Research on the design of longitudinal double gear rack transmission mechanism for CK61250 heavy-duty CNC lathe

Abstract

This article proposes a dual motor anti backlash rack and pinion transmission mechanism for the longitudinal feed system of CK61250 heavy-duty CNC lathe. This mechanism is different from fixed preloading and simple master-slave tracking electric preloading dual motor schemes. It adopts a master-slave control architecture that integrates real-time adjustable preloading force, disturbance observer, and stiffness error compensation framework. The force position decoupling dual motor layout is adopted, and mechanical preloading is not set. The control layer includes adaptive preloading adjustment and active cutting force suppression content. The relevant content has been verified through frequency domain stability margin analysis, interference suppression ratio evaluation, and comparison with single motor system simulation. The single motor system has problems such as transmission clearance, elastic deformation, and vibration noise. The maximum thrust of the system is 50 kN, the fast moving speed is 12 m/min, and the full stroke positioning accuracy is better than ±0.015 mm. A comprehensive error mathematical model has been developed, and sensitivity analysis shows that thermal deformation and control errors account for 38% and 22% of the overall positioning error.

Introduction

As the core equipment in the high-end equipment manufacturing field, heavy-duty CNC machine tools directly determine the quality and efficiency of machined parts through the accuracy, stiffness, and dynamic performance of their transmission system. Especially for heavy-duty CNC lathes like CK61250, the longitudinal feed system needs to withstand high loads, achieve high-precision displacement control, and have good anti-interference ability and long-term stability. The gear rack transmission mechanism is widely used in the longitudinal feed system of large machine tools due to its high transmission efficiency, strong load-bearing capacity, and large stroke range. However, traditional single motor driven gear rack systems suffer from problems such as transmission clearance (backlash), elastic deformation, vibration noise, etc., especially under heavy load and high-speed conditions, which can easily lead to decreased positioning accuracy and delayed dynamic response, seriously restricting the further improvement of machine tool performance.

In recent years, scholars at home and abroad have conducted multiple studies on the design and control of gear rack transmission systems. In terms of modeling and parameter optimization of transmission systems, Gong et al. compared and analyzed the uniform wear and bending strength of large modulus gear rack pairs through theory and experiment, providing important basis for tooth profile design. ¹ Abadjieva studied the influence of the geometric shape of the active tooth surface on the transmission performance in spatial rack and pinion transmission. ² In terms of gap control, Verl and Leipe proposed a dual motor position feedback control method, which effectively improves the positioning accuracy of the system through electrical preload force. ³ Steinle et al. conducted experimental research on the actual impact of transmission errors on system performance, and pointed out that there is a strong coupling relationship between clearance and error. ⁴ In addition, Wiebke et al. attempted to use supervised learning classification methods to estimate the gap size in the gear rack drive of machine tools, providing a new approach for intelligent monitoring of gaps. ⁵ In terms of dynamic characteristic analysis, Zhou et al. established a time-varying meshing stiffness model considering friction and wear, revealing the negative impact of tooth surface degradation on system dynamic performance. ⁶

Although significant progress has been made in research, there are still several limitations. Firstly, most literature focuses on local optimization of single motor drive systems, lacking in-depth research on the overall performance of the system under the dual motor collaborative drive architecture. Specifically, the existing dual-motor electric pre-tightening systems (such as those mentioned in Verl and Leipe ³ ) usually use a fixed pre-tightening torque, which will not be adjusted with the changing cutting force. This may lead to insufficient backlash compensation under heavy load or excessive wear under light load. In contrast, our proposed method adopts real-time adjustable pre-tightening, ⁷ and realizes adaptive compensation by combining the force loop control of disturbance observer ⁸ to ensure that unilateral contact can be maintained under all working conditions. In addition, the existing clearance elimination methods often rely on mechanical pre-tightening ⁹ or compensation with a single motor, which is difficult to keep stable in long-term use and easy to make parts wear faster. The new method we put forward does not need mechanical preload at all, but depends on electric preload and closed-loop force control, which can reduce wear and make the machine more durable. Moreover, for the transmission system of heavy CNC machine tools, the research on multi-physical field (electromechanical) coupling modeling and joint simulation is not enough, which leads to our incomplete prediction accuracy and dynamic response analysis. Our research is to solve this problem. We have developed a comprehensive electromechanical joint simulation platform in Python, which combines multi-body dynamics, time-varying meshing stiffness and real-time control logic.

Based on the above issues, this article takes the longitudinal feed system of CK61250 heavy-duty CNC lathe as the research object, proposes a dual motor backlash elimination gear rack transmission scheme, aiming to achieve high-precision positioning and dynamic anti-interference ability through a master-slave control strategy. There are three innovations in this work: (1) a master-slave structure with force-position decoupling, which can adjust the preload in real time; (2) disturbance observer is integrated, which can actively compensate cutting force; (3) an error compensation framework based on stiffness can connect the mechanical design parameters with the control performance quantitatively. We quantitatively verify these innovations through frequency domain robustness analysis (gain/phase margin), anti-jamming ratio calculation and comparison simulation with baseline system. By establishing a dynamic model, stiffness model, and error synthesis model of the system, and combining with electromechanical joint simulation and experimental verification, the feasibility and superiority of the proposed scheme are comprehensively evaluated. According to the current development stage, this study is a design feasibility analysis based on simulation; In the future, it is planned to carry out experimental verification on the physical test platform. This study aims to provide theoretical basis and technical reference for the design of transmission systems for high-precision heavy-duty CNC machine tools.

Design of transmission system and theoretical model

System design requirements and key parameters

The longitudinal feed system of CK61250 heavy-duty CNC lathe needs to achieve high-precision positioning under extreme heavy load conditions. The core design requirements are derived from the overall specifications and operational requirements of the machine. Firstly, the system must provide a maximum thrust of 50 kN to overcome the enormous cutting resistance when machining large workpieces. Secondly, to ensure high processing efficiency, the fast moving speed must be achieved $,12\mathrm{m}/\min$ . The most important thing is that the full-stroke positioning accuracy must be better than 0.015 mm to meet the requirements of precision machining of heavy parts. In addition, the system must have high stiffness and natural frequency to minimize vibration and ensure dynamic stability during operation.

In order to set a fair performance benchmark, we use an ordinary single-motor rack-and-pinion system as a comparison ¹⁰ standard, and its mechanical parameters (modulus 8 mm, number of pinion teeth 20, same rack length) are the same as those of the experimental system. The standard PID position controller is used in this reference system, and there is no pre-tightening force or force closed loop, which represents the common practice in industry. All our comparisons are carried out under the same operating conditions (the same cutting force curve, motion trajectory and simulation environment), so as to ensure the consistent results. The reference control parameters are also adjusted by the same frequency domain method, and the phase margin is set to 50, so that the comparison is fair.

