In modern manufacturing, the dynamic accuracy of feed systems in CNC machines plays a crucial role in determining machining precision. This study focuses on the helical rack and pinion gear system, commonly used in large-scale applications like mirror milling machines, due to its high load capacity and smooth operation. We develop an electromechanical coupling model that integrates the dynamics of the helical rack and pinion gear with a vector-controlled AC synchronous servo motor. By analyzing time-varying meshing stiffness, internal excitations from meshing errors, and motor control dynamics, we investigate the factors influencing dynamic transmission errors in the feed system. Through Simulink simulations and experimental validation, we demonstrate that the rack and pinion gear errors are the primary source of dynamic inaccuracies. This work provides insights for designing high-precision feed systems with improved dynamic performance.
The rack and pinion mechanism is widely employed in long-stroke CNC machines because it offers advantages such as high stiffness and efficient power transmission. However, dynamic errors arising from mechanical and electrical interactions can significantly impact contour accuracy. In this paper, we address these issues by modeling the coupled system and evaluating its behavior under various conditions. Our approach combines potential energy methods for stiffness calculation, slice integration for helical gear analysis, and double closed-loop control for motor dynamics. The results highlight the dominance of rack and pinion gear-induced errors and offer practical guidelines for optimization.
Mathematical Modeling of Helical Rack and Pinion Gear System
The helical rack and pinion gear system in the x-direction feed system of a large mirror milling machine consists of an AC synchronous servo motor, a reducer, a load, and the helical rack and pinion. The motor provides the driving torque, which is transmitted through the reducer to the pinion. As the pinion rotates and engages with the fixed rack, it translates, moving the load. We use a lumped parameter method to simplify this system into a dynamic model.
The equations of motion for the system are derived as follows:
$$ J_{mx} \ddot{\theta}_{mx} = T_{ex} – k_{rx} (\theta_{mx} – i_x \theta_{gx}) – c_{rx} (\dot{\theta}_{mx} – i_x \dot{\theta}_{gx}) $$
$$ J_{gx} \ddot{\theta}_{gx} = i_x \left( k_{rx} (\theta_{mx} – i_x \theta_{gx}) + c_{rx} (\dot{\theta}_{mx} – i_x \dot{\theta}_{gx}) \right) – f_{nx} r_{gx} $$
$$ m_{gx} \ddot{x}_{gx} = f_{nx} – k_{sx} (x_{gx} – x_{lx}) – c_{sx} (\dot{x}_{gx} – \dot{x}_{lx}) $$
$$ m_{lx} \ddot{x}_{lx} = k_{sx} (x_{gx} – x_{lx}) + c_{sx} (\dot{x}_{gx} – \dot{x}_{lx}) – f_{lx} $$
Here, \( T_{ex} \) is the motor’s input torque, \( k_{rx} \) is the torsional stiffness of the reducer, \( c_{rx} \) is the torsional damping, \( i_x \) is the reduction ratio, \( f_{nx} \) is the dynamic meshing force of the rack and pinion gear, \( r_{gx} \) is the pitch radius of the pinion, \( \alpha_n \) is the normal pressure angle, \( k_{sx} \) is the support stiffness, \( c_{sx} \) is the support damping, \( f_{lx} \) is the load resistance, \( J_{mx} \) is the motor inertia, \( J_{gx} \) is the pinion inertia, \( m_{gx} \) is the pinion mass, and \( m_{lx} \) is the load mass.
The dynamic meshing force \( f_{nx} \) in the tangential direction is given by:
$$ f_{nx} = k_{nx} \cos \alpha_n (r_{gx} \theta_{gx} – x_{gx} – e_{nx} \cos \alpha_n) + c_{nx} \cos \alpha_n (r_{gx} \dot{\theta}_{gx} – \dot{x}_{gx} – \dot{e}_{nx} \cos \alpha_n) $$
where \( k_{nx} \) is the normal meshing stiffness, \( c_{nx} \) is the normal meshing damping, \( \beta \) is the helix angle, and \( e_{nx} \) is the meshing error of the rack and pinion gear.
