What is Inertia? Useful calculations for machine design and how to determine the correct driving force

Here, we will discuss the inevitable elements of mechanical design.Inertia.　This section notes useful calculations and how to determine the correct driving force for the

In the design of automatic machines,Selection of motor capacity　is an extremely important process that determines the performance and reliability of the equipment.In practice, selection is based solely on static load torque, which can potentially lead to problems such as inability to keep up with acceleration during actual operation, or severe vibration during stopping, which increases the settling time.

These challenges are not simply a lack of specifications, but rather the nature of an object's attempt to maintain its state of motion, i.e.Most often arising from a lack of understanding of inertia.　It is.

General technical explanation websites often remain limited to a list of physical formulas, and the actual designers are faced with questions such as "How much safety factor should be expected?" and "How to balance rigidity to avoid hunting?Practical criteria tend to be lacking　It is.　　In this article, in addition to my experience in the design field, I have thoroughly researched the latest technical materials provided by major Japanese manufacturers and compiled them as comprehensive information.

In this article, we will organize the basic definition and unit system of inertia used as standard in design practice in Japan, and further explain the series as a learning flow, including how to use low-inertia motors and automatic selection tools to achieve efficient design.　　We hope that by reading this article, you will be able to acquire solid selection skills based on evidence, from the calculation of theoretical values to troubleshooting in the field.

Understand what inertia (moment of inertia) is, the basis of machine design

Definitions based on JIS standards and treatment of SI units

In Japanese machine design, the criteria for defining the resistance to movement of rotating parts are clearly defined by standards such as JIS (Japanese Industrial Standards) Z 8202.The current standard International System of Units (SI units) uses the kilogram-square meter (kg-m2) as the unit of inertia moment.

There was a time when the gravity unit system was used, but all the latest catalogs of major manufacturers such as Mitsubishi Electric and Yaskawa Electric are unified in these SI units.　　When performing calculations, kilograms (kg) must be strictly applied to units of mass and meters (m) to units of length.

If calculations are made in millimeters (mm), which are frequently used in design drawings, there is a risk of enormous errors in the results.　　The first step toward correct calculation of driving force is to maintain the correctness of this standard unit system.

Conversion based on Japan's own GD2 and the springboard effect

In old Japanese design documents and even today in some heavy industry fields, the GD2 (G.D. Square) index, known as the "spring wheel effect," is sometimes used.　　This is a value obtained by multiplying the weight of an object (G) by the square of its rotating diameter (D), and was once a widespread practice unique to Japan.

There is a clear conversion relationship between the current mainstream moment of inertia (J) and this GD2, J = GD2 / 4, due to differences in physics definitions.

This relationship is derived from the background that the definition of moment of inertia is based on radius, whereas GD2 is based on diameter.　　When designing updates to older equipment or meeting with experienced designers, always check to see which definition is used for the values presented.

Care should be taken to avoid the risk of under- or over-torquing the motor to be selected if the unit conversion is incorrect.

Inertia and moment of inertia differences not lost in practice (slightly different nuances)

In the field, the two terms inertia and moment of inertia are used interchangeably,As a designer, it is wise to know exactly what the nuances are.　　As the original meaning of the word,Inertia is a conceptual term that refers to the very property of an object to maintain its current state of motionIt is.

Moment of inertia, on the other hand, refers to a concrete physical quantity, a numerical value that can be assigned to a formula.　　The term moment of inertia is commonly used when referring to catalogs or preparing strength calculation sheets.

On the other hand,In field conversations and conceptual explanations, we often refer to inertia as a generic term for the difficulty of turning.　　Understanding that these terms are treated as nearly synonymous in practice will help avoid confusion in communication.　　In short, the key is to distinguish between the use of properties as concepts and physical quantities as numbers.

Calculation theory of acceleration torque derived from the equation of motion

In order for a motor to raise a stationary object to a target speed, it must continue to apply a constant force.　　The acceleration torque is required at this point and is derived by the equation of motion for rotational motion, T = J・α.

WHEREAS,T refers to torque, J to moment of inertia, and α to angular acceleration.　　As this formula indicates, the force required for acceleration increases in direct proportion to the magnitude of the moment of inertia.