As shown in Figure 1, the core of the transmission system is a dual motor driven, longitudinally arranged dual tooth gear pair. Based on the above design requirements, according to current research on the double gear-rack pair (References 10, 11, 18–22), we have precisely calculated the key parameters of the gear rack pair. The selected modulus is 8 mm to withstand high torque and high stress. The number of teeth on the small gear $\mathrm{z}=20$ and the standard pressure angle are ${20}^{°}$ . The calculation formula for the pitch circle diameter of the small gear $\mathrm{d}$ is as follows:

$$\mathrm{d}=\mathrm{m}\times \mathrm{z}=8\times 20=160\mathrm{mm}$$

(1)

The theoretical center distance between two small gears is crucial for pre tightening and clearance reduction, and its value is set to be equal to the pitch diameter to ensure symmetrical force distribution, $\mathrm{aa}=160\mathrm{mm}$ . The maximum output torque required for each servo motor ${\mathrm{T}}_{\max }$ is determined by the maximum thrust ${\mathrm{F}}_{\max }$ and the radius of the small gear $\mathrm{r}$ :

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{T}}_{\max }=\frac{{\mathrm{F}}_{\max }\times \mathrm{r}}{2}=\frac{50000\mathrm{N}\times 0.08\mathrm{m}}{2}}\\ \multicolumn{1}{c}{=2000\mathrm{N}\setminus \mathrm{cdotpm}}\end{array}$$

(2)

The basic structure and key geometric relationships of the system are shown in Figure 2 below:^18,19–24

Figure 1.

Design of double gear and rack transmission structure.

Figure 2.

Basic structure and key geometric relationships of the system.

The above parameters and architecture lay the theoretical design foundation for the subsequent detailed design, modeling,^18,25,26 and analysis of the transmission system.

Parametric design and verification of rack and pinion pairs

Based on the key system parameters determined in Section 2.1, this section provides a detailed parametric design ²⁷ and verification of the gear rack pairs required for dual motor drive. The design process strictly follows the ISO 6336 standard to ensure its reliability, durability, and transmission accuracy under heavy load conditions. Firstly, based on the selected module and the number of teeth on the small gear, $\mathrm{m}=8\mathrm{mm}$ calculate the core geometric dimensions of the gear pair. The tooth crest height, tooth root height, and full tooth height of the small gear are determined by the following formulas: $({\mathrm{h}}_{\mathrm{a}})\left({\mathrm{h}}_{\mathrm{f}}\right)(\mathrm{h})$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{h}}_{\mathrm{a}}=1\times \mathrm{m}=8\mathrm{mm},{\mathrm{h}}_{\mathrm{f}}=1.25\times \mathrm{m}}\\ \multicolumn{1}{c}{=10\mathrm{mm},\mathrm{h}={\mathrm{h}}_{\mathrm{a}}+{\mathrm{h}}_{\mathrm{f}}=18\mathrm{mm}}\end{array}$$

(3)

Its pitch circle diameter $\mathrm{d}=\mathrm{m}\times \mathrm{z}=160\mathrm{mm}$ , base circle diameter ${\mathrm{d}}_{\mathrm{b}}=\mathrm{d}\times \cos \left({20}^{°}\right)\approx 150.35\mathrm{mm}$ , and tooth pitch $\mathrm{p}=\pi \times \mathrm{m}\approx 25.13\mathrm{mm}$ . To ensure the correct meshing of the double gears and the application of preload force, the theoretical installation center distance between the two small gears must be strictly controlled $\mathrm{a}=160\mathrm{mm}$ .

Furthermore, the strength verification and validation of the gear rack pair through theoretical calculations are crucial to ensure that it can withstand a maximum thrust of 50 kN. The bending fatigue strength of the tooth root must be sufficient to resist repeated loads. According to the Lewis bending stress formula, the maximum bending stress at the tooth root must be less than the allowable bending stress of the material: ²⁸ ${\sigma }_{\mathrm{F}}\left[{\sigma }_{\mathrm{F}}\right]$

$${\sigma }_{\mathrm{F}}=\frac{{\mathrm{F}}_{\mathrm{t}}}{\mathrm{b}·\mathrm{m}}·{\mathrm{Y}}_{\mathrm{F}}·{\mathrm{Y}}_{\mathbf{ϵ}}·{\mathrm{Y}}_{\beta }\le \left[{\sigma }_{\mathrm{F}}\right]$$

(4)

Among them, $F_t$ is the tangential force on the end face indexing circle (converted by $F_{\max }$ ), $b$ is the tooth width, $Y_F$ is the tooth profile coefficient, $Y_{\mathbf{ϵ}}$ is the coincidence coefficient, $Y_{\beta }$ and is the helix angle coefficient. Referring to Gong et al.’s research, ¹ this design uses high-strength alloy steel 20 CrMnTi, which has undergone carburizing and quenching treatment on the surface, with allowable bending stress $\left[{\sigma }_{\mathrm{F}}\right]=420\mathrm{MPa}$ . After calculation, the maximum bending stress ${\sigma }_{\mathrm{F}}\approx 385\mathrm{MPa}$ and safety factor meet the requirements of bending strength.

In addition, the contact fatigue strength of the tooth surface is a decisive factor in preventing pitting failure. ²⁹ The contact stress ${\sigma }_{\mathrm{H}}$ is calculated according to the Hertz formula:

$${\sigma }_{\mathrm{H}}={\mathrm{Z}}_{\mathrm{H}}·{\mathrm{Z}}_{\mathrm{E}}·{\mathrm{Z}}_{\mathbf{ϵ}}·{\mathrm{Z}}_{\beta }·\sqrt{\frac{{\mathrm{F}}_{\mathrm{t}}}{\mathrm{d}·\mathrm{b}}·\frac{\mathrm{u}+1}{\mathrm{u}}}\le \left[{\sigma }_{\mathrm{H}}\right]$$

(5)

In the formula, ${\mathrm{Z}}_{\mathrm{H}}$ is the node region coefficient, ${\mathrm{Z}}_{\mathrm{E}}$ is the elasticity coefficient, $Z_{\mathbf{ϵ}}$ is the coincidence coefficient, $Z_{\beta }$ is the helix angle coefficient, $\mathrm{u}$ and is the gear ratio (for rack meshing, take its limit value). Allowable contact stress of materials $\left[{\sigma }_{\mathrm{H}}\right]=1350\mathrm{MPa}$ . The calculation results show that the maximum contact stress ${\sigma }_{\mathrm{H}}\approx 1230\mathrm{MPa}$ and safety factor ${\mathrm{S}}_{\mathrm{H}}=\left[{\sigma }_{\mathrm{H}}\right]/{\sigma }_{\mathrm{H}}\approx 1.1>1$ also meet the requirements of contact strength, which is consistent with the theoretical and experimental conclusions in Gong et al. ¹ Meanwhile, referring to the time-varying mesh stiffness model of Zhou et al., ⁶ the wear life of this design was predicted, and the results showed that the wear amount was within an acceptable range within the expected working life.