Calculation of Meshing Stiffness for Rack and Pinion Gear
The helical rack and pinion gear exhibits time-varying meshing stiffness due to the helix angle, which causes gradual engagement and disengagement. We employ the potential energy method and slice integration to compute this stiffness. For simplicity, the rack is等效 as a helical gear with an infinite radius. The bending stiffness \( k_b \), shear stiffness \( k_s \), axial compression stiffness \( k_a \), Hertzian contact stiffness \( k_h \), and gear body stiffness \( k_f \) are calculated as follows:
Bending stiffness:
$$ k_b = \sum_{i=1}^{N} \frac{1}{\int_{-\alpha’_1}^{\alpha_2} \frac{3 \cos \alpha (\alpha_2 – \alpha) (1 + \cos \alpha’_1 ((\alpha_2 – \alpha) \sin \alpha – \cos \alpha))^2}{2E \Delta z (\sin \alpha + (\alpha_2 – \alpha) \cos \alpha)^3} d\alpha} $$
Shear stiffness:
$$ k_s = \sum_{i=1}^{N} \frac{1}{\int_{-\alpha’_1}^{\alpha_2} \frac{1.2 (1 + \nu) (\alpha_2 – \alpha) \cos \alpha \cos^2 \alpha’_1}{E \Delta z (\sin \alpha + (\alpha_2 – \alpha) \cos \alpha)} d\alpha} $$
Axial compression stiffness:
$$ k_a = \sum_{i=1}^{N} \frac{1}{\int_{-\alpha’_1}^{\alpha_2} \frac{(\alpha_2 – \alpha) \cos \alpha \sin^2 \alpha’_1}{2E \Delta z (\sin \alpha + (\alpha_2 – \alpha) \cos \alpha)} d\alpha} $$
Hertzian contact stiffness:
$$ k_h = \frac{\pi E L_{ct}}{4 (1 – \nu^2)} $$
Gear body stiffness (based on Sainsot et al.):
$$ k_f = \frac{E \Delta z}{\cos^2 \alpha’_1} \times \frac{1}{L \left( \frac{\mu_f}{s_f} \right)^2 + M \left( \frac{\mu_f}{s_f} \right) + P (1 + Q \tan \alpha’_1)} $$
In these equations, \( \alpha’_1 \) is the angle between the meshing force and the tangential direction, \( \alpha_2 \) is the half-base-tooth angle, \( \nu \) is Poisson’s ratio, \( E \) is Young’s modulus, \( L_{ct} \) is the contact line length, \( \Delta z = L_{ct} \cos \beta_b / N \) is the projected length per slice, \( \beta_b \) is the base helix angle, and \( N \) is the number of slices. Parameters \( \mu_f, s_f, L, M, P, Q \) are obtained from reference studies.
The total meshing stiffness \( k_{nx} \) is the combination of these components in series. For the rack and pinion gear parameters listed in Table 1, we compute the time-varying meshing stiffness. The pinion moves at 50 mm/s, and the stiffness variation over a single tooth pair engagement period is shown in Figure 4. The average meshing stiffness is compared with ISO standards in Table 2, showing a small error of 1.67%, validating our approach.
| Parameter | Value |
|---|---|
| Module \( m_n \) (mm) | 2 |
| Normal pressure angle \( \alpha_n \) (°) | 20 |
| Helix angle \( \beta \) (°) | 19.5283 |
| Number of teeth \( N_{1(2)} \) | 30 |
| Face width \( B \) (mm) | 28 |
| Young’s modulus \( E \) (Pa) | 2.06 × 1011 |
| Poisson’s ratio \( \nu \) | 0.25 |
| Mass \( m \) (kg) | 0.7 |
| Method | Stiffness (N/m) | Error (N/m) | Error Rate (%) |
|---|---|---|---|
| ISO Standard | 5.009 × 108 | – | – |
| Potential Energy Method | 5.0925 × 108 | 0.0835 × 108 | 1.67 |
Meshing Error in Rack and Pinion Gear
The meshing error \( e(t) \) for the rack and pinion gear is modeled as a superposition of sinusoidal functions at the shaft frequency and meshing frequency:
$$ e(t) = 0.5 F_p \sin(2\pi \omega_a t + \phi_a) + 0.5 f’ \sin(2\pi \omega_n t + \phi_n) $$
where \( F_p \) is the total cumulative pitch deviation, \( f’ \) is the single tooth tangential composite deviation, \( \omega_a \) is the shaft frequency, \( \omega_n \) is the meshing frequency, and \( \phi_a, \phi_n \) are initial phases. For a Grade 6 machining accuracy, \( F_p = 26 \, \mu m \) and \( f’ = 10.24 \, \mu m \). This error introduces internal excitations that affect dynamic performance.