If a motor is selected by calculating only the load torque to resist friction and gravity, the actual equipment may not accelerate in time, resulting in a situation where the target tact time cannot be achieved. Therefore,To correctly estimate the dynamic driving force, it is essential to add up not only the torque during steady-state driving, but also the load during acceleration based on this equation of motion.

Reference source: Mitsubishi Electric FA What is J and GD2 of moment of inertia (inertia)? (https://fa-faq.mitsubishielectric.co.jp/faq/show/10824?site_domain=default）

Reference source: Miki Pulley Technical Data (Japanese only)https://www.mikipulley.co.jp/JP/Services/Tech_data/tech24.html）

Accurate calculation of driving force different from inertia (moment of inertia)

How to calculate inertia without getting lost in complex mechanisms

Although the parts that make up an automatic machine have a wide variety of shapes, the overall value can be calculated by combining basic geometric formulas.　　For example, in the case of a disk or cylinder, which appear most frequently, the formula J = 1/8 - m - D2 is calculated using the mass m and outer diameter D.

If a rectangular arm rotates about its central axis, the equation J = 1/12 ⋅ m ⋅ (a2 + b2) is used.If the shape of the part is too complex to calculate manually, the most reliable means is to use the physical property value calculation function of 3DCAD software.

Once the material (density) is specified in CAD, the exact moment of inertia value is calculated along with the center of gravity.　It is an efficient procedure in modern design practice to make a rough bite by manual calculation, and then use tools to finalize the detailed values.

Determination of the load inertia of the objects and parts to be transported

When designing the entire drive system, all elements to be moved by the motor must be added together as load inertia.　　This includes not only the workpiece or other object being conveyed, but also all moving parts such as tables, pulleys, gears, and bearing rotors.

Particular attention should be paid to cases where the values vary greatly depending on the presence or absence of a workpiece.　　For example, the moment of inertia value may differ several times between carrying an empty pallet and loading a workpiece of maximum weight.　　For this reason,It is recommended that values be calculated for both minimum and maximum load patterns during design.

Calculating the driving force for worst-case scenarios is an important preparation to prevent unexpected stoppages in the field.

How to utilize motor shaft conversion and reduction ratio squared

If the load-side mechanism is connected via a reduction gear, the apparent weight from the motor is significantly different from the actual value.　At this time,To convert the load side moment of inertia to motor shaft equivalent, the rule of dividing by the square of the reduction ratio applies.

If the reduction ratio is i, the motor shaft equivalent inertia Jm = JL / i^2.　　For example, a gear head with a reduction ratio of 1/5 (i=5) reduces the weight of the load from the motor to 1/25.　　This is a physical phenomenon based on the law of conservation of energy and is a very effective means of controlling a huge device with a small motor.　　Conversely, when using a speed increaser, inertia is drastically increased, so great care must be taken in the selection process.

Calculation procedure for moving machine element ball screws

In a ball screw mechanism that converts linear motion into rotational motion, the driving force is calculated using a unique conversion formula.　　Assuming the total mass of the moving table and workpiece to be m and the lead of the ball screw to be P, the conversion value for the linear motion component can be obtained by m ・ (P / 2π)2 .

This value, plus the rotational inertia of the ball screw shaft itself, is the total load on the motor.　　If a large lead value is selected here, the load on the motor shaft will rapidly become heavier due to the square effect.

When selecting a large-lead screw to achieve high-speed transfer, be sure to double-check the motor capacity to account for this increase in inertia.　　Correctly replacing linear motion with rotational loading is the key to achieving stable positioning control.

Reference source: Oriental Motor Motor Motor Selection Formula (https://www.orientalmotor.co.jp/ja/tech/calculation/sizing-motor04）

Manage a different index than inertia for stability control

Inertia Ratio to stabilize control Guideline and Criteria

Even if the required driving force can be calculated, smooth operation cannot be expected if there is an imbalance between the motor's own inertia and that of the load.　　ThisRatio is the inertia ratio (inertia ratio)　and is one of the most important indicators for stable servo motor operation.