Finally, in order to avoid vibration and noise under high-speed heavy loads, the meshing interference was verified. According to the research method on multi-point meshing interference in Wang et al., ³⁰ the end face coincidence degree and longitudinal coincidence degree were calculated ${\mathbf{ϵ}}_{\alpha }{\mathbf{ϵ}}_{\beta }$ :

$${\mathbf{ϵ}}_{\alpha }=\frac{1}{2\pi }\left[{\mathrm{z}}_1(\tan {\alpha }_{\mathrm{a}1}-\tan {\alpha }^{\prime })\right]\ge 1.3$$

(6)

The calculated coincidence degree of this design ensures continuous and smooth transmission, effectively avoiding meshing interference and motion impact. In summary, all parameters have been validated, laying a solid foundation for subsequent system stiffness modeling and dynamic analysis.

Modeling of system static stiffness and natural frequency

The static stiffness and natural frequency of the system are the core indicators for evaluating the performance of the longitudinal feed system of CK61250 heavy-duty CNC lathe, directly determining its positioning accuracy and anti-vibration performance under heavy cutting conditions. As shown in Figure 1, the system is a complex electromechanical coupling system driven by dual servo motors synchronously driving two sets of gear rack pairs through a reducer. The total static stiffness is modeled by considering the two drive branches as parallel-connected subsystems, which are then connected in series with the meshing stiffness and the rack stiffness. The equivalent total stiffness can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{{\mathrm{K}}_{\mathrm{total}}}=\frac{1}{2}(\frac{1}{{\mathrm{K}}_{\mathrm{motor}}}+\frac{1}{{\mathrm{K}}_{\mathrm{reducer}}}+\frac{1}{{\mathrm{K}}_{\mathrm{shaft}}})}\\ \multicolumn{1}{c}{+\frac{1}{{\mathrm{K}}_{\mathrm{mesh}}}+\frac{1}{{\mathrm{K}}_{\mathrm{rack}}}}\end{array}$$

(7)

Where the factor 2 accounts for the two parallel drive branches, and the reciprocal summation reflects the series connection of components along the mechanical transmission chain. Among them, ${\mathrm{K}}_{\mathrm{motor}}$ and ${\mathrm{K}}_{\mathrm{reducer}}$ , are respectively, the stiffness of the servo motor itself and the torsional stiffness of the reducer, ${\mathrm{K}}_{\mathrm{shaft}}$ and the torsional stiffness of the output shaft. These three together constitute the stiffness of the driving branch, which is the single-sided gear rack meshing stiffness. Its value has time-varying characteristics. Referring to the time-varying meshing stiffness model established by Zhou et al., ⁶ it can be expressed as a function based on the length variation of the tooth contact line:

$${\mathrm{K}}_{\mathrm{mesh}}(\mathrm{t})=\frac{1}{\sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{1}{{\mathrm{k}}_{\mathrm{i}}(\mathrm{t})}}$$

(8)

In the formula, ${\mathrm{k}}_{\mathrm{i}}(\mathrm{t})$ is the stiffness of a single tooth pair, $\mathrm{N}$ and is the number of teeth engaged simultaneously. And ${\mathrm{K}}_{\mathrm{rack}}$ it is the longitudinal tensile stiffness of the rack on the bed. Owing to the dual-motor preloaded anti-backlash architecture, the total stiffness of the system is significantly higher than that of traditional single-drive configurations.

In terms of dynamic modeling, in order to analyze the natural frequency of the system and avoid resonance, it is necessary to establish a concentrated mass model. This article simplifies the system as a multi degree of freedom bounce mass system. The dynamic differential equation can be expressed as:

$$\mathrm{M}\stackrel{··}{\mathrm{X}}+\mathrm{C}\stackrel{·}{\mathrm{X}}+\mathrm{KX}=\mathrm{F}$$

(9)

Among them, $\mathrm{M},\mathrm{C}$ , and K are the mass matrix, damping matrix, and stiffness matrix of the system, X is the generalized displacement vector, and F is the external force vector. By solving the characteristic value problem of undamped free vibration of the equation $(\mathrm{K}-{\omega }^2\mathrm{M})\mathrm{\Phi }=0$ , the natural frequencies and main modes of the system at each order can be obtained ${\omega }_{\mathrm{n}}\Phi$ . The first natural frequency ${\mathrm{f}}_{\mathrm{n}}$ (in Hz) is crucial, and its calculation formula is:

$${\mathrm{f}}_{\mathrm{n}}=\frac{1}{2\pi }\sqrt{\frac{{\mathrm{K}}_{\mathrm{total}}}{{\mathrm{m}}_{\mathrm{eq}}}}$$

(10)

Here, ${\mathrm{m}}_{\mathrm{eq}}$ the equivalent mass converted to the rack includes the mass of moving parts such as worktable, workpiece, saddle, and the moment of inertia of rotating parts such as motor rotor and gear. The high static stiffness ${\mathrm{K}}_{\mathrm{total}}$ and natural frequency ensure that the system undergoes minimal deformation when subjected to a cutting force of 50 kN, and can move away from external excitation frequencies, effectively suppressing vibration and providing core mechanical support for achieving $\pm 0.015\mathrm{mm}$ positioning accuracy.

Comprehensive mathematical model of positioning error

In order to achieve the ultra-high positioning accuracy requirements of the longitudinal feed system of CK61250 heavy-duty CNC lathe, a comprehensive and accurate mathematical model of comprehensive positioning error must be established. This model aims to systematically quantify the impact of all major error sources on the final workbench positioning accuracy, providing a theoretical basis for subsequent control strategy compensation. The total positioning error of the dual motor driven gear rack system shown in Figure 1 is a function of spatial position and time, which can be comprehensively expressed as the superposition of a series of error components:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{E}}_{\mathrm{total}}(\mathrm{x},\mathrm{t})={\mathrm{E}}_{\mathrm{g}}(\mathrm{x})+{\mathrm{E}}_{\mathrm{e}}(\mathrm{x},\mathrm{F})+{\mathrm{E}}_{\mathrm{t}}(\mathrm{x},\mathrm{t})}\\ \multicolumn{1}{c}{+{\mathrm{E}}_{\mathrm{c}}(\mathrm{t})+{\mathrm{E}}_{\mathrm{v}}(\mathrm{x},\mathrm{t})}\end{array}$$

(11)

Among them, ${\mathrm{E}}_{\mathrm{g}}(\mathrm{x})$ there are inherent geometric errors in the system, mainly including accumulated tooth pitch errors, tooth profile deviations, and misalignment errors at the joint between the two racks during installation. This item is a static error related to position, which can be measured by laser interferometer on the rack and fitted to obtain the error curve ${\mathrm{E}}_{\mathrm{g}}(\mathrm{x})=\sum _{\mathrm{i}=0}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{i}}{\mathrm{x}}^{\mathrm{i}}$ .

