Is this an ISO motor-sizing calculation?

No. It is a general-mechanics constant-net-torque calculation based on net torque equals effective inertia times angular acceleration. It does not select a motor, drive, brake, coupling, gearbox or protection system and does not claim ISO conformity.

What torque belongs in the worksheet?

Use the signed net torque at the same shaft and about the same positive axis as the entered speeds and effective inertia. It must already include the applicable driving and resisting torques and must remain constant over the whole transition.

May I use average motor torque when torque varies with speed?

Not in general. For a monotonic transition with constant inertia and speed-dependent net torque, time is the integral of constant J divided by net torque(omega) with respect to omega. Time-, state- or configuration-dependent systems require the governing differential equations and drive limits.

How is load inertia reflected through a gearbox?

Reflect every inertia to the calculation shaft by kinetic-energy equivalence: J reflected equals J source times the square of source angular speed divided by calculation-shaft angular speed. If i equals motor speed divided by load speed, load inertia reflected to the motor is J load divided by i squared.

Single-axis general mechanics · constant net torque

Documented Constant-Net-Torque Speed-Transition Worksheet

Calculate the time for one signed angular-speed transition from a documented total effective inertia and a constant signed net torque at the same shaft. The result is not a motor, drive or brake selection and is not an ISO conformity calculation.

τ_net = Jαt = J(ω₂ − ω₁)/τ_netSigned speeds and powerNo equipment verdict

Documented model inputs

Total effective/reflected inertia J

About the calculation shaft and axis

Constant signed net torque τ_net

Driving minus all resisting torque at the same shaft

Angular-speed unitChanging a unit clears its affected numeric field(s)

Initial signed speed ω₁Negative values mean the opposite chosen rotation

Final signed speed ω₂May cross zero if the same constant torque remains valid

Machine, shaft and positive-axis convention

Inertia source and reflection record

Net-torque source, operating range and load record

I confirm that J and τ_net are expressed at the same calculation shaft and about the same signed axis; J is positive and constant; τ_net is the signed sum after all applicable driving and resisting torques and remains constant from ω₁ to ω₂; the entered torque has the sign needed for the transition; and I will not treat the result as proof that a motor, drive, brake, coupling, gearbox, thermal duty or protection system is adequate.

Constant-net-torque result

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Transition time t

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Signed angular acceleration α

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Signed speed change Δω

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Signed rotational-energy change ΔE_腐敗

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Average signed net mechanical power

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Endpoint net mechanical powers P₁ / P₂

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Normalized inertia and torque

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Normalized signed speeds

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モデルの適用境界: this is the rigid, single-axis, constant-J, constant-net-torque solution only. It does not model a torque-speed curve, current/voltage limits, field weakening, slip, compliance/backlash, resonances, traction loss, thermal limits, regenerative capacity, brake heat, control ramps or transient load events. Signed mechanical energy and power are not electrical input power or dissipated heat.

Equations implemented

τ_net = Jα
Δω = ω₂ − ω₁
t = Δω/α = JΔω/τ_net > 0
ΔE_腐敗 = ½J(ω₂² − ω₁²)
P̄_net = ΔE_腐敗/t = τ_net(ω₁ + ω₂)/2
P₁ = τ_netω₁; P₂ = τ_netω₂

The positive axis is chosen by the user. Angular speeds, angular acceleration, torque, energy change and power retain their signs. The worksheet accepts the transition only when non-zero Δω and non-zero τ_net have the same sign, so calculated time is positive. A negative energy change is ordinary mechanical-energy removal, not an error.

Dimensions close: (kg·m²)(rad/s)/(N·m) reduces to seconds because the radian is dimensionless and N·m = kg·m²/s². Likewise ½Jω² is joules and τω is watts.

Reversal through zero: the arithmetic can represent reversal if the same signed net torque is valid throughout. Mechanical power changes sign when ω crosses zero. Verify the drive, brake, transmission and control behavior separately.

Entered unit	Conversion to the internal SI quantity	Important distinction
lbm·ft²	1 lbm·ft² = 0.45359237 × 0.3048² = 0.0421401100938048 kg·m²	Mass moment using the avoirdupois pound mass
スラグ·ft²	1 slug·ft² = 1.3558179483314004 kg·m²	Not interchangeable with lbm·ft²; it is 32.174048556… times larger
lbf·ft	1 lbf·ft = 4.4482216152605 × 0.3048 = 1.3558179483314004 N·m	Torque conversion; the same number as slug·ft² conversion has a different dimension
回転数	ω = rpm × 2π/60 rad/s	Signed revolutions per minute
Hz (rev/s)	ω = Hz × 2π rad/s	Here Hz explicitly means signed revolutions per second
ラド／秒	No scale change	Internal angular-speed unit

NIST SP 811 Appendix B.8 lists pound-foot-squared, pound-force-foot, slug and rpm conversion factors. The prior page used approximately 1.35582 for an input labelled lb·ft²; that is the slug·ft² scale, not the lbm·ft² scale.

