0 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.
1. Suitability analysis of the load inertia ratio of the servo motor
1.1 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.
1.2 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 moment of inertia ratio must be adopted. Table 1 lists the characteristics of different types of reduction mechanisms.
Different recommended ranges for the ratio. (Note: This refers to the load inertia attributed to the motor shaft after adding the reduction gear. JL refers to the load inertia without adding the reduction gear.)
Table 1. Recommended Inertia Ratios for Commonly Used Transmission Mechanisms (Recommended Range)
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.
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 large, the system will generally oscillate or even run out of control. If the transmission rigidity of the system is infinite, the load inertia ratio can theoretically also be set to infinity. The torsional rigidity of the coupling method and the load inertia ratio will affect the frequency and amplitude of the oscillation, but the influence of the torsional rigidity of the coupling method is much greater. If you want to eliminate the oscillation phenomenon, you should focus on increasing the transmission rigidity of the system. Increasing the torsional rigidity of the coupling between the motor and the load can increase the oscillation frequency and reduce the oscillation amplitude. The research and test results in reference [4] show that if the torsional rigidity of the coupling method is low, even under the condition of load inertia matching (load inertia ratio of 1), the speed response may still experience unstable oscillation.
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 under conditions of 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.
1.4 Strategies for addressing 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:
1) The first step is to increase the stiffness of the mechanical system. (For example, increasing the torsional stiffness of the coupling can increase the oscillation frequency of the response while reducing the amplitude of the oscillation.)
2) Secondly, adopt a suitable reduction mechanism (such as a gearbox, synchronous belt pulley reduction, etc.) to minimize the mismatch between the load inertia and the motor inertia. (Recommended suitable inertia ratios are shown in Table 1)
3) If the above measures are ineffective, increasing the motor's inertia can be used to minimize the mismatch between the load inertia and the motor's inertia. Increasing the motor's rotational inertia can significantly reduce the resonance amplitude at the resonant point and improve the stability of the servo system. However, it should be noted that the size and cost of the motor generally increase with the increase of the rated torque. Therefore, when selecting a motor with a larger inertia for a reasonable inertia ratio, it is best to choose a motor with the same torque rating but a larger inertia. This will prevent a significant increase in motor size and cost.
4) Another solution is to use a high-order servo drive system that takes into account the torsion of the coupling. This system requires position feedback from both the motor itself and the load, forming a fully closed-loop system. This type of system can provide faster and more stable transient response.
2. Examples of load inertia calculation and motor selection
Leadshine's AC servo motors are generally available in models with different inertia values, such as the 60 and 80 frame motors, which are offered in both medium and low inertia versions. The following two common case studies demonstrate how to calculate load inertia and select the appropriate motor to reduce inertia mismatch.
2.1 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.
Solution: 1) Calculate the load inertia referred to the motor shaft.
Moment of inertia of the weight referred to the motor shaft
2) Calculate the motor speed
The required motor speed N = V / PB = 30 / 0.02 = 1500 rpm
3) Calculate the torque required for the motor to drive the load.
Torque required to overcome friction
=(200 * 9.8 * 0.002 * 0.02)/ 2π / 0.9 = 0.01387 Nm
Torque required for acceleration of heavy objects
= 200 * (30 / 60 / 0.2) * 0.02 / 2π / 0.9 = 1.769 Nm
Torque required for screw acceleration
= 0.0125 * (1500 * 6.28 / 60 / 0.2) / 0.9 = 10.903 Nm
Required maximum torque
Selected motor: The total inertia of the motion system is 145.29 kg*cm², and the required maximum torque is 12.686 Nm. The Leadsai ACM13030M2E-51-B motor has a rated speed of 2500 RPM, a rated torque of 12 Nm, and a rotor inertia of 29 kg*cm². The load inertia ratio is approximately 145.29/29 ≈ 5 times, which meets the recommended load inertia ratio range in Table 1.
2.2 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.
Selected motor option:
Based on the above calculation results, the Leadshine ACM6006L2H servo motor (rated torque 1.9NM, rated speed 3000RPM, motor inertia 0.6 kg.cm^2) can be selected, with an inertia ratio of 5 / 0.6 = 8.3 times.
During my observations at some customer sites, I found that some users were using the following motor model: ACM6004L2H (rated torque 1.27 Nm, peak torque 3.81 Nm, rated speed 3000 RPM, motor inertia 0.42 kg.cm^2). If this solution is chosen, the system inertia ratio is 5/0.42 = 11.9 times. The dynamic response performance and positioning completion time will be worse than if the ACM6006L2H servo solution is used. A reasonable inertia ratio significantly improves the dynamic performance of the entire motion system.
3. Conclusion
When the ratio of the load inertia on the motor shaft to the motor's inertia is 1, we call it inertia matching. However, in practical applications, the vast majority of load inertias are mismatched. Therefore, our research is not about achieving inertia matching, but rather about ensuring that the ratio of load inertia to motor inertia is within a reasonable range.
To improve the fast response characteristics of a servo system, it is essential to first increase the resonant frequency of the mechanical transmission components, i.e., increase the rigidity of the mechanical transmission components and reduce their inertia. Secondly, increasing damping to reduce the resonant peak value can also create conditions for improved fast response characteristics. Thirdly, if the load inertia is large, a reduction gear can be considered to achieve an inertia ratio between the load inertia and the motor inertia within a suitable range (see Table 1 for details). In some applications, a motor with even greater inertia can be selected to meet the requirements of reducing the inertia ratio and improving acceleration performance and stability. Finally, the successful application of many new servo drive control algorithms has also made it possible for servo systems to operate with higher precision and greater stability.
Lei Sai Introduction
Leadsun is a world-renowned brand and industry leader in the field of motion control for intelligent equipment.
Since its establishment in 1997, Leadshine has been committed to "replacing human manual labor" as its corporate mission, focusing on the research, development, production, sales and service of a series of high-quality products such as servo motor drive systems, stepper motor drive systems, motion control cards and motion controllers. Through persistent and meticulous efforts, Leadshine strives to help customers achieve their dreams and realize common growth.
Through two decades of unwavering commitment to product innovation, market expansion, and application services, Leadshine has become a leading global provider of motion control products and solutions in terms of production and sales volume. Due to the dual advantages of stability, reliability, and superior performance, Leadshine products are used by tens of thousands of excellent equipment manufacturers in industries such as electronics, robotics, machine tools, lasers, medical, and textiles, and are exported to more than 60 countries including the United States, Germany, and India.