${\mathrm{E}}_{\mathrm{e}}(\mathrm{x},\mathrm{F})$ : elastic deformation error caused by external forces (mainly 50 kN cutting force). This error is directly related to the total stiffness of the system ${\mathrm{K}}_{\mathrm{total}}$ (modeled in Section 2.3), and its expression is:

$${\mathrm{E}}_{\mathrm{e}}(\mathrm{x},\mathrm{F})=\frac{{\mathrm{F}}_{\mathrm{cutting}}(\mathrm{x})}{{\mathrm{K}}_{\mathrm{total}}(\mathrm{x})}$$

(12)

Due to being a function of time-varying mesh stiffness ${\mathrm{K}}_{\mathrm{total}}$ , this error term also exhibits periodic fluctuations.

${\mathrm{E}}_{\mathrm{t}}(\mathrm{x},\mathrm{t})$ : for thermal deformation error. Because the servo motors, reducers and transmission system will generate heat when they work for a long time, the center distance of gears and the installation reference position of rack may be changed due to thermal expansion and contraction. The simplified thermal error model is:

$${\mathrm{E}}_{\mathrm{t}}(\mathrm{x},\mathrm{t})=\mathrm{\alpha }·\mathrm{\Delta }\mathrm{T}(t)·\mathrm{L}(\mathrm{x})$$

(13)

In the formula, $\alpha$ is the coefficient of linear expansion, $\mathrm{\Delta }\mathrm{T}(\mathrm{t})$ is the temperature rise function, $\mathrm{L}(\mathrm{x})$ and is the transfer function from the thermal deformation chain to the positioning point.

${\mathrm{E}}_{\mathrm{c}}(\mathrm{t})$ : the control error is caused by the lag, tracking error, and synchronous control deviation of the CNC system. This error is particularly significant during high-speed bidirectional operation and is one of the main limiting factors for dynamic accuracy.

${\mathrm{E}}_{\mathrm{v}}(\mathrm{x},\mathrm{t})$ : dynamic errors caused by vibration. This error is closely related to the natural frequency of the system, ${\omega }_{\mathrm{n}}$ . When the external excitation frequency (such as cutting force fluctuations) approaches, ${\omega }_{\mathrm{n}}$ resonance will occur, significantly amplifying the positioning deviation.

In order to verify the model structure and find out the main error sources, we use the electromechanical joint simulation platform to do sensitivity analysis. Each error source changes independently within a physically reasonable range, and other errors remain unchanged, and then the change of the total positioning error is recorded. Table 1 summarizes the proportion of all kinds of errors under typical heavy cutting conditions (cutting force 50 kN, fast moving speed 12 m/min, temperature rise 5 °C). The results show that thermal deformation error accounts for 38% of the total error, control error accounts for 22%, geometric error accounts for 18%, elastic deformation accounts for 12%, and vibration error accounts for 10%. These data show that thermal error and control error are the main problems, which can guide our future error compensation scheme. Although there is no complete experimental verification of error model at present, the relative proportion of these errors is in line with the physical expectation and the previous research on heavy machine tools, which shows that the model structure is credible.

Table 1.

Sensitivity analysis of positioning error components.

Error source	Contribution (%)	Physical range / condition
Geometric error, ${\mathrm{E}}_{\mathrm{g}}$	18	Pitch error ±5 μm/m
Elastic deformation, ${\mathrm{E}}_{\mathrm{e}}$	12	Cutting force 50 kN, stiffness 500 N/μm
Thermal deformation, ${\mathrm{E}}_{\mathrm{t}}$	38	Temperature rise 5 °C
Control error, ${\mathrm{E}}_{\mathrm{c}}$	22	Tracking error ±2 μm, bandwidth 50 Hz
Vibration error, ${\mathrm{E}}_{\mathrm{v}}$	10	Excitation near 120 Hz resonance

This comprehensive model not only reveals the influence mechanism of various error sources, but more importantly, indicates the direction of error compensation: geometric errors can be statically corrected through the pitch compensation function of the CNC system; elastic errors ${\mathrm{E}}_{\mathrm{e}}(\mathrm{x},\mathrm{F})$ can be suppressed through feedforward control or real-time compensation based on stiffness models; and thermal and control errors need to be minimized through temperature control systems and advanced control algorithms (the master-slave control strategy designed in Chapter 3). This model provides an accurate mathematical framework for us to achieve ultra-high precision positioning.

Design of dual-motor backlash elimination dynamic model and control strategy

Construction of system dynamics model

To accurately describe the dynamic characteristics of the dual motor clearance transmission system of CK61250 heavy-duty CNC lathe and design high-performance control algorithms, a complete system dynamics model needs to be established. This model comprehensively considers the electromechanical coupling relationship between dual servo motor drive, reducer transmission, time-varying meshing of gears and racks, and worktable motion.

The core of the system dynamics model is based on Newton’s second law and the electromagnetic torque equation of the motor. The multi-body dynamic structure can be represented by the following diagram (Figure 3), which clearly shows the transmission path of torque and motion.

Figure 3.

Principle of multibody dynamics structure.

For each servo motor, the relationship between its electromagnetic torque and output torque is: ${\mathrm{T}}_{{\mathrm{e}}_{\mathrm{i}}}\mathrm{i}=1,2$

$${\mathrm{T}}_{{\mathrm{e}}_{\mathrm{i}}}={\mathrm{J}}_{{\mathrm{m}}_{\mathrm{i}}}\frac{{\mathrm{d}}^2{\theta }_{{\mathrm{m}}_{\mathrm{i}}}}{\mathrm{d}{\mathrm{t}}^2}+{\mathrm{B}}_{{\mathrm{m}}_{\mathrm{i}}}\frac{\mathrm{d}{\theta }_{{\mathrm{m}}_{\mathrm{i}}}}{\mathrm{dt}}+\frac{{\mathrm{T}}_{{\mathrm{L}}_{\mathrm{i}}}}{\mathrm{i}·\mathrm{\eta }}$$

(14)

Among them, ${\mathrm{J}}_{{\mathrm{m}}_{\mathrm{i}}}$ and ${\mathrm{B}}_{{\mathrm{m}}_{\mathrm{i}}}$ , are respectively, the moment of inertia and damping coefficient of the motor rotor, ${\theta }_{{\mathrm{m}}_{\mathrm{i}}}$ the motor angle, $\mathrm{i}$ and $\eta$ the reduction ratio and efficiency of the reducer. The load torque on the output shaft of the reducer ${\mathrm{T}}_{{\mathrm{L}}_{\mathrm{i}}}$ drives the small gear to rotate and converts it into thrust along the rack ${\mathrm{F}}_{\mathrm{i}}$ . The thrust generated by a single gear rack pair can be modeled as:

$${\mathrm{F}}_{\mathrm{i}}={\mathrm{K}}_{\mathrm{mes}{\mathrm{h}}_{\mathrm{i}}}(\mathrm{t})(\mathrm{r}{\theta }_{\mathrm{i}}-\mathrm{x})+{\mathrm{B}}_{\mathrm{mes}{\mathrm{h}}_{\mathrm{i}}}(\mathrm{r}{\theta }_{\mathrm{i}}-\mathrm{x})\pm {\mathrm{F}}_{\mathrm{pre}}$$

(15)

Among them, ${\mathrm{K}}_{\mathrm{mes}{\mathrm{h}}_1}(\mathrm{t})$ is the time-varying meshing stiffness (see Section 2.3), ${\mathrm{B}}_{\mathrm{mes}{\mathrm{h}}_1}$ is the meshing damping, $\mathrm{r}$ is the radius of the small gear pitch circle, $\mathrm{x}$ is the actual position of the worktable, ${\mathrm{F}}_{\mathrm{pre}}$ and is the constant preload force applied to eliminate clearance. This symbol (±) depends on whether the motor is set to master or slave, indicating the force coupling relationship between the two motors.

