BIZOMAT MAKEOS · Power Transmission
Gear Ratio for Inertia Matching (Optimum Ratio and Motor Torque)
What gear ratio lets this motor accelerate the load with the least torque, what ratio can I actually use within the motor speed, and does the motor have the torque at my chosen ratio?
Gear ratio selection for accelerating a load inertia: reflected inertia, inertia ratio, motor speed, acceleration torque, load torque at the motor and peak motor torque at your chosen ratio (as the reflected inertia tool), the ratio that minimises acceleration torque (load inertia reflected equal to the motor inertia), the best ratio allowed by the motor’s top speed, the motor torque at that ratio, and checks of the chosen ratio against the motor speed and peak torque.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load inertia | kg·m² | Inertia of the load at the gearbox output. |
| Motor inertia | kg·m² | Inertia of the motor rotor (and brake). |
| Chosen ratio | :1 | Gear ratio being considered. |
| Gearbox efficiency | % | Gearbox efficiency. |
| Load speed to reach | rpm | Speed the load must reach. |
| Acceleration time | s | Time to reach it. |
| Load torque | N·m | Steady load torque at the output. |
| Motor top speed | rpm | Highest motor speed. |
| Motor peak torque | N·m | Peak torque the motor and drive can give. |
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
- What it is
- The mass moment of inertia of everything driven, referred to the gearbox output shaft: table, rotating load, couplings and pulleys on that side.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 100000 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 and the optimum ratio both rise with it.
- Get it right
- Refer any load driven through belts or screws to the gearbox output before entering it.
What this means: Motor inertia
- What it is
- The mass moment of inertia of the motor rotor, plus any brake fitted, from the motor data sheet.
- Units and format
- Enter a number in kg·m²: kilogram square metres (mass moment of inertia). The form accepts 0.000001 kg·m² to 100 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.
- Get it right
- Data sheets often give kg·cm²: divide by 10 000 for kg·m².
What this means: Chosen ratio
- What it is
- The gear reduction you are considering: 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 1 :1 to 1000 :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
- The load inertia seen by the motor falls with the square of the ratio, while motor speed rises in proportion.
- Get it right
- Enter the reduction as one number, for example 5 for a 5:1 gearbox.
What this means: Gearbox efficiency
- What it is
- The share of motor power that reaches the gearbox output after gear losses, from the gearbox maker’s data.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 30% 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 both the reflected inertia and the load torque the motor sees.
- Get it right
- Enter a percentage, for example 95, not 0.95.
What this means: Load speed to reach
- What it is
- The speed the load must reach at the gearbox output at the end of the acceleration.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 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
- Motor speed and acceleration torque rise in direct proportion.
What this means: Acceleration time
- What it is
- The time allowed to accelerate the load from rest to its target speed, taken as constant acceleration.
- Units and format
- Enter a number in s: seconds. The form accepts 0.000001 s to 600 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
- Halving the time doubles the acceleration torque.
What this means: Load torque
- What it is
- The steady torque the load needs at the gearbox output from friction, gravity or process forces, on top of the torque to accelerate it.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m to 10000000 N·m. 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.
What this means: Motor top speed
- What it is
- The highest speed the motor may run at, from its data sheet or drive setting. It caps the ratio you can use.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3000 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Motor peak torque
- What it is
- The peak torque the motor and its drive can deliver together during acceleration, from the motor and drive combination data.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0.000001 N·m to 1000000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the peak, not the continuous, torque, and the lower of the motor and drive limits.
Method and formulas
- Reflected inertia J_r = J_L ÷ (i² η); motor speed = n_L i; α = 2π n_m ÷ (60 t) (as the reflected inertia tool).
- Motor torque T(i) = (J_m + J_r) α + T_L ÷ (i η).
- Optimum ratio for acceleration i_opt = √[J_L ÷ (η J_m)]; best usable ratio = min(i_opt, n_max ÷ n_L).
- Example: at 10:1 the motor needs 9.5805 N·m; the optimum would be 43.033:1 but the motor speed allows 20:1, needing 5.3558 N·m.
Limitations
- Constant acceleration and rigid transmission; compliance and backlash change real tuning.
- Load torque is taken as constant during acceleration.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Choose a gearbox ratio for a servo axis or indexer
- See how far a ratio is from the optimum
- Check a motor’s peak torque at the ratio
Who it is for
- Automation engineers
- Machine designers
- Controls engineers
What you get
- Reflected inertia, ratio and motor torque at the chosen ratio
- Optimum and best usable ratio
- Speed and torque checks
Interpreting the result
- Near the optimum ratio the torque curve is flat: a ratio within ±30% of it costs little extra torque.
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
Choosing the ratio
In Inertia and Flywheels by the Numbers
What it covers: How the gear ratio trades motor torque against speed, why accelerating torque is least when reflected and rotor inertia are equal, and how top speed and real gearboxes set the ratio.
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Open the Gear Ratio for Inertia Matching (Optimum Ratio and Motor Torque) calculator