Moment of inertia is a physical quantity that indicates how easily a rotating object can be set in motion or brought to a stop, and it is an essential factor in motor design calculations. During motor selection, it is necessary to correctly calculate the load’s moment of inertia and estimate the required torque and acceleration/deceleration performance.
If the moment of inertia is estimated incorrectly, depending on the motor type and operating conditions, this can lead to overshoot or undershoot during startup and stopping, control instability, or even operational failure.
In this article, we will explain step-by-step everything from the basic definition of moment of inertia and its relationship to torque, to calculation formulas for typical motor configurations, and key points to consider during motor selection. We hope this serves as a useful reference for engineers involved in motor design and equipment development.
| Supervised by: C.I. TAKIRON Corporation Electronic Devices Sales Group This article has been supervised based on the advanced technical expertise and insights we have cultivated since our founding in 1919 as a leading company in plastic processing. Our department continuously analyzes market trends and the latest technologies in ultra-compact, high-precision micro motors, focusing on providing high-value-added information to designers and developers. As a team of experts with in-depth knowledge of product characteristics, we support our customers’ problem-solving and technological innovation by delivering accurate and practical content. |
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Basics of Moment of Inertia and Its Relationship to Torque
We will explain the definition, relationships, and units of the moment of inertia, comparing them to linear motion.
Contents of This Section
- Definition of Moment of Inertia in Rotational Motion
- Relationship Between Torque and Angular Acceleration
- The Difference Between the Units kg·m² and GD²
The moment of inertia J (sometimes denoted as I in academic literature, but we will use J consistently in this article) is a physical quantity that indicates how difficult it is to rotate or stop a rotating body. It corresponds to mass in linear motion and, in rotational motion, is described by the relationship between the moment of inertia, torque, and angular acceleration.
In motor design calculations, accurately determining this value serves as the starting point for calculating the required torque and performing motor selection.
Definition of Moment of Inertia in Rotational Motion
The moment of inertia in rotational motion is defined as the sum of the products of the mass M of each particle and the square of its distance R from the axis of rotation. The basic equation is expressed as J = MR², and even if the mass M is the same, the value increases as the mass is concentrated farther from the axis of rotation.
In linear motion, there is a tendency that “the greater the mass, the harder it is to move”; in rotational motion, the moment of inertia plays the same role. The larger the moment of inertia of a rotating body, the greater the torque required to start or stop its rotation. For a system of point masses, it is expressed as J = ΣMiRi², and for a continuous body, as J = ∫R²dm; however, in practice, it is common to use pre-derived formulas specific to different shapes.
The more mass is concentrated away from the axis of rotation, the greater the value of the moment of inertia. For example, for disks of the same mass, if the diameter doubles, the moment of inertia quadruples.
While linear motion is characterized by the principle that “the greater the mass, the harder it is to move,” in rotational motion, the moment of inertia determines how difficult it is to rotate the object. In motor selection, accurately calculating the moment of inertia based on the shape and mass distribution of the object to be driven is the starting point of the design process.
Relationship Between Torque and Angular Acceleration
The relationship between torque and angular acceleration is given by the equation of motion for rotational motion:
T = Jα (T: torque [N·m], J: moment of inertia [kg·m²], α: angular acceleration [rad/s²])
.
Equation of Motion for Linear Motion
F = Ma
.
This equation shows that the larger the moment of inertia, the greater the torque required to achieve the same angular acceleration. For example, accelerating a rotating body with twice the moment of inertia in the same amount of time requires twice the torque.
Understanding T = Jα is fundamental to design calculations when estimating the torque required for motor acceleration and deceleration. Correctly determining the load’s moment of inertia and using this equation to calculate the required torque leads to motor selection.
The Difference Between kg·m² and GD²
Let’s clarify the difference between the units kg·m² and GD². In motor technical documentation, the moment of inertia may be expressed in the International System of Units (SI) as J (kg·m²) or in the gravitational unit system as GD² (kgf·m²).
| Unit | kg·m² (International System of Units) | GD² (Gravitational Unit System) |
| Symbol | J | GD² |
| Definition | J=M×R² | GD²=W×D² |
| Units | kg·m² | kgf·m² |
| Conversion Relationship | J=GD²/4 | GD²=4J |
| Applications | General current technical documentation | Legacy motor catalogs |
GD² is also known as the “flywheel effect” and was once a widely used unit in motor design calculations. While the International System of Units (kg·m²) is now the standard, motor manufacturers’ catalogs may still list both units. Since misidentifying the unit during selection can result in a fourfold error in the calculation results, it is essential to correctly apply the conversion formula J = GD²/4.