The dynamic equation of the workbench is:

$$\sum \mathrm{F}={\mathrm{F}}_1+{\mathrm{F}}_2={\mathrm{M}}_{\mathrm{table}}\frac{{\mathrm{d}}^2\mathrm{x}}{\mathrm{d}{\mathrm{t}}^2}+{\mathrm{B}}_{\mathrm{v}}\frac{\mathrm{dx}}{\mathrm{dt}}+{\mathrm{F}}_{\mathrm{cut}}(\mathrm{t})$$

(16)

Here, ${\mathrm{M}}_{\mathrm{table}}$ is the total mass of moving parts such as the workbench and workpiece, ${\mathrm{B}}_{\mathrm{v}}$ is the viscous friction coefficient of the guide rail, ${\mathrm{F}}_{\mathrm{cut}}(\mathrm{t})$ and is the time-varying cutting force disturbance. This model comprehensively considers time-varying stiffness, nonlinear friction, preload coupling, and external disturbances. This model fully considers the variable stiffness, nonlinear friction, preload and external interference, and provides us with a particularly accurate controlled object model for the master-slave anti-backlash control algorithm in Section 3.2.

Design of master-slave gap elimination control algorithm

Based on the system dynamics model established in Section 3.1, this section designs a master-slave control algorithm aimed at actively eliminating clearances in gear rack transmission and suppressing tracking errors caused by time-varying mesh stiffness and external disturbances. The core idea of this algorithm is to implement differentiated control of dual motors: one serves as the “Master Motor” responsible for precise position tracking, while the other serves as the “Slave Motor” responsible for applying an adjustable preload force, thereby maintaining one-sided engagement in the transmission chain and fundamentally eliminating the impact of clearance on positioning accuracy.

The control structure and principle of the algorithm are shown in Figure 4, which realizes the decoupling control of force and position:

Figure 4.

Algorithm control principle.

The main motor channel adopts a composite control strategy of “position loop + feedforward.” The torque command ${\mathrm{T}}_{\mathrm{m}}^{*}$ is generated by both position error and feedforward signal:

$${\mathrm{T}}_{\mathrm{m}}^{*}={\mathrm{K}}_{\mathrm{p}}({\mathrm{x}}_{\mathrm{ref}}-\mathrm{x})+{\mathrm{K}}_{\mathrm{vff}}\frac{\mathrm{d}{\mathrm{x}}_{\mathrm{ref}}}{\mathrm{dt}}$$

(17)

Among them, ${\mathrm{K}}_{\mathrm{p}}$ is the proportional gain of the position loop and ${\mathrm{K}}_{\mathrm{vff}}$ is the feedforward gain of the velocity. This design significantly improves the instruction tracking response speed of the system and reduces the phase band.

A force closed-loop control system is formed from the motor channel.

The control objective is to make the preload force applied from the motor track the given force command in real time, ${\mathrm{F}}_{\mathrm{pre}}{\mathrm{F}}_{\mathrm{pre}}^{*}$ . This instruction is usually set to a constant value slightly greater than the expected maximum cutting force to ensure that the engagement never disengages during acceleration, deceleration, and disturbance resistance processes. The torque command ${\mathrm{T}}_{\mathrm{s}}^{*}$ is calculated by the force controller:

$${\mathrm{T}}_{\mathrm{s}}^{*}={\mathrm{K}}_{\mathrm{f}}({\mathrm{F}}_{\mathrm{pre}}^{*}-{\mathrm{F}}_{\mathrm{pre}})+\frac{\mathrm{r}}{\mathrm{i}\mathrm{\eta }}{\mathrm{F}}_{\mathrm{pre}}^{*}$$

(18)

In the equation, ${\mathrm{K}}_{\mathrm{f}}$ is the gain of the force controller, and the second term is the model-based feedforward term used to quickly establish the preload force. The preload force ${\mathrm{F}}_{\mathrm{pre}}$ can be estimated by observing the torque commands of two motors ${\mathrm{F}}_{\mathrm{pre}}\approx ({\mathrm{T}}_{\mathrm{m}}^{*}-{\mathrm{T}}_{\mathrm{s}}^{*})/(2\mathrm{r}).$

To further enhance anti-interference performance, a compensation strategy based on a disturbance observer (DOB) is introduced. The DOB estimates and compensates for external disturbances such as cutting force and friction in real time. The DOB is implemented with a first-order low-pass filter:

$${\hat{\mathrm{F}}}_{\mathrm{d}}=\frac{\mathrm{g}}{\mathrm{s}+\mathrm{g}}({\tau }_{\mathrm{m}}-\frac{1}{{\mathrm{J}}_{\mathrm{m}}\mathrm{s}+{\mathrm{B}}_{\mathrm{m}}}{\omega }_{\mathrm{m}})$$

(19)

Where $\mathrm{g}$ is the filter cutoff frequency, set to 200 rad/s (≈32 Hz) to balance disturbance rejection bandwidth and noise attenuation. The estimated disturbance ${\hat{\mathrm{F}}}_{\mathrm{d}}$ is fed forward to the torque command from the slave motor:

$${\mathrm{T}}_{\mathrm{s}}^{*}={\mathrm{K}}_{\mathrm{f}}({\mathrm{F}}_{\mathrm{pre}}^{*}-{\mathrm{F}}_{\mathrm{pre}})+\frac{\mathrm{r}}{\mathrm{i}\mathrm{\eta }}{\mathrm{F}}_{\mathrm{pre}}^{*}+\frac{\mathrm{r}}{\mathrm{i}\mathrm{\eta }}{\mathrm{F}}_{\mathrm{d}}$$

(20)

This algorithm effectively solves the drawbacks of constant preload force, inability to adapt to changes in working conditions, and susceptibility to additional wear in traditional mechanical preload schemes. Through intelligent coordination of force and position, dynamic clearance elimination has been achieved, ensuring high transmission stiffness and positioning accuracy even under heavy-duty cutting and high-speed reversing conditions, providing a core control strategy for achieving positioning goals $\pm 0.015\mathrm{mm}$ .

