BIZOMAT MAKEOS · Power Transmission
Reflected Inertia and Motor Acceleration Torque
What torque must the motor give to accelerate this load through the gearbox, and is the inertia ratio reasonable?
Load inertia reflected through a reduction, inertia ratio, acceleration torque and total motor torque to reach a speed in a given time against a load torque.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load inertia (at the load shaft) | kg·m² | Inertia of everything on the load side, about the load shaft (see the mass moment of inertia tool). |
| Motor rotor inertia | kg·m² | Rotor inertia from the motor data sheet (add gearbox input inertia). |
| Reduction ratio (motor ÷ load speed) | :1 | Gear ratio: motor speed divided by load speed. |
| Gearbox efficiency | % | Gearbox efficiency from the maker. |
| Load speed to reach | rpm | Speed the load must reach. |
| Acceleration time | s | Time allowed to reach that speed from rest. |
| Load torque at the load shaft | N·m | Steady torque at the load shaft from friction, gravity or process. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Load inertia (at the load shaft)
- What it is
- The mass moment of inertia of everything on the load side, taken about the load shaft: tables, rolls, couplings and payload.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0 kg·m² to 1000000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Reflected inertia rises in proportion and falls with the square of the ratio.
- Get it right
- Enter inertia in kg·m², not mass or weight; convert from kg·cm² if a catalogue uses that.
What this means: Motor rotor inertia
- What it is
- The inertia of the motor rotor from the motor data sheet, plus the gearbox input inertia if there is one.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 10000 kg·m². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0012 kg·m² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger rotor inertia lowers the inertia ratio but adds to the acceleration torque.
- Get it right
- Data sheets often list kg·cm²; divide by 10 000 to get kg·m².
What this means: Reduction ratio (motor ÷ load speed)
- What it is
- The speed reduction between motor and load: motor speed divided by load speed.
- Units and format
- Enter a number in :1: a ratio to one (type 3 for 3:1). The form accepts 0.01 :1 to 100000 :1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 :1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Reflected inertia falls with the square of the ratio, and load torque at the motor falls in proportion.
- Get it right
- Enter it as a reduction greater than 1 for a gearbox that slows the load.
What this means: Gearbox efficiency
- What it is
- The efficiency of the gearbox between motor and load, from the gearbox maker.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lower efficiency raises the reflected inertia and the load torque seen by the motor.
- Get it right
- Type the percentage itself, not a fraction.
What this means: Load speed to reach
- What it is
- The speed the load shaft must reach at the end of the acceleration, starting from rest.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Acceleration torque rises in proportion.
- Get it right
- Enter load speed, not motor speed; the tool multiplies by the ratio.
What this means: Acceleration time
- What it is
- The time allowed to accelerate the load from rest to the target speed, at constant acceleration.
- Units and format
- Enter a number in s: seconds. The form accepts 0.000001 s to 100000 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.5 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Acceleration torque falls in proportion as the time is lengthened.
- Get it right
- Enter seconds, not milliseconds.
What this means: Load torque at the load shaft
- What it is
- The steady torque the load needs at the load shaft from friction, gravity or the process, apart from accelerating inertia.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds to the peak motor torque after dividing by ratio and efficiency.
- Get it right
- Take it at the load shaft; the tool reflects it to the motor through the ratio and efficiency.
Method and formulas
- Reflected inertia = J_L ÷ (i² η); inertia ratio = reflected ÷ J_m.
- Motor angular acceleration α = 2π n_L i ÷ (60 t).
- Motor torque = (J_m + J_L ÷ (i² η)) α + T_L ÷ (i η).
- Example: 2 kg·m² through 10:1 reflects as 0.022222 kg·m² (ratio 18.519); peak motor torque 9.5805 N·m.
Limitations
- Constant acceleration from rest; deceleration and duty cycle (RMS torque) need separate checks.
- Coupling and gearbox inertia should be added to the motor inertia.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a servo or induction motor and gearbox
- Check acceleration time of a conveyor or rotary table
- Choose a gear ratio for inertia matching
Who it is for
- Automation engineers
- Machine designers
What you get
- Reflected load inertia and inertia ratio
- Motor speed
- Acceleration and load torque at the motor
- Peak motor torque
Interpreting the result
- Many servo makers recommend inertia ratios of about 10:1 or less for good tuning; follow your drive maker’s guidance.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 180–186, 2569–2580. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Loads on a winch drive
In Clutches and Brakes by the Numbers
What it covers: How to turn a winch's duty into the forces, torques, speeds, inertias and energies that its clutch and brakes must handle, on the shaft where each one sits.
Method explained here · BIZOMAT original book
Inertia through gears and belts
In Inertia and Flywheels by the Numbers
What it covers: How inertia and load torque look from the motor through a gearbox, belt or screw: the ratio squared, linear masses as m r², efficiency, inertia ratio and accelerating torque.
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