Issue 
Mechanics & Industry
Volume 24, 2023



Article Number  31  
Number of page(s)  14  
DOI  https://doi.org/10.1051/meca/2023020  
Published online  28 August 2023 
Regular Article
Inertia matching of CNC cycloidal gear form grinding machine servo system
^{1}
School of Mechatronics Engineering, Henan University of Science and Technology, Luoyang, Henan 471003, PR China
^{2}
Advanced manufacturing of mechanical equipment Henan Collaborative Innovation Center, Henan University of Science and Technology, Luoyang, Henan 471003, PR China
^{3}
First Tractor Company Limited, Luoyang, Henan 471004, PR China
^{4}
Zhong Bang Superhard tools Co. Ltd, Zhengzhou, Henan 450001, PR China
^{*} email: 9903437@haust.edu.cn
Received:
9
September
2022
Accepted:
14
June
2023
Reasonable ratio between the load inertia and servo motor inertia plays a decisive role for the dynamic performance and stability of the servo system, as well as the machining accuracy of the whole CNC machine. In order to improve the control performance and contour machining accuracy of the servo system of the CNC cycloidal gear form grinding machine, an optimization design method of the inertia matching for the CNC cycloidal gear form grinding machine servo system is proposed. The twomass servo driving closedloop PID control system is constructed, the influence of the different inertia ratios on the dynamic performance and contour errors of the servo system are deeply analyzed, and the inertia ratio is optimized to satisfied with the servo system performance requirements. Finally, the feasibility and practicability of the optimization design method of inertia matching are verified through the inertia ratio optimization grinding experiments of the cycloid gear in the CNC gear form grinding machine. This inertia matching optimization design method provides a valuable reference for the further design of CNC machine servo system.
Key words: Servo system / CNC cycloidal gear form grinding machine / inertia matching / inertia ratio
© J. Li et al., Published by EDP Sciences 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
With the rapid development of the numerical control and computer technologies, the modern precision grinding technology has put forward more requirements for the computer numerical control (CNC) gear grinding machine, such as more sensitive response, higher motion precision, and stronger reliability. After being manufactured, the entire operation performance of the CNC gear grinding machine usually needs to be test. If the test is conducted directly on the machine, it is not only low efficiency, but also may cause damage even safety accidents to the machine due to the unreasonable parameter settings. In the practical engineering application, the ratio of the load inertia to servo motor inertia is very important for the CNC gear grinding machine servo system. If the ratio of inertia exceeds reasonable control range, the servo system may generate oscillation or control failure, which will have a direct effect on the dynamic performance, stability, and machining accuracy of the servo system.
The CNC cycloidal gear form grinding machine is a key CNC equipment to realize the highefficiency and highprecision grinding of the rotary vector reducer cycloid gear. Therefore, it is urgent to carry on the inertia matching design of the CNC cycloidal gear form grinding machine so as to keep the inertia ratio within an appropriate range, which not only guarantees the fast response and stable operation of the servo driving system, but also satisfies the ideal geometric accuracy of the gear machining. Aiming at the inertia matching design of the CNC machine, relevant scholars at home and abroad have carried out relevant research.
For instance, Richard [1] analyzed and demonstrated the Kolmorgen motor with known torque, and concluded through mathematical formula derivation and experiment that when the inertia ratio of load was set to 1, the feed system could obtain the maximum load acceleration capacity. Li et al. [2] studied the speed adaptive control technology of the permanent magnet synchronous motor system with varied load inertia. Zhang and Furusho [3] studied the dynamic performance of a twomass system with different inertia ratios by using three pole assignment methods. Younkin et al. [4] concluded through theoretical analysis and experiments that reducing the moment of inertia of the motor shaft would not only affect the acceleration performance and followability of the system, but also affect the robustness of the controller, system bandwidth, dynamic stiffness, etc. Pritschow [5] used the motorload isolation single inertiaspring model to give the approximate relationship between the load mass and the upper limit of control gain and antijamming stiffness, and pointed out that increasing the load inertia will improve the antijamming ability. By means of closedloop frequency response discussion, Moscrop et al. [6] analysed the effects of high motorload inertia mismatch on servo system performance by using closed loop frequency responses. On this basis, presented the fast feedback control methods to improve the system response and verified the effectiveness of these control methods through the comprehensive experiment. Benath et al. [7] proposed a new design rule for servo driving applications to select a servo motor with an optimal energy transmission ratio. Boscariol et al. [8] proved that the most common size criteria based on inertia matching could not obtain the most energyefficient design through the parameterized analysis based on motor size, deceleration ratio and inertia ratio, as well as the energy consumption analysis for each design sample.