Formulas for Calculating Moment of Inertia and Methods for Determining It by Shape
Although the moment of inertia is technically calculated using integration, in practice it is common to use pre-derived formulas specific to each shape.
Topics Covered in This Section
- Calculating the Moment of Inertia for Cylinders and Discs
- Calculating the Moment of Inertia for Rods and Rectangular Prisms
- Calculation Procedure for Motor Shaft Equivalents, Including Deceleration Mechanisms
Calculation formulas are available for typical rotating bodies such as cylinders, disks, rods, and rectangular prisms; the moment of inertia can be calculated once the mass, dimensions, and position of the rotation axis of the object are known.
Furthermore, in actual mechanical systems—such as belt conveyors and reduction mechanisms—it is necessary to sum the moments of inertia of multiple rotating bodies and convert them to the motor shaft.
Calculating the Moment of Inertia for Cylinders and Discs
The moments of inertia for cylinders and disks are used in calculations for motor rotors, cylindrical parts, and similar components. If the mass M and diameter D (and inner diameter d) are known, they can be calculated using the following formulas.
| Shape | Formula | Application Examples |
| Solid Cylinder (Disk) | J = 1/8 × M × D² | Rotor, Flywheel |
| Hollow Cylinder | J = 1/8 × M × (D² + d²) | Couplings, Pipes |
As a calculation example, consider a hollow cylindrical part with mass M = 0.5 kg, outer diameter D = 60 mm (0.06 m),and an inner diameter d = 30 mm (0.03 m), the moment of inertia is calculated as J = 1/8 × 0.5 × (0.06² + 0.03²) = 1/8 × 0.5 × 0.0045 ≈ 2.8 × 10⁻⁴ kg·m².
These formulas are used to calculate the moment of inertia for cylindrical components such as rotors and couplings.
Calculating the Moment of Inertia for Rods and Rectangular Prisms
The moment of inertia for rods and rectangular prisms is necessary when calculating rod-shaped or rectangular prism-shaped components included in rotational mechanisms, such as arms and sliders. The moment of inertia of a rod varies depending on the position of the rotation axis, so the following two formulas are used as appropriate.
[Formulas for the Moment of Inertia of a Rod]
- When the rotation axis is at one end: J = 1/3 × M × L²
- When the rotation axis is at the center: J = 1/12 × M × L²
The value is four times larger when the rotation axis is at the end. In cases where the center of rotation differs from the component’s center of mass—such as at the joints of a robot arm—it is necessary to accurately determine where the mechanism’s rotation axis is located and select the corresponding formula.
Calculation Procedure for Motor-Shaft Equivalence Including a Reduction Gear
Calculations for motor-shaft equivalent values involving a reduction gear are necessary for mechanical systems where a reduction gear is interposed between the motor and the load. The formula for converting the load’s moment of inertia to the motor shaft is Jm = JL/i² (where Jm is the motor-shaft equivalent load moment of inertia, JL is the load moment of inertia, and i is the reduction ratio).
For example, if the load moment of inertia is 1.0 × 10⁻³ kg·m² and the reduction ratio is 2, the value converted to the motor shaft is Jm = 1.0 × 10⁻³ / 4 = 2.5 × 10⁻⁴ kg·m². When the reduction ratio is 2, the load moment of inertia converted to the motor shaft is one-fourth of the load-side value.Note that since the definition of the reduction ratio i varies by manufacturer, both Jm = JL / i² and Jm = JL × i² represent the same concept expressed differently. Care must be taken regarding the basis for the reduction ratio i.
You must check the notation for the reduction ratio in each catalog and apply the conversion formula according to the definition provided by each manufacturer.
Three Steps for Motor Selection Based on Moment of Inertia
After correctly calculating the value of the moment of inertia, the next step is motor selection based on that value.