In general, the ideal load inertia should be within 5 to 10 times the motor rotor inertia.　　For semiconductor manufacturing equipment, for example, which requires high frequency and agile operation, it should be less than 5 times, and for general transport equipment, 10 to 15 times is considered to be within the practical range.

If this ratio is overstated by a factor of more than 30, overshooting is likely to occur during stopping, making it difficult to shorten takt time.　　To avoid such a situation,Evaluate at an early stage whether the calculated values fall within the manufacturer's recommended range.

Reference source: Mitsubishi Electric Corporation Guideline for moment of inertia ratio (https://fa-faq.mitsubishielectric.co.jp/faq/show/10690?site_domain=default）

Reference Source: Oriental Motor Selection Points (https://www.orientalmotor.co.jp/system/files/document/products/selection-results-sample.pdf）

How to determine the safety factor essential to drive force selection

Selecting a motor based on theoretical calculations of torque alone is extremely risky in practice.Since there are uncertainties such as the assembly condition of the machine, changes in grease viscosity, and increased friction due to aging, it is always necessary to project an appropriate safety factor.

It is common practice in the design field in Japan to provide a margin of 1.2 to 1.5 times for acceleration torque and 1.5 to 2.0 times for instantaneous maximum torque.。

A higher safety factor is desirable, especially for motors with brakes that hold vertical axes or are used in environments with high ambient temperatures.　　Allowing too much leeway will lead to higher costs, but a shortage will mean rework after the equipment is delivered. Designers must be willing to set this safety factor strategically to ensure room for adjustment in the field.

Low and medium inertia types and machine rigidity

In most motor lineups, two types of motors are available: low-inertia types that emphasize response and medium-inertia types that emphasize stability.　　The low-inertia type has a long, narrow rotor that enables quick acceleration and deceleration, but on the other hand, it is easily out of balance with the load and can cause vibration in equipment with low mechanical rigidity.

If the frame and coupling of the equipment are not sufficiently rigid, the machine side cannot follow the force of the motor trying to move, and resonance will occur.　　In such a case, it is effective to choose a medium-inertia type to suppress disturbance and vibration by physical weight to achieve stable operation.　　It is necessary to make a judgment to select a low-inertia type for speed and a medium-inertia type for stable quality, depending on the purpose of the equipment.

Use of anti-hunching measures and capacity selection software

Hunting is a phenomenon in which a motor roars or vibrates continuously during operation.　　This is mainly caused by an excessive inertia ratio or inadequate gain adjustment, but many of these problems can be prevented by using the capacity selection software provided by each manufacturer at the design stage.

Mitsubishi Electric's "Motorizerand Oriental Motor's "Selection Toolsand other major domestic manufacturers have released advanced calculation software free of charge.　　By entering the dimensions and operating pattern of the mechanism into these tools, they automatically perform not only inertia calculations, but also torque utilization ratio, regenerative power, and even determination of the optimal inertia ratio.

A system of cross-checking the final validity with these software, while respecting the basis for hand calculations, will result in more reliable design results.

Summary: What is inertia in automatic machine design?

• A term referring to the tendency of an object to maintain its current state of motion
• Physical quantity that quantifies the difficulty of turning in rotational motion
• JIS (Japanese Industrial Standard) defines kg/m2 as SI unit
• The relationship with the old unit system's spring-wheel effect GD2 is a quarter of the value of J
• Calculate the torque required for acceleration/deceleration by the equation of motion T = Jα
• Load inertia is determined by adding up the inertia of all conveyed objects and moving parts
• By passing through the reducer, the inertia of the load is drastically reduced by the square of the reduction ratio.
• Care must be taken when increasing the lead of the ball screw to avoid a sudden increase in inertia.
• Ideally, the inertia ratio, which determines control stability, should be within 5 to 15 times.
• Compliance with recommended motor inertia ratio to prevent deterioration of stopping accuracy and vibration
• Safety factor of around 1.5 times considering friction and aging
• Selection between low-inertia type with high responsiveness and medium-inertia type with high resistance to disturbances
• Consider reviewing the structure as hunting is likely to occur if the machine rigidity is low.
• Confirm the validity of calculation results with capacity selection software provided by each manufacturer
• Comprehensive understanding of the overall picture and the best options for calculating driving force

That's it.