Besides, Shao et al. [9] studied a parallel robot, comprehensively considered the resonance frequency, acceleration torque and dynamic performance of the machine tool, and combined with simulation, obtained the inertia index and the appropriate matching range of inertia, which has been popularized and applied. Zhang et al. [10] proposed a new method to study the design law of inertia ratio. By eliminating the influence of servo control, they studied the matching characteristics of motion process and inertia ratio under high speed and high acceleration, and proved that the inertia ratio must be strictly limited within a certain range to effectively improve the system performance, which provided theoretical support for the design of highspeed machine tools. Zhang et al. [11] researched on the effect of load inertia on system performance and contour errors of 5coordinate cross beam mobile gantry machining center in time and frequency domain, and provided a reasonable selection range of inertia ratio of CNC machine tools in different machining forms, which provided theoretical basis for the inertia matching of heavy CNC machine tools. Liu et al. [12] employed the dual inertia model to analyze the relationship between energy matching efficiency and inertia ratio, and put forward the general design steps and basic methods of the load inertia ratio of servo system. Yuan et al. [13] proposed a method to obtain the positioning accuracy and synchronization control accuracy of servo motion system. Wang et al. [14] took servo motor selection of threeaxis transmission system in a CNC machine tools for example, established a servo motor selection mathematical model of transmission system in CNC machine tool, deeply analyzed the parameter selection and calculation of horizontal, inclined and vertical transmission system, and proved the rationality of the method by engineering test and motor control software.
All of the above researches have obtained very important research results, which provide the theory and foundation for the inertia matching research of servo system. However, there have been few scholars or actual papers that have studied the inertia matching technology of CNC cycloidal gear form grinding machine servo system.
In view of this, an optimization design method of inertia matching for the CNC cycloidal gear form grinding machine servo system is proposed to improve its control performance and contour machining accuracy in this paper. The closedloop PID control system of twomass servo driving is constructed, the dynamic performance of the singleaxis driving system and the doubleaxes driving system under different inertia ratios are compared, and the influence of inertia ratio on the servo system contour errors under the doubleaxes simultaneous control is analyzed. Finally, the inertia ratio satisfying the servo system performance requirements is optimized, and the feasibility and practicability of the optimization design method of inertia matching are verified through the inertia ratio optimization grinding experiments of the cycloid gear in the CNC gear form grinding machine.
2 Design and optimization of servo system inertia matching
According to the actual CNC machining needs of cycloidal gear, the designed YK7350B form grinding machine model and the form grinding model of the cycloidal gear are shown in Figures 1 and 2, respectively.
In Figure 2, Oxyz is the righthand Cartesian coordinate system, the subscript “J” represents the coordinate system of grinding wheel dressing wheel, subscript “G” indicates grinding wheel coordinate system, and subscript “2” indicates the workpiece coordinate system. “E” refers to the distance between the center of grinding wheel and the center of cycloidal gear in the workpiece moving direction.
Xaxis and Zaxis are the two servo driving axes of the cycloidal gear form grinding. In the cycloidal gear grinding processing, the corresponding machining motion is mainly completed through the coordination of Xaxis and Zaxis, and the relevant design parameters of each axis are shown in Table 1.
According to the different inertia ratios of cycloidal gear grinding machines and their dynamics relationships, the simulation analysis model is built to analyze the step response, gain upper limit, following error, frequency characteristics and antiinterference stiffness of the singleaxis doubledriving servo system and the singleaxis singledriving servo system under different inertia ratios. At the same time, when the Xaxis and Zaxis are simultaneously controlled, the influence laws of different inertia ratios on the contour errors of the twoaxis simultaneous motion can be analyzed, and then the accurate inertia ratio range can be optimized under the premise of meeting various performance requirements.
