside this pure inertia transformation; account for losses in the applicable torque model and follow the manufacturer's sizing convention.

J_trans,screw

translating mass referred to the screw shaft, kg·m²

m

translated mass, kg

L

screw lead, m/rev

Example

For m = 20 kg and L = 0.010 m/rev, J = 20(0.010/2π)² = 5.066×10⁻⁵ kg·m², agreeing with the diagram's rounded 5.07×10⁻⁵ kg·m².

Direct-drive equivalent inertia and acceleration torque

J_total = J_motor + J_external; J_external = J_screw + J_coupling + J_trans + …; T_acc = J_total α_motor

The motor's electromagnetic torque accelerates its own rotor as well as the external mechanism. Keep J_motor separate when reporting the external-load-to-rotor inertia ratio, but include it when calculating total acceleration torque. Catalogue units such as kg·cm², g·cm² or legacy inertia units must be converted to kg·m² before addition.

J_total

motor plus external inertia referred to the motor shaft, kg·m²

J_motor

motor rotor inertia, kg·m²

J_external

all external mechanism inertia referred to the motor shaft, kg·m²

T_acc

ideal inertial acceleration torque, N·m

α_motor

motor angular acceleration, rad/s²

Example

If two components rotate at different speeds, their catalogue inertias cannot simply be added; each must first be reflected through the intervening transmission.

Apply gearbox and screw transformations consistently

Ideal inertia reflection through a reduction

i = ω_motor ÷ ω_output; J_output→motor = J_output ÷ i²

For i greater than one, the motor turns faster than the output. The squared reduction can greatly lower downstream reflected inertia, but it simultaneously raises motor speed and introduces gearbox inertia, efficiency loss, compliance and possibly backlash. Add input-side inertia directly; reflect each downstream or intermediate inertia by the square of its own speed ratio to the motor.

i

reduction ratio defined here as motor angular speed divided by output angular speed

J_output

inertia about the output shaft, kg·m²

J_output→motor

same inertia referred to the motor shaft, kg·m²

Example

An output inertia of 0.009 kg·m² behind a 3:1 ideal reduction appears as 0.009/3² = 0.001 kg·m² at the motor, before adding gearbox and motor-side inertias.

Translating mass through both screw and reduction

J_trans→motor = m[L ÷ (2πi)]²

This is the direct screw result divided by i². It is valid only for the ratio definition shown above. Some catalogues define ratio as output speed over input speed; copying their symbol without checking the definition reverses the transformation.

J_trans→motor

translated mass referred to the motor shaft, kg·m²

m

translated mass, kg

L

screw lead, m/rev

i

motor-speed-to-screw-speed ratio

Example

For the illustrated 20 kg and 0.010 m/rev load behind a 3:1 reduction, the ideal translated-mass term becomes 5.066×10⁻⁵/9 = 5.63×10⁻⁶ kg·m².

Recalculate the cycle after every mechanism change

A practical peak–RMS–inertia workflow

1. 1. Create signed speed and load profiles.Use the complete repeated motion, including reverse travel, settling, holds and credible worst-case loads.
2. 2. Refer the mechanism to the motor shaft.Transform torque, speed and every inertia with one declared ratio convention; add transmission losses in the correct energy direction.
3. 3. Calculate motor torque by interval.Combine load torque with J_totalα and other documented terms without counting translated-mass acceleration twice.
4. 4. Check every peak at speed.Confirm torque-speed availability, drive current, overload duration and repetition rules for the exact motor-drive-supply combination.
5. 5. Calculate RMS or the required equivalent value.Use all intervals in the manufacturer's method, then apply stated ambient, mounting, altitude and cooling derating.
6. 6. Check energy and mechanics.Verify regeneration, DC-bus or resistor capacity, gearbox ratings, screw critical speed and life, coupling, bearings and braking.
7. 7. Measure the commissioned machine.Capture current or torque estimate, speed, following error, vibration and temperatures during worst defined production sequences.

Use Sizing a Motor and Screw as One System for the full calculation loop, Understanding Motor Torque–Speed Curves for peak and continuous envelopes, and Gear Ratio, Speed and Torque in Real Machines for ratio and efficiency conventions. Keep the interval ledger, manufacturer files and commissioning traces together so later motion changes trigger a visible recalculation.

Selection boundary and evidence to keep with the decision

Treat “Duty Cycle, RMS Torque and Reflected Inertia” 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. Measure the commissioned machine. Capture current or torque estimate, speed, following error, vibration and temperatures during worst defined production sequences. 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 Duty Cycle, RMS Torque and Reflected Inertia?

1. Create signed speed and load profiles. Use the complete repeated motion, including reverse travel, settling, holds and credible worst-case loads.

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?

Do not replace the torque profile with a simple on-time percentage. Two cycles can have the same duty percentage and very different heating because loss grows roughly with current squared in the normal torque-producing region. Record torque magnitude, speed and duration for every interval.

How do I verify the result in the real system?

7. Measure the commissioned machine. Capture current or torque estimate, speed, following error, vibration and temperatures during worst defined production sequences.

Sources and further reading

1. Servo Motor Sizing ConceptsYaskawa America, Inc. · 2026-10-12Primary technical document reviewed for the independent roles of inertia ratio, speed, maximum torque at speed and RMS torque at speed.
2. Sigma-7-Series AC Servo Drive Rotary Servomotor with 400 V-Input Power Product ManualYaskawa Electric Corporation · 2026-10-12Primary product manual reviewed for RMS/effective torque examples, motor-shaft load inertia, gear-ratio transformations and separate rated and instantaneous torque checks.
3. Sigma-X Series Rotary Servomotor Product ManualYaskawa Electric Corporation · 2026-10-12Current primary manual reviewed for product-specific torque, inertia and operating-condition constraints; it supports using exact matched-product limits rather than a universal inertia ratio.
4. Studying the Driving MotorTHK · 2026-10-12Primary ball-screw selection reference reviewed for maximum torque, effective torque, speed and manufacturer-dependent inertial-moment checks.
5. Ball / Lead Screw Sizing ToolOriental Motor U.S.A. Corp. · 2026-10-12Manufacturer tool documentation reviewed for segment RMS torque, rotor inertia, gear ratio, load inertia, speed and acceleration-torque inputs.
6. Using SigmaSelect Sizing Software to Size a Ballscrew ApplicationYaskawa America, Inc. · 2026-10-12Primary manufacturer application paper reviewed for the complete ball-screw sizing workflow and the need to evaluate speed, torque and inertia together.