Inertia issues are frequently encountered during the selection and commissioning of servo systems. Specifically, when selecting a servo system, in addition to considering factors such as motor torque and rated speed, it is necessary to first calculate the inertia of the mechanical system as a function of the motor shaft. Then, based on the actual motion requirements of the machinery and the quality requirements of the machined parts, a motor with a suitable inertia value must be selected. During commissioning, correctly setting the inertia ratio parameter is a prerequisite for fully realizing the optimal performance of the machinery and servo system. This point is particularly prominent in systems requiring high speed and high precision, thus leading to the issue of inertia matching. Let's take a look at this together!
I. What is "Inertia Matching"?
1. According to Newton's second law: "The torque required by the feed system T = system inertia J × angular acceleration θ". The acceleration θ affects the dynamic characteristics of the system. The smaller θ is, the longer the time from when the controller issues a command to when the system completes execution, and the slower the system response. If θ changes, the system response will fluctuate, affecting machining accuracy. Since the maximum output T value remains constant after the motor is selected, if a small change in θ is desired, J should be as small as possible.
2. The total inertia of the feed axis, J, equals the rotational inertial momentum of the servo motor , JM, plus the load inertial momentum converted from the motor shaft, JL. The load inertia JL is composed of the inertia of the worktable and the fixtures, workpiece, screw, coupling, and other linear and rotary moving parts mounted on it (taking a flatbed metal cutting machine as an example), which is then converted to the inertia of the motor shaft. JM is the rotor inertia of the servo motor, a constant value once the servo motor is selected, while JL varies with the load, such as the workpiece load. To minimize the rate of change of J, it's best to keep the proportion of JL small. This is what is commonly referred to as "inertia matching."
II. How is "Inertia Matching" Determined?
Transmission inertia affects the accuracy, stability, and dynamic response of a servo system. A large inertia results in a high mechanical constant, slow response, and a decrease in the system's natural frequency, making it prone to resonance. This limits the servo bandwidth and impacts servo accuracy and response speed. Appropriately increasing inertia is only beneficial for improving low-speed crawling. Therefore, in mechanical design, inertia should be minimized as much as possible without affecting system stiffness. When evaluating the dynamic characteristics of a mechanical system, the smaller the inertia, the better the system's dynamic response; the larger the inertia, the greater the motor load and the more difficult it is to control. However, the inertia of the mechanical system must be matched with the inertia of the motor. Different mechanisms have different choices of inertia matching principles, resulting in different effects. Different mechanism actions and machining quality requirements have different requirements for the ratio of JL to JM, but most require the ratio of JL to JM to be less than ten. In short, the determination of inertia matching needs to be based on the mechanical process characteristics and machining quality requirements. For basic metal cutting machine tools, for servo motors , it is generally recommended that the load inertia should be less than 5 times the motor inertia.
Inertia matching is crucial for motor selection. For motors of the same power, some brands offer light, medium, or large inertia. Ideally, the load inertia should be calculated using formulas. Common formulas for calculating the inertia of shapes are readily available in textbooks (you can consult mechanical design handbooks). We once conducted an experiment where a large inertia disk was added to the shaft extension of a servo motor for testing. The result was that the servo motor couldn't stop at low speeds, constantly oscillating and unable to calm down. Later, we changed the method: a coupling was installed between the shaft extensions of two servo motors. One servo motor was energized as the driving motor, while the other acted as the driven motor, serving as a small load. The previously oscillating servo motor started, moved, and stopped normally!
III. Theoretical Calculation Formula for 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 in terms of flywheel torque, 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.
IV. 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 designed for testing. As for its effectiveness, I haven't used it myself, but I imagine it is.
There have been instances where foreign machines were copied verbatim during the design phase, with the mechanical parts supposedly identical and the motor power increased by 50%, only to find that the motor wouldn't turn. This was because the prototype's machining and assembly precision was too poor; while the load inertia was similar, the frictional resistance differed significantly, 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.
V. 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. Maximum motor speed > maximum required system speed
2. The rotor inertia of the motor is matched with the load inertia.
3. Continuous load operating torque ≤ motor rated torque
4. The motor's maximum output torque > the system's maximum required torque (torque during acceleration).
(2) Selection calculation
1. Inertia matching calculation (JL/JM)
2. Rotation speed calculation (load end speed, motor end speed)
3. Load torque calculation (continuous load working torque, acceleration torque)