Fig. 1 Designed YK7350B form grinding machine model. 
Fig. 2 Form grinding model of cycloidal gear. 
Xaxis and Zaxis design parameters of gear forming machine YK7350B.
2.1 Inertia matching design of servo system
In the design of CNC cycloidal gear form grinding machine, it can be obtained that the weight of the load (including workbench and workpiece) of the CNC cycloidal gear form grinding machines is about m_{1} = 300 kg, the friction coefficient between the sliding table and the guide rail is μ = 0.002, and the pitch, diameter, weight of the ball screw are about S = 26mm , D = 80mm and m_{2} = 40kg, respectively μ = 0.002. Assume that the mechanical transmission efficiency is η = 95%, the maximum moving speed of the sliding table is v = 0.8m/s, and the acceleration and deceleration time is t = 0.5.s. In the selection design and quantitative calculation of servo motor, the calculation of servo motor speed, torque required to overcome the friction, torque required for heavy acceleration, and torque required for screw acceleration are shown in equations (1)−(4), respectively.
According to equations (2)−(4), the maximum driven torque required for the motor is shown in equation (5).
The maximum loading inertia converted to the motor shaft is shown in equation (6).
Thus, according to the structure design parameters of cycloidal gear grinding machine and the design parameters of CNC servo system, the corresponding parameters of servo motor can be preliminarily selected based on the servo motor selection manual, such as the rated speed is 2000 r/min, the rated torque is 16 N ⋅ m, and the rotor inertia is 75kg ⋅ cm^{2}. Theoretically, the loading inertia ratio of the motor can be calculated, which is about 371.42/75 ≈ 4.95 and satisfies the principle of inertia matching.
2.2 Servo system inertia matching optimization
In order to optimize the inertia matching, the concept of twomass servo system is introduced and the twomass servo driving closedloop system is constructed for simulation analysis. Generally, the servo motors and loads are seen as a whole and called the singlemass servo system. However, for a real servo system, the rigidity of mechanical driving components is finite. Under the action of motor driving torque, the mechanical shaft will be deformed to some extent. in the qualitative calculation, the inertia of driving shaft J_{a} is usually added to the maximum load inertia on the motor shaft J_{L} to simplify the motor driving system into a twomass servo system composed of motor and load [15]. The twomass servo system model is shown in Figure 3.
In Figure 3, u is the power supply voltage, θ_{m} is the motor rotation angle, θ_{L} is the loading rotation angle, T_{m} is the motor torque, T_{L} is the load torque, J_{a} is the rotational inertia of the transmission shaft, J_{L} is the maximum load inertia on the motor shaft, and K_{L} is the coupling stiffness coefficient, and the servo motor control model of the twomass servo system is shown in Figure 4.
According to Figure 4, the electric equation of the servo motor can be obtained, which is shown in equation (7).
where, L is the armature inductance, R is the armature resistance, C_{e} is the motor viscous damping coefficient, K_{i} is the current feedback coefficient, K_{m} is the motor torque coefficient, b_{m} is the motor viscous damping coefficient, J_{m} is the motor inertia, J_{L} is the load inertia, and T_{d} is the disturbance torque which including the friction torque and coupling torque. Meanwhile, the load model of the twomass servo system is shown in Figure 5.
According to Figure 5, the loading dynamics equation is shown in equation (8).
where, b_{L} is the loading viscous damping factor.
Combining equations (7) and (8), the loading motor inertia ratio can be obtained, which is shown in equation (9).
According to Figures 3−5, the structural model of the motorload twomass servo system can be established, as shown in Figure 6. Meanwhile, the simulation model of the twomass servo PID control system can be constructed with the help of the Simulink, which is shown in Figure 7.
Fig. 3 Twomass servo system model. 
Fig. 4 Servo motor control model of the twomass servo system. 
Fig. 5 Load model of the twomass servo system. 
Fig. 6 Structural model of the motorload twomass servo system. 
Fig. 7 Simulation model of the twomass servo PID control system. 
3 Influences of the inertia ratio on the servo system control performances
In order to better match the inertia ratio of the designed servo system, the influence laws of the inertia ratio on the servo system performances (e.g., unit step response performance, upper limit of control gain, following error, antiinterference stiffness, and closedloop frequency characteristics) are analyzed respectively.
3.1 Influence on the unit step response performance