Mechanical System Simulation
Using the parameters in Table 3, we implement the dynamic equations in Simulink. The motor torque is set to 0.64 N·m, and the load force is balanced. The ode45 solver is used for simulation. The load displacement error, shown in Figure 5, exhibits periodic micro-level variations. Spectral analysis in Figure 6 reveals dominant frequencies at the shaft frequency \( f_a = 0.25 \, \text{Hz} \) and meshing frequency \( f_n = 7.5 \, \text{Hz} \), aligning with internal excitations from the rack and pinion gear.
| Parameter | Value |
|---|---|
| Motor inertia \( J_{mx} \) (kg·m²) | 1.7 × 10-5 |
| Pinion inertia \( J_{gx} \) (kg·m²) | 3.7703 × 10-4 |
| Pinion mass \( m_{gx} \) (kg) | 0.7 |
| Load mass \( m_{lx} \) (kg) | 300 |
| Reducer stiffness \( k_{rx} \) (N·m/rad) | 24,064 |
| Support stiffness \( k_{sx} \) (N/m) | 4.843 × 109 |
| Reducer damping \( c_{rx} \) (N·m·s/rad) | 0.041 |
| Support damping \( c_{sx} \) (N·s/m) | 3,489.7 |
| Reduction ratio \( i_x \) | 50 |
Modeling of AC Synchronous Servo Motor
The AC synchronous servo motor employs vector control to manage its multi-variable, nonlinear dynamics. By transforming three-phase stator currents into a two-dimensional rotating reference frame (d-q axis), we simplify control. The control model includes Clark and Park transformations.
The Clark transformation matrix \( T_{\text{Clark}} \) is:
$$ T_{\text{Clark}} = \frac{2}{3} \begin{bmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & -\frac{\sqrt{3}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} $$
The Park transformation matrix \( T_{\text{Park}} \), dependent on the rotor angle \( \theta_e \), is:
$$ T_{\text{Park}} = \begin{bmatrix} \cos \theta_e & \sin \theta_e \\ -\sin \theta_e & \cos \theta_e \end{bmatrix} $$
The inverse transformations are used to revert to the natural coordinate system. The motor’s torque balance equation is:
$$ J \frac{d\omega_m}{dt} = T_e – T_L $$
where \( J \) is the motor inertia, \( \omega_m \) is the motor speed, \( T_e \) is the electromagnetic torque, and \( T_L \) is the load torque. The electromagnetic torque in the d-q frame is:
$$ T_e = \frac{3}{2} p_n i_q (i_d (L_d – L_q) + \psi_f) $$
Here, \( p_n \) is the pole pair number, \( i_d \) and \( i_q \) are d-q axis currents, \( L_d \) and \( L_q \) are inductances, and \( \psi_f \) is the flux linkage.
Motor Control Simulation
We simulate the motor using parameters from the Panasonic A6 series, as listed in Table 4. The initial speed is 500 r/min with a load of 0.2 N·m, stepping to 750.73 r/min at 0.4 s and the load to 0.64 N·m at 0.6 s. The output speed and torque, shown in Figures 8 and 9, demonstrate rapid response with minimal overshoot (2.6%) and settling time (~20 ms). Spectral analysis of the steady-state speed in Figure 10 reveals a base frequency of 150 Hz and its harmonics, indicating motor-induced frequencies.
