Abstract : There are significant differences in the understanding of inertia matching in servo systems both domestically and internationally. This article proposes the meaning of inertia matching in engineering applications. In practical applications in the equipment manufacturing industry, most systems are not designed with inertia matching in mind. The article also analyzes the impact of severe inertia mismatch on servo systems. It emphasizes that in servo systems, the focus should not be on achieving load inertia matching, but rather on ensuring that the ratio of load inertia to motor inertia is within a reasonable range, guaranteeing both rapid system response and stable operation. Finally, it provides countermeasures that can be taken in applications with a high degree of mismatch between load inertia and motor inertia.
introduction
Moment of inertia is a measure of the inertia (the property of a rotating object to maintain its uniform circular motion or remain at rest) of a rigid body rotating about an axis, denoted by the letters I or J. In rotational dynamics, moment of inertia plays a role similar to mass in linear dynamics; it can be figuratively understood as the inertia of an object with respect to rotational motion. Moment of inertia significantly impacts the accuracy, stability, and dynamic response of servo systems. In servo system applications, the ratio of the load inertia referred to the motor shaft to the motor's inertia cannot be too large and must be appropriately chosen; otherwise, the system will generally experience oscillations or even loss of control. However, why a suitable inertia ratio is needed, and how to reasonably determine this recommended inertia ratio in practice, are questions that often perplex engineers.
Suitability analysis of servo motor load inertia ratio
1. Inertia matching – optimal power transfer and maximum acceleration
All mechanical systems possess a certain degree of elasticity (meaning rigidity cannot be infinitely large), while some mechanical systems exhibit backlash. When either of these reaches a certain level, it will lead to extremely poor system response performance. Therefore, the so-called problem caused by inertia mismatch is actually due to insufficient mechanical rigidity, potentially resulting in significant elasticity or backlash, leading to motion instability. In servo systems, the motion quantities we need to control are the position or speed of the load end. However, in reality, the position or speed signals detected by feedback devices mounted on the motor are used to replace the target load control quantity. Due to the finite nature of rigidity, this control method is highly likely to experience instability under certain conditions, especially when the inertia ratio is too large.
To improve the system's rapid response, it is essential to first increase the resonant frequency of the mechanical transmission components, i.e., increase their rigidity and reduce their inertia. Secondly, increasing damping to lower the resonant peak value can also contribute to improved rapid response. In many equipment applications, insufficient rigidity and excessive inertia in mechanical transmission components are common. Therefore, while meeting the strength and rigidity requirements of the components, the inertia of moving parts should be minimized as much as possible.
For a specific electric motor, if a reduction mechanism is used to match the load inertia referred to the motor shaft with the motor inertia (the load inertia equals the motor inertia, i.e., the inertia ratio is 1), the system can achieve optimal power transmission and obtain the maximum load acceleration without ignoring the inertia and efficiency loss added by the reducer. This is the meaning of inertia matching. Reference [5] also has a similar interpretation.
However, in China, the concept of "inertia matching" is often used instead of "appropriate inertia ratio". In foreign inertia ratio research, the concept of "inertia matching" is basically not mentioned. Instead, the concept of "inertia mismatch" is mentioned. For example, references [2], [3], and [4] are based on the fact that most servo system applications are "inertia mismatched", and study how to achieve fast response of servo system without causing instability.
Reference [1] derived that when the relationship between load inertia, motor inertia and reduction ratio conforms to formula (1), load inertia matching can be achieved, based on the principle of maximizing acceleration.
For a system where the load is determined and the motor is selected, if the reduction ratio of the reduction mechanism is selected according to formula (1), it is called the optimal reduction ratio. In this case, the load inertia referred to the motor shaft and the motor inertia achieve the so-called inertia matching (i.e., the inertia ratio is 1).
Reference [5] also deduced that when the load inertia is equal to the motor inertia, inertia matching is achieved, based on the principle of maximizing the load power change rate.
However, in practical applications, considering factors such as the inertia of the reduction mechanism itself, its inefficiency, the maximum speed limitations of the input shaft and motor, mechanical space constraints, and cost, the selection of reducers for servo drive systems used in most equipment manufacturing industries is not based on the optimal reduction ratio. In other words, the load inertia and motor inertia are generally mismatched. Therefore, in engineering applications, the focus should not be on achieving load inertia matching, but rather on ensuring that the ratio of load inertia to motor inertia is within a reasonable range, guaranteeing both rapid system response and stable operation.