The unit step response can directly reflect the dynamic performance of the servo system. The dynamic response characteristics of the servo system can be reflected by the parameters (e.g., delay time _{td}, rise time t_{x}, peak time _{tp}, overshoot σ and adjustment time _{ts}) of the response curve. Where, t_{x} can reflect the sensitivity and the transient process speed of the servo system, σ can reflect the smoothness of the servo system transient process, t_{s} can reflect the damping and response speeds of the servo system. Due to the frequent startup and braking, acceleration and deceleration, the speediness and overshoot under the step signal response are the two important indicators for evaluating the dynamic performance of the servo system. The response curve of the servo system under the step signal is shown in Figure 8.
Under the same output torque T_{L}, the smaller the inertia of the mechanical structure, the greater the angular acceleration, and the faster the response speed of the servo system. For the secondorder mechanical systems, the inherent frequency ω_{n}, the damping ratio ξ, the overshoot σ, and the adjustment time t_{s} can be expressed, which are shown in equations(10) and (11), respectively.
According to equations (10) and (11), it is can be demonstrated that the damping of the system B_{L} is proportional to the damping ratio ξ, the smaller the damping B_{L}, the longer the adjustment time of the servo system t_{s}, and the larger the overshoot σ. For the highorder system composed of servo feed system and mechanical system, the overshoot σand adjustment time t_{s} are expressed in equations (12) [16].
In equation (12), it can be seen that in the highorder systems, the overshoot is related to the position of
polezero of transfer function, and the adjustment time t_{s} is inversely proportional to the inherent frequency ω_{n} and the damping ratio ξ. In this paper, the Simulink is emploied to simulate highorder systems. Under the initial condition, the loading inertia ratio n is modified by changing the loading inertia of the servo feed system, the unit step response curves of the servo system under the singleaxis driving and double axes driving are shown in Figure 9.
From Figure 9, it can be seen that with the increase of the load inertia ratio n, the maximum overshoot of the servo system gradually increases, both the rising time tr and the adjustment time ts increase, the response speed and the stability of the servo system decreases. In addition, under the same load inertia condition, the response speed of the doubleaxes driving is faster, and the overshoot is relatively small. Therefore, in the initial condition, in order to make the system response quickly and stable, the preferred inertia ratio is the n ≤ 3.
Fig. 8 Response curve of the servo system under the step signal. 
Fig. 9 Unit step response curves under the singleaxle. driving and doubleaxle drivingg. 
3.2 Influence on the gain upper limit of servo system
The position loop gain is closely related to the fast responsiveness, stability and positioning accuracy of the servo system. The higher the upper limit of the position loop gain, the faster the dynamic response and the smaller the steadystate errors of the servo system. However, if the gain of the position loop is too large, the oscillation will occur. In this connection, the doubleaxes driving control model is simplified as the position loop transfer function, which is shown in Figure 10.
The steadystate errors is defined as , and transformed by Laplace transformation, which is as shown in equations (13), (14), (15) and (16), respectively..
If G_{k}(s) is set as the openloop transfer function of the servo system, which can be expressed as equation (17).
where, v is the number of series integration link. Define v= 0, 1, 2, corresponding to the type 0, type I and type II, respectively, which can be expressed as equation (18) [17].
Then
From equation (21), it can be seen that the magnitude of the servo system steadystate errors is related to the type of system, mechanical structure parameters, servo parameters, system equivalent gain and input signal. By changing the load inertia and adjusting the gain of the system position loop, taking the no overshoot of the position loop as the constraints of step response performance, the relationship between the load inertia ratio and the gain upper limit of position loop under the singleaxis driving and doubleaxes driving are studied, and the simulation results are shown in Figure 11.
It can be seen that with the loading inertia ratio gradually increasing, the gain upper limit of position loop decreases and the descending speed decreases, the static errors of the system increases and the dynamic response speed of the system decreases both in singleaxis driving and doubleaxes driving. In addition, even if the load inertia ratio is same, the gain upper limit of the position loop of doubleaxes driving is slightly higher than that of the singleaxis.
Fig. 10 Position loop transfer function of the simplified doubleaxes driving control model. 
Fig. 11 Relationship between the load inertia. ratio and the gain upper limit of position loop. 
3.3 Influence on the servo system following errors