| Parameter | Value |
|---|---|
| d-axis inductance \( L_d \) (H) | 0.02682 |
| q-axis inductance \( L_q \) (H) | 0.02682 |
| Resistance \( R \) (ohm) | 18.7 |
| Motor inertia \( J \) (kg·m²) | 1.7 × 10-5 |
| Pole pairs \( p_n \) | 2 |
| Flux linkage \( \psi_f \) (V·s/rad) | 0.1717 |
| Speed controller gain \( P_\omega \) | 0.013 |
| q-axis current controller gain \( P_{iq} \) | 200 |
| d-axis current controller gain \( P_{id} \) | 200 |
| Speed integrator gain \( I_\omega \) | 1.3 |
| d-axis current integrator gain \( I_{id} \) | 10,000 |
| q-axis current integrator gain \( I_{iq} \) | 10,000 |
Electromechanical Coupling System Simulation
We integrate the rack and pinion gear dynamics with the motor control model to form an electromechanical coupling system. The input is the motor speed command, and the output is the load velocity. Feedback loops include motor speed and load torque from the mechanical system to the motor module, creating a closed-loop control. The overall system model is depicted in Figure 11.
In Simulink, we set the motor speed to 750.73 r/min and include a friction load of 313.6 N. The load displacement over 10 s, shown in Figure 12, matches the theoretical value of 500 mm at 50 mm/s. The displacement error between command and simulation, with and without meshing errors, is plotted in Figure 13. Errors are on the order of 10-5 m, with larger fluctuations when meshing errors are present, underscoring the impact of rack and pinion gear inaccuracies.
Spectral analysis of load acceleration (Figure 14) and displacement (Figure 15) shows frequencies including shaft frequency, meshing frequency, motor base frequency (150 Hz), and their modulations. The displacement spectrum is dominated by mechanical frequencies, while motor frequencies have lesser influence, confirming that the rack and pinion gear is the primary error source.
Experimental Analysis of Feed System Dynamic Characteristics
We conduct experiments on a CNC machine with a rack and pinion gear feed system. The setup includes the machine, an electrical control cabinet, and a host computer for motion commands. The host sends instructions to the controller, which executes motion planning and drives the feed system. The experimental setup is illustrated in Figure 17.
For testing, we command a load velocity of 50 mm/s from an initial position of 100 mm to 400 mm. The experimental displacement, compared to the command in Figure 18, shows a lag due to system inertia. The displacement error in Figure 19 oscillates around 4.04 × 10-4 m, decreasing over time as the system stabilizes. This error is larger than in simulation due to real-world imperfections in the rack and pinion gear.
Spectral analysis of the filtered displacement in Figure 20 reveals key frequencies: shaft frequency \( f_a \), meshing frequency \( f_n \), 0.5\( f_n \), 2\( f_n \), motor frequency \( f_m \), and 2\( f_m \). The largest amplitudes are at \( f_a \) and \( f_n \), consistent with simulations. After adjusting the static meshing error coefficient to one-quarter of the maximum value, the spectrum in Figure 21 better matches experimental ratios.
Additional tests at 30 mm/s, 40 mm/s, 60 mm/s, and 70 mm/s confirm that shaft and meshing frequencies remain dominant across velocities, as summarized in Table 5. This reinforces the conclusion that the rack and pinion gear errors are the main contributors to dynamic inaccuracies.
| Velocity (mm/s) | Primary Frequency (Hz) | Secondary Frequency (Hz) |
|---|---|---|
| 30 | 4.471 (meshing) | 0.149 (shaft) |
| 40 | 5.998 (meshing) | 0.1996 (shaft) |
| 50 | 7.481 (meshing) | 0.2494 (shaft) |
| 60 | 9.063 (meshing) | 0.3021 (shaft) |
| 70 | 10.41 (meshing) | 0.3717 (shaft) |
Conclusion
In this study, we developed a comprehensive electromechanical model for a helical rack and pinion gear feed system in CNC machines. Our analysis demonstrates that meshing errors in the rack and pinion gear are the primary source of dynamic transmission errors, with dominant frequencies at the shaft and meshing frequencies. Motor dynamics, while present, have a lesser impact. The integration of potential energy methods for stiffness calculation, vector control for motor modeling, and experimental validation provides a robust framework for assessing dynamic performance. These findings emphasize the importance of optimizing rack and pinion gear design and manufacturing to enhance feed system accuracy. Future work could explore advanced control strategies or material improvements to further reduce errors in high-precision applications.