Topics Covered in This Section
- Calculating Load Inertia and Required Torque
- Verifying the Inertia Ratio and Ensuring Control Stability
- Addressing Load Inertia by Utilizing Geared Motors
When performing motor selection, you must proceed through the following three steps in order: calculating the load moment of inertia, calculating the required torque, and verifying the inertia ratio or allowable load moment of inertia. In applications where the rotational speed varies, it is necessary to consider not only the torque during steady-state operation but also the torque associated with acceleration and deceleration; therefore, following these steps in order leads to an appropriate selection
Calculating Load Inertia and Required Torque
Calculating the load moment of inertia and the required torque is the first step in motor selection. To determine the total load moment of inertia of the entire mechanism to be driven, calculate the moments of inertia for each mechanism component (such as pulleys and tables) and the load itself, then sum them to calculate the total load moment of inertia, JL.
Once the total load moment of inertia JL has been determined, calculate the angular acceleration α based on the acceleration time and target rotational speed, and then use the equation T = Jα to determine the acceleration torque. The total torque required of the motor is the sum of the acceleration torque and the load torque required during steady-state operation, such as friction torque. For operations involving startup and acceleration, the required torque must be calculated by taking into account both the load torque and the acceleration torque.
Verification of the Inertia Ratio and Ensuring Control Stability
The inertia ratio is the load moment of inertia divided by the motor rotor’s moment of inertia. If this ratio exceeds the motor’s recommended range, the motor may be unable to follow acceleration commands, potentially leading to loss of synchronization or unstable operation.
Since the recommended inertia ratio and the upper limit of allowable moment of inertia vary depending on the motor type, the basic principle of motor selection is to check the recommended values listed in the catalog or technical documentation for the motor to be used and ensure that the system remains within those limits.
Even if there is a margin in acceleration and deceleration torque, exceeding the recommended inertia ratio may lead to loss of synchronization or unstable operation. On the other hand, setting the safety factor too high results in overspecification, which is disadvantageous in terms of motor size and cost. It is essential to select a motor within the appropriate range for the specific application.
Addressing Load Inertia Through the Use of Geared Motors
Using Geared motors is an effective solution when the load’s moment of inertia exceeds the motor’s allowable limit. By incorporating a reduction gear, the moment of inertia on the motor shaft is reduced by the square of the reduction ratio, enabling even compact motors to drive heavy loads. For example, with a reduction ratio of 2, the load’s moment of inertia, as calculated relative to the motor shaft, becomes one-fourth of that on the load side.
However, increasing the reduction ratio lowers the motor’s output speed, so the design must strike a balance with the target speed. When the load inertia moment is large, incorporating a reduction mechanism is also a viable option.
Product specifications and application conditions must be verified on each manufacturer’s product page or through their customer service department. C.I. Takiron Corporation offers Geared motors that combine gearheads with Coreless or Brushless motors, and we can also propose configurations tailored to specific applications and requirements.
Summary
Moment of inertia is a physical quantity that indicates how easily a rotating body can be set in motion or brought to a stop; it is a key factor in motor design calculations, directly affecting the determination of required torque and the assurance of control stability. By following the correct procedure—from understanding the basic formula T = Jα to applying calculation formulas for different shapes, and further verifying the inertia ratio and utilizing Geared motors—it is possible to perform appropriate motor selection.
In operations involving changes in rotational speed, the acceleration torque and deceleration torque increase in proportion to the magnitude of the moment of inertia. If you have any questions regarding motor design or selection that takes moment of inertia into account, please feel free to contact C.I. Takiron Corporation.
Product Information & Inquiries
For more details on C.I. Takiron’s micro motor products, please visit the website below.
- Product Site: https://cik-ele.com/en/
- Coreless Motors: https://cik-ele.com/en/products/list/coreless_motor/
- Brushless Motors: https://cik-ele.com/en/products/list/brushless_motor/
- Geared Motors: https://cik-ele.com/en/products/list/gearhead/
- Encoders: https://cik-ele.com/en/products/list/encoder/
If you are having trouble selecting a small motor for your product development, please feel free to contact us via the inquiry form. Our technical staff will discuss your application and requirements with you and propose the optimal solution.
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