Due to inertia, the output of CNC machining servo system lags behind input to some extent, and the difference between them is following errors. The servo system following errors are the proportional to the speed command and inversely proportional to the gain of the position loop. By changing the loading inertia, taking the sine signal as the position input signal and the position loop gain as the servo system control parameters, the change laws of the servo system following errors with different loading inertia ratios n under the singleaxis driving and doubleaxes driving are simulated and analyzed, and the simulation results are shown in Figure 12.
As illustrated in Figure 12, with the increase of the loading inertia ratio n, the servo system following errors increases, however, the following accuracy of the servo system decreases, and the following errors changes together with the input signal speed. In addition, under the same loading inertia, compared with the singleaxis driving system, the doubleaxes driving system can obtain higher position gain and relatively smaller tracking errors.
Fig. 12 Change laws of the servo system following errors with different loading inertia ratios n under the singleaxis driving and doubleaxes driving. 
3.4 Influence on the interferencefree stiffness of servo system
The interferencefree dynamic stiffness of the servo system is the unit output angle caused by the feed mechanism (i.e. ball screw) under the different frequency interference signals, which is shown in equation (22).
It reflects the ability of the servo system to resist position errors under the influence of the external interference signals. In this paper, the antiinterference dynamic stiffness of the servo system under different load inertia ratios is shown in Figure 13.
As shown in Figure 14, the antiinterference stiffness of the system decreases with the increase of the inertia ratio in the low frequency range and increases together with the inertia ratio in the medium and high frequency, which having a better inhibition action to the high frequency disturbance. In addition, under the same load inertia, the antiinterference stiffness of doubleaxes driving system is higher than that of the singleaxis driving. So, in the initial condition, in order to improve the interferencefree dynamic stiffness of the servo system, the preferred inertia ratio is 3≤n≤5.
Fig. 13 Antiinterference dynamic stiffness of the servo system under different load inertia ratios. 
Fig. 14 Amplitudefrequency characteristic curve of the servo system under the different load inertia ratios. 
3.5 Influence on the closed loop frequency characteristics of servo system
The closedloop frequency characteristics of the servo system are described as equation (23).
The closedloop frequency characteristics of the servo system position loop under the different loading inertia ratios are calculated, and the amplitudefrequency characteristic curve of the servo system is shown in Figure 14.
As illustrated in Figure 14, when the inertia ratio is 0.5 ≤ n ≤ 2, the control bandwidth of the doubleaxes driving increases together with the load inertia ratio, while when the inertia ratio is 2<n ≤ 5, the control bandwidth of the doubleaxes driving decreases with the increase of the load inertia ratio. In addition, with the increase of load inertia, the resonance frequency of the system decreases gradually, and the position loop of the doubleaxes driving system can obtain a higher control bandwidth at the same inertia ratio. Therefore, for the servo system with large load inertia, in order to have a high control bandwidth and avoid resonance, the speed regulation performance of the system will be reduced. It is preferred that the inertia ratio of the doubleaxes driving system be less than n< 2 and that of the singleaxis driving system be less than n < 3.
4 Influence of the inertia ratio on the servo system contour errors
The CNC servo system accurately controls the speed and position of each axis according to the command signals to obtain the different motion trajectories. Due to the influence of the mechanical structure of the system on the steadystate and dynamic performance of the servo system, the contour errors are inevitably generated during the synchronous driving process of each axis, which thereby affects the machining accuracy of the CNC machine. In this connection, it is essential to analyze the influence of the inertia ratio on the servo system contour errors.
4.1 Influence on the linear contour errors of servo system
Perfect inertia matching between the Xaxis and Zaxis is significant to guarantee the excellent gear grinding quality. Thus, it is necessary to analyze the influence of inertia ratio on the servo system contour errors during the synchronous driving of Xaxis and Zaxis. Assuming the input speed of synchronous driving of Xaxis and Zaxis is ν, the angle between ν and the positive Xaxis is θ, the theoretical desired positions of the two axes are shown in equation (24).
During the liner motion of the CNC machine, the contour errors of the synchronous driving trajectory of the two axes is generated due to the following errors between the two axes, and the schematic diagram is shown in Figure 15.
As shown in Figure 15, Xaxis and Zaxis move along a straight line in the X−Z plane. The angle between the theoretical desired trajectory and the positive Xaxis is θ, P^{*} is the theoretical position and P is the corresponding actual position. The following errors of Xaxis and Zaxis are △x and △z, respectively, and the contour errors ε can be expressed in equation (25).
The following errors of two axes are recorded in equation (26).
Substitute equation (26) into equation (25), then ε can be obtained, which is shown in equation (27).
For the ideal liner trajectory, there
So, combined with equation (28), equation (27) can be simplified to equation (29).
Set other structural parameters be invariant, the input speed signal ν is 30 mm/s, the angle θ between ν and the positive Xaxis is 45°, only change the load inertia ratio, and the linear contour errors change rules of the servo system is simulated and analyzed. Maintaining Zaxis inertia ratio unchanged, changing Xaxis load inertia ratio, the simulated liner contour errors is shown in Figure 16.
From Figure 16, it can be seen that the linear contour errors of Xaxis fluctuate greatly during the acceleration phase of linear motion and tend to stabilize gradually within 0.35 s. When n = 0.5, the contour errors are mainly positive (maximum:2.4 × 10^{−5}). When load inertia ratio is within 1 ≤ n ≤ 5, the fluctuation degree of the line contour errors and the maximum contour errors increase together with the inertia ratio. When the load inertia ratio n = 5, the max contour errors is −3.76 × 10^{−4}. So, it is preferred that the load inertia ratio of the doubleaxes driving servo system Xaxis is taken as n ≤ 2.
Similarly, Maintaining Xaxis inertia ratio unchanged, changing Zaxis inertia ratio, the simulated linear contour errors is shown in Figure 17.
As seen Figure 17, the linear contour errors of Zaxis fluctuate greatly and tend to be stable over time during the acceleration stage of linear motion. Whereas, when 0.5 ≤ n ≤ 1, the maximum contour errors decreases with the increase of load inertia, but the difference is not obvious. When 2≤n≤5, the contour errors increases together with the load inertia [18, 19]. So, the load inertia ratio of the doubleaxes driving servo system Zaxis is preferred 0.5 ≤ n ≤ 2 to realize the excellent linear machining accuracy.
Fig. 15 Schematic diagram of contour errors of the linear motion trajectory. 
Fig. 16 Servo system linear trajectory contour errors when changing Xaxis inertia ratio separately. 
Fig. 17 Servo system linear trajectory contour errors when changing Zaxis inertia ratio separately. 
4.2 Influence on the circular contour errors of servo system
During the circular motion of the CNC machine, doubleaxes position commands are set as equation (30).
where, R is the arc radius, ν/R is the angular velocity. In actual motion, contour errors inevitably exist in the circular motion due to the following errors △x and △z of the Xaxis and Zaxis, which is shown in Figure 18.
In Figure 18, θ is the angle between the theoretical reference point A and Xaxis, R is the arc radius, A(A_{x},A_{z})is the theoretical reference point, B(B_{x},B_{z})is the corresponding actual point, C(C_{x},C_{z})is the center of the circular trajectory, then the contour errors ε of the arc trajectory can be expressed in equation (31).
The location of actual point B is:
Substitute equation (32) and (33) into equation (31), the contour errors ε can be obtained, which is shown in equation (34).
Assuming Δx and Δy are much larger than ϵ, and R is large enough, equation (34) can be expanded by applying Taylor formula.
If higher order items of equation (35) are omitted, equation (34) can be simplified to equation (36).
For any curvilinear motion, the contour can be infinitely approximated by an arc, which substituting its curvature radius for the arc radius at a certain point. Therefore, the contour errors can be expressed by equation (36) during the doubleaxes curve synchronous driving. The calculation model built in Simulink is shown in Figure 19.
Change the load inertia ratio and keep other structural parameters unchanged, the circular contour errors change rules of the servo system is simulated and analyzed. The maximum input speed signal νis 5 m/min and the arc radius is 0.05 m. The input position command signals of Xaxis and Zaxis are shown in Figures 20 and 21.
Maintaining the inertia ratio of one axis unchanged, the circular contour errors of the simulation calculation results is shown in Figure 22. It can be seen that the circular contour errors of CNC machine change periodically during the doubleaxes arc motion, and the maximum contour errors occurs at the commutating time of each axis. Changes of the inertia ratio have less influence on the arc motion contour errors of servo system.
Therefore, combined with the influence analysis of the inertia ratio on the servo system control performance and contour errors, in order to reduce the linear motion contour errors of the servo system and make the servo system have higher machining accuracy, the precise range of the inertia ratio can be optimized as 0.5 ≤ n ≤ 2.
Fig. 18 Schematic diagram of the contour errors of the arc. 
Fig. 19 Computational model of contour errors of circular arc motion. 
Fig. 20 Xaxis input signal. 