2. Recommended range of suitable inertia ratio for commonly used transmission mechanisms
In applications, the type of mechanical mechanism driven by the motor must be carefully considered (because different transmission mechanisms have different rigidities) and an appropriate inertia ratio must be adopted. Table 1 lists the recommended range of ratio values for different types of reduction mechanisms. (Note: This refers to the load inertia attributed to the motor shaft after adding the reduction mechanism. JL refers to the load inertia without adding the reduction mechanism.)
Type - Recommended range of inertia ratio
Ball screw — ≤ (2～10) (related to the length of the screw)
Harmonic gears — ≤ (3～10)
Planetary gears — ≤ (4～10)
Gears and racks — ≤ (1～8)
Synchronous pulleys — ≤ (1 to 8) (depending on belt type, tension, length, etc.)
Table 1 Recommended range of suitable inertia ratio for commonly used transmission mechanisms
Supplementary explanation for the application of Table 1:
When frequent and rapid starts and stops are required, in order to ensure sufficient acceleration for rapid system response and to meet system stability requirements, the inertia ratio should be selected closer to the lower limit. It is advisable to select an inertia ratio that does not exceed half of the maximum value in Table 1.
3. Analysis of the adverse effects of a large load inertia ratio and experimental conclusions
When the load-inertia ratio of a servo drive system is too high, the system will generally experience oscillations or even runaway. Theoretically, if the system's transmission rigidity is infinite, the load-inertia ratio can also be set to infinity. Both the torsional rigidity of the coupling and the load-inertia ratio affect the frequency and amplitude of oscillations, but the torsional rigidity of the coupling has a much greater impact. To eliminate system oscillations, efforts should primarily focus on increasing the system's transmission rigidity. Increasing the torsional rigidity of the coupling between the motor and the load can increase the oscillation frequency while reducing the oscillation amplitude.
The research and test results in reference [4] show that if the torsional stiffness of the coupling method is low, even under the condition of load inertia matching (load inertia ratio of 1), the speed response may exhibit oscillation instability.
In order to study the effect of excessive load-inertia ratio on servo drive system, two types of tests were conducted in reference [3] on motors with different inertia loads. The conclusions of the two types of tests are as follows:
1) In both tests, under a certain inertia ratio, the system operated stably with no overshoot or oscillation in speed response. (Note that even with an inertia ratio of 5, rather than inertia matching, the servo system response remained very stable.)
2) Compared to the initial inertia value when the driver parameters were properly tuned, the load response becomes very poor as the load inertia increases or decreases significantly. For example, in Test 1, after the inertia ratio increases to a certain extent, the speed suffers severe overshoot, or even oscillation, with a low oscillation frequency. Moreover, the settling time becomes increasingly longer as the load inertia increases. In Test 2, compared to the initial inertia value when the driver parameters were properly tuned, as the load inertia decreases, the system becomes unstable when it decreases to less than half of the initial inertia value, and the oscillation frequency is high.
4. Strategies to address excessively high load inertia ratio
To eliminate system instability caused by a large mismatch between load inertia and motor inertia, the following measures can be taken:
Load inertia calculation and motor selection examples
1. Screw structure
Given: load weight m = 200 kg, screw pitch Pb = 20 mm, screw diameter Db = 50 mm, screw weight mb = 40 kg, coefficient of friction µ = 0.002, mechanical efficiency η = 0.9, load moving speed V = 30 m/min, total moving time t = 1.4 s, acceleration/deceleration time t1 = t3 = 0.2 s, and stationary time t4 = 0.3 s. Please select the minimum power servo motor that meets the load requirements.
2. Synchronous pulley structure
Given: In the rapid positioning motion model, the load weight M = 5kg, the diameter of the synchronous pulleys D = 60mm, D1 = 90mm, D2 = 30mm, the coefficient of friction between the load and the machine tool µ = 0.003, the maximum speed of the load is 2m/s, and the time for the load to accelerate from a standstill to the maximum speed is 100ms. Ignore the weight of each conveyor pulley and select a servo motor.
1) Calculate the load inertia referred to the motor shaft.
2) Calculate the torque required for the motor to drive the load.
Torque required to overcome friction
Torque required during acceleration
3) Required torque
4) Calculate the required speed of the motor
Selected motor option:
in conclusion:
