I. The theoretical formula for calculating inertia?
There are formulas for calculating inertia. For multiple loads, such as gears with gears or worm gear drives, you only need to calculate the inertia of each rotating component separately and then add them together to get the system inertia. When selecting a motor, it is recommended to choose one based on the specific motor being selected. The load's rotational inertia must definitely be calculated during the design phase. Without this value, the motor selection will certainly be unreasonable or problematic; this is one of the most important parameters for selecting a servo motor. The motor inertia is usually indicated in the motor's manual. Of course, for some servos, the load's inertia can be measured by adjusting the servo and used as a reference for calculations in theoretical design. After all, during the design phase, many parameters, such as the coefficient of friction, can only be guessed based on experience and cannot be accurate. The formula for calculation in theoretical design is as follows (for reference only): The rotational inertia J is usually expressed using the flywheel torque GD2, and the relationship between them is...
J=mp^2= GD^2/4g
In the formula
m and G – the mass (kg) and weight (N) of the rotating part;
D – Radius of inertia and diameter (m);
g = 9.81 m/s² - gravitational acceleration; flywheel inertia = rate of change of velocity * flywheel pitch / 375.
Of course, there will always be discrepancies between theory and practice. In some regions (such as Europe), intermediate values are generally obtained through actual testing. This is more accurate than our empirical formulas. However, calculations are still necessary at present, and there are fixed formulas that can be found in mechanical design handbooks.
II. Regarding the coefficient of friction?
Regarding the coefficient of friction, motor selection typically only considers this coefficient in the calculation process and usually doesn't take it into account during motor adjustment. However, if this factor is significant, or even enough to affect motor adjustment, some Japanese general-purpose servo motors reportedly have a parameter specifically for testing. Whether it's effective or not, I haven't used it myself, but I imagine it is. A netizen posted that someone encountered a situation where they copied a foreign machine during design, claiming the mechanical parts were identical, and even increased the motor power by 50%, but the motor wouldn't turn. This was because the prototype's machining and assembly precision was too poor; the load inertia was similar, but the frictional resistance differed too much, indicating insufficient consideration of the specific operating conditions.
Of course, viscous damping and the coefficient of friction are not the same issue. The coefficient of friction is a constant value, which can be compensated for by motor power. However, viscous damping is a variable value. While increasing motor power can alleviate this, it's actually unreasonable. Moreover, without design justification, this is best addressed through mechanical condition analysis. Without proper mechanical condition analysis, servo adjustment is meaningless. Furthermore, viscous damping is related to mechanical structure design, manufacturing, and assembly, all of which must be considered during motor selection. It is also closely related to the coefficient of friction. Insufficient manufacturing quality leads to inconsistent coefficients of friction, with significant differences at different points. Even variations in the assembly skills of skilled workers can cause substantial differences, all of which must be considered during motor selection. This ensures a safety margin, but ultimately, it still comes down to motor power.
III. Simplification of Fine-tuning Correction After Theoretical Calculation of Inertia
Some readers might think: This is too complicated! The reality is that various parameters of a brand's product are already determined. Given that the power, torque, and speed requirements are met, and the product model is fixed, if the inertia still isn't sufficient, can the power be increased to meet the inertia requirement? The answer is: if increasing the power can lead to increased acceleration, then it should be possible.
IV. Servo Motor Selection
After selecting the mechanical transmission solution, it is necessary to select and confirm the model and size of the servo motor.
1. Selection criteria: Generally, the following conditions must be met when selecting a servo motor:
(1) The maximum speed of the motor is greater than the maximum moving speed required by the system.
(2) The rotor inertia of the motor is matched with the load inertia.
(3) Continuous load working torque ≤ motor rated torque.
(4) The maximum output torque of the motor is greater than the maximum torque required by the system (torque during acceleration).
2. Selection Calculation:
(1) Inertia matching calculation (JL/JM).
(2) Calculation of rotational speed (load end speed, motor end speed).
(3) Calculation of load torque (continuous load working torque, torque during acceleration).
V. The difference between low-inertia and high-inertia servo motors
Moment of inertia = radius of rotation * mass
Low inertia motors are typically designed to be flat and elongated, with a small spindle inertia. When the motor performs high-frequency reciprocating motion, the low inertia results in less heat generation. Therefore, low inertia motors are suitable for high-frequency reciprocating motion. However, their torque is generally relatively lower. High inertia servo motors, on the other hand, are larger and have higher torque, making them suitable for applications requiring high torque but not rapid reciprocating motion. This is because stopping at high speeds requires the driver to generate a large reverse drive voltage to stop the high inertia, resulting in significant heat generation.
Inertia is a measure of the inertia of a rigid body rotating about an axis. Rotational moment of inertia is a physical quantity characterizing the magnitude of a rigid body's rotational inertia. It is related to the mass of the rigid body and its mass distribution relative to the axis of rotation. (A rigid body refers to an object that does not change under ideal conditions.) When selecting a servo motor, inertia is an important parameter. It refers to the inertia of the servo motor rotor itself, which is crucial for the motor's acceleration and deceleration. If the inertia is not properly matched, the motor's operation will be very unstable.
Generally, motors with low inertia have good braking performance, quick start-up, acceleration, and stopping response, and good high-speed reciprocating properties, making them suitable for light-load, high-speed positioning applications, such as linear high-speed positioning mechanisms. Medium- and large-inertia motors are suitable for heavy-load applications with high stability requirements, such as circular motion mechanisms and machine tool applications. If the load is very large or the acceleration characteristics are very high, choosing a motor with low inertia may cause excessive damage to the motor shaft. The selection should be based on factors such as the load size and acceleration magnitude; relevant energy calculation formulas are usually available in selection manuals.
For servo motor drivers to control the response of servo motors, the optimal ratio of load inertia to motor rotor inertia is one, and should not exceed five times. Through the design of the mechanical transmission device, the ratio of load inertia to motor rotor inertia can be made close to one or smaller. When the load inertia is indeed very large, and the mechanical design makes it impossible to reduce the ratio of load inertia to motor rotor inertia to less than five times, a motor with a larger rotor inertia, i.e., a high-inertia motor, can be used. To achieve a certain response when using a high-inertia motor, the driver capacity should be larger.