Fig. 21 Zaxis input signal. 
Fig. 22 Simulation calculation results of the circular contour errors. 
5 Grinding experiment of the cycloidal gear
To verify the rationality of the optimized inertia ratio, the inertia ratio optimization grinding experiments were conducted on the CNC cycloidal gear form grinding machine YK7350B is independently developed by the Gear Manufacturing Henan Engineering Technology Research Center. The tooth profile of the grinded cycloidal gear is measured on Gleason 650 gear measuring machine. The cycloidal gear form grinding is shown in Figure 23, and the cycloidal gear measuring is shown in Figure 24. The measuring results of the cycloidal gear tooth profile before and after the inertia ratio optimized in grinding process are shown in Figures 25 and 26, respectively.
From Figures 25 and 26, it can be seen that the maximum tooth profile deviations of the cycloidal gear before inertia ratio optimized is 0.037288 mm, the minimum tooth profile deviations is −0.017133 mm and the difference between them is 0.020155 mm.
Comparatively, the maximum tooth profile deviations of cycloidal gear after the inertia ratio optimized is 0.019536 mm, the minimum tooth profile deviations is −0.007253 mm and the difference between them is 0.012283 mm. The comparison shows that the cycloidal tooth profile deviations is reduced and satisfied with the requirements of grade 4 accuracy of tooth profile. After optimizing the inertia ratio, the cycloidal gear profile is more stable, the variation range of tooth curve is much less, so as to the roughness of cycloidal gear tooth surface is improved and the tooth surface accuracy is ensured to a certain extent.
Fig. 23 Cycloidal gear form grinding. 
Fig. 24 Cycloidal gear measuring. 
Fig. 25 Measuring results of the cycloidal gear tooth profile before the inertia ratio optimized. 
Fig. 26 Measuring results of the cycloidal gear. tooth profile after the inertia ratio optimized 
6 Conclusion
Through the inertia matching analysis of the CNC cycloidal gear form grinding machine servo system, the twomass servo driving closedloop PID control system is constructed. The influence laws of different inertia ratios on the system control performance and the system profile errors are studied and analyzed. The feasibility and practicability of the inertia matching design for the CNC gear form grinding machine servo system is verified by the simulation analysis and machining experiment of different inertia ratios. The main research conclusions are as follows:
With the increase of the inertia ratio, the response time and stability time of singleaxis and doubleaxes driving systems increase, the following errors of the servo system increases, and the antiinterference stiffness in high frequency band of the servo system increases. Meanwhile, the upper limit of the gain of closedloop position control of the servo system decreases, the resonant frequency of the system decreases gradually, the fast response speed of the system decreases, the speed regulation performance of the system decreases, and the antiinterference stiffness of the system in low frequency band decreases. Under the same load inertia, the dynamic performance indexes of the doubleaxes driving system are more superior.
In the doubleaxes simultaneous motion process of the CNC cycloidal gear form grinding machine, the increase of the load inertia both Xaxis and Zaxis will cause a change in the linear contour errors of the servo system, but it has little influence on the circular contour errors of the servo system. Therefore, in the doubleaxes CNC machine tool with cross sliding table and other structures, the preferred inertia ratio is 0.5≤n≤2, so as to reduce the linear contour errors of servo system and ensure the machining accuracy.
The inertia matching optimization design method proposed in this paper provides a valuable reference for improving the control performance and contour machining precision of the CNC cycloidal gear form grinding machine servo system, as well as the development of the cycloidal gear machining quality. Strictly speaking, in studying the influence of inertia ratio on the dynamic adjustment performance of the servo system, the model is simplified. The established servo driving model is an idealized model, which without considering the changes in ball screw diameter, stiffness, and other aspects with the load inertia changing. The research results may have some deviations from the actual results, so the next efforts is to establish a more accurate servo system model and further research on the problem of inertia matching.
Conflict of Interest
The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article. This work was supported by the National Key R&D Program of China (No. 2020YFB2006800) and National Natural Science Foundation of China (No. 51405135, No. 51775171).
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Cite this article as: J. Li, J. Su, M. Gao, D. Zhao, L. Zhang, D. Wang, F. Shi, Inertia matching of CNC cycloidal gear form grinding machine servo system, 24, 31 (2023)
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All Figures
Fig. 1 Designed YK7350B form grinding machine model. 