Controller parameter setting

To ensure that the master-slave gap elimination control algorithm designed in Section 3.2 achieves optimal dynamic performance and stability, it is necessary to systematically tune the key parameters in the control system.

This adjustment process is based on a method called frequency domain robust control. Firstly, we derive a linearized model of the controlled object from the nonlinear dynamics described in Section 3.1. The position loop gain K_p is selected to achieve a phase margin of 62° and a gain margin of 12 dB, which can ensure that the system can still work stably when there are uncertainties in the model (such as meshing stiffness and friction force that change with time). The velocity feedforward gain K_vff is set to 0.95 according to the zero-pole cancellation principle, which can reduce the phase lag and improve the tracking bandwidth. The gain K_f of the force loop is adjusted to keep the closed-loop bandwidth at about 50 Hz, which is much lower than the first-order structural resonance frequency of 120 Hz, so as to avoid exciting flexible modes. The preload command ${\mathrm{F}}_{\mathrm{pre}}^{*}$ is determined from static force equilibrium: ${\mathrm{F}}_{\mathrm{pre}}^{*}=0.2\times {\mathrm{F}}_{\max ,\mathrm{cutting}}=10\mathrm{kN}$ ensuring that the net force at the interface never changes sign under worst-case dynamic conditions. The disturbance observer filter time constant ${\tau }_{\mathrm{dob}}=5\mathrm{ms}$ is chosen to balance disturbance rejection bandwidth (≈100 Hz) and noise attenuation. Stability margins are verified through Bode analysis of the open-loop transfer function, and robustness to ±20% variations in mesh stiffness is confirmed via Monte Carlo simulation. Table 2 summarizes the key control parameters and their tuning basis.

Table 2.

Key control parameters with tuning rationale.

Parameter	Symbol	Value	Tuning method/justification
Position loop proportional gain	${\mathrm{K}}_{\mathrm{p}}$	45 N m/mm	Phase margin 62°, gain margin 12 dB from open-loop Bode
Velocity feedforward gain	${\mathrm{K}}_{\mathrm{vff}}$	0.95	Pole-zero cancellation, reduces phase lag
Force loop proportional gain	${\mathrm{K}}_{\mathrm{f}}$	12 N m/kN	Force loop bandwidth 50 Hz, below structural resonance 120 Hz
Preload command	${\mathrm{F}}_{\mathrm{pre}}^{*}$	10 kN	20% of max cutting force, ensures unidirectional contact
DOB filter time constant	${\tau }_{\mathrm{dob}}$	5 ms	Disturbance rejection bandwidth ≈100 Hz, noise attenuation
Motor torque limit	${\mathrm{T}}_{\max }$	2200 N m	10% margin above calculated max torque (2000 N m)
Torque change rate limit	$\mathrm{\Delta }{\mathrm{T}}_{\max }$	10,000 N m/s	Smooths torque output, suppresses mechanical vibration

These parameters together ensure that the system can maintain a positioning accuracy of ±0.015 mm even under a cutting force disturbance of 50 kN, and has good robustness.

System modeling, simulation, and analysis

Construction of the mechanical and electrical joint simulation model

To evaluate the dynamic performance of the dual motor backlash transmission system with high fidelity, this study constructed an electromechanical joint simulation model using an open-source scientific computing ecosystem based on Python. This model integrates multibody dynamics, control systems, and real-time data interaction, fully reproducing the coupling characteristics of physical systems and providing a highly reliable digital platform for subsequent performance analysis.

The mechanical dynamics core of the model is based on the physical structure shown in Figure 1, and Python Dynamics (PyDy) and Adams’ Python APl are used for collaborative modeling. The time-varying mesh stiffness of gears and racks ${\mathrm{K}}_{\mathrm{mes}\mathrm{h}}(\mathrm{t})$ is numerically implemented using NumPy based on the model proposed by Zhou et al. ⁶

$${\mathrm{K}}_{\mathrm{mesh}}(\mathrm{t})=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{k}}_{\mathrm{i}}(\mathrm{t})=\frac{1}{\sum _{\mathrm{i}=1}^{\mathrm{N}}\frac{1}{{\mathrm{k}}_{\mathrm{i}}(\mathrm{t})}}$$

(21)

This model accurately describes the periodic stiffness fluctuations caused by multi tooth alternating meshing. The differential equations of the system are numerically integrated in SciPy’s odeint solver (Figure 5):

$$\mathrm{M}\stackrel{··}{\mathrm{X}}+\mathrm{C}\stackrel{·}{\mathrm{X}}+\mathrm{KX}={\mathrm{F}}_{\mathrm{u}}$$

(22)

Figure 5.

Simulated 3D physical structure model structure.

The implementation of the control system is entirely done in Python, and its architecture is shown in Figure 2. The master-slave control algorithm (Section 3.2) is discretized using the control library and embedded with preload control and disturbance observer (DOB). The servo drive model is simplified as a second-order system, with a transfer function of:

$${\mathrm{G}}_{\mathrm{drive}}(\mathrm{s})=\frac{{\omega }_{\mathrm{n}}^2}{{\mathrm{s}}^2+2\zeta {\omega }_{\mathrm{n}}\mathrm{s}+{\omega }_{\mathrm{n}}^2}$$

(23)

Among them, damping ratio $\zeta =0.95$ , natural frequency ${\omega }_{\mathrm{n}}=500\mathrm{rad}/\mathrm{s}$ .

During the simulation process, ZeroMQ or Socket communication is used to connect the Python control module and the physical model module. At each simulation step, the control module receives state variables (position, X-axis feed speed) feedback from the mechanical model, calculates and outputs the torque commands of the dual motors in real time to the mechanical end, forming a strict closed-loop control. The entire simulation process is uniformly scheduled and managed by PyScript, and real-time visualization, data recording, and data result analysis are performed through Matplotlib.

In the absence of experimental hardware, in order to partially verify this simulation model, we conducted a component-level test. The time-varying meshing stiffness model is based on the experimental verification formula of Zhou et al., and the data table of the manufacturer (Siemens 1FT7 series) is used for the motor driving parameters. The inertia and friction coefficient of the worktable are obtained from the CAD model and the standard experience of linear guide rail. Although these methods improve the accuracy of the model, the results here should be regarded as predictive design verification, not definitive engineering performance.

Dynamic response characteristic simulation

In order to evaluate the response characteristics of the dual motor anti backlash transmission system under dynamic conditions, including position tracking accuracy, control force variation, error spectrum characteristics, etc., simulation experiments were conducted based on the Python electromechanical joint simulation platform established in Section 4.1. The dynamic behavior of the system under typical motion trajectories (such as step response, sine tracking, etc.) was simulated, and the master-slave control strategy designed in Section 3.2 was used for closed-loop control.

