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  1. Engineering Information
  2. Rotary Components
  3. Stepping motors

Motor Torque Calculation Formulas

Calculating moment of inertia, load torque, and required torque is essential for proper motor selection. To support your design process, we have compiled the relevant formulas for various drive mechanisms, including discs, cylinders, and ball screws.

Moment of Inertia Calculation

Solid Discs and Cylinders

When a disc-shaped load is mounted directly on the motor shaft, the moment of inertia is calculated as follows:

J = 1 2 ( W g ) R 0 2 = 1 2 ( ρ π R 0 2 ℓ g ) R 0 2 · 10 − 3 [ kgf·cm·s 2 ]
  • ・ W: Mass [kg]
  • ・ g: Gravitational acceleration = 980 [cm/s2]
  • ・ R0: Outer radius [cm]
  • ・ ρ: Specific gravity [g/cm2]
  • ・ ℓ: Length [cm]
Solid Discs and Cylinders

Hollow Discs and Cylinders

For loads with a hollow center, such as gears and pulleys, the moment of inertia is calculated as follows:

J = 1 2 ( W g ) · ( R 0 2 + R 1 2 ) = 1 2 ( ρ π ℓ g ) ( R 0 4 + R 1 4 ) · 10 − 3 [ kgf·cm·s 2 ]
  • ・ W: Mass [kg]
  • ・ g: Gravitational acceleration = 980 [cm/s2]
  • ・ R0: Outer radius [cm]
  • ・ R1: Inner radius [cm]
  • ・ ρ: Specific gravity [g/cm2]
  • ・ ℓ: Length [cm]
Hollow Discs and Cylinders

Linear Motion Mechanisms

For applications where the motor's rotary motion is converted into linear motion, such as with ball screws, the moment of inertia is calculated as follows:

J = ( W g ) · ( v ω ) 2 = ( W g ) · ( P 2 π ) 2 = ( W g ) · ( 180 · α π θχ ) 2 [ kgf·cm·s 2 ]
  • ・ W: Mass of workpiece and table [kg]
  • ・ g: Gravitational acceleration = 980 [cm/s2]
  • ・ P: Pitch [cm]
  • ・ α: Travel distance per pulse [cm]
  • ・ θx: Step angle per pulse [deg]
  • ・ n: Rotational speed [rpm]
Linear Motion Mechanisms

Hoisting and Lowering

For mechanisms that lift or lower a load vertically using a drum, the moment of inertia is calculated as follows:

J = ( W g ) · R 0 2 [ kgf·cm·s 2 ]
  • ・ W: Suspended mass [kg]
  • ・ g: Gravitational acceleration = 980 [cm/s2]
  • ・ R0: Drum radius [cm]
Hoisting and Lowering

Load Torque Calculation

Rotational Load Torque

T L = R · F [ kgf·cm·s 2 ]
  • ・ R: Measurement radius [cm]
  • ・ F: Pulling force [kg]
Rotational Load Torque

Ball Screw-Driven Tables

T L = μ 0 F a P 2 π + P 2 π η ( F + μ 1 ( W 1 + W 2 ) ) [ kgf·cm ]
  • ・ μ0: Ball screw friction coefficient
  • ・ Fa: Preload [kg]
  • ・ P: Pitch [cm]
  • ・ η: Ball screw efficiency = 0.85–0.95
  • ・ F: External force [kg]
  • ・ μ1: Specific gravity [g/cm2]
  • ・ W1: Workpiece mass [kg]
  • ・ W2: Table mass [kg]
Ball Screw-Driven Tables

Rack and Pinion-Driven Tables

T L = D p 2 ( F + μ 1 ( W 1 + W 2 ) ) [ kgf·cm ]
  • ・ Dp: Pinion pitch diameter [cm]
  • ・ F: External force [kg]
  • ・ μ1: Guide surface friction coefficient
  • ・ W1: Workpiece mass [kg]
  • ・ W2: Table mass [kg]
Rack and Pinion-Driven Tables

Square Thread-Driven Tables

● When using the square thread friction coefficient:

T L = D p 2 [ F + μ 1 ( W 1 + W 2 ) ] · ℓ + μ π D p μ D p − μ ℓ [ kgf·cm ]
  • ・ Dp: Square thread pitch diameter [cm]
  • ・ μ: Square thread friction coefficient = 0.1–0.3
  • ・ W1: Workpiece mass [kg]
  • ・ W2: Table mass [kg]

● For trapezoidal threads (threads other than square threads)

T = D p 2 [ ( W 1 + W 2 ) ] · π d p μ ℓ + cos α / 2 μ d p − μ ℓ cos α / 2 [ kgf·cm ]
  • ・ Dp: Square thread pitch diameter [cm]
  • ・ μ: Square thread friction coefficient = 0.1–0.3
  • ・ W1: Workpiece mass [kg]
  • ・ W2: Table mass [kg]
  • ・ α: Thread angle (flank angle) [deg]

Required Torque Calculation

T m = ( J m + J 1 ) × π × θ 1 180 × f 2 − f 1 t 1 + T 1 [ kgf·cm ]
  • ・ Tm: Required torque [kgf・cm]
  • ・ Jm: Motor moment of inertia [kgf・cm s2] (Inertia / 980 cm/s2)
  • ・ J1: Load moment of inertia [kgf・cm s2] (Inertia / 980 cm/s2)
  • ・ θ1: Step angle [deg]
  • ・ f1: Starting pulse speed (starting frequency) [pps]
  • ・ f2: Operating pulse speed (operating frequency) [pps]
  • ・ t1: Acceleration/deceleration time [s]
  • ・ T1: Load torque [kgf・cm]

Related Page

Engineering Information for Stepping motors

Technical Column

Comparing PM and Hybrid Stepping Motors｜ Structural Basics for Selection and Use

Technical Data

Motor Torque Calculation Formulas

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