In the text 
Fig. 2 Form grinding model of cycloidal gear. 

In the text 
Fig. 3 Twomass servo system model. 

In the text 
Fig. 4 Servo motor control model of the twomass servo system. 

In the text 
Fig. 5 Load model of the twomass servo system. 

In the text 
Fig. 6 Structural model of the motorload twomass servo system. 

In the text 
Fig. 7 Simulation model of the twomass servo PID control system. 

In the text 
Fig. 8 Response curve of the servo system under the step signal. 

In the text 
Fig. 9 Unit step response curves under the singleaxle. driving and doubleaxle drivingg. 

In the text 
Fig. 10 Position loop transfer function of the simplified doubleaxes driving control model. 

In the text 
Fig. 11 Relationship between the load inertia. ratio and the gain upper limit of position loop. 

In the text 
Fig. 12 Change laws of the servo system following errors with different loading inertia ratios n under the singleaxis driving and doubleaxes driving. 

In the text 
Fig. 13 Antiinterference dynamic stiffness of the servo system under different load inertia ratios. 

In the text 
Fig. 14 Amplitudefrequency characteristic curve of the servo system under the different load inertia ratios. 

In the text 
Fig. 15 Schematic diagram of contour errors of the linear motion trajectory. 

In the text 
Fig. 16 Servo system linear trajectory contour errors when changing Xaxis inertia ratio separately. 

In the text 
Fig. 17 Servo system linear trajectory contour errors when changing Zaxis inertia ratio separately. 

In the text 
Fig. 18 Schematic diagram of the contour errors of the arc. 

In the text 
Fig. 19 Computational model of contour errors of circular arc motion. 

In the text 
Fig. 20 Xaxis input signal. 

In the text 
Fig. 21 Zaxis input signal. 

In the text 
Fig. 22 Simulation calculation results of the circular contour errors. 

In the text 
Fig. 23 Cycloidal gear form grinding. 

In the text 
Fig. 24 Cycloidal gear measuring. 

In the text 
Fig. 25 Measuring results of the cycloidal gear tooth profile before the inertia ratio optimized. 

In the text 
Fig. 26 Measuring results of the cycloidal gear. tooth profile after the inertia ratio optimized 

In the text 
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