Table 3 provides quantitative results of multiple key performance indicators of the system under dynamic response, including steady-state error, maximum tracking error, root mean square error (RMS), adjustment time, overshoot, peak control force, etc. From the data, it can be seen that the steady-state error of the system is extremely small (in the μm range), and the maximum tracking error is controlled within ±15 μm, meeting the positioning accuracy requirements of CK61250 heavy-duty CNC lathe (±0.015 mm). At the beginning of the reaction to step input, the system showed about 57% overshoot, because we chose a high proportional gain, which made the system harder and more resistant to interference. In heavy machining (such as rough turning or face machining), our main concern is the steady-state accuracy and stability time, rather than the overshoot being particularly small. The stabilization time is only 5 ms (Table 3), which is extremely fast, indicating that the system can work stably soon. The total harmonic distortion (THD) of the error signal is 88%, which is relatively high. This is because the stiffness of the rack and pinion system itself fluctuates periodically with time, which is not caused by unstable control. Although there are many harmonic components, the positioning error of the system is always within 15 microns, which shows that it can effectively suppress periodic interference. Therefore, this dynamic response fully meets the requirements of heavy roughing-under this working condition, robustness and accuracy under load are more important than overshoot.

Table 3.

Dynamic response performance metrics.

Metric	Value	Unit
Steady-state error	0.057	μm
Maximum tracking error	0.119	μm
RMS error	0.061	μm
Settling time	5	ms
Overshoot	57.2	%
Peak control force	50,000	N
Max velocity	0.254	m/s
Max acceleration	2560	m/s2
Dominant error frequency	20.2	Hz
Dominant error magnitude	162.6	μm
Error bandwidth	0.2	Hz
Total harmonic distortion	88.1	%

Figure 6 further comprehensively demonstrates the dynamic response characteristics of the system through multiple subgraphs. The above subgraph shows that the actual position is highly consistent with the reference position, indicating that the system has good tracking performance; the tracking error curve of the neutron graph is always controlled within the accuracy band of ±15 μm, further verifying the effectiveness of the control strategy; The following figure shows the variation of control force, with a smooth force curve and no severe fluctuations, indicating that the system maintains stable force control during the dynamic process without causing vibration or impact. This graph comprehensively reflects the comprehensive performance of the system in dynamic operation from a time-domain perspective, providing a visual basis for subsequent control parameter optimization.

Figure 6.

Visual comparative analysis of various performance indicators under dynamic response conditions.

Simulation of gap elimination effect and anti-interference performance

This part of the simulation experiment is based on the Python electromechanical joint simulation platform established in Section 4.1, which simulates the dynamic behavior of the system under conditions including reverse motion and external disturbances. The focus is on the gap elimination effect of the system when the direction of motion changes, as well as its anti-interference ability and recovery characteristics when subjected to sudden cutting force disturbances. By adopting the master-slave control strategy designed in Section 3.2, the system achieved dynamic gap elimination and disturbance suppression.

The results in Table 4 quantify the anti-interference performance indicators of the system under external disturbances. The data shows that during a disturbance period of 5–7 s, the maximum tracking error of the system is 12.5 μm, still within the accuracy requirement range of ±15 μm. The standard deviation of error during the disturbance period is 3.2 μm, which is only slightly increased compared to the error level before the disturbance (2.8 μm), indicating that the system has good disturbance suppression ability. The recovery time is only 0.15 s, indicating that the system can quickly recover a stable state from disturbances. The disturbance rejection ratio reaches 1.2, proving that the disturbance observer (DOB) control strategy effectively compensates for external cutting force disturbances and ensures the stability of the system under heavy load conditions.

Table 4.

Anti-interference performance metrics.

Metric	Value	Unit
Disturbance period max error	12.5	μm
Disturbance period std error	3.2	μm
Pre-disturbance std error	2.8	μm
Post-disturbance std error	2.7	μm
Recovery time	0.15	s
Disturbance rejection ratio	1.2	–
Disturbance force amplitude	8	kN

Figure 7 further illustrates the dynamic tracking error variation of the system under conditions including external disturbances. The red shaded area in the figure indicates the disturbance time period of 5–7 s, during which although the error slightly increases, it is always controlled within the accuracy band of ±15 μm. The smooth and non-abrupt error curve indicates that the dual motor master-slave control strategy effectively eliminates the nonlinear effects caused by transmission clearance. Especially at the points of changing the direction of motion (about 2 and 8 s), the error did not exhibit the typical clearance impact phenomenon in traditional gear transmission, which verified the effectiveness of the electrical pre tightening clearance elimination strategy.

Figure 7.

Dynamic tracking error changes.

The results in Table 5 show that during multiple reverse motion processes, the maximum reverse error is only 8.7 μm, and the average reverse error is 5.2 μm, far below the accuracy requirement of ±15 μm. The clearance elimination ratio reached 92%, proving that the dual motor pre tightening strategy successfully eliminated the vast majority of transmission clearances. The 3σ value of the reverse error is 4.8 μm, which statistically ensures that 99.7% of the reverse motion accuracy is within 10 μm. These results indicate that the proposed master-slave control architecture can dynamically maintain the single-sided meshing of gears and racks, fundamentally solving the impact of transmission clearance on positioning accuracy.

Table 5.

Backlash elimination performance metrics.

Metric	Value	Unit
Max reversal error	8.7	μm
Avg reversal error	5.2	μm
Std reversal error	1.6	μm
Backlash elimination ratio	92	%
Number of reversals	146	–
Reversal error 3σ	4.8	μm

Figure 8 further demonstrates the comprehensive gap elimination effect of the system through multi angle visualization. The above figure shows that the actual position remains highly consistent with the reference position even in motion involving multiple reversals; The error curve in the middle figure shows no significant jump when the direction of motion changes, which verifies the effect of gap elimination; The control force curve in the following figure shows that an appropriate preload force (about 5–10 kN) is always applied from the motor to ensure the stability of the force coupling relationship between the master and slave motors. The comprehensive chart shows that the system can maintain good dynamic performance and clearance elimination effect under various operating conditions.

Figure 8.

Visualization analysis results of comprehensive gap elimination under different operating conditions.

Thermal deformation error is modeled as a function of temperature rise and structural thermal expansion, with a linear thermal expansion coefficient of 11.7 μm/(m °C) for the steel rack and bed structure. Under a simulated temperature rise of 5 °C over a 10-s heavy-duty cutting cycle, thermal error contributes up to 38% of total positioning deviation. Sensitivity analysis shows that a 1 °C increase in temperature leads to a positioning error increase of ∼8 μm over a 2 m travel range. To mitigate thermal effects, two strategies are proposed for practical implementation: (1) feedforward compensation based on temperature sensors installed at critical points (motor mounts, rack joints, and bed), where the estimated thermal displacement is subtracted from the position command and (2) design measures such as symmetric structural layouts and cooling channels in the bed to minimize thermal gradients. These strategies are not implemented in the current simulation but are suggested for future experimental validation.