Reflect all inertias to one shaft

Total J is the sum of rotor, coupling, gear and load inertias after every component has been expressed as kinetic-energy-equivalent inertia at the calculation shaft. For a component at angular speed ω_ソース and a calculation shaft at ω_calc:

J_ref,calc = J_ソース(ω_ソース/ω_calc)²

If i = ω_モーター/ω_負荷, load inertia reflected to the motor is J_負荷/i². State the ratio convention explicitly and include the inertias of relevant gears, pulleys and couplings. Oriental Motor’s official engineering guidance likewise notes that gearing reduces load inertia reflected to the motor by the gear ratio squared. Compliance, backlash and losses require a fuller model.

Net torque means the signed remainder at that same shaft

τ_net is not merely motor nameplate or stall torque. It is the signed sum after the applicable motor/prime-mover torque, process/load torque, friction, windage, transmission losses and other torques have been brought to the same shaft and sign convention. The torque must remain constant over the entire speed interval for this closed-form time to apply.

Variable torque: when J remains constant and τ_net varies only with speed over a monotonic interval, use t = ∫[J/τ_net(ω)]dω. A simple arithmetic or time average of torque is not generally equivalent. If torque depends on time, temperature, current, voltage, controller state or another dynamic state—or if effective inertia changes—solve the governing differential equations and apply the real drive limits.

General mechanics, not an ISO-prescribed formula

MIT OpenCourseWare, Chapter 17: Two Dimensional Rotational Dynamics, equation 17.3.23, states the fixed-axis relationship between total external torque, moment of inertia and angular acceleration. This worksheet applies that general-mechanics relationship to a documented constant-J and constant-net-torque interval. No ISO clause, acceptance limit or equipment-selection rule is claimed.

Published university example reproduced

The University of Maryland Physics 121 wire-spool worked solution uses J = 2.25 kg·m² and τ = 9 N·m, obtains α = 4 rad/s², and after 2 s obtains Δω = 8 rad/s. Entering 0 to 8 rad/s reproduces t = 2 s. The same documented model gives ΔE_腐敗 = 72 J, average net mechanical power 36 W, and endpoint powers 0 W and 72 W.

The University of Tennessee physics module independently explains that torque is directional and that net torque is the vector sum of torques. NIST SP 811 supplies the unit conversion basis. These are public sources; no closed normative coefficient or table is embedded.

Former content or behavior	Problem and correction
Input labelled lb·ft² multiplied by 1.35582	That scale corresponds to slug·ft², not lbm·ft². The former conversion overstated literal pound-mass-foot-squared inertia by about 32.174. Separate explicit lbm·ft² and slug·ft² units now use distinct NIST-based factors.
Metric/imperial toggle reinterpreted existing numbers	Changing a unit could silently change the physical case. Each unit is now explicit, and changing it clears the affected numeric field(s) and prior result.
Only non-negative speeds and absolute speed change	Direction, braking and reversal were lost. Initial/final speeds, torque, acceleration, energy and power now retain a documented sign convention.
Absolute energy and power outputs	Absolute values hid whether rotational energy was added or removed. The replacement reports signed ΔE and signed net mechanical power and does not relabel them as electrical input or brake heat.
“Net motor torque minus load torque; include friction”	The shaft, direction and transmission boundary were not controlled. The replacement requires a signed same-shaft net-torque record including all applicable driving and resisting contributions.
Variable torque: “integrate numerically or use time-averaged torque”	A generic time-average torque is not a valid substitute. The replacement shows the speed-domain integral and directs state-dependent systems to their differential equations and real drive limits.
Ambiguous reflected-inertia ratio	The old N-motor/N-load notation could mean speeds or tooth counts and reverse the result. The replacement states energy-equivalent reflection and defines i = motor speed/load speed before giving J_load,ref = J_負荷/i².
Defaults, presets, auto-calculation, partial parsing, persistence and dynamic innerHTML	The page could display an apparently authoritative result without a controlled load record. It now starts blank, requires provenance and confirmation, validates complete finite input, uses explicit submit and writes results with textContent.

No. Check the complete torque-speed envelope, current and voltage limits, controller ramps, duty cycle, temperatures, regenerative or dissipation capacity, transmission ratings, couplings, protection and the governing equipment/application requirements.

Do not enter a guessed average. With constant J, integrate J/τ_net(ω) over a monotonic interval using controlled data, or solve the state equations when torque depends on time, current, voltage, temperature, controller state or changing configuration.

Yes. For positive rotation slowing down, Δω is negative and net torque must also be negative. The energy and average net mechanical power will normally be negative. This still does not calculate brake temperature or electrical regeneration.

They are different inertia units by a factor of about 32.174. A field labelled only lb·ft² is unsafe unless the mass convention is explicit.