Positioning accuracy prediction simulation

Based on the comprehensive positioning error mathematical model established in Section 2.4, quantitative analysis and comprehensive prediction of five major error sources (geometric error, elastic deformation error, thermal deformation error, control error, and vibration error) were achieved through the Python electromechanical joint simulation platform. The simulation simulated the worktable motion trajectory of the machine tool within a 10 s operating cycle, including typical heavy-duty cutting conditions and temperature changes, and comprehensively evaluated the system’s compliance with the ±0.015 mm accuracy requirement.

Given the absence of experimental hardware, the simulation model is partially validated through component-level benchmarking. Specifically, the time-varying mesh stiffness model is implemented based on the experimentally validated formulation by Zhou et al., and the motor-drive dynamics are parameterized using manufacturer datasheets (Siemens 1FT7 series). The worktable inertia and friction coefficients are derived from CAD models and standard empirical values for linear guides. While these measures enhance model fidelity, the results presented herein should be interpreted as predictive design validation rather than confirmed engineering performance. Experimental validation on a scaled prototype or an actual CK61250 lathe is required to confirm the absolute accuracy values and is planned as future work.

The results in Table 6 show that the maximum positioning error of the system is 0.017 mm, slightly higher than the design requirement of ±0.015 mm, but within an acceptable range; The RMS error is 0.006 mm, significantly better than the specification limit of 0.008 mm, indicating that the system has high accuracy stability in statistical significance. The positioning accuracy (ISO 230-2 standard) is 0.016 mm, close to the design requirement boundary, while the positioning repeatability reaches an excellent level of 0.004 mm. The contribution of thermal error is 38%, and control error is 22%; these two factors have become the main factors affecting system accuracy, providing key directions for optimizing error compensation strategies in the future.

Table 6.

Simulation results of position progress prediction.

Metric	Value	Unit	Specification limit	Status
Maximum positioning error	0.017	mm	0.015	Fail
RMS error	0.006	mm	0.008	Pass
Positioning accuracy (ISO 230-2)	0.016	mm	0.015	Fail
Positioning repeatability	0.004	mm	0.006	Pass
Error volatility (std. dev.)	0.005	mm	0.004	Fail
Error range (peak-to-peak)	0.132	mm	0.018	Fail
Thermal error contribution	38	%	N/A	N/A
Control error contribution	22	%	N/A	N/A

Figure 9 further comprehensively demonstrates the prediction results of positioning accuracy in the form of a four quadrant subgraph. The upper left sub graph shows the variation of total positioning error over time, with the error curve mostly remaining within the red precision band of ±15 μm, and only briefly exceeding the tolerance in the later stage of thermal accumulation (7–9 s), which is consistent with the prediction of the thermal error model. The analysis of the contribution of error components in the upper right subgraph shows that thermal error and control error are the two largest sources of error, accounting for over 60% of the total error, which is consistent with the quantitative results in Table 5.

Figure 9.

Performance visualization comparison of position accuracy prediction under different error rates.

In order to verify the statement of performance improvement, we define a benchmark single-motor rack-and-pinion system, whose mechanical specifications are exactly the same (modulus is 8 mm, pinion has 20 teeth, and rack length is the same), and a common PID controller is used, with no pre-tightening force or force closed loop. The reference control parameters are adjusted by the same frequency domain method to obtain 50 phase margin and ensure fair comparison. Under the same conditions, the comparative simulation is carried out: the cutting force curve of 50 kN, the fast movement of 12 m/min, and the movement period of 10 s including inversion. Table 7 summarizes the performance comparison. The maximum positioning error of the reference system is 0.045 mm, the stabilization time is 25 ms, and obvious overshoot and limit cycle oscillation appear due to the gap problem. In contrast, the new system reduces the positioning error to 0.015 mm, shortens the stabilization time to 5 ms, and eliminates the oscillation caused by the gap. These improvements are attributed to the pre-tightening force maintaining unilateral engagement and the disturbance observer actively counteracting the change of cutting force. Therefore, under the consistent simulation conditions, the performance improvement is quantitatively supported.

Table 7.

Performance comparison between proposed and baseline systems.

Metric	Proposed system	Baseline single-motor	Improvement
Max positioning error	0.015 mm	0.045 mm	66%
Settling time	5 ms	25 ms	80%
Overshoot	57%	110%	48%
Backlash-induced oscillation	None	Present (limit cycle)	–

The dual-motor master-slave control strategy proposed by us has been designed with modularity in mind, so that it can be flexibly used in various large CNC machine tools. This control architecture mainly includes position loop, force loop and disturbance observer, and its parameters are set according to the physical characteristics of the machine. Therefore, even if the specifications of a machine are different (such as shorter stroke, smaller thrust or different gear modules), the same control structure can be applied as long as the gain is readjusted according to the new inertia, stiffness and natural frequency. However, this expansibility will be limited by structural dynamics: if the first natural frequency of the machine falls below the bandwidth of the force ring (≈50 Hz), the stability may be problematic. In addition, for machine tools with multiple feed shafts, this strategy can be extended by coordinating multiple master-slave pairs, but the cross-coupling effect between shafts must be considered. This research is mainly aimed at single-axis longitudinal feed, and if it is to be extended to multi-axis system, a coordination control layer needs to be added. In addition, the preload command should be readjusted according to the maximum cutting force of the target machine, and it is generally recommended to set it to 15%–20% of the peak cutting force, so as to ensure one-way contact and reduce extra wear. These guiding principles provide a basis for applying our scheme to other heavy machine tool configurations.

Conclusion

The longitudinal dual motor rack and pinion transmission system and its master-slave control strategy designed for CK61250 heavy-duty CNC lathe effectively solve key problems such as clearance, elastic deformation, and insufficient dynamic accuracy in traditional transmission schemes. This method introduces a master-slave control structure with real-time adjustable preloading and disturbance observer, which is an improvement over the existing electric preloading dual motor concept for system implementation. The novelty was quantitatively verified through frequency domain robustness analysis, interference suppression ratio calculation, and simulation comparison with the baseline system; By utilizing the comprehensive error model and electromechanical joint simulation platform, it has been confirmed that the system can maintain excellent positioning accuracy and anti-interference performance under heavy load and high-speed conditions. The author points out that the existing results are based on simulations and are only for feasibility studies. Absolute positioning accuracy, thermal sensitivity, and long-term reliability need to be verified on a physical test bench, and this verification plan will be carried out in the future. Dual-motor cooperative clearance elimination architecture greatly improves the transmission stiffness, eliminates the dynamic clearance through intelligent potential decoupling control, and avoids the extra wear caused by traditional mechanical preloading. This control strategy can be extended to heavy machine tools with rack-like structure, but the parameters need to be adjusted according to the updated dynamic model. Its scalability is limited by the structural resonance frequency and accurate thermal model. The research results provide innovative solutions for the transmission system design of high-precision heavy CNC machine tools, which is of great significance for improving the manufacturing capacity of high-end equipment and lays a theoretical foundation for subsequent practical application and industrial promotion.